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I am reading a logic book which discussed the differences between Aristotelian Logic and Boole-Russell (modern) Logic. If the Boole-Russell logic leaves 5 valid moods out, which Aristotelian Logic covers, why do we continue to use Boole-Russell logic if it is "incomplete" per se?

October 27, 2005

 Response from Daniel J. Velleman on October 28, 2005
I'm not sure what you (or your book) are referring to when you say that modern logic "leaves 5 valid moods out". But modern logic is complete. To explain what this means, the following terminology is helpful: We say that a conclusion is a semantic consequence of a collection of premises if, in every situation in which the premises are true, the conclusion is also true. Then it is possible to prove the following statement: Whenever a conclusion is a semantic consequence of a collection of premises, it is possible to derive the conclusion from the premises using the rules of modern logic. To put it another way: If you cannot derive a conclusion from a collection of premises using the rules of modern logic, then there must be some possible situation in which all the premises would be true and the conclusion false. This is the completeness theorem of logic, proven by Godel in his doctoral dissertation in 1929.
 Response from Richard Heck on October 31, 2005

There are some syllogistic figures that at least some Aristotleans regarded as valid that are not treated as valid by modern logic. An example would be: All Fs are G; all Gs are H; therefore, some Fs are H. This is valid if, but only if, one supposes that "univeral judgements are existentially committed", as it is sometimes put, that is, if one supposes that, if "All Fs are G" is to be true, there must be some Fs. That assumption is not usually made in modern logic, and so the contemporary translation of this syllogism: ∀x(Fx → Gx); ∀x(Gx → Hx); therefore, ∃x(Fx ∧ Hx), is not valid. However, if one does think that "All F are G" is existenally committal, one can perfectly well define a new quantifier, "∀+x", that incorporates that assumption. And then the inference can be shown to be valid.

Whether the English statement "All Fs are G" is existentially committed is not for a logician (qua logician) to decide. That's an empirical question about natural language.

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