3 Understanding First-order Logic Propositional logic can only formalize some patterns of reasoning, but it cannot grasp the logical structure or the truth behavior of very simple sentences such as: • “x + is greater than 5;” • “there exists y such that y = 2;” • “for every real number x, if x is greater than then there exists a real number y such that y is less than and y equals x;” or, for a non-mathematical example; • “every man loves a woman.” Indeed, note that an expression such as “x + is greater than 5” is not a proposition, for it can be true or false depending on the choice of x Neither is “there exists y such that y = 2” a proposition until the range of possible values of y is specified: if y is an integer, then the statement is false; but if y can be any real number, then it is true As for the third sentence above, it is a proposition but its truth depends heavily on its internal logical structure and mathematical meaning of all phrases involved, and those are not tractable on a propositional level All these sentences take us out of the simple world of propositional logic into the realm of first-order logic (the term “first-order” will be explained soon), also known as classical predicate logic or just classical logic, the basic concepts of which we introduce and discuss here First-order logic (just like every formal logical system) has two major aspects: • precise syntax, involving a formal language called a first-order language, that enables us to express statements in a uniform way by means of logical formulae; • formal logical semantics, specifying the meaning of all components of the language by means of their interpretation into suitable models called first-order structures and formal truth definitions, extending the truth tables for the propositional connectives1 Here we discuss the basic components of the syntax and semantics of first-order logic and their relevance to mathematical reasoning The reader who has some experience with programming languages should find the concepts of formal syntax and semantics familiar Logic as a Tool: A Guide to Formal Logical Reasoning, First Edition Valentin Goranko © 2016 John Wiley & Sons, Ltd Published 2016 by John Wiley & Sons, Ltd