Part I The art of logic 1 Chapter 1 Grammar After some preliminary grammatical considerations in this chapter, we collect the material on truth-functional logic in chapter 2; the material on quantifiers and iden- tity in chapter 3 (proofs), chapter 5 (symbolization), and chapter 6 (semantics); some applications in chapter 7 (logical theory, arithmetic, set theory), and a discus- sion of definitions in chapter 8. Chapter 13 is an unfulfilled promise of a discussion of the chief theorems about modern logic. 1A Logical Grammar Rigorous logic ought to begin with rigorous grammar; otherwise bungle is almost certain. Here by a “logical” grammar we mean one of the sort that guides the thinking of most logicians; such a grammar is a more or less language-independent family of grammatical categories and rules clearly deriving from preoccupation with formal systems but with at least prospective applications to natural languages. A logical grammar is accordingly one which is particularly simple, rigorous, and tidy, one which suppresses irregular or nonuniform or hard to handle details (hence differentiating itself from a “linguistic” grammar whether of the MIT type or an- other), one which idealizes its subject matter, and one which by ignoring as much as possible leads us (perhaps wrongly) to the feeling of “Aha; that’s what’s really going on!” Among such logical grammars we think a certain one lies at the back of most logicians’ heads. Our task is to make it explicit, not in order to criticize it—indeed though it would be dangerous to suppose it eternally ordained, we think very well of it—but because only when it is brought forth in all clarity can we sen- 2 1A. Logical Grammar 3 sibly discuss how it ought to be applied. The version here presented derives from Curry and Feys (1958) and Curry (1963). Of course here and hereafter application of any logical grammar will be far more straightforward and indisputable in the case of formal languages than in the case of English; for in the case of the former, possession of a tidy grammar is one of the design criteria. But English is like Topsy, and we should expect a fit only with a “properly understood” or “preprocessed” English—the preprocessing to remove, of course the hard cases. So read on with charity. 1A.1 Sentence and term “Logical grammar,” as we understand it, begins with three fundamental grammat- ical categories: the “sentence,” the “term,” and the “functor.” The first two are taken as primitive notions, hence undefined; but we can say enough to make it clear enough how we plan to apply these categories to English and to the usual formal languages. By a sentence is meant a declarative sentence in pretty much the sense of tradi- tional grammar, thus excluding—for expository convenience—interrogatives and imperatives. A sentence is of a sort to express the content of an assertion or of a conjecture, and so on. Semantically, a sentence is of a kind to be true or false, although typically its truth or falsity will be relative to various items cooked up by linguists and logicians. Examples: The truth or falsity of “It is a dog” often depends on what the speaker is pointing at, and that of the symbolic sentence “Fx” depends on how “F” is interpreted and on the value assigned to “x.” Traditional grammar gives us less help in articulating the concept of a term, al- though the paradigm cases of both traditional nouns and logical terms are proper names such as “Wilhelm Ackermann.” Some more examples of terms from English: 4 Notes on the art of logic Terms What some linguists call them Wilhelm Ackermann proper noun the present king of France noun phrase your father’s mustache noun phrase triangularity (abstract) noun Tom’s tallness (abstract) noun phrase that snow is white that clause; factive nominal; nominalized sentence what John said factive nominal; nominalized sentence his going gerund; nominalized sentence he pronoun We add some further examples from formal languages. Terms What some logicians call them 3+4 closed term x variable 3+x open term fx: x is oddg set abstract ιxFx definite description What the logical grammarians contrast with these are so-called “common nouns” such as “horse,” as well as plural noun phrases such as “Mary and Tom.” And although we take the category of terms to be grammatical, it is helpful to heighten the contrast by a semantic remark: The terms on our list purport—at least in con- text or when fully interpreted (“assigned a value”)—to denote some single entity, while “horse” and “Mary and Tom” do not. Perhaps the common noun is the most important English grammatical category not represented anywhere in logical gram- mar. Contemporary logicians (including us) uniformly torture sentences containing common nouns, such as “A horse is a mammal,” into either “The-set-of-horses is included in the-set-of-mammals” or “For-anything-you-name, if it is-a-horse then it is-a-mammal,” where the role of the common noun “horse” is played by either the term (our sense) “the-set-of-horses” or the predicate (see below) “ is-a-horse.” Exercise 1 (Logical-grammar terms) 1A. Logical Grammar 5 Taking “term” in the sense given to it by logical grammar, give several fresh ex- amples of terms, trying to make them as diverse as possible. Include both English examples and formal examples—perhaps even an example from advanced mathe- matics. ........................................................................ / 1A.2 Functors So much for sentence and term.1 By a functor is meant a way of transforming a given ordered list of grammatical entities (i.e., a list the members of which are terms, sentences, or functors) into a grammatical entity (i.e., into either a term, a sentence, or a functor). That is to say, a functor is a function—a grammatical function—taking as inputs (arguments) lists of items from one or more grammati- cal categories and yielding uniquely as output (value) an item of some grammatical category. For each functor, as for any function, there is defined its domain, that is, the set of its input lists and its range, which is the set of its outputs. For our limited purposes, however, we can give a definition of a functor which, while not as accurate as the foregoing, is both easier to understand and adequate for our needs: 1A-1 DEFINITION. (Functor) A functor is a pattern of words with (ordered) blanks, such that when the blanks are filled with (input) any of terms, sentences, or functors, the result (output) is itself either a term, sentence or functor.2 We wish henceforth to ignore the cases in which functors are used as either inputs or outputs. For this reason, we define an “elementary functor.” 1A-2 DEFINITION. (Elementary functor) An elementary functor is a functor such that either all of its inputs are terms or all of its inputs are sentences; and whose output is either a term or a sentence. 1There is in fact a little more to say about terms, but you will understand it more easily if we delay saying it until after introducing the idea of a functor. 2This is the first official (numbered and displayed) definition in this book. Don’t bother contin- uing your study of logic unless you commit yourself to memorizing each and every such definition. Naturally learning how to use these definitions is essential as well; but you cannot learn how to use them unless you first memorize them. 6 Notes on the art of logic Hence, “input-output analysis” leads us to expect four kinds of elementary func- tors, depending exclusively on whether the inputs are terms or sentences, and whether the outputs are terms or sentences. The following is intended as a defi- nition of “operator,” “predicate,” “connective,” and “subnector.” 1A-3 DEFINITION. (Four kinds of elementary functor) Inputs Output Name Examples Terms Term Operator + ; ’s father Terms Sentence Predicate < ; is nice Sentences Sentence Connective and ; John wonders if Sentences Term Subnector “ ” ; that For example, the table tells us that a predicate is a pattern of words with blanks such that when the blanks are filled with terms, the result is a sentence. Exercise 2 (Kinds of elementary functors) 1. Write out a definition of each of “operator,” “connective,” and “subnector” that is parallel to the foregoing definition of “predicate.” 2. Use an example of each of the four, filling its blanks with short and simple words. 3. Give a couple of examples such that the output of some functor is used as the input of a different functor. We will discuss these in class. ........................................................................ / Here are some more examples. Connectives 1A. Logical Grammar 7 and ; both and ;( & )( ^ ). or ; either or ;( _ ). if then ; if ; only if ; only if , ;( ! ). if and only if ;( $ ). it is not the case that ; ∼ . The above are often studied in logic; the following, however, are also connectives by our definition. John believes (knows, wishes) that ; it is possible (necessary, certain) that ; that implies (entails) that ; if snow is red then either or if Tom is tall then . We note in passing that in English an expression can often have an analysis supple- mentary to the one given above; for example, English allows us to fill the blanks of “ and ” with the terms “Mary” and “Tom,” obtaining “Mary and Tom,” which is a plural noun phrase and hence not a creature of logical grammar. We there- fore only intend that our English examples have at least the analysis we give them, without excluding others. Predicates. hit ; John hit ;( = ); was the first person to notice that ’s mother had stolen the pen of from .