Hypothetical Syllogistic and Stoic Logic (Philosophia Antiqua 87)

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Arguments of this mood also seem irreducible in the Alexandrian or the basic Simplician system. Sextus Empiricus states an argument of this mood by the Pyrrhonian sceptic Aenesidemus A. She comments that anyone using such a reconstruction would almost certainly come up with the analysis only after working out the complete hidden structure by another method, which as she says is a kind of cheating.

But her comment applies only to Fredes Alexandrian reconstruction, not to the one proposed here. I worked the proof out, without cheating, in the Alexandrian reconstruction. The heuristic key is to see that you need an even number of occurrences of each of p, q, r and s. But there are an odd number of occurrences of p, so you need to add another one by using the second or fourth thema. To do so without at the same time creating an odd number of occurrences of q, r or s, you have to contrapose the argument, putting the contradictory of x s x p in the conclusion of the new argument, because that contradictory follows from p along with a simple unstated premiss s.

The parenthetical remark following each mood indicates why it is invalid. Any mood with one premiss Instances of such moods are not arguments, since an argument must have at least two premisses. That they are not syllogistic can be shown by construing the logical operators as truth-functions with more than one distinguished value, in.

Milne , 47 uses the indemonstrability of arguments of this form as evidence of the incompleteness of what he calls Basic Symbolic Logic.

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But he does not use Chrysippus concept of validity as his gold standard against which to measure what is demonstrable. The exception is in the inability of the Alexandrian reconstruction to demonstrate S11 and S It is possible that the transition from the Alexandrian version to the Simplician version was motivated partly by noticing this deficiency. Such a double overlap never occurs when. Erik Krabbe pointed out in November that this mood is invalid through redundancy.

I have left the text as it was written in The Stoics had to provide for it because it occurs and because repetition counted among their premisses. In particular, it is possible to prove in the Alexandrian reconstruction proposed here the validity of arguments which Bobzien produced as a challenge, without mentally using heuristics of a Simplician sort. And there are useful heuristics, for example, that one will need the second thema or fourth thema if any simple proposition occurs an odd number of times.

Two explanations of this discrepancy immediately come to mind: perhaps Galen over-generalized, or perhaps the conjectured reconstruction of the second non-differently concluding argument is incorrect.

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Note that, if the simple proposition occurs only once, the argument is not demonstrable in the system. For each simple proposition in an undemonstrated argument occurs at least twice in it, and the themata preserve this property. Duplicated arguments are first undemonstrated arguments, and so not in need of demonstration; this fact is consistent with Alexanders lumping them in the same category of useless Stoic innovations as the second thema. The distinction Galen makes between arguments analysed using the first and second themata and arguments analysed using the third and fourth themata is the distinction between two-premissed arguments analysable in the system and multi-premissed arguments analysable in the system.

It is a plausible conjecture that some later Stoic changed the method of search for component arguments from one which focused first on how the conclusion could be proved to one which focused first on what could be inferred from some pair of premisses.

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The dialectical theorem used by Sextus Empiricus to analyse two syllogisms is an informal statement of the method licensed by the Simplician version of the themata. The second thema deals with the case where the subordinate argument shares a premiss with the superordinate argument and has just one external premiss. The third thema deals with the case where the subordinate argument shares no premiss with the superordinate argument and. The fourth thema deals with the combined case where the subordinate argument both shares a premiss with the superordinate argument and has more than one external premiss.

The proof is straightforward.

In a first undemonstrated argument, the contradictory of the conclusion is the contradictory of the consequent of the arguments leading premiss, and the conjunction of the premisses is the conjunction of a conditional and the antecedent of that conditional, which is true if and only if both the antecedent is true and the contradictory of the consequent is incompatible with the antecedent. Thus the contradictory of the conclusion cannot be true at the same time as the conjunction of the premisses; given the truth of the antecedent and the incompatibility of the These three conditions may not be sufficient for conflict, whose definition by the Stoics is not preserved.


Anicius Manlius Severinus Boethius

But they are individually necessary. In the proofs which follow, I shall take the three necessary conditions as jointly sufficient. If the three individually necessary conditions are not jointly sufficient, the proofs which follow need to be expanded to take account of whatever additional necessary conditions are required for joint sufficiency.

Further, in general, the contradictory of the conclusion does not conflict with a proper part of the conjunction of the premisses. Similarly for the other undemonstrated arguments. Further, each thema generates a Chrysippean-valid argument when it is applied to one or two Chrysippean-valid arguments. In the case of the basic first thema, for example, suppose that p, q r is Chrysippean-valid. Hence it cannot be that both propositions are true, i. Further, r does not conflict with p and r does not conflict with q.

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From the first consequence, we can infer that it cannot be that p is true, q is true, and r is true, i. Similarly for the other themata. There is however such a conflict when the conditional is true. Sextus used this fact to argue quite cogently that on Chrysippus own assumptions a first undemonstrated argument either is invalid through redundancy or has a false leading premiss A. Antipaters admission of one-premissed arguments A. The additional assumption is necessary to accommodate the case in which the connection is between q and r, which both end up in the premisses of the contraposed argument so that theoretically there might be no connection between q and r on the one hand and p on the other.

Completeness: A Chrysippean-complete system would allow one to analyse into one or more of the five Stoic indemonstrables every argument which is an instance of a mood whose component premisses and conclusion contain only propositional variables and one or more of the four logical operators of Chrysippus system and whose instances are all Chrysippean-valid arguments. That is, the mood should have at least two premisses. No instance of the mood should be able to have true premisses and a false conclusion.

Some instance of the mood should have true premisses. Some instance of the mood should have a false conclusion. In at least some instances of the mood no premiss should be redundant; that is, the contradictory of the conclusion should not conflict with any proposition which is a proper part of the conjunction of the premisses.

The requirements for completeness outlined in the preceding paragraph require a much less powerful system than the requirement imposed by some interpreters e. Becker, Kneale and Kneale that all classically valid arguments using only conjunction and negation which are expressible in the system be provable in it. Such anachronistic requirements have led to historically ungrounded reconstructions of Stoic propositional logic which are in fact unsound on Chrysippus conception of a valid argument; they allow one to prove things which are not valid arguments by Chrysippus criteria.

On the other hand, they require in some respects a more powerful system than the requirements laid down by Bobzien , ; , n. She proposes that Stoic propositional logic would be complete if in it any argument with two or more premisses is either undemonstrated or reducible to undemonstrated arguments which: 1 because of its form can never have true premisses and a false conclusion, and. The first four conditions are accommodated in my proposal in the preceding paragraph.

The fifth and sixth of these requirements seem unnecessarily restrictive.

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They seem motivated by the fact that no mood of an undemonstrated argument is wholly hypothetical and that each undemonstrated argument is self-evidently valid. But nothing in Chrysippus conception of a valid argument licenses their inclusion. Thus, they function more as excuses for the incompleteness of the system than as necessary conditions for completeness.

On the other hand, Bobzien does not take into account the requirement of a connection between premisses and conclusion which is central to Chrysippus concept of validity. Thus, she seems to impose on Chrysippus system a requirement to demonstrate, for example, ex falso quodlibet p, p q , unless this argument is not self-evidently valid or composed of self-evidently valid arguments.

Milnes counter-example RV1 shows that Chrysippus system of propositional logic is not complete, as does RV2. It is however surprisingly difficult to come up with Chrysippeanvalid moods with only propositional variables and one or more of the four logical operators which cannot be demonstrated within the system. The difficulty is surprising because the system at first glance has glaring deficiencies. You can prove valid arguments from a conditional but not arguments to a conditional, arguments to a conjunction but not from a conjunction, and.

Hypothetical Syllogistic and Stoic Logic (Philosophia Antiqua 87)

The reasons for these deficiencies are various. Valid arguments from a conjunction need no other premiss, so either are not arguments by Chrysippus criterion or are invalid because of redundancy. Valid arguments to conditionals or disjunctions require an assumption of some conflict between two propositions, an assumption which cannot be expressed simply in terms of the propositions and the logical operators of the system.

Thus, Chrysippus system of propositional logic came closer to being Chrysippean-complete than we might imagine, because of the restrictions imposed in his concept of a valid argument. It is nevertheless noteworthy that his system allows one to prove the validity of those arguments with formally valid moods expressible in the system which we are inclined to use in real reasoning and argument.

From this point of view, the Chrysippean restrictions on valid arguments may not be as unmotivated as we might suppose. Acknowledgement The present paper is essentially unaltered from a paper presented at a conference entitled 'Mistakes of Reason' in honour of John Woods held at the University of Lethbridge, April , Some corrections and additions were made in November , in response to annotations by Erik Krabbe, for which I am grateful to him. Part of this paper formed the basis of my chapter entitled The peculiarities of Stoic logic in Andrew D.

Irvine and Kent A. Peacock, eds.

Abbreviations A. Commentarium Alexander In Top. Maximiloian Wallies, Commentaria in Aristotelem Graeca vol. First published c. Alexander of Aphrodisias Maximilian Wallies, Commentaria in Aristotelem Graeca vol. Ammonius In Aristotelis De Interpretatione Commentarius, ed. Adolf Busse, Commentaria in Aristotelem Graeca vol. Apollonius Dyscolus De Coniunctionibus, in Richard Schneider, ed.

First published in the second century.