# 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.