Frege's Begriffsschrift ( 1879 ) introduced both a complete propositional calculus and what is essentially modern predicate logic.

* Extension ( predicate logic )

* Extension ( predicate logic ), the set of tuples of values that satisfy the predicate

The process of abstract axiomatization as exemplified by Hilbert's axioms reduces geometry to theorem proving or predicate logic.

This is the case of the Mycin and Dendral expert systems, and of, for example, fuzzy logic, predicate logic ( Prolog ), symbolic logic and mathematical logic.

It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic ( a less precise term ).

The adjective " first-order " distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.

In first-order logic, however, the sentences can be expressed in a more parallel manner using the predicate Phil ( a ), which asserts that the object represented by a is a philosopher.

The language ’ s grammar is based on predicate logic, which is why it was named Loglan, an abbreviation for " logical language ".

It also became clear that such clauses could be restricted to definite clauses or Horn clauses, where < tt > H </ tt >, < tt > B < sub > 1 </ sub ></ tt >, …, < tt > B < sub > n </ sub ></ tt > are all atomic predicate logic formulae, and that SL-resolution could be restricted ( and generalised ) to LUSH or SLD-resolution.

The analogue of conjunction for a ( possibly infinite ) family of statements is universal quantification, which is part of predicate logic.

Logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic.

Just as propositional logic can be seen as an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic was an advancement from the earlier propositional logic.

Also in first-order predicate logic:

The main difference between truth-value semantics and the standard semantics for predicate logic is that there are no domains for truth-value semantics.

Whereas in standard semantics atomic formulas like Pb or Rca are true if and only if ( the referent of ) b is a member of the extension of the predicate P, resp., if and only if the pair ( c, a ) is a member of the extension of R, in truth-value semantics the truth-values of atomic formulas are basic.

What categories, or kinds of thing, can be the subject or the predicate of a proposition?

In his Categories, Aristotle identifies ten possible kinds of thing that can be the subject or the predicate of a proposition.

An " extensional stance " and restriction to a second-order predicate logic means that a propositional function extended to all individuals such as " All ' x ' are blue " now has to list all of the ' x ' that satisfy ( are true in ) the proposition, listing them in a possibly infinite conjunction: e. g. x < sub > 1 </ sub > V x < sub > 2 </ sub > V.

In the mathematical model, reasoning about such data is done in two-valued predicate logic, meaning there are two possible evaluations for each proposition: either true or false ( and in particular no third value such as unknown, or not applicable, either of which are often associated with the concept of NULL ).

The configurations of background knowledge which he considers are those which are provided by a sample proposition, namely a proposition which is a conjunction of atomic propositions, each of which ascribes a single predicate to a single individual, with no two atomic propositions involving the same individual.

Normally, to decide whether a proposition of the standard subject-predicate form is true or false, one checks whether the subject is in the extension of the predicate.

The proposition is then true if and only if the subject is in the extension of the predicate.

Aristotelian logic identifies a proposition as a sentence which affirms or denies a predicate of a subject.

As noted above, in Aristotelian logic a proposition is a particular kind of sentence, one which affirms or denies a predicate of a subject.

A proposition is ( i ) a predicate symbol applied to the number of terms required by its arity, ( ii ) an operator applied to the number of propositions required by its arity, or ( iii ) a quantifier applied to a proposition.

For example, if = is a binary predicate symbol and ∀ is a quantifier, then ∀ x, y, z = y ) → ( x + z = y + z ) is a proposition.

Because the elimination of functional predicates is both convenient for some purposes and possible, many treatments of formal logic do not deal explicitly with function symbols but instead use only relation symbols ; another way to think of this is that a functional predicate is a special kind of predicate, specifically one that satisfies the proposition above.

The antecedent is the subject and the consequent is the predicate of a universal affirmative proposition.

If X is a domain of x and P ( x ) is a predicate dependent on x, then the universal proposition is expressed in Boolean algebra terms as

which shows that to disprove a " for all x " proposition, one needs no more than to find an x for which the predicate is false.

to disprove a " there exists an x " proposition, one needs to show that the predicate is false for all x.

A categorical proposition is a simple proposition containing two terms, subject and predicate, in which the predicate is either asserted or denied of the subject.

A proposition consists of at least three ideas, namely: a subject idea, a predicate idea and the copula ( i. e. ' has ', or another form of to have ).

The fact that the two hypotheses incorporate different kinds of testing procedures is expressed in the formal language by prefixing the operator '*' to a different predicate.

Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic by defining the formula

This " gives rise to artificial problems as to the nature of truth, which disappear at once when they are expressed in logical symbolism ..." According to Ramsey, it is only because natural languages lack, what he called, pro-sentences ( expressions that stand in relation to sentences as pronouns stand to nouns ) that the truth predicate cannot be defined away in all contexts.

*( vi ) An atomic sentence F ( x1 ,..., xn ) is true ( relative to an assignment of values to the variables x1, ..., xn )) if the corresponding values of variables bear the relation expressed by the predicate F.

The syntax by means of which these two sub-parts are combined can be expressed in first-order predicate calculus.

The thought " John is tall " is clearly composed of two sub-parts, the concept of John and the concept of tallness, combined in a manner that may be expressed in first-order predicate calculus as a predicate ' T ' (" is tall ") that holds of the entity ' j ' ( John ).

This judgment is typically expressed in the form of a specific truth predicate, whose positive application to a sign, or so on, asserts that the sign is true.

The truth predicate of interest in a typical correspondence theory of truth tells of a relation between representations and objective states of affairs, and is therefore expressed, for the most part, by a dyadic predicate.

The T-schema is often expressed in natural language, but it can be formalized in many-sorted predicate logic or modal logic ; such a formalisation is called a T-theory.

These models can be expressed in modal predicate logic from which code in the Prolog artificial intelligence language can be derived.

Aristotle intended them to enumerate everything that can be expressed without composition or structure, thus anything that can be either the subject or the predicate of a proposition.

Thus Mentalese is best expressed through predicate and propositional calculus.

The use of such clauses can be considered analogous to existential quantification in predicate logic ( often expressed with the phrase " There exist ( s )...").

Namely, a propositional formula can be expressed in first-order logic by replacing each propositional variable with a predicate of zero arity ( i. e., a predicate with no arguments ).