Contents

  • 7.1 Introduction
  • 7.2 Predicates, Individual Constants, and Quantity Terms of English
  • 7.3 Introduction to PL
  • 7.4 Quantifiers Introduced
  • 7.5 The Formal Syntax of PL
  • 7.6 A-, E-, I-, and O- sentences
  • 7.7 Symbolization Techniques
  • 7.8 Multiple Quantifiers with Overlapping Scope
  • 7.9 Identity, Definite Descriptions, and Properties of Relations

Notes

The language PL 'PL' stands for 'Predicate Logic', the language studied in this chapter.

PL vocabulary

PL expression A sequence of not necessarily distinct elements of the vocabulary of PL.

PL atomic formula Every expression of PL that is either a sentence letter of PL or an $n$-place predicate of PL followed by $n$ individual terms of PL is an atomic formula of PL.

quantifier of PL A quantifier of PL is an expression of PL of the form $(\forall x)$ or $(\exists x).$ An expression of the first form is a universal quantifier, and one of the second form is an existential quantifier.

PL logical operator An expression of PL that is either a quantifier or a truth-functional connective.

PL formula

  1. Every atomic formula of PL is a formula of PL.
  2. If $\mathscr{P}$ is a formula of PL, so is $\sim\mathscr{P}.$
  3. If $\mathscr{P}$ and $\mathscr{Q}$ are formula of PL, so are $(\mathscr{P}\&\mathscr{Q}),$ $(\mathscr{P}\lor\mathscr{Q}),$ $(\mathscr{P}\supset\mathscr{Q}),$ and $(\mathscr{P}\equiv\mathscr{Q}).$
  4. If $\mathscr{P}$ is a formula of PL that contains at least one occurrence of $x$ and no $x$-quantifier, then $(\forall x)\mathscr{P}$ and $(\exists x)\mathscr{P}$ are both formula of PL.
  5. Nothing is a formula of PL unless it can be formed by repeated applications of one of these clauses.

PL main logical operator

  1. If $\mathscr{P}$ is an atomic formula of PL, then $\mathscr{P}$ contains no logical operator, and hence no main logical operator, and $\mathscr{P}$ is the only subformula of $\mathscr{P}$.
  2. If $\mathscr{P}$ is a formula of PL of the form $\sim\mathscr{Q},$ then the tilde $('\sim')$ that precedes $\mathscr{Q}$ is the main logical operator of $\mathscr{P}$, and $\mathscr{Q}$ is the immediate subformula of $\mathscr{P}$.
  3. If $\mathscr{P}$ is a formula of PL of the form $(\mathscr{Q}\&\mathscr{R}),$ $(\mathscr{Q}\lor\mathscr{R}),$ $(\mathscr{Q}\supset\mathscr{R}),$ or $(\mathscr{Q}\equiv\mathscr{R}),$ then the binary connective between $\mathscr{Q}$ and $\mathscr{R}$ is the main logical operator of $\mathscr{P}$, and $\mathscr{Q}$ and $\mathscr{R}$ are the immediate subformulas of $\mathscr{P}$.
  4. If $\mathscr{P}$ is a formula of PL of the form $(\forall x)\mathscr{Q}$ or $(\exists x)\mathscr{Q},$ then the quantifier that occurs before $\mathscr{Q}$ is the main logical operator of $\mathscr{P}$, and $\mathscr{Q}$ is the immediate subformula of $\mathscr{P}$.
  5. If $\mathscr{P}$ is a formula of PL, then every subformula (immediate or not) of a subformula of $\mathscr{P}$ is a subformula of $\mathscr{P}$, and $\mathscr{P}$ is a subformula of itself.

scope of a quantifier The scope of a quantifier in a formula $\mathscr{P}$ of PL is the subformula $\mathscr{Q}$ of $\mathscr{P}$ of which that quantifier is the main logical operator.

PL bound variable An occurrence of a variable $x$ in a formula $\mathscr{P}$ of PL is bound if and only if that occurrence is within the scope of an $x$-quantifier.

PL free variable An occurrence of a variable $x$ in a formula $\mathscr{P}$ of PL is free if and only if it is not bound.

PL sentence A formula of PL is a sentence of PL if and only if no occurrence of a variable in $\mathscr{P}$ is free.

singular term A singular term is any word or phrase that designates or purports to designaate (or denote or refer to) some one thing. Singular terms are of two sorts, proper names and definite descriptions.

tense logic

modal logic