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?

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.

There are some syllogistic figures that at least some Aristotleansregarded as valid that are not treated as valid by modern logic. Anexample 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 "univeraljudgements are existentially committed", as it is sometimes put, thatis, if one supposes that, if "All Fs are G" is to be true, there mustbe some Fs. That assumption is not usually made in modern logic, and sothe contemporary translation of this syllogism: ∀x(Fx → Gx); ∀x(Gx →Hx); therefore, ∃x(Fx ∧ Hx), is not valid. However, if one does thinkthat "All F are G" is existenally committal, one can perfectly welldefine a new quantifier, "∀+x", that incorporates that assumption. Andthen 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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