Chapter 04 Summary

Chapter 4: Monadic Predicate Logic

Summary

 

  • A predicate logic builds on propositional logic by introducing quantifiers and by dividing propositions into singular terms and predicates.
  • Predicate logics may be monadic or relational
  • In a monadic predicate logic, predicates are followed by exactly one singular term.
  • In a relational predicate logic, predicates are followed by one or more singular terms.
  • M is the formal object language of monadic predicate logic.
  • The syntax of M specifies its vocabulary and rules for making formulas.
  • The vocabulary of M includes lower-case letters, uppercase letters, the five propositional operators (~, •, ˅, ⊃, ≡), two quantifier symbols (∃, ∀), and punctuation marks (), [], {}.
  • Singular terms are lower-case letters that follow predicates. a, b, c, . . . u stand for specific objects and are called constants. v, w, x, y, z are used as variables.
  • A predicate is an uppercase letter that precedes a singular term and stands for a property.
  • A predicate of M followed by a constant is called a closed sentence.
  • A predicate of M followed by a variable is called an open sentence.
  • Quantifiers are operators that work with variables to stand for terms like ‘something’, ‘everything’, ‘nothing’, and ‘anything’.
  • The two quantifiers of M are the existential, represented by the symbol ∃ together with a variable, for example ‘(∃x)’, and the universal, represented by the symbol ∀ together with a variable, for example ‘(∀w)’.
  • The subject of a sentence is what is discussed. The attribute of a sentence is what is said about the subject. Both may contain multiple logical predicates.
  • The scope of an operator is its range of application.
  • The scope of a quantifier is whatever formula immediately follows the quantifier.
  • A variable is bound to a quantifier when it is in the scope of the quantifier with that variable.
  • A variable is free when it is not bound by a quantifier.
  • A closed sentence has no free variables. An open sentence has at least one free variable.
  • M has five formation rules: (1) a predicate (capital letter) followed by a singular term (lower-case letter) is a wff; (2) for any variable β, if α is a wff that does not contain either ‘(∃β)’ or ‘(∀β)’, then ‘(∃β)α’ and ‘(∀β)α’ are wffs; (3) if α is a wff, so is ~α; (4) if α and β are wffs, then so are (α • β), (α ˅ β), (α ⊃ β), (α ≡ β); and (5) these are the only ways to make wffs.
  • An atomic formula of M is formed by a single use of rule 1: a predicate followed by a singular term.
  • A complex formula of M is a wff formed in any way besides a single use of rule 1.
  • A subformula is a formula that is part of another formula.
  • Our system of inference for constructing derivations in M uses the twenty-five rules for constructing derivations in PL, along with four new rules governing the removal and addition of quantifiers, and a rule for exchanging the universal and existential quantifiers.
  • Universal instantiation (UI) is a rule of inference in predicate logic that allows for the removal of the universal quantifier when it is the main operator.
  • Universal generalization (UG) is a rule of inference in predicate logic that allows for the addition of a universal quantifier.
  • Existential generalization (EG) is a rule of inference in predicate logic that allows for the addition of an existential quantifier.
  • Existential instantiation (EI) is a rule of inference in predicate logic that allows for the removal of the existential quantifier when it is the main operator.
  • Hasty generalization is the logical fallacy of inferring a universal claim from an existential one.
  • Quantifier exchange (QE) is a rule of equivalence in predicate logic that allows us to manage the interactions between negations and quantifiers.
  • Conditional and indirect proofs in M work much the same way they do in PL, with one addendum: within the scope of an assumption for conditional or indirect proof, never UG on a variable that is free in the assumption.
  • All lines of an indented sequence are within the scope of the assumption in the first line.
  • Proof theory is the study of axioms (if any) and rules for a formal theory.
  • The semantics of M specifies the rules for interpreting the symbols and formulas of the language.
  • An interpretation of a formal language describes the meanings or truth conditions of its components.
  • To interpret predicates and quantifiers requires basic set theory in our metalanguage.
  • A set is an unordered collection of objects.
  • A subset of a set is a collection, all of whose members are in the other set.
  • A domain of interpretation is a set of objects to which we apply a theory.
  • An interpretation of a theory of M proceeds in four steps: (1) specify a set to serve as a domain of interpretation; (2) assign a member of the domain to each constant; (3) assign some set of objects in the domain to each predicate; and (4) use the customary truth tables for the interpretation of the propositional operators.
  • An object satisfies a predicate if it is in the set which interprets that predicate.
  • An existentially quantified sentence is satisfied when it is satisfied by some object in the domain. A universally quantified sentence is satisfied when it is satisfied by all objects in the domain.
  • A model of a theory is an interpretation on which all of the sentences of the theory are true.
  • An invalid argument of M can be shown to be invalid by constructing a counterexample in a finite domain.
  • A counterexample is a valuation that makes the premises true and the conclusion false.
  • A wff of M can be proven to be logically true either proof-theoretically, using conditional or indirect proof, or semantically by showing that it is true for every interpretation.
  • A wff of M can be proven not to be logically true by providing a valuation that makes it false in a finite domain.

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