Chapter 7 Predicate Logic: Symbolization and Syntax
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
- Every atomic formula of PL is a formula of PL.
- If $\mathscr{P}$ is a formula of PL, so is $\sim\mathscr{P}.$
- 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 $(\mathscr{P}\equiv\mathscr{Q}).$
-
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. - Nothing is a formula of PL unless it can be formed by repeated applications of one of these clauses.
- 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 $\mathscr{P}$.
- 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 $\mathscr{P}$.
- 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 $\mathscr{P}$.
- 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 $\mathscr{P}$.
- 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.