# first-order logic

Also found in: Acronyms, Wikipedia.

## first-order logic

(language, logic)
The language describing the truth of mathematical formulas. Formulas describe properties of terms and have a truth value. The following are atomic formulas:

True False 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 arrow),

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 constraints.

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)].
This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)
References in periodicals archive ?
Insofar as the teaching of argumentation is implemented by means of formal logic, one of the instruments would be a textbook on first-order logic. There are a number of textbooks that fill this role, and they are in the business of presenting their subject matter in as accessible a way for a non-mathematical audience as possible.
A key point of this article is that current knowledge representation (KR) formalisms, particularly those of first-order logic, are good for specifying the space of conjectures.
Some members of classical logic which are still being used up until now are Propositional Logic [22,23,24], First-Order Logic [25,26,27,28], and Second-Order Logics [29,30,31] among others.
It uses formal logic to encode the deep lexical semantics of the full breadth of psychological words and phrases, providing fourteen hundred axioms of first-order logic organized into twenty-nine commonsense psychology theories and sixteen background theories.
MLN  is an interface layer in artificial intelligence, which defines a first-order knowledge base in terms of first-order logic formulae and associated weights.
Kit Fine has reawakened a puzzle about variables with a long history in analytic philosophy, labeling it "the antinomy of the variable." Fine suggests that the antinomy demands a reconceptualization of the role of variables in mathematics, natural language semantics, and first-order logic. The difficulty arises because: (1) the variables x and' cannot be synonymous, since they make different contributions when they jointly occur within a sentence, but (2) there is a strong temptation to say that distinct variables x and y are synonymous, since sentences differing by the total, proper substitution of x for y always agree in meaning.
Seven chapters are: preliminaries; sets, relations, orders; propositional logic; first-order logic; number theory; combinatorics; graph theory.
Markov Logic Networks refer to a learning method of statistical relation that is combining the Markov network and first-order logic together.
The first piece was Montague grammar (named after philosopher and mathematician Richard Montague), which uses a formal system of first-order logic, a systematic method of machine learning where a programmer assigns rigid meanings based on syntax and each word's definition.
Kharlampovich will update ICM attendees on the progress of their collaboration to develop a framework using first-order logic that allows mathematicians to approach previously unsolvable problems.
The syntax of the first-order logic language with intensional abstraction [??], denoted by L, is as follows:

Site: Follow: Share:
Open / Close