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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.

Ali Caglayan
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1 Answers1

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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.

Peter Smith
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  • 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? –  Dec 30 '14 at 11:07
  • Also I have read and tried with books but I really struggle with predicate logic and I need to learn intuitively I think. –  Dec 30 '14 at 11:10
  • '$f(x, y)$' 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. – Peter Smith Dec 30 '14 at 11:17
  • 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? –  Dec 30 '14 at 11:25
  • Yes ... "arity" talk is just fancy talk for the number of places! :-) – Peter Smith Dec 30 '14 at 11:37
  • oh ok :) I am starting to really understand this topic now after struggling with it for weeks so thanks for your help :) Above you said that "....p(x)=y is ill formed because you can only plus terms into the identity predicate...." What does that mean and is that the reason F(x,x)=y is not a term? –  Dec 30 '14 at 11:51
  • Oops sorry that was a typo too late to correct now: read "you can only plug terms ...". – Peter Smith Dec 30 '14 at 14:06
  • $f(x,x)=y$ is formed by plugging two terms, $f(x,x)$ and $y$ into a two-place predicate (not into another term), so you get an (open) sentence. – Peter Smith Dec 30 '14 at 14:08