First-order logic

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Short description: Formal system of notation and reasoning

First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable.[1] This distinguishes it from propositional logic, which does not use quantifiers or relations;[2] in this sense, propositional logic is the foundation of first-order logic.

A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold about them. Sometimes, "theory" is understood in a more formal sense as just a set of sentences in first-order logic.

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 quantification over predicates or functions, or both, are permitted.[3]:56 In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets.

There are many deductive systems for first-order logic which are both sound (i.e., all provable statements are true in all models) and complete (i.e. all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

First-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axiom systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.

The foundations of first-order logic were developed independently by Gottlob Frege and Charles Sanders Peirce.[4] For a history of first-order logic and how it came to dominate formal logic, see José Ferreirós (2001).

Introduction

While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification.

A predicate takes an entity or entities in the domain of discourse and evaluates to true or false. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated, and might be denoted, for example, by variables such as p and q. The predicate "is a philosopher" occurs in both sentences, which have a common structure of "a is a philosopher". The variable a is instantiated as "Socrates" in the first sentence, and is instantiated as "Plato" in the second sentence. While first-order logic allows for the use of predicates, such as "is a philosopher" in this example, propositional logic does not.[5]

Relationships between predicates can be stated using logical connectives. Consider, for example, the first-order formula "if a is a philosopher, then a is a scholar". This formula is a conditional statement with "a is a philosopher" as its hypothesis, and "a is a scholar" as its conclusion. The truth of this formula depends on which object is denoted by a, and on the interpretations of the predicates "is a philosopher" and "is a scholar".

Quantifiers can be applied to variables in a formula. The variable a in the previous formula can be universally quantified, for instance, with the first-order sentence "For every a, if a is a philosopher, then a is a scholar". The universal quantifier "for every" in this sentence expresses the idea that the claim "if a is a philosopher, then a is a scholar" holds for all choices of a.

The negation of the sentence "For every a, if a is a philosopher, then a is a scholar" is logically equivalent to the sentence "There exists a such that a is a philosopher and a is not a scholar". The existential quantifier "there exists" expresses the idea that the claim "a is a philosopher and a is not a scholar" holds for some choice of a.

The predicates "is a philosopher" and "is a scholar" each take a single variable. In general, predicates can take several variables. In the first-order sentence "Socrates is the teacher of Plato", the predicate "is the teacher of" takes two variables.

An interpretation (or model) of a first-order formula specifies what each predicate means, and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in an interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato.

Syntax

There are two key parts of first-order logic. The syntax determines which finite sequences of symbols are well-formed expressions in first-order logic, while the semantics determines the meanings behind these expressions.

Alphabet

See also: Alphabet (formal languages)

Unlike natural languages, such as English, the language of first-order logic is completely formal, so that it can be mechanically determined whether a given expression is well formed. There are two key types of well-formed expressions: terms, which intuitively represent objects, and formulas, which intuitively express statements that can be true or false. The terms and formulas of first-order logic are strings of symbols, where all the symbols together form the alphabet of the language. As with all formal languages, the nature of the symbols themselves is outside the scope of formal logic; they are often regarded simply as letters and punctuation symbols.

It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and non-logical symbols, whose meaning varies by interpretation. For example, the logical symbol [math]\displaystyle{ \land }[/math] always represents "and"; it is never interpreted as "or", which is represented by the logical symbol [math]\displaystyle{ \lor }[/math]. However, a non-logical predicate symbol such as Phil(x) could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate depending on the interpretation at hand.

Logical symbols

Logical symbols are a set of characters that vary by author, but usually include the following:[6]

  • Quantifier symbols: for universal quantification, and for existential quantification
  • Logical connectives: for conjunction, for disjunction, for implication, for biconditional, ¬ for negation. Some authors[7] use Cpq instead of and Epq instead of , especially in contexts where → is used for other purposes. Moreover, the horseshoe may replace ; the triple-bar may replace ; a tilde (~), Np, or Fp may replace ¬; a double bar [math]\displaystyle{ \| }[/math], [math]\displaystyle{ + }[/math], or Apq may replace ; and an ampersand &, Kpq, or the middle dot may replace , especially if these symbols are not available for technical reasons. (The aforementioned symbols Cpq, Epq, Np, Apq, and Kpq are used in Polish notation.)
  • Parentheses, brackets, and other punctuation symbols. The choice of such symbols varies depending on context.
  • An infinite set of variables, often denoted by lowercase letters at the end of the alphabet x, y, z, ... . Subscripts are often used to distinguish variables: x0, x1, x2, ... .
  • An equality symbol (sometimes, identity symbol) = (see § Equality and its axioms below).

Not all of these symbols are required in first-order logic. Either one of the quantifiers along with negation, conjunction (or disjunction), variables, brackets, and equality suffices.

Other logical symbols include the following:

  • Truth constants: T, V, or for "true" and F, O, or for "false" (V and O are from Polish notation). Without any such logical operators of valence 0, these two constants can only be expressed using quantifiers.
  • Additional logical connectives such as the Sheffer stroke, Dpq (NAND), and exclusive or, Jpq.

Non-logical symbols

Non-logical symbols represent predicates (relations), functions and constants. It used to be standard practice to use a fixed, infinite set of non-logical symbols for all purposes:

  • For every integer n ≥ 0, there is a collection of n-ary, or n-place, predicate symbols. Because they represent relations between n elements, they are also called relation symbols. For each arity n, there is an infinite supply of them:
    Pn0, Pn1, Pn2, Pn3, ...
  • For every integer n ≥ 0, there are infinitely many n-ary function symbols:
    f n0, f n1, f n2, f n3, ...

When the arity of a predicate symbol or function symbol is clear from context, the superscript n is often omitted.

In this traditional approach, there is only one language of first-order logic.[8] This approach is still common, especially in philosophically oriented books.

A more recent practice is to use different non-logical symbols according to the application one has in mind. Therefore, it has become necessary to name the set of all non-logical symbols used in a particular application. This choice is made via a signature.[9]

Typical signatures in mathematics are {1, ×} or just {×} for groups, or {0, 1, +, ×, <} for ordered fields. There are no restrictions on the number of non-logical symbols. The signature can be empty, finite, or infinite, even uncountable. Uncountable signatures occur for example in modern proofs of the Löwenheim–Skolem theorem.

Though signatures might in some cases imply how non-logical symbols are to be interpreted, interpretation of the non-logical symbols in the signature is separate (and not necessarily fixed). Signatures concern syntax rather than semantics.

In this approach, every non-logical symbol is of one of the following types:

  • A predicate symbol (or relation symbol) with some valence (or arity, number of arguments) greater than or equal to 0. These are often denoted by uppercase letters such as P, Q and R. Examples:
    • In P(x), P is a predicate symbol of valence 1. One possible interpretation is "x is a man".
    • In Q(x,y), Q is a predicate symbol of valence 2. Possible interpretations include "x is greater than y" and "x is the father of y".
    • Relations of valence 0 can be identified with propositional variables, which can stand for any statement. One possible interpretation of R is "Socrates is a man".
  • A function symbol, with some valence greater than or equal to 0. These are often denoted by lowercase roman letters such as f, g and h. Examples:
    • f(x) may be interpreted as "the father of x". In arithmetic, it may stand for "-x". In set theory, it may stand for "the power set of x".
    • In arithmetic, g(x,y) may stand for "x+y". In set theory, it may stand for "the union of x and y".
    • Function symbols of valence 0 are called constant symbols, and are often denoted by lowercase letters at the beginning of the alphabet such as a, b and c. The symbol a may stand for Socrates. In arithmetic, it may stand for 0. In set theory, it may stand for the empty set.

The traditional approach can be recovered in the modern approach, by simply specifying the "custom" signature to consist of the traditional sequences of non-logical symbols.

Formation rules