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

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?

• It might be helpful if you gave more context, as well as some of those examples. – vadim123 Jun 25 '13 at 1:46
• @vadim123 , there is no example in the text about the equality symbol to give . just say that equality is a symbol of first order language of set theory but not a symbol of first order language of number theory . – Fawzy Hegab Jun 25 '13 at 1:49

## 1 Answer

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.

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.

• the text is , Mathematical Introduction to Logic , by Herbert Enderton , there is no definition for the word parameter in the text . i now understand what the author mean by equality in the context ( FOL with Identity ) . cont. – Fawzy Hegab Jun 25 '13 at 2:08
• but in FOL without identity , $=$ is a $2$-place predicate,so, what does $x=y$ mean in this case? – Fawzy Hegab Jun 25 '13 at 2:12
• Yes, I suspect that Enderton may be distinguishing the use of $=$ as a binary predicate, from its use as a logical symbol. – Namaste Jun 25 '13 at 2:13
• yes , it seems he does such distinguishing , but when we consider $=$ as predicate , here is it like and other symbol ? so it requires a definition to what does it mean ? or it already has a specific meaning by convention ? – Fawzy Hegab Jun 25 '13 at 2:16
• I think the predicate = applies strictly to members of a domain, and is a relation. It also returns the truth value of the predicate $a = b$. Either it is true that to members a, b are such that a = b, true, or not (false)...some properties of predicate equality are shared by logical equality, but not vice versa. – Namaste Jun 25 '13 at 2:18