5.1 Predicates, Individual Constants, and Variables

Predicates, individual constants, and variables form the building blocks of predicate logic. These elements allow us to represent complex relationships and make general statements about objects in a given domain, enabling more nuanced logical reasoning.

Understanding how to construct well-formed formulas and evaluate their truth values is crucial. By combining atomic formulas with logical connectives and quantifiers, we can express sophisticated logical arguments and analyze their validity in various interpretations.

Predicates, Individual Constants, and Variables

Predicates and constants in logic

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• Predicates represent properties or relationships between objects in logic
  • Denoted by uppercase letters (P, Q, R)
  • Take one or more arguments to form statements
    • Unary predicates take one argument (P(x), "x is prime")
    • Binary predicates take two arguments (R(x, y), "x is greater than y")
    • n-ary predicates take n arguments (Q(x, y, z), "x, y, and z are collinear")
• Individual constants represent specific objects or entities
  • Denoted by lowercase letters (a, b, c)
  • Refer to a particular object in the domain of discourse (a = Socrates, b = Plato)
• Variables represent arbitrary objects or entities
  • Denoted by lowercase letters (x, y, z)
  • Used to quantify over a domain of objects
  • Enable general statements about properties or relationships (∀x P(x), "for all x, P(x) holds")

Construction of well-formed formulas

• Atomic formulas consist of a predicate followed by the appropriate number of arguments
  • Arguments can be individual constants or variables
  • Examples: P(a), R(x, y), Q(a, b, c)
• Complex formulas are formed by combining atomic formulas using logical connectives
  • Negation: $\neg$ (not)
  • Conjunction: $\land$ (and)
  • Disjunction: $\lor$ (or)
  • Implication: $\rightarrow$ (if...then)
  • Biconditional: $\leftrightarrow$ (if and only if)
• Quantifiers can be used to bind variables in formulas
  • Universal quantifier: $\forall$ (for all)
  • Existential quantifier: $\exists$ (there exists)
• Examples of well-formed formulas:
  1. $\forall x (P(x) \rightarrow Q(x))$ (for all x, if P(x) then Q(x))
  2. $\exists y (R(a, y) \land S(y))$ (there exists a y such that R(a, y) and S(y))

Truth values of atomic formulas

• An interpretation assigns meaning to the symbols in a predicate logic formula
  • Specifies a domain of objects and the truth values of predicates for those objects
  • Example: Domain = {1, 2, 3}, P(x) is true if x is even
• Evaluating the truth value of an atomic formula:
  1. Replace variables with objects from the domain
  2. Determine the truth value of the resulting statement based on the interpretation
• Examples:
  • Given the interpretation above, P(1) is false and P(2) is true
  • R(a, b) is true if the relation R holds between the objects denoted by a and b

Syntax vs semantics in predicate logic

• Syntax concerns the formal structure and rules for constructing well-formed formulas
  • Specifies the allowed symbols and their arrangement
  • Examples:
    • P(x) is syntactically correct
    • P(x, y) is syntactically incorrect if P is a unary predicate
• Semantics concerns the meaning and interpretation of well-formed formulas
  • Assigns truth values to formulas based on the interpretation of symbols
  • Examples:
    • $\forall x (P(x) \rightarrow Q(x))$ is semantically true if, for every object in the domain, if P(x) is true, then Q(x) is also true
    • $\exists y (R(a, y) \land S(y))$ is semantically true if there exists at least one object in the domain for which both R(a, y) and S(y) are true