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I was reading Anthony Andres' article, "ARISTOTLE AND THE CONVENTIONAL LOGICIANS ON THE FOURTH FIGURE".

The author explains why according to him the introduction of the fourth figure by his students and in today's logic books is a misinterpretation of Aristotle's view on syllogisms.

The author remarks that nothing is wrong with today's different interpretation given to the syllogism. In fact it arises naturally from our current, developed understanding of logic. So it's even more correct.

The article ends with a quote from John Locke on his consideration of the value and importance of the syllogism, which, under its current interpretation, had lost its deductive power:

Hence it is that men, in their inquiries after truth, never use syllogisms .... Because before they can put [ideas] into a syllogism, they must see the connection that is between the intermediate idea and the two other ideas it is set between and applied to, to show their agreement; and when they see that, they see whether the inference be good or no; and so syllogism comes too late to settle it.

So now I'm asking myself, is the syllogism to be considered (still considered) an inference rule?

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Not exactly.

We can consider the propositional valid argument called Hypothetical syllogism as a (derived) rule of inference.

We call it "derived", because in standard presentations of propositional logic we can derive it from more basic ones, like Modus Ponens.

In modern terms, syllogism is a fragment of first-order logic, the so-called Monadic predicate calculus.

But we can present logic in "rules only" form; see Natural Deduction.

In this case, we can re-write A's syllogistic figures as rules of a modern calculus.

See: Jan von Plato, Elements of Logical Reasoning, Cambridge UP (2013), Ch.14.1 Aristotle’s deductive logic.


For modern studies, see at least:

For a good overview, see:

  • Hi! Thank you always for you answer Mauro! I'm taking a course in logic and I came up on that article. So, I didn't get the distinction. What makes the hypothetical syllogism an inference rule? and What makes the categorical syllogism not an inference rule? Maybe we should start with what exactly is a rule of inference and then derive what is not, but I looked it up on wikipedia and it didn't clarify at all – Gabriele Scarlatti Oct 19 '17 at 11:26
  • @GabrieleScarlatti - very useful: John Corcoran, Three logical theories (1969). We may have rules only, like Natural Deduction, or axioms+rules, like in Hilbert-style systems. In this second case we have many different combinations, but at least one rule; usually Modus Ponens. – Mauro ALLEGRANZA Oct 19 '17 at 11:35
  • An inference rule has the form: from blah blah, derive buh buh. Having propositional axioms an MP, we can derive theorems (the tautologies); one is: (P→Q)→((Q→R)→(P→R)). With two applications of *MP we can immediately derive from it the "derived rule": (P→Q), (Q→R) ⊢ (P→R). – Mauro ALLEGRANZA Oct 19 '17 at 11:38
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"So now I'm asking myself, is syllogism to be considered (still considered) an inference rule?"

What exactly Locke means to have us consider is not in every way obvious. Generally speaking, the problem is that one must make sure what you put in the syllogistic form is sound. This is surely the general meaning of Locke's comment. The move from the world, to the formal machine for deriving syllogisms, he perhaps says, is further in trouble because we do not know if the human mind is sound. The mind that makes the leap, draws the inference. Or, put another way, that inference as such is a sound thing.

The Symbolic Logic people got out of the real problems, so to speak, by claiming to be doing something sheerly "logical" rather than psychological. That's a bit like the question of non-euclidean maths. They were historicaly held in the grips of a powerful suspicion, and thereby suppressed, since they didn't correspond to ordinary experience. But it turned out the most imaginary maths could be made to return practical results. Something like that is true in the modern "logics" (which are really forms of math), insofar as Locke's worries are set aside in the face of the support practical applications offer the new "logics".

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The classical syllogism was a linguistic entity, and mathematical logic did not exist until the period 1845-1850.

The classical syllogism refers to a form of an argument depending on the placement of the terms. The placement of the terms made up what is called the argument figure.

Mood is another term that referred to the kind of propositions that were used in the syllogism. So when we speak about classical syllogisms we are expressing a type of argument that has repeating patterns. The patterns are the placement of terms. The placement of terms expresses the figure of the argument pattern which can be used to determine the validity of the argument along with the mood.

Mood refers to the type of proposition: universal affirmative, universal negative, particular affirmative, and particular negative. Certain moods with figure are known to be valid regardless of the terms. So studying syllogisms requires some concepts and terminology that is not emphasized in maths. I would say the context is different and the purpose of syllogistic logic is different from symbolization.

Propositional logic is supposed to carry on where the linguistics style left off and add things that using syllogisms alone would be difficult to express. We can say the same thing why we need predicate logic over propositional logic. Each step is supposed to be backward compatible and add something the previous system could not express easily.

All of that above is different from an inference rule. An inference rule is a set of steps that guarantee the certainty of the conclusion provided the premises are true. In this way the ideal thinking should begin with true premises and end with true conclusions. It is not acceptable to start with truth and end with a false conclusion.

  • Thank you for your answer! But I didn't get if the syllogism is an inference rule or not...it's not clear...If it's not, why it isn't? What makes it different from an inference rule? Providing some examples would be useful to the purpose of explaining – Gabriele Scarlatti Oct 24 '17 at 7:58
  • An inference rule is a method to go from true proposition to another and different true proposition. A classical syllogism is a pattern of argument that is classified as I stated above by mood and figure. As I stated before mathematical logic did not exist during the times of Aristotle. – Logikal Oct 24 '17 at 13:11
  • According to wikipedia, rules of inference are logical forms that "typically preserve truth"...this means that 1) there could be some rules of inference that doesn't preserve truth (?) 2) the 15 aristotelian valid syllogisms can be considered rule of inferences – Gabriele Scarlatti Oct 24 '17 at 14:25
  • All inference rules preserve truths. Arisrtotellian forms are lost in the translation of symbols. So no the valid forms are not incerence rules themselves. The forms have truth tables before the creation of mathematical logic. The best symbolization can do is sxpress equivalent propositions to the original. This is not equal. – Logikal Oct 24 '17 at 14:58
  • thank you for your help! by the way... what do you mean by "forms are lost in translation"?....."best symbolization you can do is express equivalent proposition"? – Gabriele Scarlatti Oct 24 '17 at 15:46
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Locke's views on the syllogism are nuanced. out of four possibilities he selects two roles that it can fulful. Both are inferential :

He does not totally reject the deductive mode of reasoning. In fact, such a rejection would have been inconsistent with his recommendation of mathematics as the appropriate model for scientific inquiry.

But in Chapter 17 of Book IV, entitled "Of Reason," he does discredit the syllogism, the Aristotelian paradigm of deductive reasoning. His attack on the syllogism is grounded on his notion of the four degrees of reasoning: "the first and highest is the discovering and finding out of proofs; the second, the regular and methodical disposition of them and laying them in a clear and fit order to make their connexion and force be plainly and easily perceived; the third is the perceiving their connexion; and the fourth, the making a right conclusion" (IV, xvii, 3). Fundamentally, he contends that the syllogism applies only to the third and fourth of these degrees and that even there, the syllogism was not so much a means of establishing the connections between propositions as a device for testing the connections. Those who are interested in a more detailed summary of Locke's attack on the syllogism can consult pp. 285-289 of Wilbur Samuel Howell's Eighteenth-Century British Logic and Rhetoric. (Edward P. J. Corbett, 'John Locke's Contributions to Rhetoric', College Composition and Communication, Vol. 32, No. 4 (Dec., 1981), pp. 423-433 : 428.)

If inference is reasoning from premises to conclusion ('To infer is nothing but by virtue of one Proposition laid down as true, to draw in another as true, i.e. to see or suppose such a connexion of the two Ideas, of the inferr'd Proposition' : Locke, An Essay concerning Human Understanding, IV.xvii.4), then there appears to be nothing in Locke's text, quoted here, that rules out 'the perceiving of their connexion' and 'the making a right conclusion' as inferential.


Reference

John Locke, An Essay Concerning Human Understanding, ed. P. Nidditch, ISBN 10: 0198245955 / ISBN 13: 9780198245957 Published by Oxford University Press, USA, 1979.

Edward P. J. Corbett, 'John Locke's Contributions to Rhetoric', College Composition and Communication, Vol. 32, No. 4 (Dec., 1981), pp. 423-433 : 428.

Wilbur Samuel Howell,Eighteenth-Century British Logic and Rhetoric, ISBN 10: 069106203X / ISBN 13: 9780691062037 Published by Princeton University Press, 1971.

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