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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.
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. 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. Although the logical consequence relation is only semidecidable , much progress has been made in automated theorem proving in first-order logic.
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. 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. 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.
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. 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. 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 vary by author, but usually include the following: [6]. 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. 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:. 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. 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. There are no restrictions on the number of non-logical symbols. The signature can be empty , finite, or infinite, even uncountable.
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.
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. The formation rules define the terms and formulas of first-order logic. These rules are generally context-free each production has a single symbol on the left side , except that the set of symbols may be allowed to be infinite and there may be many start symbols, for example the variables in the case of terms.
The set of terms is inductively defined by the following rules:. Only expressions which can be obtained by finitely many applications of rules 1 and 2 are terms. For example, no expression involving a predicate symbol is a term. The set of formulas also called well-formed formulas [13] or WFFs is inductively defined by the following rules:. Only expressions which can be obtained by finitely many applications of rules 1—5 are formulas.
The formulas obtained from the first two rules are said to be atomic formulas. The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way—by following the inductive definition i. This property is known as unique readability of formulas. There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted.
Each author's particular definition must be accompanied by a proof of unique readability. This definition of a formula does not support defining an if-then-else function ite c, a, b , where "c" is a condition expressed as a formula, that would return "a" if c is true, and "b" if it is false.
This is because both predicates and functions can only accept terms as parameters, but the first parameter is a formula. For convenience, conventions have been developed about the precedence of the logical operators, to avoid the need to write parentheses in some cases.
These rules are similar to the order of operations in arithmetic. A common convention is:. Moreover, extra punctuation not required by the definition may be inserted—to make formulas easier to read.
Thus the formula. In some fields, it is common to use infix notation for binary relations and functions, instead of the prefix notation defined above.
It is common to regard formulas in infix notation as abbreviations for the corresponding formulas in prefix notation, cf. This convention is advantageous in that it allows all punctuation symbols to be discarded. As such, Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read. In Polish notation, the formula.
In a formula, a variable may occur free or bound or both. The free and bound variable occurrences in a formula are defined inductively as follows.
A formula in first-order logic with no free variable occurrences is called a first-order sentence. These are the formulas that will have well-defined truth values under an interpretation. For example, whether a formula such as Phil x is true must depend on what x represents. The axioms for ordered abelian groups can be expressed as a set of sentences in the language.
An interpretation of a first-order language assigns a denotation to each non-logical symbol predicate symbol, function symbol, or constant symbol in that language. It also determines a domain of discourse that specifies the range of the quantifiers.
The result is that each term is assigned an object that it represents, each predicate is assigned a property of objects, and each sentence is assigned a truth value. In this way, an interpretation provides semantic meaning to the terms, predicates, and formulas of the language. The study of the interpretations of formal languages is called formal semantics. What follows is a description of the standard or Tarskian semantics for first-order logic.
It is also possible to define game semantics for first-order logic , but aside from requiring the axiom of choice , game semantics agree with Tarskian semantics for first-order logic, so game semantics will not be elaborated herein.
The most common way of specifying an interpretation especially in mathematics is to specify a structure also called a model ; see below. The structure consists of a domain of discourse D and an interpretation function I mapping non-logical symbols to predicates, functions, and constants.
The domain of discourse D is a nonempty set of "objects" of some kind.
Logic pro x 7.1 free -
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