If so, does it mean we should always use quantificational logic since it combines both syllogistic and proportional logic? Also, Is there any other logic which further builds on quantificational logic?

  • There are many logics out there, and none of them replace the others-- the choice of which logic is most appropriate depends on the context. What is the context of your asking the question? Do you have a proof of some kind that you would like to formalize? – Michael Dorfman Oct 15 '12 at 11:04
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    @MichaelDorfman: Actually, I was reading this page on syllogism It says that "syllogism was superseded by first-order predicate logic" So, I was wondering if that is right. – user102313 Oct 15 '12 at 11:11
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    Yes, that also says "but syllogisms remain useful", and "[citation needed]"; as always, take pronouncements on Wikipedia with a grain of salt. – Michael Dorfman Oct 15 '12 at 11:29
  • I think that wiki page is referring to the fact that for a long time (basically until Frege) philosophers thought that all propositions could be expressed in syllogistic form, if only one worked hard enough at it. That is no longer the belief. – Xodarap Oct 17 '12 at 17:41

Syllogistic logic is one of the things that one does with quantificational logic, just as one occasionally works with arithmetic of integers when working in real analysis or group theory (both of which generalize integer arithmetic). We do not take special note of the structure of syllogisms, because they are only a special case of the usual rules of inference; because we have more flexible tools, we do not put as much effort into memorizing what syllogistic formulae are valid or invalid, except as examples of valid or invalid reasoning in general.

In the following, I will let x:A denote that an object x takes a property A.

All A are B: ∀x: (x:A) ⇒ (x:B)

Some A are B: ∃x: (x:A) & (x:B);

Some A are not B: ∃x: (x:A) & ¬(x:B);

No A is B: ∀x: (x:A) ⇒ ¬(x:B).

Syllogistic reasoning can be performed as it was traditionally, albeit with extra instantiation and generalization steps, and often using rules of inference such as modus ponens and modus tollens. Here are two typical examples of syllogistic reasoning, where the first two lines are classic syllogistic premisses, and the final line is the classic syllogistic conclusion.

Example 1.

P1. ∀x: (x:Human) ⇒ (x:Mortal)
Premise — gloss: all humans are mortal

P2. Socrates:Human
Premise — gloss: Socrates is human

UI. (Socrates:Human) ⇒ (Socrates:Mortal)
universal instantiation — gloss: if Socrates in particular is human, then he is mortal

MP. Socrates:Mortal
modus ponens — gloss: therefore, Socrates is mortal

Example 2.

P1. ∀x: (x:Rabbit) ⇒ (x:Furry)
Premise — gloss: all rabbits have fur

P2. ∃x: (x:Rabbit) & (x:Pet)
Premise — gloss: some rabbits are pets

EI. (Flopsy:Rabbit) & (Flopsy:Pet)
existential instantiation — gloss: consider Flopsy, a (generic) pet rabbit

UI. (Flopsy:Rabbit) ⇒ (Flopsy:Furry)
universal instantiation — gloss: if Flopsy in particular is a rabbit, then Flopsy is furry

MP. Flopsy:Furry
modus ponens (with a conjunctive simplification) — gloss: therefore, Flopsy is furry

CI. (Flopsy:Pet) & (Flopsy:Furry)
conjunctive introduction (with a conjunctive simplification) — gloss: therefore, Flopsy is a furry pet

EG. ∃x: (x:Pet) & (x:Furry)
existential generalization — therefore, some pets have fur

Syllogistic reasoning is therefore a special case of quantificational logic; it is a framework in which one can confidently reason syllogistically, without being limited only to syllogisms. But one should not feel absolutely restricted to using quantificational logic, or any particular logic; it suffices to use some logic which is both effective and reliable for coming to usable conclusions.

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