Logic:
Organon
Prior Analytics: Chapters 1-5
January 13, 2007
Prior Analytics: Chapters 1-5
January 13, 2007
Chapter
1:
"We
must first state the subject of our inquiry and the faculty to which it
belongs: its subject is demonstration and the faculty that carries it
out demonstrative science. We must next define a premiss, a term, and a
syllogism, and the nature
of a perfect and of an imperfect syllogism;
and after that, the inclusion or non-inclusion of one term in another
as in a whole, and what we mean by predicating one term of all, or
none, of another."
He stated from the very beginning that logical inquiry must be approached scientifically. Along the way, there is no question that he made significant contributions to the development of scientific methodology.
"A premiss then is a sentence affirming or denying one thing of another. This is either universal or particular or indefinite. By universal I mean the statement that something belongs to all or none of something else; by particular that it belongs to some or not to some or not to all; by indefinite that it does or does not belong, without any mark to show whether it is universal or particular, e.g. 'contraries are subjects of the same science', or 'pleasure is not good'. The demonstrative premiss differs from the dialectical, because demonstrative premiss is the assertion of one of two contradictory statements (the demonstrator does not ask for his premiss, but lays it down), whereas the dialectical premiss depends on the adversary's choice between two contradictories. But this will make no difference to the production of a syllogism in either case; for both the demonstrator and the dialectician argue syllogistically after stating that something does or does not belong to something else. Therefore a syllogistic premiss without qualification will be an affirmation or denial of something concerning something else in the way we have described; it will be demonstrative, if it is true and obtained through the first principles of its science; while a dialectical premiss is the giving of a choice between two contradictories, when a man is proceeding by question, but when he is syllogizing it is the assertion of that which is apparent and generally admitted, as has been said in the Topics. The nature then of a premiss and the difference between syllogistic, demonstrative, and dialectical premisses, may be taken as sufficiently defined by us in relation to our present need, but will be stated accurately in the sequel."
The first observation I have is a simple, perhaps trite one: he used "either" and offers 3 choices: "This is either universal or particular or indefinite". Perhaps this is grammatically permissible in the ancient Greek language; but in English, "either" implies one of two choices. Perhaps this is the relic of a mistranslation. Regardless, Aristotle gave clear, concise definitions of these 3 choices.
Regarding his definition of the dialectical process, he argued that it involved a choice between two contradictory statements in response to a line of questioning. In more modern times (via Hegel and Marx), the dialectical process involved a synthesis of contraries. Hence, this process has undergone quite an evolution, a revolution, or a degeneration, depending upon one's point of view.
"I call that a term into which the premiss is resolved, i.e. both the predicate and that of which it is predicated, 'being' being added and 'not being' removed, or vice versa.
A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. I mean by the last phrase that they produce the consequence, and by this, that no further term is required from without in order to make the consequence necessary."
The basic form of a syllogism is: If A = B, and B = C, then A = C.
"I call that a perfect syllogism which needs nothing other than what has been stated to make plain what necessarily follows; a syllogism is imperfect, if it needs either one or more propositions, which are indeed the necessary consequences of the terms set down, but have not been expressly stated as premisses.
That one term should be included in another as in a whole is the same as for the other to be predicated of all of the first. And we say that one term is predicated of all of another, whenever no instance of the subject can be found of which the other term cannot be asserted: 'to be predicated of none' must be understood in the same way."
Again this is basic set theory. If set A is included in set B, then we can say that set B is predicated upon set A. No element in A can be excluded from set B; or stated in a positivist manner, every element of A must be inclusive in B.
Chapter 2:
"Every premiss states that something either is or must be or may be the attribute of something else; of premisses of these three kinds some are affirmative, others negative, in respect of each of the three modes of attribution; again some affirmative and negative premisses are universal, others particular, others indefinite. It is necessary then that in universal attribution the terms of the negative premiss should be convertible, e.g. if no pleasure is good, then no good will be pleasure; the terms of the affirmative must be convertible, not however universally, but in part, e.g. if every pleasure is good, some good must be pleasure; the particular affirmative must convert in part (for if some pleasure is good, then some good will be pleasure); but the particular negative need not convert, for if some animal is not man, it does not follow that some man is not animal."
Let us start with analyzing the universal declaration: every pleasure is good. Assign set A = goodness, and assign set B = pleasure. Then "every pleasure is good" necessitates that B be completely included in A; in terms of predication, A is predicated upon B. The contrary of this declaration is "no pleasure is good". This would necessite that B be completely excluded from A, or that the two sets be disjoint. Now, in order for the converse of the contrary to be true, it would necessitate that "no good is a pleasure" hold true. Since the two sets are disjoint, this is true. Nothing interior to the "good set" (set A) is included in the "pleasure set" (set B); however, "every non-good is a pleasure" would be a false assertion, for not everything exterior to set A is included in set B.
Now let's analyze the converse of the positive assertion (the "affirmative premiss"), "every pleasure is good". The converse doesn't hold true because "not every good is a pleasure". This is because while B is completely included in A, A is not included in B. So, there are some good things (inside set A) that are not pleasurable (outside of set B) while there are other good things (inside set A) that are pleasurable (inside set B). Hence, Aristotle was correct when he in effect stated that the converse of the affirmative universal premiss must partially be true and partially false.
"First then take a universal negative with the terms A and B. If no B is A, neither can any A be B. For if some A (say C) were B, it would not be true that no B is A; for C is in B. But if every B is A, then some A is B. For if no A were B, then no B could be A. But we assumed that every B is A. Similarly too, if the premiss is particular. For if some B is A, then some of the As must be B. For if none were, then no B would be A. But if some B is not A, there is no necessity that some of the As should not be B; e.g. let B stand for animal and A for man. Not every animal is a man; but every man is an animal."
Let's analyse the "universal negative": no B is in A. If this is the case, then the two sets are disjoint, meaning that no A is in B. So he was correct so far! He illustrated this by introducing some subset C contained in A. If C were in B, then A and B would have some common elements (an intersection) and thus would not be disjoint.
Now let's analyse the "particular premiss": some B is in A. Then some A will be in B (represented by the intersection of the sets). For if not, A and B would be disjoint sets.
Chapter 3:
"The same manner of conversion will hold good also in respect of necessary premisses. The universal negative converts universally; each of the affirmatives converts into a particular. If it is necessary that no B is A, it is necessary also that no A is B. For if it is possible that some A is B, it would be possible also that some B is A. If all or some B is A of necessity, it is necessary also that some A is B: for if there were no necessity, neither would some of the Bs be A necessarily. But the particular negative does not convert, for the same reason which we have already stated."
His assertions about converses (reversals) were correct:
1) The converse of the universal negative premiss is also universal: "No B is contained in A" ==> "no A is contained in B". This is true because the two sets are disjoint.
2) The converse of the particular premise is also particular: "some B in A" ==> "some A in B". This is true because the two sets have an intersection area, the region where they overlap, that can be described as either "some x in y" or "some y in x" (same thing).
3) The particular negative doesn't have a converse: "some B is not in A" =//=> "some A is not in B". This will only be true if set A is completely contained in set B. Otherwise, B and A intersect, and "some B is not in A" ==> "some A is not in B", because both sides of this "equation" describe a unique non-intersection area. In the beginning of Chapter 2, Aristotle stated that "the particular negative need not convert". That is, it's not necessary that it has a converse, but it's possible. Here, at the beginning of Chapter 3, he contended this is impossible when he states "the particular negative does not convert". This is another instant where he wasn't consistent.
"In respect of possible premisses, since possibility is used in several senses (for we say that what is necessary and what is not necessary and what is potential is possible), affirmative statements will all convert in a manner similar to those described. For if it is possible that all or some B is A, it will be possible that some A is B. For if that were not possible, then no B could possibly be A. This has been already proved. But in negative statements the case is different. Whatever is said to be possible, wither because B necessarily is A, or because B is not necessarily A, admits of conversion like other negative statements, e.g. if one should say, it is possible that man is not horse, or that no garment is white. For in the former case the one term necessarily does not belong to the other; in the latter there is no necessity that it should: and the premiss converts like other negative statements. For if it is possible for no man to be a horse, it is also admissible for no horse to be a man; and if it is admissible for no garment to be white, it is also admissible for nothing white to be a garment. For if any white thing must be a garment, then some garment will necessarily be white. This has been already proved."
First, let's take a look at the the converse of possible affirmative premises:
1) The premise = "it's possible that all of B is A" . The converse = "it's possible that all of A is B".
a) The converse is true if A is a subset of B (A
B) OR if A
= B.
b) The premise and its converse are both true if A = B.
c) Since the premise and converse are stated in terms of possibilities, then for the converse it's also possible that only some of A is in B, represented by the intersection region (A
B), or
none of A is in B, represented by disjoint sets (in logical notation,
disjoint sets are represented by A 
B). Note
that if the
premise and converse were stated in terms of only, i.e. "it's possible that only
all of B is A" & "it's possible that only all of A is B", then the
"some" and "none" cases would have been excluded as possibilities.
2) The premise = "it's possible that some of B is A". The converse = "it's possible that some of A is B".
a) The converse is true if A B exists
in some
region. If A and B are disjoint sets, then it's not true.
b) The premise and its converse are both true in the same intersection region (A B).
c) Since the premise and converse are stated in terms of possibilities, then for the converse it's also possible that all of A is in B, represented by A B OR A = B,
or none of A is in B, represented by disjoint
sets (A B). Note
that if the premise and converse were stated in terms of only,
i.e. "it's possible that only some of B is A" & "it's possible that
only
all of A is B", then the "all" and "none" cases would have been
excluded as possibilities.
Though the primary converses of 1a and 2a are different, if we add the secondary converses of 1c and 2c, #1 and #2 are essentially the same. In other words, 1a + 1c = 2a + 2c.
Next, let's take a look at the the converse of possible negative premises:
1) The premise = "it's possible that all of B is not A". The converse = "it's possible that all of A is not B".
a) The converse is true if A and B are disjoint sets (A B).
b) The premise and its converse are both true if A B.
c) Since the premise and converse are stated in terms of possibilities, then for the converse it's also possible that only some of A is not in B, represented by the region of A that doesn't intersect with B (A - (A B)), or
none of A is not in B (meaning all of A is in B), represented by A
being a subset of B (A B) or by A
= B. The same "only" conditions apply here as well.
2) The premise = "it's possible that some of B is not A". The converse = "it's possible that some of A is not B".
a) The converse is true in the region of A that doesn't intersect with B: A - (A B).
b) The premise and its converse are both true in the non-intersecting region of A and B. This is equal to the union of A and B (A
B) minus
the intersection of A and B (A B), or
simply (A B) - (A B).
c) Since the premise and converse are stated in terms of possibilities, then for the converse it's also possible that none of A is not in B (meaning that all of A is in B), represented either by A B or by A =
B, or that all of A is not in B (meaning that none of A is in B),
represented by disjoint sets (A B). The
same "only" conditions apply
here as well.
Again, 1a + 1c = 2a + 2c, and essentially #1 and #2 are the same.
Like possible affirmative statements, the converse of possible negative statements are true. So I'm somewhat puzzled he said "But in negative statements the case is different". How is it different? What he described after this statement, however, is coherent and definitely true.
Just for fun, let's look at the converse of impossible positive premises:
1) The premise = "it's impossible that all of B is A". The converse = "it's impossible that all of A is B".
Since the premise and converse are stated in terms of impossibilities, and since semantically "all"
"some" "none",
then "impossibility of
all" = "possibility of some" + "possibility of none".
a) The converse is true in the region where A B
("possibility of some" case) or where A B
("possibility of none" case); but
not in the case where A = B.
b) The premise and the converse are both true in the same regions described in 1a.
2) The premise = "it's impossible that some of B is A". The converse = "it's impossible that some of A is B".
Since the premise and converse are stated in terms of impossibilities, and since semantically "some""all" "none",
then "impossibility of some" = "possibility of all" + "possibility of
none".
a) The converse is true if A B or if A =
B ("possibility of all" cases) or if A B (
"possibility of none" case).
b) The premise and the converse are both true if A B (only
"possibility of none" case). We must exclude B A for the
premise ("possibility of all" case) because its converse states that
some of A can't be in B, which contradicts B A since
some of A would overlap with B. Likewise, we must exclude A B for the
converse ("possibility of all" case) because the premise states that
some of B can't be in A, which contradicts A B since
some of B would overlap with A.
Finally, again for fun, let's look at the converse of impossible negative premises:
1) The premise = "it's impossible that all of B is not A". The converse = "it's impossible that all of A is not B". Since these are double negatives, we can restate the premise = "it's possible that all of B is not A", and the converse = "it's possible that all of A is B".
See #1 of possible affirmative premises above for the solutions.
2) The premise = "it's impossible that some of B is not A". The converse = "it's impossible that some of A is not B". Since these are double negatives, we can restate the premise = "it's possible that some of B is not A", and the converse = "it's possible that some of A is B".
See #2 of possible affirmative premises above for the solutions.
"The particular negative also must be treated like those dealt with above. But if anything is said to be possible because it is the general rule and natural (and it is in this way we define the possible), the negative premisses can no longer be converted like the simple negatives; the universal negative premiss does not convert, and the particular does. This will be plain when we speak about the possible. At present we may take this much as clear in addition to what has been said: the statement that it is possible that no B is A or some B is not A is affirmative in form: for the expression 'is possible' ranks along with 'is', and 'is' makes an affirmation always and in every case, whatever the terms to which it is added, in predication, e.g.. 'it is not-good' or 'it is not-white' or in a word 'it is not-this'. But this also will be proved in the sequel. In conversion these premisses will behave like the other affirmative propositions."
* Regarding "At present we may take this much as clear in addition to what has been said: the statement that it is possible that no B is A or some B is not A is affirmative in form: for the expression 'is possible' ranks along with 'is', and 'is' makes an affirmation always and in every case ...". A possibility is a would-be "is" insofar that it can assume the "is" when given the *chance*. A possibility is a probability that once actualized becomes an observable; the event becomes "real" when it's observed. This is one of the core tenets of the Copenhagen Interpretation of Quantum Mechanics.
Chapter 4:
"After these distinctions we now state by what means, when, and how every syllogism is produced; subsequently we must speak of demonstration. Syllogism should be discussed before demonstration, because syllogism is the more general: the demonstration is a sort of syllogism, but not every syllogism is a demonstration.
Whenever three terms are so related to one another that the last is contained in the middle as in a whole, and the middle is either contained in, or excluded from, the first as in or from a whole, the extremes must be related by a perfect syllogism. I call that term middle which is itself contained in another and contains another in itself: in position also this comes in the middle. By extremes I mean both that term which is itself contained in another and that in which another is contained. If A is predicated of all B, and B of all C, A must be predicated of all C: we have already explained what we mean by "predicated of all". Similarly also, if A is predicated of no B, and B of all C, it is necessary that no C will be A."
The syllogism is the foundation of deductive logic. In order for a syllogism to be formed, two conditions must be met:
1) The conclusion of an argument must necessarily follow from the premises, and the conclusion must be unique.
2) The relationship between the premises must meet the requirements stated by Aristotle (last term in the middle, middle term in or not in the first).
One form of the syllogism is: All of B is in A. All of C is in B. Therefore, All of C is in A. Here, the last term (C) is contained in the middle term (B), commensurate with Aristotle's description of "the last is contained in the middle as in a whole". The middle term (B) is contained in the first term (A), commensurate with Aristotle's description of "the middle is either contained in, or excluded from, the first as in or from a whole". The conclusion follows from the premises because C is completely contained in B which is completely contained in A.
* Regarding "If A is predicated of all B, and B of all C, A must be predicated of all C". This is the same as the example above. Translating the predication into syllogism form: All B are in A. All C are in B. Therefore, all C are in A. In terms of sets: BA
and C
B ==> C A. This is
true because B is completely contained in A, and C is completely
contained in B.
* Regarding "If A is predicated of no B, and B of all C, it is necessary that no C will be A.". Translated into syllogism form:
No B are in A. All C are in B. Therefore, no C are in A. In terms of sets, B A and C B ==> C A. This
is
true because B and A are disjoint, and since C is completely contained
in B, A and C are disjoint as well.
"But if the first term belongs to all the middle, but the middle to none of the last term, there will be no syllogism in respect of the extremes; for nothing necessary follows from the terms being so related; for it is possible that the first should belong either to all or to none of the last, so that neither a particular nor a universal conclusion is necessary. But if there is no necessary consequence, there cannot be a syllogism by means of these premisses. As an example of a universal affirmative relation between the extremes we may take the terms animal, man, horse; of a universal negative relation, the terms animal, man, stone. Nor again can a syllogism be formed when neither the first term belongs to any of the middle, nor the middle to any of the last. As an example of a positive relation between the extremes take the terms science, line, medicine: of a negative relation science, line, unit."
When he stated that "for nothing necessary follows from the terms being so related", Aristotle in effect stated that the conclusion must follow from the premises, and this conclusion must be unique. By "necessary consequence", he meant there could not be a broken chain in the stream of logic. If the statements in an alleged syllogism don't follow from each other out of necessity, then it isn't a syllogism.
Now, let us take a look at his examples:
1) Universal Affirmative relation: animal, man, horse. Let A = animals, B = man, C = horse. Then we have the relation "All men are animals" (All B in A), "All horses are animals" (All C in A), but "No horse is a man" (no C in B). This is represented in logical notation by: BA, C A ==> C B. For
the given example, both B
and C are contained in A, but B and C are disjoint sets. Since the last
term C is not in the middle term B, no syllogism is formed.
However, while his example is not a syllogism (since the last term C is not in the middle term B), he failed to present the other representative example of why no syllogism is formed. The general underlying logic is that if the major premise is BA, and the
minor premise is C A, then no
conclusion is derived out of necessity because
possible conclusions are C B OR C
and B intersect (C B). He
should have mentioned this
latter possibility (C B) in his
discussion above.
2) Universal Negative relation: animal, man, stone. Let A = animals, B = man, C = stone. Then we have the relation "All men are animals" (All B in A), "No stone is an animal " (No C in A) ==> "No stone is a man" (no C in B). This is represented in logical notation by: BA, C A ==>
C B.
For the given example, this makes perfect sense, for if A and C are
disjoint, then B and C
must be disjoint since B is contained in A. Again, since the last term
C is not in the middle term B, no syllogism is formed; and there are no
other possible conclusions to consider.
"If then the terms are universally related, it is clear in this figure when a syllogism will be possible and when not, and that if a syllogism is possible the terms must be related as described, and if they are so related there will be a syllogism.
But if one term is related universally, the other in part only, to its subject, there must be a perfect syllogism whenever universality is posited with reference to the major term either affirmatively or negatively, and particularity with reference to the minor term affirmatively: but whenever the universality is posited in relation to the minor term, or the terms are related in any other way, a syllogism is impossible. I call that term the major in which the middle is contained and that term the minor which comes under the middle. Let all B be A and some C be B. Then if 'predicated of all' means that was said above, it is necessary that some C is A. And if no B is A, but some C is B, it is necessary that some C is not A. (The meaning of 'predicated of none' has also been defined.) So there will be a perfect syllogism. This holds good also if the premiss BC should be indefinite, provided that it is affirmative: for we shall have the same syllogism whether the premiss is indefinite or particular."
Aristotle continued to lay down the conditions for the syllogism, discussing the relations between major premise, minor premise, and conclusion. Here, he introduces additional syllogism forms where "some C is in B". In order for this to hold true, for a given ordering of {A, B, C}, the "new" syllogism must take this form: the last term (C) must be *partially* be contained in the middle term (B), while the middle term (B) must be completely contained in or disjoint from the first term (A). It takes a while to absorb all of this.
* Regarding "Let all B be A and some C be B ... it is necessary that some C is A". In set form, BA and C B ==>
C A.
This is true because the part of B that intersects with C must also
intersect with A. Now, it's possible that C A, in which
case all C is A. But the conclusion states that it's *necessary* that
some C be in A, meaning that *at least* some C must be in A with a
possibility that all C could be in A. By his additional syllogism
forms, the last term of the ordering {A, B, C} is indeed partially in
the middle term B, while the middle term is completely in the first
term. Hence, it's a syllogism.
* Regarding "And if no B is A, but some C is B, it is necessary that some C is not A.". In set form, BA and C B ==>
C
not-A. This is true because since A and B are disjoint sets (from the
1st line of the syllogism), and C is at least partially contained in B,
then at least that part of B won't be a part of A. It's still possible
that C
A, for *some* of C would still be outside of A.
Again, by his additional syllogism forms, the last term of the ordering {A, B, C} is indeed partially in the middle term B, while the middle term is disjoint from the first term. Hence, it's a syllogism.
"But if the universality is posited with respect to the minor term either affirmatively or negatively, a syllogism will not be possible, whether the major premiss is positive or negative, indefinite or particular: e.g. if some B is or is not A, and all C is B. As an example of a positive relation between the extremes take the terms good, state, wisdom: of a negative relation, good, state, ignorance. Again if no C is B, but some B is or is not A, or not every B is A, there cannot be a syllogism. Take the terms white, horse, swan: white, horse, raven. The same terms may be taken also if the premiss BA is indefinite."
* Regarding "Again if no C is B, but some B is or is not A, or not every B is A, there cannot be a syllogism". Let's take a look at all of these combinations:
1) If no C is B, and some B is A. C and B are disjoint sets, and B intersects with A. In logical notation, C B and B A.
However, from the major premise (C B) and
the minor premise (B A), the
relationship between C and A can't be uniquely determined. The
conclusion is valid if either C A OR C A. Since
the conclusion can't be uniquely determined (deduced), and at least
some of the last term C is not in the middle term B, this is not a
syllogism.
2) If no C is B, and some B is not A. C and B are disjoint sets, and B intersects with not-A. In logical notation, C B and B not-A.
Again, the conclusion is valid if either C A OR C A. Since
the conclusion can't be uniquely determined (deduced), and at least
some of the last term C is not in the middle term B, this is not a
syllogism.
3) If no C is B, and not every B is A. In logical notation, C B and B A (but B
isn't a subset of A). This is the same case as #1.
"Nor when the major premiss is universal, whether affirmative or negative, and the minor premiss is negative and particular, can there be a syllogism, whether the minor premiss be indefinite or particular: e.g. if all B is A, and some C is not B, or if not all C is B. For the major term may be predicable both of all and of none of the minor, to some of which the middle term cannot be attributed. Suppose the terms are animal, man, white: next take some of the white things of which man is not predicated -- swan and snow: animal is predicated of all of the one, but of none of the other. Consequently there cannot be a syllogism. Again let no B be A, but let some C not be B. Take the terms inanimate, man, white: then take some white things of which man is not predicated -- swan and snow: the term inanimate is predicated of all of the one, of none of the other."
Again, let's look at these possibilities:
1) All B is in A, some C is not in B. In logical notation, BA and C not-B;
and since *some* of C is not in B, some of C must also be in B, meaning
C B.
Hence, the major premise remains B A, but the
minor premise is transformed to C B.
Regarding the conclusion, which must express a unique relation between
A and C in order for the syllogism to hold true, it can only be A C; since
because B and C intersect, and B is completely contained in A, A must
intersect with C. This is commensurate with the stipulation that it's "necessary that some C is A" as he
asserted above, meaning that A C covers
both the "some" and "all" conditions (A C and C A
respectively). For this ordering of the terms, {A, B, C}, the last term
(C) is partially contained in the middle term (B), while the middle
term (B) is completely contained in the first term (A). Hence, by his
own "new" definition above, it is a syllogism. So I'm not certain why
he stated it is not a syllogism! Is it because the minor premise takes
the form of "some x is NOT in y" instead of "some x is in y"? The only
way I can see that it doesn't form a syllogism is if "some C is not in
B" is interpreted as "no C is in B".
Using Aristotle's example of "animal, man, white", if A = animal, B = man, and C = white, we see that the major premise BA is true
(since all men are animals!), and the minor premise C B is also
true (since some men are white, but not all). The conclusion A C is also
true (since some animals are white, but not all).
2) All B is in A, not all C is in B. In logical notation, BA and not-C B ==>
C A.
This is the same as B A and C B (some C
intersects with B) ==> C A. Hence,
this is the same case as #1.
3) No B is in A, not all C is in B. In logical notation, B A and
not-C
B; and since *not all* (i.e. some) of C is not in B, some of C must
also be in B, meaning C B. Again,
the major premise remains B A, but
the minor premise is transformed to C B.
Regarding the conclusion, there isn't a unique relationship between A
and C, since either A C or A C are
valid conclusions. However, it is commensurate with the stipulation
that "it is necessary that some C is
not A" as he asserted above, meaning that A C covers
both the "some" and "no" conditions (A C and A C
respectively). For this ordering of the terms, {A, B, C}, the last term
(C) is partially contained in the middle term (B), while the middle
term (B) is completely disjoint from the first term (A). Hence, by his
own "new" definition above, it is a syllogism. So again, I'm not
certain why
he stated it is not a syllogism! The only
way I can see that it doesn't form a syllogism is if "not all C is in
B" is interpreted as "no C is in B".
Using Aristotle's example of "inanimate, man, white", on the other hand, paints a different picture. If we assign A = inanimate, B = man, and C = white, certainly B A (since
no man is inanimate, although some do leave that lasting impression!),
and certainly C B (since
some men are white). Now, it's certainly true that at least some
inanimate objects are white (i.e. A C), but
this is NOT deduced from the pure logical premises. It's derived from
the world of experience! Here's an example where Aristotle's example is
a syllogism, but the logical premises it's purportedly based upon is
not.
Once he introduced "swan and snow", I'm lost! I can see how "animals" are predicate upon the first but not the second. And I can see how this wouldn't form a syllogism as a result. However, I don't see how it's related to the examples he gave.
"Further since it is indefinite to say some C is not B, and it is true that some C is not B, whether no C is B, or not all C is B, and since if terms are assumed such that no C is B, no syllogism follows (this has already been stated), it is clear that this arrangement of terms will not afford a syllogism: otherwise one would have been possible with a universal negative minor premiss. A similar proof may also be given if the universal premiss is negative."
Since we've already analyzed the cases where the minor premise is "some C is not B" and "not all C is B", let's take a look at the final possibility of "no C in B":
4) All B is in A, no C in B. In logical notation, BA and C B. Again,
regarding the conclusion, which must express a unique relation between
A and C, it is not uniquely deduced, since the conclusion can either be
C A
(where C B inside
of A) OR C A (where
C is partially outside of A) OR C A. Also,
the last term (C) is neither fully nor partially contained in the
middle term (B).
this is
not a syllogism.
"Nor can there in any way be a syllogism if both the relations of subject and predicate are particular, either positively or negatively, or the one negative and the other affirmative, or one indefinite and the other definite, or both indefinite. Terms common to all the above are animal, white, horse: animal, white, stone.
It is clear then from what has been said that if there is a syllogism in this figure with a particular conclusion, the terms must be related as we have stated: if they are related otherwise, no syllogism is possible anyhow. It is evident also that all the syllogisms in this figure are perfect (for they are all completed by means of the premisses originally taken) and that all conclusions are proved by this figure, viz. universal and particular, affirmative and negative. Such a figure I call the first."
And so Aristotle laid down the conditions for one of the most powerful constructs in the foundations of logic.
Chapter 5:
"Whenever the same thing belongs to all of one subject, and to none of another, or to all of each subject or to none of either, I call such a figure the second; by middle term in it I mean that which is predicated of both subjects, by extremes the terms of which this is said, by major extreme that which lies near the middle, by minor that which is further away from the middle. The middle term stands outside the extremes, and is first in position. A syllogism cannot be perfect anyhow in this figure, but it may be valid whether the terms are related universally or not."
Sounds good. Read on.
"If then the terms are related universally a syllogism will be possible, whenever the middle belongs to all of one subject and to none of another (it does not matter which has the negative relation), but in no other way. Let M be predicated of no N, but of all O. Since, then, the negative relation is convertible, N will belong to no M: but M was assumed to belong to all O: consequently N will belong to no O. This has already been proved. Again if M belongs to all N, but to no O, then N will belong to no O. For if M belongs to no O, O belongs to no M: but M (as was said) belongs to all N: O then will belong to no N: for the first figure has again been formed. But since the negative relation is convertible, N will belong to no O. Thus it will be the same syllogism that proves both conclusions.
It is possible to prove these results also by reduction ad impossibile."
* Regarding "Since, then, the negative relation is convertible, N will belong to no M: but M was assumed to belong to all O: consequently N will belong to no O.", this can be interpreted two different ways depending upon what "belonging to all" actually means. Let's look at each case:
1) "M belongs to all O" means that M is a subset of O (MO). Then
using logical notation, N M, M O ==> N O. This
can only be true if N
completely lies outside of O; but there are the
other
conclusions that can be deduced from the premises and which formulate a
relation between sets O and N: O N or N O (see
figures below). Thus, his example isn't a syllogism (certainly not a
perfect one) because the conclusion isn't a necessary consequence of
the premises, but merely a possible consequence. Also, with the
ordering of terms, {M, N, O}, it can't possibly follow the form of a
syllogism because the last term (O) is not contained in the middle term
(N); and even if the terms are rearranged, it doesn't satisfy the
conditions to form a syllogism. So why does he refer
to the "same syllogism that proves both conclusions"?! It could only be
an imperfect one, and if so, he would have needed to describe the
supporting statements to reach his stated conclusion.
_________
_______________|____ N | _____________________
| O |____ |____ | | O |
| __________ | | _______ ______ |
| | M | | | | M | | N | |
| | | | | |_______ | |_____ | |
| |__________| | | |
|____________________| |____________________|
2) "M belongs to all O" means that O is a subset of M (OM). Then
using logical notation, N M, O M ==> N O. This
makes complete sense and meets the criteria for a syllogism (since the
middle term M belongs to all of another set O, but belongs to none of
the first set N). See the diagram below.
_________
____________________ | N |
| M | |____ ____|
| __________ |
| | O | |
| | | |
| |_________ | |
|____________________|
If we give Aristotle the benefit of the doubt, namely that he had impeccable logician credentials, then we must assume that the second interpretation holds true. He essentially was equating "M belongs to all O" with "M is predicated upon all O". We will use that interpretation henceforth. He would have made this clearer by using "subset" terminology (i.e. O is a subset of M); or that "M interects with ALL of O". There would have been no ambiguity in what he was describing.
* Regarding "if M belongs to all N, but to no O, then N will belong to no O.", let's analyze this using the following logical notation: N M and M O ==>
N O.
This is quite true and meets the criteria for a syllogism. Again, see
the diagram below.
___________________
| M |
| __________ | _______
| | N | | | O |
| | | | |______|
| |__________| |
|___________________|
"It is clear then that a syllogism is formed when the terms are so related, but not a perfect syllogism; for necessity is not perfectly established merely from the original premisses; others are so needed.
But if M is predicated of every N and O, there cannot be a syllogism. Terms to illustrate a positive relation between the extremes are substance, animal, man; a negative relation, substance, animal, number -- substance being the middle term."
Let's take a look at this assertion. If M is predicated of every N and O, then NM and O M. It's true
that this doesn't lead to a syllogism, since the relationship between N
and O can't be deduced from the premises.
I don't understand how his examples are related to this assertion, especially since he said that "substance" is the middle term (and if it is the middle, why didn't he place it there instead of the beginning of the trio?!).
Nor is a syllogism possible when M is predicated neither of any N nor of any O. Terms to illustrate a positive relation are line, animal, man: a negative relation, line, animal, stone.
Let's take a look at this assertion. If M is predicated neither of any N nor of any O, then M N and M O. It's
true that this doesn't lead to a syllogism, since the relationship
between N and O can't be deduced from the premises.
Again, I don't understand how his examples are related to this assertion.
"It is clear then that if a syllogism is formed when the terms are universally related, the terms must be related as we stated at the outset: for if they are otherwise related no necessary consequence follows.
If the middle terms is related universally to one of the extremes, a particular negative syllogism must result whenever the middle term is related universally to the major whether positively or negatively, and particularly to the minor and in a manner opposite to that of the universal statement: by 'an opposite manner' I mean, if the universal statement is negative, the particular is affirmative: if the universal is affirmative, the particular is negative. For if M belongs to no N, but to some O, it is necessary that N does not belong to some O. For since the negative statement is convertible, N will belong to no M: but M was admitted to belong to some O: therefore N will not belong to some O: for the result is reached by means of the first figure. Again if M belongs to all N, but not to some O, it is necessary that N does not belong to some O: for if N belongs to all O, and M is predicated also of all N, M must belong to all O: but we assumed that M does not belong to some O. And if M belongs to all N but not to all O, we shall conclude that N does not belong to all O: the proof is the same as the above. But if M is predicated of all O, but not of all N, there will be no syllogism. Take the terms animal, substance, raven; animal, white, raven. Nor will there be a conclusion when M is predicated of no O, but of some N. Terms to illustrate a positive relation between the extremes are animal, substance, unit: a negative relation, animal, substance, science."
Let's analyze these assertions:
1) Regarding "For if M belongs to no N, but to some O, it is necessary that N does not belong to some O.", this is represented by this logical notation: M N, M O ==>
N
non-O = O N or N O. A
valid conclusion is reached with either set O being a subset of set N
or the two sets intersecting. This is still a valid syllogism because
only one
conclusion is reached (N non-O);
it's just that this conclusion can be *represented* two different ways
(N
O or N
O).
2) Regarding "For since the negative statement is convertible, N will belong to no M: but M was admitted to belong to some O: therefore N will not belong to some O: for the result is reached by means of the first figure", this is represented by this logical notation: N M, M O ==>
N
non-O = O N or N O. Same
as #1.
3) Regarding "Again if M belongs to all N, but not to some O, it is necessary that N does not belong to some O: for if N belongs to all O, and M is predicated also of all N, M must belong to all O: but we assumed that M does not belong to some O.", this is represented by this logical notation: NM, M non-O
==> N not-O. If
we strictly interpret "N does not belong to *some* O", then we must
repudiate the possible interpretation of "N does not belong to *any*
O). Hence, by strict interpretation, N O, and a
syllogism is formed. But if we interpret "necessary" as meaning that N
must *at least* exclude some O, which includes both "some" or "any",
then either "N does not belong to some O" or "N does not belong to any
O" is an acceptible solution. In this case, there are two possible
solutions, either of which forms a syllogism.
____________________ _____________________
| M | | M |
| __________ | | __________ |
| | N ___ |_____|___ | | N | ____ |____
| | | | | O | | | | | | O |
| |_____ |____| | | | |_________| |____|____|
| |_________|____| | |
|___________________| |_____________________|
4) Regarding his examples "Take the terms animal, substance, raven; animal, white, raven".
The first relation forms a syllogism:
Major premise: All animals have substance.
Minor premise: A raven is an animal.
Conclusion: A raven has substance.
The second relation doesn't form a syllogism:
Major premise: Some animals are white.
Minor premise: A raven is an animal.
Conclusion: We can't conclude that some ravens are white because the Sets of ravens and whiteness might be disjoint.
"If then the universal statement is opposed to the particular, we have stated when a syllogism will be possible and when not: but if the premisses are similar in form, I mean both negative or both affirmative, a syllogism will not be possible anyhow. First let them be negative, and let the major premiss be universal, e.g. let M belong to no N, and not to some O. It is possible then for N to belong either to all O or to no O. Terms to illustrate the negative relation are black, snow, animal. It is not possible to find terms of which the extremes are related positively and universally, if M belongs to some O, and does not belong to some O. For if N belonged to all O, but M to no N, then M would belong to no O: but we assumed that it belongs to some O. In this way then it is not admissible to take terms: our point must be proved from the indefinite nature of the particular statement. For since it is true that M does not belong to some O, even if it belongs to no O, and since if it belongs to no O a syllogism is (as we have seen) not possible, clearly it will not be possible now either."
Let's analyze these claims:
* Regarding "First let them be negative, and let the major premiss be universal, e.g. let M belong to no N, and not to some O. It is possible then for N to belong either to all O or to no O", this is represented by the logical notation:
major premise: M N
minor premise: M not-O
(this
implies that some of M must belong to O)
conclusion: Either O N or N O (N
either belongs to all O or to no O).
The first conclusion is not possible, but I think Aristotle meant to say "it's possible for O to belong to all N". The second conclusion is possible. But it's also possible that N O (see
figure below). Why is this possibility excluded?
____________________
| O |
____ |___ _____ | ____
| M | | N |
|_______ | |__________|
| |
|____________________|
Also, for the relations "black, snow, animal", if M = black, N = snow, and O = animal, if M N, then
no snow will be black. This is positively false, for some snow gets
very dirty!
* Regarding "For if N belonged to all O, but M to no N, then M would belong to no O", the logical notation is:
Major premise: O N
Minor premise: M N
Conclusion: M O
This makes perfect sense.
_____________________
| N |
| _______ | ________
| | O | | | M |
| |_______| | |________|
| |
|_____________________|
"Again let the premisses be affirmative, and let the major premiss as before be universal, e.g. let M belong to all N and to some O. It is possible then for N to belong to all O or to no O. Terms to illustrate the negative relation are white, swan, stone. But it is not possible to take terms to illustrate the universal affirmative relation, for the reason already stated: the point must be proved from the indefinite nature of the particular statement. But if the minor premiss is universal, and M belongs to no O, and not to some N, it is possible for N to belong either to all O or to no O. Terms for the positive relation are white, animal, raven: for the negative relation, white, stone, raven. If the premisses are affirmative, terms for the negative relation are white, animal, snow; for the positive relation, white, animal, swan. Evidently then, whenever the premisses are similar in form, and one is universal, the other particular, a syllogism cannot be formed anyhow. Nor is one possible if the middle term belongs to some of each of the extremes, or does not belong to some of either, or belongs to some of the one, not to some of the other, or belongs to neither universally, or is related to them indefinitely. Common terms for all the above are white, animal, man,: white, animal, inanimate."
Analyzing his propositions:
1) Premises affirmative, major premise universal.
Major premise: NM (M belongs
to all N)
Minor premise: M O (M
belongs to some O)
Conclusion: O N
or N O. (N
belongs to all O or to no O)
Huh? Clearly, the second conclusion is possible since the part of O intersecting with M might not intersect with O. But the first conclusion could only hold true if the minor premise was "M belongs to all O", because if we strictly interpret the existing minor premise "M belongs to some O", then these two sets must intersect with neither containing the other.
Regarding "Terms to illustrate the negative relation are white, swan, stone", I don't see how this can work here. If M = white, N = swan, and O = stone, certain the major premise NM is untrue,
for not all swans are white (he probably
didn't know about black
swans). Yes, some stones are white, so the the minor premise M O is
true. And finally for the conclusion, only N O holds
true since no swan is a stone. But this example does NOT follow from
his stated logic; his argument is invalid (the set relations disprove
his argument). And regarding the converse,
assuming we were to assign M = swan, and N = white (i.e. not all things
white are swans), it certainly can't be true that N M.
2) Minor premise is universal.
Major premise: M O. (M
belongs to no O).
Minor premise: M not-N (M
doesn't belong to some N) = M N
Conclusion: O N
or N O (N
belongs to all O or to no O)
This makes sense. Either conclusion works:
Regarding "Terms for the positive relation are white, animal, raven", if M = white, N = animal, and O = raven, it's not certain the major premise is true, for is M and O are disjoint, there would be no white ravens. It's not clear if white ravens actually existed (were they a mere mythological creature?), but it was probably permissible for Aristotle to assume no ravens were white because at least they weren't present in Greece. Yes, the minor premise M not-N is
true because not all things white are animals, but "some" things are
white (M N). The
first possible conclusion O N is true
since all ravens are animals. The converse is true, assuming
we assigned N = raven, and O = animal; but again that would change the
ordering of his example. Certainly, N O is not
a possible conclusion because at least some animals are ravens (unless
he thought the raven species was extinct!).
"It is clear then from what has been said that if the terms are related to one another in the way stated, a syllogism results of necessity; and if there is a syllogism, the terms must be so related. But it is evident also that all the syllogisms in this figure are imperfect: for all are made perfect by certain supplementary statements, which either are contained in the terms of necessity or are assumed as hypotheses, i.e. when we prove per impossibile. And it is evident that an affirmative conclusion is not attained by means of this figure, but all are negative, whether universal or particular."
Note: Some logical symbols:
=
inequality
=
subset
=
union
=
intersection
=
set membership
=
therefore
= disjoint sets
Last updated: February 24, 2007He stated from the very beginning that logical inquiry must be approached scientifically. Along the way, there is no question that he made significant contributions to the development of scientific methodology.
"A premiss then is a sentence affirming or denying one thing of another. This is either universal or particular or indefinite. By universal I mean the statement that something belongs to all or none of something else; by particular that it belongs to some or not to some or not to all; by indefinite that it does or does not belong, without any mark to show whether it is universal or particular, e.g. 'contraries are subjects of the same science', or 'pleasure is not good'. The demonstrative premiss differs from the dialectical, because demonstrative premiss is the assertion of one of two contradictory statements (the demonstrator does not ask for his premiss, but lays it down), whereas the dialectical premiss depends on the adversary's choice between two contradictories. But this will make no difference to the production of a syllogism in either case; for both the demonstrator and the dialectician argue syllogistically after stating that something does or does not belong to something else. Therefore a syllogistic premiss without qualification will be an affirmation or denial of something concerning something else in the way we have described; it will be demonstrative, if it is true and obtained through the first principles of its science; while a dialectical premiss is the giving of a choice between two contradictories, when a man is proceeding by question, but when he is syllogizing it is the assertion of that which is apparent and generally admitted, as has been said in the Topics. The nature then of a premiss and the difference between syllogistic, demonstrative, and dialectical premisses, may be taken as sufficiently defined by us in relation to our present need, but will be stated accurately in the sequel."
The first observation I have is a simple, perhaps trite one: he used "either" and offers 3 choices: "This is either universal or particular or indefinite". Perhaps this is grammatically permissible in the ancient Greek language; but in English, "either" implies one of two choices. Perhaps this is the relic of a mistranslation. Regardless, Aristotle gave clear, concise definitions of these 3 choices.
Regarding his definition of the dialectical process, he argued that it involved a choice between two contradictory statements in response to a line of questioning. In more modern times (via Hegel and Marx), the dialectical process involved a synthesis of contraries. Hence, this process has undergone quite an evolution, a revolution, or a degeneration, depending upon one's point of view.
"I call that a term into which the premiss is resolved, i.e. both the predicate and that of which it is predicated, 'being' being added and 'not being' removed, or vice versa.
A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. I mean by the last phrase that they produce the consequence, and by this, that no further term is required from without in order to make the consequence necessary."
The basic form of a syllogism is: If A = B, and B = C, then A = C.
"I call that a perfect syllogism which needs nothing other than what has been stated to make plain what necessarily follows; a syllogism is imperfect, if it needs either one or more propositions, which are indeed the necessary consequences of the terms set down, but have not been expressly stated as premisses.
That one term should be included in another as in a whole is the same as for the other to be predicated of all of the first. And we say that one term is predicated of all of another, whenever no instance of the subject can be found of which the other term cannot be asserted: 'to be predicated of none' must be understood in the same way."
Again this is basic set theory. If set A is included in set B, then we can say that set B is predicated upon set A. No element in A can be excluded from set B; or stated in a positivist manner, every element of A must be inclusive in B.
Chapter 2:
"Every premiss states that something either is or must be or may be the attribute of something else; of premisses of these three kinds some are affirmative, others negative, in respect of each of the three modes of attribution; again some affirmative and negative premisses are universal, others particular, others indefinite. It is necessary then that in universal attribution the terms of the negative premiss should be convertible, e.g. if no pleasure is good, then no good will be pleasure; the terms of the affirmative must be convertible, not however universally, but in part, e.g. if every pleasure is good, some good must be pleasure; the particular affirmative must convert in part (for if some pleasure is good, then some good will be pleasure); but the particular negative need not convert, for if some animal is not man, it does not follow that some man is not animal."
Let us start with analyzing the universal declaration: every pleasure is good. Assign set A = goodness, and assign set B = pleasure. Then "every pleasure is good" necessitates that B be completely included in A; in terms of predication, A is predicated upon B. The contrary of this declaration is "no pleasure is good". This would necessite that B be completely excluded from A, or that the two sets be disjoint. Now, in order for the converse of the contrary to be true, it would necessitate that "no good is a pleasure" hold true. Since the two sets are disjoint, this is true. Nothing interior to the "good set" (set A) is included in the "pleasure set" (set B); however, "every non-good is a pleasure" would be a false assertion, for not everything exterior to set A is included in set B.
Now let's analyze the converse of the positive assertion (the "affirmative premiss"), "every pleasure is good". The converse doesn't hold true because "not every good is a pleasure". This is because while B is completely included in A, A is not included in B. So, there are some good things (inside set A) that are not pleasurable (outside of set B) while there are other good things (inside set A) that are pleasurable (inside set B). Hence, Aristotle was correct when he in effect stated that the converse of the affirmative universal premiss must partially be true and partially false.
"First then take a universal negative with the terms A and B. If no B is A, neither can any A be B. For if some A (say C) were B, it would not be true that no B is A; for C is in B. But if every B is A, then some A is B. For if no A were B, then no B could be A. But we assumed that every B is A. Similarly too, if the premiss is particular. For if some B is A, then some of the As must be B. For if none were, then no B would be A. But if some B is not A, there is no necessity that some of the As should not be B; e.g. let B stand for animal and A for man. Not every animal is a man; but every man is an animal."
Let's analyse the "universal negative": no B is in A. If this is the case, then the two sets are disjoint, meaning that no A is in B. So he was correct so far! He illustrated this by introducing some subset C contained in A. If C were in B, then A and B would have some common elements (an intersection) and thus would not be disjoint.
Now let's analyse the "particular premiss": some B is in A. Then some A will be in B (represented by the intersection of the sets). For if not, A and B would be disjoint sets.
Chapter 3:
"The same manner of conversion will hold good also in respect of necessary premisses. The universal negative converts universally; each of the affirmatives converts into a particular. If it is necessary that no B is A, it is necessary also that no A is B. For if it is possible that some A is B, it would be possible also that some B is A. If all or some B is A of necessity, it is necessary also that some A is B: for if there were no necessity, neither would some of the Bs be A necessarily. But the particular negative does not convert, for the same reason which we have already stated."
His assertions about converses (reversals) were correct:
1) The converse of the universal negative premiss is also universal: "No B is contained in A" ==> "no A is contained in B". This is true because the two sets are disjoint.
2) The converse of the particular premise is also particular: "some B in A" ==> "some A in B". This is true because the two sets have an intersection area, the region where they overlap, that can be described as either "some x in y" or "some y in x" (same thing).
3) The particular negative doesn't have a converse: "some B is not in A" =//=> "some A is not in B". This will only be true if set A is completely contained in set B. Otherwise, B and A intersect, and "some B is not in A" ==> "some A is not in B", because both sides of this "equation" describe a unique non-intersection area. In the beginning of Chapter 2, Aristotle stated that "the particular negative need not convert". That is, it's not necessary that it has a converse, but it's possible. Here, at the beginning of Chapter 3, he contended this is impossible when he states "the particular negative does not convert". This is another instant where he wasn't consistent.
"In respect of possible premisses, since possibility is used in several senses (for we say that what is necessary and what is not necessary and what is potential is possible), affirmative statements will all convert in a manner similar to those described. For if it is possible that all or some B is A, it will be possible that some A is B. For if that were not possible, then no B could possibly be A. This has been already proved. But in negative statements the case is different. Whatever is said to be possible, wither because B necessarily is A, or because B is not necessarily A, admits of conversion like other negative statements, e.g. if one should say, it is possible that man is not horse, or that no garment is white. For in the former case the one term necessarily does not belong to the other; in the latter there is no necessity that it should: and the premiss converts like other negative statements. For if it is possible for no man to be a horse, it is also admissible for no horse to be a man; and if it is admissible for no garment to be white, it is also admissible for nothing white to be a garment. For if any white thing must be a garment, then some garment will necessarily be white. This has been already proved."
First, let's take a look at the the converse of possible affirmative premises:
1) The premise = "it's possible that all of B is A" . The converse = "it's possible that all of A is B".
a) The converse is true if A is a subset of B (A
B) OR if A
= B.b) The premise and its converse are both true if A = B.
c) Since the premise and converse are stated in terms of possibilities, then for the converse it's also possible that only some of A is in B, represented by the intersection region (A
B), or
none of A is in B, represented by disjoint sets (in logical notation,
disjoint sets are represented by A B). Note
that if the
premise and converse were stated in terms of only, i.e. "it's possible that only
all of B is A" & "it's possible that only all of A is B", then the
"some" and "none" cases would have been excluded as possibilities.2) The premise = "it's possible that some of B is A". The converse = "it's possible that some of A is B".
a) The converse is true if A
B exists
in some
region. If A and B are disjoint sets, then it's not true. b) The premise and its converse are both true in the same intersection region (A
B).c) Since the premise and converse are stated in terms of possibilities, then for the converse it's also possible that all of A is in B, represented by A
B OR A = B,
or none of A is in B, represented by disjoint
sets (A B). Note
that if the premise and converse were stated in terms of only,
i.e. "it's possible that only some of B is A" & "it's possible that
only
all of A is B", then the "all" and "none" cases would have been
excluded as possibilities.Though the primary converses of 1a and 2a are different, if we add the secondary converses of 1c and 2c, #1 and #2 are essentially the same. In other words, 1a + 1c = 2a + 2c.
Next, let's take a look at the the converse of possible negative premises:
1) The premise = "it's possible that all of B is not A". The converse = "it's possible that all of A is not B".
a) The converse is true if A and B are disjoint sets (A
B). b) The premise and its converse are both true if A
B.c) Since the premise and converse are stated in terms of possibilities, then for the converse it's also possible that only some of A is not in B, represented by the region of A that doesn't intersect with B (A - (A
B)), or
none of A is not in B (meaning all of A is in B), represented by A
being a subset of B (A B) or by A
= B. The same "only" conditions apply here as well.2) The premise = "it's possible that some of B is not A". The converse = "it's possible that some of A is not B".
a) The converse is true in the region of A that doesn't intersect with B: A - (A
B).b) The premise and its converse are both true in the non-intersecting region of A and B. This is equal to the union of A and B (A
B) minus
the intersection of A and B (A B), or
simply (A B) - (A B).c) Since the premise and converse are stated in terms of possibilities, then for the converse it's also possible that none of A is not in B (meaning that all of A is in B), represented either by A
B or by A =
B, or that all of A is not in B (meaning that none of A is in B),
represented by disjoint sets (A B). The
same "only" conditions apply
here as well.Again, 1a + 1c = 2a + 2c, and essentially #1 and #2 are the same.
Like possible affirmative statements, the converse of possible negative statements are true. So I'm somewhat puzzled he said "But in negative statements the case is different". How is it different? What he described after this statement, however, is coherent and definitely true.
Just for fun, let's look at the converse of impossible positive premises:
1) The premise = "it's impossible that all of B is A". The converse = "it's impossible that all of A is B".
Since the premise and converse are stated in terms of impossibilities, and since semantically "all"
"some" "none",
then "impossibility of
all" = "possibility of some" + "possibility of none".a) The converse is true in the region where A
B
("possibility of some" case) or where A B
("possibility of none" case); but
not in the case where A = B. b) The premise and the converse are both true in the same regions described in 1a.
2) The premise = "it's impossible that some of B is A". The converse = "it's impossible that some of A is B".
Since the premise and converse are stated in terms of impossibilities, and since semantically "some"
"all" "none",
then "impossibility of some" = "possibility of all" + "possibility of
none".a) The converse is true if A
B or if A =
B ("possibility of all" cases) or if A B (
"possibility of none" case).b) The premise and the converse are both true if A
B (only
"possibility of none" case). We must exclude B A for the
premise ("possibility of all" case) because its converse states that
some of A can't be in B, which contradicts B A since
some of A would overlap with B. Likewise, we must exclude A B for the
converse ("possibility of all" case) because the premise states that
some of B can't be in A, which contradicts A B since
some of B would overlap with A.Finally, again for fun, let's look at the converse of impossible negative premises:
1) The premise = "it's impossible that all of B is not A". The converse = "it's impossible that all of A is not B". Since these are double negatives, we can restate the premise = "it's possible that all of B is not A", and the converse = "it's possible that all of A is B".
See #1 of possible affirmative premises above for the solutions.
2) The premise = "it's impossible that some of B is not A". The converse = "it's impossible that some of A is not B". Since these are double negatives, we can restate the premise = "it's possible that some of B is not A", and the converse = "it's possible that some of A is B".
See #2 of possible affirmative premises above for the solutions.
"The particular negative also must be treated like those dealt with above. But if anything is said to be possible because it is the general rule and natural (and it is in this way we define the possible), the negative premisses can no longer be converted like the simple negatives; the universal negative premiss does not convert, and the particular does. This will be plain when we speak about the possible. At present we may take this much as clear in addition to what has been said: the statement that it is possible that no B is A or some B is not A is affirmative in form: for the expression 'is possible' ranks along with 'is', and 'is' makes an affirmation always and in every case, whatever the terms to which it is added, in predication, e.g.. 'it is not-good' or 'it is not-white' or in a word 'it is not-this'. But this also will be proved in the sequel. In conversion these premisses will behave like the other affirmative propositions."
* Regarding "At present we may take this much as clear in addition to what has been said: the statement that it is possible that no B is A or some B is not A is affirmative in form: for the expression 'is possible' ranks along with 'is', and 'is' makes an affirmation always and in every case ...". A possibility is a would-be "is" insofar that it can assume the "is" when given the *chance*. A possibility is a probability that once actualized becomes an observable; the event becomes "real" when it's observed. This is one of the core tenets of the Copenhagen Interpretation of Quantum Mechanics.
Chapter 4:
"After these distinctions we now state by what means, when, and how every syllogism is produced; subsequently we must speak of demonstration. Syllogism should be discussed before demonstration, because syllogism is the more general: the demonstration is a sort of syllogism, but not every syllogism is a demonstration.
Whenever three terms are so related to one another that the last is contained in the middle as in a whole, and the middle is either contained in, or excluded from, the first as in or from a whole, the extremes must be related by a perfect syllogism. I call that term middle which is itself contained in another and contains another in itself: in position also this comes in the middle. By extremes I mean both that term which is itself contained in another and that in which another is contained. If A is predicated of all B, and B of all C, A must be predicated of all C: we have already explained what we mean by "predicated of all". Similarly also, if A is predicated of no B, and B of all C, it is necessary that no C will be A."
The syllogism is the foundation of deductive logic. In order for a syllogism to be formed, two conditions must be met:
1) The conclusion of an argument must necessarily follow from the premises, and the conclusion must be unique.
2) The relationship between the premises must meet the requirements stated by Aristotle (last term in the middle, middle term in or not in the first).
One form of the syllogism is: All of B is in A. All of C is in B. Therefore, All of C is in A. Here, the last term (C) is contained in the middle term (B), commensurate with Aristotle's description of "the last is contained in the middle as in a whole". The middle term (B) is contained in the first term (A), commensurate with Aristotle's description of "the middle is either contained in, or excluded from, the first as in or from a whole". The conclusion follows from the premises because C is completely contained in B which is completely contained in A.
* Regarding "If A is predicated of all B, and B of all C, A must be predicated of all C". This is the same as the example above. Translating the predication into syllogism form: All B are in A. All C are in B. Therefore, all C are in A. In terms of sets: B
A
and C
B ==> C A. This is
true because B is completely contained in A, and C is completely
contained in B.* Regarding "If A is predicated of no B, and B of all C, it is necessary that no C will be A.". Translated into syllogism form:
No B are in A. All C are in B. Therefore, no C are in A. In terms of sets, B
A and C B ==> C A. This
is
true because B and A are disjoint, and since C is completely contained
in B, A and C are disjoint as well."But if the first term belongs to all the middle, but the middle to none of the last term, there will be no syllogism in respect of the extremes; for nothing necessary follows from the terms being so related; for it is possible that the first should belong either to all or to none of the last, so that neither a particular nor a universal conclusion is necessary. But if there is no necessary consequence, there cannot be a syllogism by means of these premisses. As an example of a universal affirmative relation between the extremes we may take the terms animal, man, horse; of a universal negative relation, the terms animal, man, stone. Nor again can a syllogism be formed when neither the first term belongs to any of the middle, nor the middle to any of the last. As an example of a positive relation between the extremes take the terms science, line, medicine: of a negative relation science, line, unit."
When he stated that "for nothing necessary follows from the terms being so related", Aristotle in effect stated that the conclusion must follow from the premises, and this conclusion must be unique. By "necessary consequence", he meant there could not be a broken chain in the stream of logic. If the statements in an alleged syllogism don't follow from each other out of necessity, then it isn't a syllogism.
Now, let us take a look at his examples:
1) Universal Affirmative relation: animal, man, horse. Let A = animals, B = man, C = horse. Then we have the relation "All men are animals" (All B in A), "All horses are animals" (All C in A), but "No horse is a man" (no C in B). This is represented in logical notation by: B
A, C A ==> C B. For
the given example, both B
and C are contained in A, but B and C are disjoint sets. Since the last
term C is not in the middle term B, no syllogism is formed.However, while his example is not a syllogism (since the last term C is not in the middle term B), he failed to present the other representative example of why no syllogism is formed. The general underlying logic is that if the major premise is B
A, and the
minor premise is C A, then no
conclusion is derived out of necessity because
possible conclusions are C B OR C
and B intersect (C B). He
should have mentioned this
latter possibility (C B) in his
discussion above.2) Universal Negative relation: animal, man, stone. Let A = animals, B = man, C = stone. Then we have the relation "All men are animals" (All B in A), "No stone is an animal " (No C in A) ==> "No stone is a man" (no C in B). This is represented in logical notation by: B
A, C A ==>
C B.
For the given example, this makes perfect sense, for if A and C are
disjoint, then B and C
must be disjoint since B is contained in A. Again, since the last term
C is not in the middle term B, no syllogism is formed; and there are no
other possible conclusions to consider. "If then the terms are universally related, it is clear in this figure when a syllogism will be possible and when not, and that if a syllogism is possible the terms must be related as described, and if they are so related there will be a syllogism.
But if one term is related universally, the other in part only, to its subject, there must be a perfect syllogism whenever universality is posited with reference to the major term either affirmatively or negatively, and particularity with reference to the minor term affirmatively: but whenever the universality is posited in relation to the minor term, or the terms are related in any other way, a syllogism is impossible. I call that term the major in which the middle is contained and that term the minor which comes under the middle. Let all B be A and some C be B. Then if 'predicated of all' means that was said above, it is necessary that some C is A. And if no B is A, but some C is B, it is necessary that some C is not A. (The meaning of 'predicated of none' has also been defined.) So there will be a perfect syllogism. This holds good also if the premiss BC should be indefinite, provided that it is affirmative: for we shall have the same syllogism whether the premiss is indefinite or particular."
Aristotle continued to lay down the conditions for the syllogism, discussing the relations between major premise, minor premise, and conclusion. Here, he introduces additional syllogism forms where "some C is in B". In order for this to hold true, for a given ordering of {A, B, C}, the "new" syllogism must take this form: the last term (C) must be *partially* be contained in the middle term (B), while the middle term (B) must be completely contained in or disjoint from the first term (A). It takes a while to absorb all of this.
* Regarding "Let all B be A and some C be B ... it is necessary that some C is A". In set form, B
A and C B ==>
C A.
This is true because the part of B that intersects with C must also
intersect with A. Now, it's possible that C A, in which
case all C is A. But the conclusion states that it's *necessary* that
some C be in A, meaning that *at least* some C must be in A with a
possibility that all C could be in A. By his additional syllogism
forms, the last term of the ordering {A, B, C} is indeed partially in
the middle term B, while the middle term is completely in the first
term. Hence, it's a syllogism.* Regarding "And if no B is A, but some C is B, it is necessary that some C is not A.". In set form, B
A and C B ==>
C
not-A. This is true because since A and B are disjoint sets (from the
1st line of the syllogism), and C is at least partially contained in B,
then at least that part of B won't be a part of A. It's still possible
that C
A, for *some* of C would still be outside of A.Again, by his additional syllogism forms, the last term of the ordering {A, B, C} is indeed partially in the middle term B, while the middle term is disjoint from the first term. Hence, it's a syllogism.
"But if the universality is posited with respect to the minor term either affirmatively or negatively, a syllogism will not be possible, whether the major premiss is positive or negative, indefinite or particular: e.g. if some B is or is not A, and all C is B. As an example of a positive relation between the extremes take the terms good, state, wisdom: of a negative relation, good, state, ignorance. Again if no C is B, but some B is or is not A, or not every B is A, there cannot be a syllogism. Take the terms white, horse, swan: white, horse, raven. The same terms may be taken also if the premiss BA is indefinite."
* Regarding "Again if no C is B, but some B is or is not A, or not every B is A, there cannot be a syllogism". Let's take a look at all of these combinations:
1) If no C is B, and some B is A. C and B are disjoint sets, and B intersects with A. In logical notation, C
B and B A.
However, from the major premise (C B) and
the minor premise (B A), the
relationship between C and A can't be uniquely determined. The
conclusion is valid if either C A OR C A. Since
the conclusion can't be uniquely determined (deduced), and at least
some of the last term C is not in the middle term B, this is not a
syllogism.2) If no C is B, and some B is not A. C and B are disjoint sets, and B intersects with not-A. In logical notation, C
B and B not-A.
Again, the conclusion is valid if either C A OR C A. Since
the conclusion can't be uniquely determined (deduced), and at least
some of the last term C is not in the middle term B, this is not a
syllogism.3) If no C is B, and not every B is A. In logical notation, C
B and B A (but B
isn't a subset of A). This is the same case as #1."Nor when the major premiss is universal, whether affirmative or negative, and the minor premiss is negative and particular, can there be a syllogism, whether the minor premiss be indefinite or particular: e.g. if all B is A, and some C is not B, or if not all C is B. For the major term may be predicable both of all and of none of the minor, to some of which the middle term cannot be attributed. Suppose the terms are animal, man, white: next take some of the white things of which man is not predicated -- swan and snow: animal is predicated of all of the one, but of none of the other. Consequently there cannot be a syllogism. Again let no B be A, but let some C not be B. Take the terms inanimate, man, white: then take some white things of which man is not predicated -- swan and snow: the term inanimate is predicated of all of the one, of none of the other."
Again, let's look at these possibilities:
1) All B is in A, some C is not in B. In logical notation, B
A and C not-B;
and since *some* of C is not in B, some of C must also be in B, meaning
C B.
Hence, the major premise remains B A, but the
minor premise is transformed to C B.
Regarding the conclusion, which must express a unique relation between
A and C in order for the syllogism to hold true, it can only be A C; since
because B and C intersect, and B is completely contained in A, A must
intersect with C. This is commensurate with the stipulation that it's "necessary that some C is A" as he
asserted above, meaning that A C covers
both the "some" and "all" conditions (A C and C A
respectively). For this ordering of the terms, {A, B, C}, the last term
(C) is partially contained in the middle term (B), while the middle
term (B) is completely contained in the first term (A). Hence, by his
own "new" definition above, it is a syllogism. So I'm not certain why
he stated it is not a syllogism! Is it because the minor premise takes
the form of "some x is NOT in y" instead of "some x is in y"? The only
way I can see that it doesn't form a syllogism is if "some C is not in
B" is interpreted as "no C is in B".Using Aristotle's example of "animal, man, white", if A = animal, B = man, and C = white, we see that the major premise B
A is true
(since all men are animals!), and the minor premise C B is also
true (since some men are white, but not all). The conclusion A C is also
true (since some animals are white, but not all).2) All B is in A, not all C is in B. In logical notation, B
A and not-C B ==>
C A.
This is the same as B A and C B (some C
intersects with B) ==> C A. Hence,
this is the same case as #1.3) No B is in A, not all C is in B. In logical notation, B
A and
not-C
B; and since *not all* (i.e. some) of C is not in B, some of C must
also be in B, meaning C B. Again,
the major premise remains B A, but
the minor premise is transformed to C B.
Regarding the conclusion, there isn't a unique relationship between A
and C, since either A C or A C are
valid conclusions. However, it is commensurate with the stipulation
that "it is necessary that some C is
not A" as he asserted above, meaning that A C covers
both the "some" and "no" conditions (A C and A C
respectively). For this ordering of the terms, {A, B, C}, the last term
(C) is partially contained in the middle term (B), while the middle
term (B) is completely disjoint from the first term (A). Hence, by his
own "new" definition above, it is a syllogism. So again, I'm not
certain why
he stated it is not a syllogism! The only
way I can see that it doesn't form a syllogism is if "not all C is in
B" is interpreted as "no C is in B".Using Aristotle's example of "inanimate, man, white", on the other hand, paints a different picture. If we assign A = inanimate, B = man, and C = white, certainly B
A (since
no man is inanimate, although some do leave that lasting impression!),
and certainly C B (since
some men are white). Now, it's certainly true that at least some
inanimate objects are white (i.e. A C), but
this is NOT deduced from the pure logical premises. It's derived from
the world of experience! Here's an example where Aristotle's example is
a syllogism, but the logical premises it's purportedly based upon is
not.Once he introduced "swan and snow", I'm lost! I can see how "animals" are predicate upon the first but not the second. And I can see how this wouldn't form a syllogism as a result. However, I don't see how it's related to the examples he gave.
"Further since it is indefinite to say some C is not B, and it is true that some C is not B, whether no C is B, or not all C is B, and since if terms are assumed such that no C is B, no syllogism follows (this has already been stated), it is clear that this arrangement of terms will not afford a syllogism: otherwise one would have been possible with a universal negative minor premiss. A similar proof may also be given if the universal premiss is negative."
Since we've already analyzed the cases where the minor premise is "some C is not B" and "not all C is B", let's take a look at the final possibility of "no C in B":
4) All B is in A, no C in B. In logical notation, B
A and C B. Again,
regarding the conclusion, which must express a unique relation between
A and C, it is not uniquely deduced, since the conclusion can either be
C A
(where C B inside
of A) OR C A (where
C is partially outside of A) OR C A. Also,
the last term (C) is neither fully nor partially contained in the
middle term (B). this is
not a syllogism."Nor can there in any way be a syllogism if both the relations of subject and predicate are particular, either positively or negatively, or the one negative and the other affirmative, or one indefinite and the other definite, or both indefinite. Terms common to all the above are animal, white, horse: animal, white, stone.
It is clear then from what has been said that if there is a syllogism in this figure with a particular conclusion, the terms must be related as we have stated: if they are related otherwise, no syllogism is possible anyhow. It is evident also that all the syllogisms in this figure are perfect (for they are all completed by means of the premisses originally taken) and that all conclusions are proved by this figure, viz. universal and particular, affirmative and negative. Such a figure I call the first."
And so Aristotle laid down the conditions for one of the most powerful constructs in the foundations of logic.
Chapter 5:
"Whenever the same thing belongs to all of one subject, and to none of another, or to all of each subject or to none of either, I call such a figure the second; by middle term in it I mean that which is predicated of both subjects, by extremes the terms of which this is said, by major extreme that which lies near the middle, by minor that which is further away from the middle. The middle term stands outside the extremes, and is first in position. A syllogism cannot be perfect anyhow in this figure, but it may be valid whether the terms are related universally or not."
Sounds good. Read on.
"If then the terms are related universally a syllogism will be possible, whenever the middle belongs to all of one subject and to none of another (it does not matter which has the negative relation), but in no other way. Let M be predicated of no N, but of all O. Since, then, the negative relation is convertible, N will belong to no M: but M was assumed to belong to all O: consequently N will belong to no O. This has already been proved. Again if M belongs to all N, but to no O, then N will belong to no O. For if M belongs to no O, O belongs to no M: but M (as was said) belongs to all N: O then will belong to no N: for the first figure has again been formed. But since the negative relation is convertible, N will belong to no O. Thus it will be the same syllogism that proves both conclusions.
It is possible to prove these results also by reduction ad impossibile."
* Regarding "Since, then, the negative relation is convertible, N will belong to no M: but M was assumed to belong to all O: consequently N will belong to no O.", this can be interpreted two different ways depending upon what "belonging to all" actually means. Let's look at each case:
1) "M belongs to all O" means that M is a subset of O (M
O). Then
using logical notation, N M, M O ==> N O. This
can only be true if N
completely lies outside of O; but there are the
other
conclusions that can be deduced from the premises and which formulate a
relation between sets O and N: O N or N O (see
figures below). Thus, his example isn't a syllogism (certainly not a
perfect one) because the conclusion isn't a necessary consequence of
the premises, but merely a possible consequence. Also, with the
ordering of terms, {M, N, O}, it can't possibly follow the form of a
syllogism because the last term (O) is not contained in the middle term
(N); and even if the terms are rearranged, it doesn't satisfy the
conditions to form a syllogism. So why does he refer
to the "same syllogism that proves both conclusions"?! It could only be
an imperfect one, and if so, he would have needed to describe the
supporting statements to reach his stated conclusion._________
_______________|____ N | _____________________
| O |____ |____ | | O |
| __________ | | _______ ______ |
| | M | | | | M | | N | |
| | | | | |_______ | |_____ | |
| |__________| | | |
|____________________| |____________________|
2) "M belongs to all O" means that O is a subset of M (O
M). Then
using logical notation, N M, O M ==> N O. This
makes complete sense and meets the criteria for a syllogism (since the
middle term M belongs to all of another set O, but belongs to none of
the first set N). See the diagram below._________
____________________ | N |
| M | |____ ____|
| __________ |
| | O | |
| | | |
| |_________ | |
|____________________|
If we give Aristotle the benefit of the doubt, namely that he had impeccable logician credentials, then we must assume that the second interpretation holds true. He essentially was equating "M belongs to all O" with "M is predicated upon all O". We will use that interpretation henceforth. He would have made this clearer by using "subset" terminology (i.e. O is a subset of M); or that "M interects with ALL of O". There would have been no ambiguity in what he was describing.
* Regarding "if M belongs to all N, but to no O, then N will belong to no O.", let's analyze this using the following logical notation: N
M and M O ==>
N O.
This is quite true and meets the criteria for a syllogism. Again, see
the diagram below.___________________
| M |
| __________ | _______
| | N | | | O |
| | | | |______|
| |__________| |
|___________________|
"It is clear then that a syllogism is formed when the terms are so related, but not a perfect syllogism; for necessity is not perfectly established merely from the original premisses; others are so needed.
But if M is predicated of every N and O, there cannot be a syllogism. Terms to illustrate a positive relation between the extremes are substance, animal, man; a negative relation, substance, animal, number -- substance being the middle term."
Let's take a look at this assertion. If M is predicated of every N and O, then N
M and O M. It's true
that this doesn't lead to a syllogism, since the relationship between N
and O can't be deduced from the premises.I don't understand how his examples are related to this assertion, especially since he said that "substance" is the middle term (and if it is the middle, why didn't he place it there instead of the beginning of the trio?!).
Nor is a syllogism possible when M is predicated neither of any N nor of any O. Terms to illustrate a positive relation are line, animal, man: a negative relation, line, animal, stone.
Let's take a look at this assertion. If M is predicated neither of any N nor of any O, then M
N and M O. It's
true that this doesn't lead to a syllogism, since the relationship
between N and O can't be deduced from the premises.Again, I don't understand how his examples are related to this assertion.
"It is clear then that if a syllogism is formed when the terms are universally related, the terms must be related as we stated at the outset: for if they are otherwise related no necessary consequence follows.
If the middle terms is related universally to one of the extremes, a particular negative syllogism must result whenever the middle term is related universally to the major whether positively or negatively, and particularly to the minor and in a manner opposite to that of the universal statement: by 'an opposite manner' I mean, if the universal statement is negative, the particular is affirmative: if the universal is affirmative, the particular is negative. For if M belongs to no N, but to some O, it is necessary that N does not belong to some O. For since the negative statement is convertible, N will belong to no M: but M was admitted to belong to some O: therefore N will not belong to some O: for the result is reached by means of the first figure. Again if M belongs to all N, but not to some O, it is necessary that N does not belong to some O: for if N belongs to all O, and M is predicated also of all N, M must belong to all O: but we assumed that M does not belong to some O. And if M belongs to all N but not to all O, we shall conclude that N does not belong to all O: the proof is the same as the above. But if M is predicated of all O, but not of all N, there will be no syllogism. Take the terms animal, substance, raven; animal, white, raven. Nor will there be a conclusion when M is predicated of no O, but of some N. Terms to illustrate a positive relation between the extremes are animal, substance, unit: a negative relation, animal, substance, science."
Let's analyze these assertions:
1) Regarding "For if M belongs to no N, but to some O, it is necessary that N does not belong to some O.", this is represented by this logical notation: M
N, M O ==>
N
non-O = O N or N O. A
valid conclusion is reached with either set O being a subset of set N
or the two sets intersecting. This is still a valid syllogism because
only one
conclusion is reached (N non-O);
it's just that this conclusion can be *represented* two different ways
(N
O or N
O). 2) Regarding "For since the negative statement is convertible, N will belong to no M: but M was admitted to belong to some O: therefore N will not belong to some O: for the result is reached by means of the first figure", this is represented by this logical notation: N
M, M O ==>
N
non-O = O N or N O. Same
as #1.3) Regarding "Again if M belongs to all N, but not to some O, it is necessary that N does not belong to some O: for if N belongs to all O, and M is predicated also of all N, M must belong to all O: but we assumed that M does not belong to some O.", this is represented by this logical notation: N
M, M non-O
==> N not-O. If
we strictly interpret "N does not belong to *some* O", then we must
repudiate the possible interpretation of "N does not belong to *any*
O). Hence, by strict interpretation, N O, and a
syllogism is formed. But if we interpret "necessary" as meaning that N
must *at least* exclude some O, which includes both "some" or "any",
then either "N does not belong to some O" or "N does not belong to any
O" is an acceptible solution. In this case, there are two possible
solutions, either of which forms a syllogism.____________________ _____________________
| M | | M |
| __________ | | __________ |
| | N ___ |_____|___ | | N | ____ |____
| | | | | O | | | | | | O |
| |_____ |____| | | | |_________| |____|____|
| |_________|____| | |
|___________________| |_____________________|
4) Regarding his examples "Take the terms animal, substance, raven; animal, white, raven".
The first relation forms a syllogism:
Major premise: All animals have substance.
Minor premise: A raven is an animal.
Conclusion: A raven has substance.
The second relation doesn't form a syllogism:
Major premise: Some animals are white.
Minor premise: A raven is an animal.
Conclusion: We can't conclude that some ravens are white because the Sets of ravens and whiteness might be disjoint.
"If then the universal statement is opposed to the particular, we have stated when a syllogism will be possible and when not: but if the premisses are similar in form, I mean both negative or both affirmative, a syllogism will not be possible anyhow. First let them be negative, and let the major premiss be universal, e.g. let M belong to no N, and not to some O. It is possible then for N to belong either to all O or to no O. Terms to illustrate the negative relation are black, snow, animal. It is not possible to find terms of which the extremes are related positively and universally, if M belongs to some O, and does not belong to some O. For if N belonged to all O, but M to no N, then M would belong to no O: but we assumed that it belongs to some O. In this way then it is not admissible to take terms: our point must be proved from the indefinite nature of the particular statement. For since it is true that M does not belong to some O, even if it belongs to no O, and since if it belongs to no O a syllogism is (as we have seen) not possible, clearly it will not be possible now either."
Let's analyze these claims:
* Regarding "First let them be negative, and let the major premiss be universal, e.g. let M belong to no N, and not to some O. It is possible then for N to belong either to all O or to no O", this is represented by the logical notation:
major premise: M
Nminor premise: M
not-O
(this
implies that some of M must belong to O)conclusion: Either O
N or N O (N
either belongs to all O or to no O).The first conclusion is not possible, but I think Aristotle meant to say "it's possible for O to belong to all N". The second conclusion is possible. But it's also possible that N
O (see
figure below). Why is this possibility excluded?____________________
| O |
____ |___ _____ | ____
| M | | N |
|_______ | |__________|
| |
|____________________|
Also, for the relations "black, snow, animal", if M = black, N = snow, and O = animal, if M
N, then
no snow will be black. This is positively false, for some snow gets
very dirty!* Regarding "For if N belonged to all O, but M to no N, then M would belong to no O", the logical notation is:
Major premise: O
NMinor premise: M
NConclusion: M
O This makes perfect sense.
_____________________
| N |
| _______ | ________
| | O | | | M |
| |_______| | |________|
| |
|_____________________|
"Again let the premisses be affirmative, and let the major premiss as before be universal, e.g. let M belong to all N and to some O. It is possible then for N to belong to all O or to no O. Terms to illustrate the negative relation are white, swan, stone. But it is not possible to take terms to illustrate the universal affirmative relation, for the reason already stated: the point must be proved from the indefinite nature of the particular statement. But if the minor premiss is universal, and M belongs to no O, and not to some N, it is possible for N to belong either to all O or to no O. Terms for the positive relation are white, animal, raven: for the negative relation, white, stone, raven. If the premisses are affirmative, terms for the negative relation are white, animal, snow; for the positive relation, white, animal, swan. Evidently then, whenever the premisses are similar in form, and one is universal, the other particular, a syllogism cannot be formed anyhow. Nor is one possible if the middle term belongs to some of each of the extremes, or does not belong to some of either, or belongs to some of the one, not to some of the other, or belongs to neither universally, or is related to them indefinitely. Common terms for all the above are white, animal, man,: white, animal, inanimate."
Analyzing his propositions:
1) Premises affirmative, major premise universal.
Major premise: N
M (M belongs
to all N)Minor premise: M
O (M
belongs to some O)Conclusion: O
N
or N O. (N
belongs to all O or to no O)Huh? Clearly, the second conclusion is possible since the part of O intersecting with M might not intersect with O. But the first conclusion could only hold true if the minor premise was "M belongs to all O", because if we strictly interpret the existing minor premise "M belongs to some O", then these two sets must intersect with neither containing the other.
Regarding "Terms to illustrate the negative relation are white, swan, stone", I don't see how this can work here. If M = white, N = swan, and O = stone, certain the major premise N
M is untrue,
for not all swans are white (he probably
didn't know about black
swans). Yes, some stones are white, so the the minor premise M O is
true. And finally for the conclusion, only N O holds
true since no swan is a stone. But this example does NOT follow from
his stated logic; his argument is invalid (the set relations disprove
his argument). And regarding the converse,
assuming we were to assign M = swan, and N = white (i.e. not all things
white are swans), it certainly can't be true that N M. 2) Minor premise is universal.
Major premise: M
O. (M
belongs to no O).Minor premise: M
not-N (M
doesn't belong to some N) = M NConclusion: O
N
or N O (N
belongs to all O or to no O)This makes sense. Either conclusion works:
|
____________________ ______|______ N | | | M | _______ | | | | | O| | |______|_____ | |______| | |___________________ | |
_______ ____|____ N| ________ | | M | | | O | |___ |____| | |________| |______ | |
Regarding "Terms for the positive relation are white, animal, raven", if M = white, N = animal, and O = raven, it's not certain the major premise is true, for is M and O are disjoint, there would be no white ravens. It's not clear if white ravens actually existed (were they a mere mythological creature?), but it was probably permissible for Aristotle to assume no ravens were white because at least they weren't present in Greece. Yes, the minor premise M
not-N is
true because not all things white are animals, but "some" things are
white (M N). The
first possible conclusion O N is true
since all ravens are animals. The converse is true, assuming
we assigned N = raven, and O = animal; but again that would change the
ordering of his example. Certainly, N O is not
a possible conclusion because at least some animals are ravens (unless
he thought the raven species was extinct!)."It is clear then from what has been said that if the terms are related to one another in the way stated, a syllogism results of necessity; and if there is a syllogism, the terms must be so related. But it is evident also that all the syllogisms in this figure are imperfect: for all are made perfect by certain supplementary statements, which either are contained in the terms of necessity or are assumed as hypotheses, i.e. when we prove per impossibile. And it is evident that an affirmative conclusion is not attained by means of this figure, but all are negative, whether universal or particular."
Note: Some logical symbols:
=
inequality =
subset
=
union
=
intersection
=
set membership =
therefore
= disjoint sets