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If $P$ is a predicate symbol with arity $1$, why is $P(x)$ not a term?

Is it because terms can only consist of function symbols applied to variables? i.e $F(x)$ where $F$ is a function symbol of arity $1$ and $x$ is a variable.

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  • $\begingroup$ Such a P(x) is usually called a "formula". Terms stand for objects, formulas for sentences (true/false). $\endgroup$
    – coffeemath
    Dec 30, 2014 at 10:59
  • $\begingroup$ Thank you @ Peter smith :) (sorry I cant comment under your comment) $\endgroup$
    – user99865
    Dec 30, 2014 at 14:09

1 Answer 1

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The basic principle: think of terms as denoting expressions whose role is to pick out objects in the domain (relative to a given assignment of objects as denotations to any lurking free variables). Terms are not sentences, i.e. not expressions which are true or false (relative to an assignment, etc.) but rather the most basic kind of sentence is formed by taking an $n$-place predicate and applying it to $n$ terms.

Thus an expression like '$2$' or '$x$' counts as a simple term, and function expressions like '$2 +3$' (or '$f(2, 3)$') and $f(x, y)$ count as terms too. The term '$2 +3$' is plainly not a sentence (it isn't the sort of thing that can be true or false). But it can feature as part of a sentence if we supply a predicate as in (informally) '$2 + 3$ is odd' or '$2 + 3 = 5$' (or, if it helps, '$=(2 +3, 5)$').

On the other hand, an expression like $P(2)$, with $P$ a one-place predicate, is sentence (a 'closed wff') not a term, and $P(x)$ is is an 'open sentence' in the jargon and again not a term.

This should be all explained in any standard text.

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  • $\begingroup$ Thanks for your explanation. It is really good and intuitive. (I asked this question before but I did not understand) If F(x,y) is a term then how can F(x,y)=x be an atomic formula? And why is P(x)=y not a atomic formula? $\endgroup$
    – user99865
    Dec 30, 2014 at 11:07
  • $\begingroup$ Also I have read and tried with books but I really struggle with predicate logic and I need to learn intuitively I think. $\endgroup$
    – user99865
    Dec 30, 2014 at 11:10
  • $\begingroup$ '$f(x, y)$' is a term; '$x$' is a term. '$=$' is a two-place predicate, which we usually write infix, i.e. between a couple of terms. Plug two terms into a two-place predicate and you get an atomic formula. Thus '$f(x, y) = x$' is an atomic wff, an atomic (open) sentence. '$P(x) = y$' is ill-formed because you can only plus terms into the identity predicate, and '$P(x)$', as explained, is not a term. $\endgroup$ Dec 30, 2014 at 11:17
  • $\begingroup$ Thank you, Great answer and I really am starting to understand it much better. When you say "is a two place predicate" do you mean it has arity 2? $\endgroup$
    – user99865
    Dec 30, 2014 at 11:25
  • $\begingroup$ Yes ... "arity" talk is just fancy talk for the number of places! :-) $\endgroup$ Dec 30, 2014 at 11:37

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