lower predicate calculus with identity
Quick Summary
The lower predicate calculus (LPC) is a branch of symbolic logic that analyzes propositions by breaking them down into predicates and individual variables, rather than treating them as unanalyzable units like propositional logic. It uses quantifiers, such as "all" (\u2200) and "some" (\u2203), to express how predicates apply to individuals or relations between them.123
For example, "Socrates is wise" can be represented as \u03d5x, where \u03d5 is the predicate "is wise" and x is the individual "Socrates." More complex statements, like "Everything is wise," are symbolized as (\u2200x)\u03d5x.14
The LPC is considered "lower" because quantifiers only apply to individual variables, not to predicates themselves. Systems that allow quantification over predicates are called higher-order calculi. The predicate calculus with identity extends the LPC by including a symbol for identity (=), allowing statements about whether two individuals are the same.2314
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metalogicLogic and metalogic…may include as well the logic of identity, symbolized “=,” which takes the ordinary properties of identity as part of logic. In this sense Gottlob Frege achieved a formal calculus of logic as early as 1879. Sometimes logic is construed, however, as including also higher-order predicate calculi, which admit variables…
Read More…may include as well the logic of identity, symbolized “=,” which takes the ordinary properties of identity as part of logic. In this sense Gottlob Frege achieved a formal calculus of logic as early as 1879. Sometimes logic is construed, however, as including also higher-order predicate calculi, which admit variables…
Read More- metalogicAxioms and rules of inference
…to be those of the first-order predicate calculus with identity.
Read More…to be those of the first-order predicate calculus with identity.
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- metalogicBackground and typical problems
…especially in that of the first-order predicate calculus with identity—i.e., in elementary logic. A first-order language is given by a collection S of symbols for relations, functions, and constants, which, in combination with the symbols of elementary logic, single out certain combinations of symbols as sentences. Thus, for example, in…
Read More…especially in that of the first-order predicate calculus with identity—i.e., in elementary logic. A first-order language is given by a collection S of symbols for relations, functions, and constants, which, in combination with the symbols of elementary logic, single out certain combinations of symbols as sentences. Thus, for example, in…
Read More - metalogicElementary logic
The development of model theory has led to a more general outlook that enabled the Swedish logician Per Lindström to prove in 1969 a general theorem to the effect that, roughly speaking, within a broad class of possible logics, elementary logic is the only one that satisfies the…
Read MoreThe development of model theory has led to a more general outlook that enabled the Swedish logician Per Lindström to prove in 1969 a general theorem to the effect that, roughly speaking, within a broad class of possible logics, elementary logic is the only one that satisfies the…
Read More