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 for first-order logic
From The Free On-line Dictionary of Computing (30 December 2018) :

  first-order logic
      The language describing the truth of
     mathematical formulas.  Formulas describe properties of
     terms and have a truth value.  The following are atomic
      p(t1,..tn)	where t1,..,tn are terms and p is a predicate.
     If F1, F2 and F3 are formulas and v is a variable then the
     following are compound formulas:
      F1 ^ F2	conjunction - true if both F1 and F2 are true,
      F1 V F2	disjunction - true if either or both are true,
      F1 => F2	implication - true if F1 is false or F2 is
     		true, F1 is the antecedent, F2 is the
     		consequent (sometimes written with a thin
      F1 <= F2	true if F1 is true or F2 is false,
      F1 == F2	true if F1 and F2 are both true or both false
     		(normally written with a three line
     		equivalence symbol)
      ~F1		negation - true if f1 is false (normally
     		written as a dash '-' with a shorter vertical
     		line hanging from its right hand end).
      For all v . F	universal quantification - true if F is true
     		for all values of v (normally written with an
     		inverted A).
      Exists v . F	existential quantification - true if there
     		exists some value of v for which F is true.
     		(Normally written with a reversed E).
     The operators ^ V => <= == ~ are called connectives.  "For
     all" and "Exists" are quantifiers whose scope is F.  A
     term is a mathematical expression involving numbers,
     operators, functions and variables.
     The "order" of a logic specifies what entities "For all" and
     "Exists" may quantify over.  First-order logic can only
     quantify over sets of atomic propositions.  (E.g. For all p
     . p => p).  Second-order logic can quantify over functions on
     propositions, and higher-order logic can quantify over any
     type of entity.  The sets over which quantifiers operate are
     usually implicit but can be deduced from well-formedness
     In first-order logic quantifiers always range over ALL the
     elements of the domain of discourse.  By contrast,
     second-order logic allows one to quantify over subsets.
     ["The Realm of First-Order Logic", Jon Barwise, Handbook of
     Mathematical Logic (Barwise, ed., North Holland, NYC, 1977)].

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