Lógica de Predicados: Sintaxis, Semántica, Cuantificadores y Validez

Predicate Logic: Syntax, Semantics, Quantifiers, and Validity

Overview

In mathematical logic, predicate logic extends propositional logic by introducing structured propositions and quantifiers. It allows for a more nuanced description of relationships within mathematical and everyday contexts. In this article, we will discuss predicate logic, focusing on its syntax, semantics, quantifiers, and validity.

Syntax

The syntax of predicate logic involves specifying the formation rules to generate well-formed formulas from a set of primitive symbols. These symbols include:

• Atomic formulas consisting of an n-ary predicate followed by n variables
• Universally quantified formulas, denoted by ∀ (read "for all"), followed by a variable and a formula
• Existentially quantified formulas, denoted by ∃ (read "there exists"), followed by a variable and a formula

For example, consider the following formulas:

• (\forall x\phi(x)): "For all (x), (\phi(x))"
• (\exists x\phi(x)): "There exists an (x) such that (\phi(x))"

Semantics

The semantics of predicate logic provides the connection between formulas and the real world. It deals with understanding the meaning of formulas and how they relate to concrete situations or interpretations. In predicate logic, semantics is based on the notion of models, which includes a domain of discourse (the set of individuals) and an interpretation for each predicate and function symbol.

An interpretation assigns a value to each predicate symbol, indicating whether the symbol holds true or false for certain arguments from the domain. Similarly, function symbols are interpreted as functions that map elements from the domain to other elements. The models determine the truth values of formulas based on their structures and interpretations.

In first-order logic, verifying the truth of universally quantified statements requires checking them for infinitely many elements in the domain, which can make direct verification difficult. However, we can still construct an algorithm that searches through possible truth functions to verify validity.

Quantifiers

Quantifiers in predicate logic are used to express properties that hold for all individuals in a domain or for at least one individual. There are two main types of quantifiers: universal ((\forall)) and existential ((\exists)).

Universal Quantifier

The (\forall) quantifier states that a property holds for every element in the domain. It is read as "for all." For example, the formula "(\forall x(P(x)\rightarrow Q(x)))" means "For all (x), if (P(x)) is true, then (Q(x)) is true."

Existential Quantifier

The (\exists) quantifier indicates that there exists at least one element in the domain for which a property holds. It is read as "there exists." For instance, "(\exists x(P(x)\wedge Q(x)))" means "There exists an (x) such that both (P(x)) and (Q(x)) are true."

Validity

Validity in predicate logic refers to an argument being sound in all possible interpretations. A valid argument preserves truth across interpretations, meaning that if its premises are true in any interpretation, its conclusion must also be true in that same interpretation.

Given a set of sentences (call it (A)), we say that a sentence (X) is semantically entailed by (A) if every model that satisfies (A) also makes (X) true. In other words, (A\models X) if and only if every model that satisfies (A) also assigns the truth value 'true' to (X).

Proof Techniques

Proof techniques in predicate logic are methods used to establish the validity of arguments or conclusions. These techniques can include various strategies like inference rules, decision procedures, or model theoretic proofs. Some common proof techniques in predicate logic include Natural Deduction, Sequent Calculus, and Tableaux. Each technique provides a systematic way to prove the validity of formulas while ensuring consistency with the given logical system.