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Equational logic
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First-order equational logic consists of quantifier-free terms of ordinary first-order logic, with equality as the only predicate symbol. The model theory of this logic was developed into universal algebra by Birkhoff, Grätzer, and Cohn. It was later made into a branch of category theory by Lawvere ("algebraic theories").[1]
The terms of equational logic are built up from variables and constants using function symbols (or operations).
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Syllogism
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Here are the four inference rules of logic. denotes textual substitution of expression for variable in expression . Next, denotes equality, for and of the same type, while , or equivalence, is defined only for and of type boolean. For and of type boolean, and have the same meaning.
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Proof
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We explain how the four inference rules are used in proofs, using the proof of [clarify]. The logic symbols and indicate "true" and "false," respectively, and indicates "not." The theorem numbers refer to theorems of A Logical Approach to Discrete Math.[2]
First, lines – show a use of inference rule Leibniz:
is the conclusion of Leibniz, and its premise is given on line . In the same way, the equality on lines – are substantiated using Leibniz.
The "hint" on line is supposed to give a premise of Leibniz, showing what substitution of equals for equals is being used. This premise is theorem with the substitution , i.e.
This shows how inference rule Substitution is used within hints.
From and , we conclude by inference rule Transitivity that . This shows how Transitivity is used.
Finally, note that line , , is a theorem, as indicated by the hint to its right. Hence, by inference rule Equanimity, we conclude that line is also a theorem. And is what we wanted to prove.[2]
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