Talk:Syllogism

Valid forms

There can only be 64 moods, as four terms arranged in groups of three have only that many combinations. Of these, only ten can be valid, as follows:

• 18 are excluded by the rule that if one of the premisses is negative, the conclusion must be negative, and if one is particular, the conclusion must be particular.
• 28 are excluded by the rule that nothing can be inferred from two negatives or two particulars. A negative proposition separates the subject from the predicate, but from the fact that two things are separated it cannot be concluded that they are or are not the same. If both premisses are particular, either both are the particular affirmative, or not. If so, we cannot infer anything when the middle term is taken as a particular twice. If not, the conclusion must be negative (see above), and then one of the premisses must be universal, which contradicts the assumption.
• 6 are excluded by the rule that a negative conclusion does not follow from two affirmatives.
• I-E-O and A-E-O

If it is possible for the premisses of any syllogism to be true, and the conclusion false, then it is not a true syllogism, and the combination of propositions is called 'useless combination'.

Some combinations are useless in all three figures, some in two, and some only in one. The ones that are useless in all figures are (a) if both premisses are particular or indefinite or (b) if both are singular or (c) if both are negative. This is because, in whatever figure they occur, it is possible for the premisses to be true and the conclusion false, and so they are 'useless'.

For example, the conclusion of the first figure syllogism 'no man is a stone, no pearl is a man, therefore no pearl is a stone' is false, even though the premisses are true. The same happens in the second and third figure. So from figures made from only negative, or particulars or indefinites or singulars, nothing follows.

Some combinations are useless in only two of the figures. There are two of these. The first applies to the first and third figure, namely when the minor premiss is negative. The second applies to the first and second figure, namely when the major proposition is particular.

Reduction

Spade (See http://pvspade.com/Logic/docs/Thoughts,%20Words%20and%20Things1_2.pdf‎).

This elaborate system of “reducing” the other moods directly or indirectly to the first figure (although not of course the names of the moods, which were a mediaeval contribution) represents the first deductive system in the history of logic. And it is a truly amazing accomplishment indeed!

The dici de omni and the first figure.

"it is denoted that nothing falls under the subject without the predicate being predicated of it", and the dici de nullo when it is denoted that nothing falls under the subject without the predicate being denied of it.

http://www.logicmuseum.com/wiki/Authors/Ockham/Summa_Logicae/Book_III-1/Chapter_2

Reduction

With two exceptions (described below), valid moods not in the first figure can be shown to be valid if we assume the four non-subalternate moods of the first figure as axioms. This is done by “reducing” those other moods to one of the four axiomatic first-figure moods. And what is a “reduction”? It is an argument showing, by the rules of conversion described on pp. 18–19 above, together with one of the four axiomatic first-figure syllogisms, that the mood in question (the one to be “reduced”) is valid.

The procedure for reducing to the first figure is also secretly there in the mnemonic names. Here is the key to unpacking it: (a) The initial letter of the mnemonic name is the same as the initial letter of the axiomatic first-figure mood to which the syllogism is to be reduced. For example, Darapti will be reduced to Darii. (b) The letter ‘s’ after a vowel (but not as the last letter of the name) means: Convert the corresponding proposition simply (that is, the proposition going with that vowel). (c) The letter ‘p’ after a vowel (but not as the last letter of the name) means: Convert the corresponding proposition per accidens. (d) When ‘s’ or ‘p’ occurs as the final letter in a name, it means that the conclusion of the first figure syllogism (the “target” of the reduction) must be converted simply or per accidens, respectively. (e) The letter ‘m’ indicates that in performing the reduction you should change (= mutare) the order of the premises. (You only have to do that if you want the major premise to come first in the “target” syllogism.) (f) Unless it is the first letter of a name (when it means that the mood can be reduced to Celarent), the letter ‘c’ means that the syllogism cannot be directly reduced to the first figure, and must be given an indirect or reductio proof. (This is good old reductio ad absurdum. Do not confuse it with "reduction" in the syllogistic sense we are now defining.) The conclusion of the new syllogism will contradict (hence the ‘c’) the premise the mnemonic vowel for which is followed by ‘c’ in the name. (There are two moods like this; they are the two exceptions mentioned at the beginning of this paragraph. See below for a further explanation.) (g) The letters ‘b’ and ‘d’ (unless they are the first letters in a name, where they indicate the “target” mood in the first figure),

The two exceptional cases that cannot be directly reduced in this way to the first figure are Baroco (second figure) and Bocardo (third figure). Note the ‘c’s in their names. Since both begin with the letter ‘B’, they will both be “reduced” to Barbara in the first figure, although this new or “indirect” kind of reduction involves more than what we have seen so far. Here is the technique: Assume the A-form premise (that is, the major premise in Baroco, the minor in Bocardo). Next, assume the contradictory of the conclusion. These two assumptions will serve as the premises of a syllogism in Barbara with a conclusion contradicting the O-form premise of the original syllogism. Hence, given the premises of the original syllogism, the contradictory of the conclusion must be false, and so the original conclusion must be true. Q. E. D. Note that there is no “conversion” involved in these two exceptional cases. The proofs rely on Barbara and reductio ad absurdum alone.41

See Dici_de_omni#Applications.

a S – M a M – P a S – P

a M – P a S – M a S – P

" In the fourth figure, the middle term is the predicate of the major premise and the subject of the minor" (with 'minor' defined how?)

Rescher

Spade says “These “subalternate” moods were apparently first discussed by Ariston of Alexandria (c. 50 BC) [See Patzig, Aristotle’s Theory of the Syllogism, p. 130 n. 29. The relevant text is translated in Bocheński, A History of Formal Logic, p. 140 (§ 24.27).]; Aristotle doesn’t mention them at all”