Advanced Mathematics

1.3Predicate Logic

Purpose of Section

This section introduces predicate logic (or first‐order logic) that is the “language” of mathematics. We will see how predicate logic extends the language of sentential (or propositional) logic, introduced in the first two sections, by adding universal and existential quantifiers, functions, and variables.

1.3.1 Introduction

Although sentential logic studied in Sections 1.1 and 1.2 is probably sufficient to get you through your daily activities, it is not sufficient for higher mathematics. This was realized in the late 1800s by the German logician Gottlob Frege, who observed that mathematics requires a more extensive language than simple logical sentences connected by ∧, ∨, ∼, ⇒, ⇔. Frege introduced what is called predicate logic¹(or first‐order logic), which are sentences which in addition to the logical connectives of sentential logic, includes quantifiers, variables, and functions called predicates (or propositions). Predicate logic allows one to express concepts we often hear in mathematics, like

for any real number x, there exists a real number y such that x < y

which is impossible to express in sentential logic.

1.3.2 Existential and Universal Quantifiers

Two phrases one hears again and again in mathematics are for all, and there exists. These expressions are called quantifiers and are necessary to describe mathematical concepts. The meaning of an expression like x < y in itself is not clear until we describe the extent ...