100 Logic as a Tool The first two quantifiers are particularly important and many others can be expressed by means of them, so they are given special names and notation: The quantifier “for every” is called the universal quantifier, denoted ∀ The quantifier “there exists” is called existential quantifier, denoted ∃ These are the only quantifiers used in first-order languages, so they are called first-order quantifiers The term “first-order” refers to the fact that variables and quantifications in these languages are only permitted to range over individuals in the universe of discourse, called “first-order objects”4 As well as the phrases above, the universal quantifier is usually represented by “all”, “for all”, and “every”, while the existential quantifier can appear as “there is”, “a”, some”, and “for some”, particularly in a non-mathematical discourse Sometimes, recognizing the correct quantification in natural language can be quite tricky, or even confusing Take for example: “a dog ate my homework”, meaning “some dog ate my homework” (existential quantification) v “a dog is an animal”, meaning “every dog is an animal” (universal quantification) Alternatively, the well-known expression5 “All that glitters is not gold” is actually meant to mean “Not all that glitters is gold”, something logically quite different, as we will see in Section 3.4 So, watch out! 3.1.3 Terms and formulae Using the symbols in a given first-order language and following certain common syntactic rules, we can compose formal expressions which allow us to symbolically represent statements, to reason about them, and to prove them in a precise, well-structured, and logically correct way There are two basic syntactic categories in a first-order language: (first-order) terms and (first-order) formulae 3.1.3.1 Terms Terms are formal expressions (think of algebraic expressions) built from constant symbols and individual variables, using functional symbols Terms are used to denote specified or unspecified individuals, that is, elements of the domain Here is the formal inductive definition of terms in a first-order language L Definition 79 (Terms) Let L be any first-order language Every constant symbol in L is a term in L Every individual variable in L is a term in L First-order logic can be extended to second-order logic, where there are second-order variables and quantifiers ranging over sets, relations, and functions; then further to third-order logic with variables and quantifiers over more complex objects definable in second-order logic, etc In this book we will not go beyond first-order logic From Shakespeare’s play The Merchant of Venice