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November 9, 2009

1:35 AM

The conclusion of a modal syllogism does not always just follow the modality of the weakest premise

In my first ten blog posts, I have endeavored to give readers a quick cross-section of my philosophy, ranging over a variety of subjects. In the present posting, and the next few ones, I aim to showcase some of the important discoveries in logic theory to be found in my works.

In chapter 63, section 2 of Future Logic, discussing modal syllogism, I point out that Aristotle apparently allowed 'drawing a possible conclusion from a possible major premise in the first figure'. This is suggested, for instance, in his Prior Analytics, book I, chapter 14, where he says: "Whenever A may possibly belong to all B, and B to all C, there will be a perfect syllogism to prove that A may possibly belong to all C". (Actually, he has previously defined "possibility" as what we would call contingency, i.e. as including possibility-not, but the result is the same.) Likewise, Theophrastus of Eresus, Aristotle’s successor as head of the Lyceum, is reported to have taught that “the conclusion follows the modality of the weakest premise” as a general principle. This was a serious error of logic on their part (that to my knowledge no one has since corrected).

In chapter 15 of Future Logic, I on the contrary class as logically invalid to first figure syllogisms of the form:

All M can be P (or nonP)

All/This/Some S can (or must) be M

ergo, All/This/Some S can be P (or nonP)

I expose this invalidity in section 3 of that chapter, saying that we cannot be sure that the circumstances referred to by the major premise include those referred to by the minor premise. The rule for modality here is similar to the rule for quantity that the middle term must be distributive. (This is said by me with regard to the ‘circumstances’ underlying natural modality, but the same applies equally well to the ‘contexts of knowledge’ underlying logical modality, which is seemingly more the intent of Aristotle’s discourse.)

I further explain this in chapter 17 of the same book, where I deal with “transitive categorical propositions”, which concern change either in the sense of alteration, e.g. “This egg has hardened (gotten to be hard)” or in the sense of mutation, e.g. “This soft egg has become hard (or a hard egg)”, as against attributive propositions such as “This egg is soft (or is hard)”. The following is an extract from section 4 of that chapter.

a. When both premises are attributive, an attributive conclusion can only be drawn (if at all) from a necessary major premise; if the major is potential, we cannot draw an attributive conclusion. However, a transitive conclusion can be drawn, showing that the modality may have an impact on the copula. The following moods are valid:

All M can be P (or nonP)

All/This/Some S can (or must) be M

ergo, All/This/Some S can get to be or become P (or nonP)

We know from the premises that what started as S, will in some circumstances be S and M; and that whatever is M, will in some further circumstances be M and P (or nonP); but be cannot predict whether the end result of this process includes or excludes S. It is conceivable that S stays on with P (or nonP), but it is also conceivable that S disappears prior to the arrival at P (or nonP). For this reason, our conclusion cannot be merely 'S can be P (or nonP), but must be open to the 'S can become P (or nonP)' outcome.

b. The same can be argued with a mutative major premise, whatever the modalities involved. Thus, the following are valid:

All M can (or must) become P (or nonP)

All/This/Some S can (or must) be M

ergo, All/This/Some S can get to be or become P (or nonP)

If one or both premises are potential, so is the conclusion, as above; but if both premises are necessary, a necessary conclusion can be drawn, as below:

All M must become P (or nonP)

All/This/Some S must be M

ergo, All/This/Some S must get to be or become P (or nonP)

c. In cases where the minor premise is mutative, whether the major is attributive or mutative, similarly disjunctive conclusions may be drawn. We know from the minor premise that S will disappear to become M, but we cannot be sure whether, in the circumstances when M is or becomes P, S reappears or stays away.

Note that in all the cases so far considered, we could view the conclusion as a logical disjunction as we did, or we could say that a more specific conclusion can be drawn if we know one or the other alternative to be excluded. This would be equivalent to having a third premise, viz. 'S cannot get to be P' or 'S cannot become P'. But formally speaking, this constitutes an additional argument (apodosis) after the syllogism as such.

d. All the above only concerns cases with both premises affirmative (whether the predicate be P or nonP). Now, the minor cannot be negative in the first figure, but what if the major premise is negative (in the sense of negating the copula, not merely the predicate)? In such case, we cannot draw a likewise negative conclusion, because we can construct a syllogism with compatible affirmative premises yielding a conflicting affirmative conclusion. Thus, for example, the following mood is invalid:

No M can become P

This S must be M

ergo, This S cannot get to be or become P

This is invalid, because it is conceivable that, though no M can become P, all M can nonetheless be P, in which case the following syllogism could be constructed, as earlier established:

All M can be P

This S must be M

ergo, This S can get to be or become P

We could interpret this to mean that, a (compound) negative conclusion is possible, if the negative major is compound, as in the following mood. Note that major premise and conclusion are conjunctions, not disjunctions, of negatives. The result is due to the attachment of S to M.

No M can be or become P

This S must be M

ergo, This S cannot get to be or become P

If either or both of these premises were potential instead of necessary, a potential conclusion would be drawn.

For more details on this topic, see FUTURE LOGIC, CHAPTER 17.

http://www.thelogician.net/2_future_logic/2_chapter_17.htm

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