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There are a lot of questions and answers on what the difference a predicate and a function in predicate logic is on this website, but there is no question/answer on what the difference between a predicate symbol and a function symbol is.

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    $\begingroup$ There's no difference in the symbols themselves. The difference is in what you have decided to use them for. $\endgroup$ Mar 7 '17 at 22:30
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    $\begingroup$ A predicate is a function that maps entities/terms to truth-values/claims rather than to other terms. $\endgroup$ Mar 7 '17 at 23:36
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Predicate logic distinguishes between terms (formal expressions denoting elements of the domain of discourse, e.g., addition of numbers in arithmetic) and predicates (formal expressions denoting relations amongst elements in domain of discourse, e.g., the less-than relation in arithmetic).

A predicate symbol is an operator that combines terms and produces a predicate. E.g., in arithmetic $=$, $<$ and $>$ are predicate symbols.

A function symbol is an operator that combines terms and produces a new term. E.g., in arithmetic $+$ and $\times$ are function symbols.

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Once we interpret the symbols:

A function applied to one or more objects gives you another object, while a predicate applied to one or more objects gives you a claim

For example, under the standard interpretation for the language of arithmetic:

$1+2$ gives you the object $3$ (and $1+2$ is not a claim)

$1<2$ gives you the claim that $1$ is smaller than $2$ (and $1<2$ is not an object from the domain)

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