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What is the difference between terms and atomic formulae? I get contradicting advice from everywhere. On one hand I have got written if $t_1,\ldots,t_n$ are terms and if $F$ is a function symbol with arity $n$ then $F(t_1,\ldots,t_n)$ is a term.

Then on the other hand I have been told that if $F$ is a function symbol with arity $2$ and given $x$ and $y$ are variables, then $F(x,y)$ is an atomic formulae.

Are terms and atomic formulae the same of different?

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  • $\begingroup$ This probably depends on whose system of logic you are using. In the ones I am most familiar with, a term is like a noun: an object or number, etc. A formula is a sentence: Sam touched the ball. In that formula, Sam and the ball are terms and F(A,B) = A touched B is a sentential formula. $\endgroup$
    – robjohn
    Jan 19, 2015 at 16:19
  • $\begingroup$ I think this could be clarified if the definition of formula, predicate and term where written out. $\endgroup$ Oct 15, 2022 at 19:58

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Warning: do not mix up terms, variables, constants, functions and predicates.

In First-Order Logic, terms function as names, and well-formed formulas (atomic or not) as statements. There is a neat inductive definition of a term:

  1. A single occurrence of a variable is a term;
  2. A single occurrence of a constant symbol is a term;
  3. If $f$ is an $n$-ary function (meaning that it takes $n$ arguments), and $t_1$, $t_2$, ..., $t_n$ are terms, then $ft_1t_2\ldots t_n$ is a term.

Now, the direct definition of an atomic formula:

  • If $P$ is an $n$-ary predicate symbol and $t_1$, $t_2$, ..., $t_n$ are terms, then $Pt_1t_2\ldots t_n$ is an atomic formula.

These two definitions look a bit similar (just a bit, though), so let me make it very clear:

  • A TERM is an ELEMENT OF THE LANGUAGE (by its inductive definition, it can be as simple as a single occurrence of a constant or a variable, or arbitrarily complicated if it is formed with a function).
  • An ATOMIC FORMULA is a LOGICAL STATEMENT, i.e. one to which you can assign a TRUTH-VALUE. From atomic formulae, you can build up, using logical connectives, any other formula.
  • A FUNCTION takes $n$ terms and is evaluated as a NEW TERM.
  • A PREDICATE takes $n$ terms and is evaluated as a TRUTH VALUE.
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  • $\begingroup$ Thank you, I understand things much clearly. But what about F(x,y) then. It is still an atomic formula because it is a term and terms can be atomic formula because terms are the building blocks of formula? $\endgroup$
    – user204450
    Jan 19, 2015 at 16:43
  • $\begingroup$ If $F$ is a function and $x,y$ are terms, then $F(x,y)$ is a term. This notation (with the brackets) is not the standard notation in logic. It looks more like something you would write in maths. And in maths, a function $F(x,y)$, with $x,y$ some numbers, returns another number (a function of terms returns a new term). $\endgroup$
    – Demosthene
    Jan 19, 2015 at 16:45
  • $\begingroup$ Why is the equality symbol an atomic formula? $\endgroup$
    – user204450
    Jan 19, 2015 at 16:48
  • $\begingroup$ However, if $F$ is a 2-place predicate, say $F(x,y)$: "$x$ is greater than "y", and $x,y$ are terms, then $F(x,y)$ is an atomic formula, and will return a truth-value (true if indeed $x>y$, false otherwise). $\endgroup$
    – Demosthene
    Jan 19, 2015 at 16:48
  • $\begingroup$ Because the equality symbol is a 2-place predicate. It takes two terms, checks a property, and returns true/false. When you say $x=y$, you mean that "it is true that $x$ equals $y$". You wouldn't use $x=y$ as a number (read, a term). $\endgroup$
    – Demosthene
    Jan 19, 2015 at 16:50

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