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  "body": "\n## Statement\n\nThe well-formed terms and propositions of ordinary [[first-order logic|086QKPF0A6]] have the following syntax:\n\nTerms:\n\n- $$t \\equiv c \\mid x \\mid f (t_{1},\\dotsc, t_{n}),$$\n\nthat is, a term is recursively defined to be a constant *c* (a named object from the domain of discourse), or a variable *x* (ranging over the objects in the domain of discourse), or an *n*-ary function *f* whose arguments are terms *t*\u003csub\u003e*k*\u003c/sub\u003e. Functions map tuples of objects to objects.\n\nPropositions:\n\n- $$A, B, ... \\equiv P (t_{1},\\dotsc, t_{n}) \\mid A \\wedge B \\mid \\top \\mid A \\vee B \\mid \\bot \\mid A \\supset B \\mid \\forall x.\\ A \\mid \\exists x.\\ A,$$\n\nthat is, a proposition is recursively defined to be an *n*-ary predicate *P* whose arguments are terms *t*\u003csub\u003e*k*\u003c/sub\u003e, or an expression composed of logical connectives (and, or) and [[quantifiers|0FYQMKFE77]] (for-all, there-exists) used with other propositions.\n\nAn **atomic formula** or **atom** is simply a predicate applied to a tuple of terms; that is, an atomic formula is a formula of the form *P* (*t*\u003csub\u003e1\u003c/sub\u003e ,…, *t*\u003csub\u003e*n*\u003c/sub\u003e) for *P* a predicate, and the *t*\u003csub\u003e*n*\u003c/sub\u003e terms.\n\nAll other well-formed formulas are obtained by composing atoms with logical connectives and quantifiers.\n\nFor example, the formula ∀*x. P* (*x*) ∧ ∃*y. Q* (*y*, *f* (*x*)) ∨ ∃*z. R* (*z*) contains the atoms\n\n- $$P (x)$$\n- $$Q (y, f (x))$$\n- $$R (z).$$\n\nAs there are no quantifiers appearing in an atomic formula, all occurrences of variable symbols in an atomic formula are free.\n",
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