What is the difference in the meaning of equality symbol "$=$" as a logical symbol vs. a parameter?

Question

In my text, the author says that he considers $=$ to be a logical symbol, and he adds that this makes the translation of the equality symbol to English different from if $=$ were a parameter.

But the examples don't explain what the difference is.

Can anyone explain, please, what the difference is? When should we use $=$ as a logical symbol and when should we use it as a parameter?

Answer 1

See the Wikipedia entry on First-Order-Logic: Equality and its Axioms. There, you'll find distinctions in the way the equality symbol is used in logics:

• FOL (First Order Logic) with identity (where = is a primitive logic symbol), and this is what your author intends when discussing $=$ as a logical symbol, which is its most common usage in first-order-logic). As a logical symbol, its incorporation adds the "Axioms of Equality" to the deductive system one is using.

  • Reflexivity.

  • Substitution for functions.

  • Substitution for formulas.

The above are "axiom schemes," each of which specifies an infinite set of axioms. The third scheme is known as Leibniz's law, "the principle of substitutivity", "the indiscernibility of identicals", or "the replacement property". The second scheme, involving the function symbol f, is (equivalent to) a special case of the third scheme, using the formula."

Many other properties of equality are consequences of the axioms above, for example: Symmetry and Transitivity.

• FOL without identity, where the equal sign does not denote "identity".

• The entry also discusses the prospect of defining equality within a theory.

It might also be the case that your author is distinguishing $=$ as a binary predicate, from $=$ as a logical symbol.

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Perhaps you can include a definition of parameter, as given in your text, so we can help disambiguate the two uses of equality you're text is introducing.