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glopez
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 Posted - Aug 02 2004 : 1:00:45 PM
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It seems that there are three so-called "laws of logic" that are touted as being fundamental to logic itself. These are the laws of noncontradiction, the excluded middle, and identity. Are there formal systems of sentential (or predicate, or another)logic which do not include the laws of noncontradiction, identity, and the excluded middle, as axioms? If so, can these laws be derived within these systems?
Thanks in advance. |
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Tom Trelogan
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Posted - Aug 02 2004 : 7:51:32 PM
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First of all, there are a great many systems of sentential logic and predicate logic that employ no axioms at all, but that use, in their place, so-called “natural deduction” rules, the advantage being that such rules are in practice much easier to use than axioms when it comes to constructing derivations.
But to address the heart of your question, while a great many systems of predicate logic do include all three of the laws you’ve mentioned among their theorems even if not as axioms, there are also systems that don’t include them all, either as axioms or as theorems.
Systems of sentential logic never include the law of identity, unless one wants to count such formulas as “P ↔ P” as expressions of
the law of identity for truth values, and there are many formulations of predicate logic in which nothing like “(x)x=x” is either an axiom or a theorem—namely, systems of first order predicate logic without identity.
And one can have systems of sentential logic and predicate logic that lack at least the law of excluded middle as well. The man from whom I first learned logic when I was myself in school, Frederic Brenton Fitch, favored systems in which it was impossible to derive the law of excluded middle, this being his preferred way of avoiding the implications of the famous Russell paradox. The price of avoiding those implications in this way is, of course, weakening considerably one’s system’s deductive apparatus. Still, in this way, Fitch was able to preserve the possibility of allowing self-reference, which is ruled out by Russell’s own preferred way of avoiding the implications of his paradox, and this seemed attractive to Fitch because of the multitude of contexts within the history of philosophy in which philosophers have been interested in formulating problems that can’t be formulated at all if one adopts Russell’s preferred strategy.
Hope this helps....
Tom Trelogan
PS: Welcome to the Philosophy Forums!
quote:
Originally posted by glopez
It seems that there are three so-called "laws of logic" that are touted as being fundamental to logic itself. These are the laws of noncontradiction, the excluded middle, and identity. Are there formal systems of sentential (or predicate, or another) logic which do not include the laws of noncontradiction, identity, and the excluded middle, as axioms? If so, can these laws be derived within these systems?
Thanks in advance.
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The Laws of Logic
