Classical Logic (original) (raw)

Typically, a logic consists of a formal or informal language together with a deductive system and/or a modeltheoretic semantics. The language has components that correspond to a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record arguments that are valid for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions for at least part of the language. The following sections provide the basics of a typical logic, sometimes called "classical elementary logic" or "classical first-order logic". Section 2 develops a formal language, with a rigorous syntax and grammar. The formal language is a recursively defined collection of strings on a fixed alphabet. As such, it has no meaning, or perhaps better, the meaning of its formulas is given by the deductive system and the semantics. Some of the symbols have counterparts in ordinary language. We define an argument to be a non-empty collection of sentences in the formal language, one of which is designated to be the conclusion. The other sentences (if any) in an argument are its premises. Section 3 sets up a deductive system for the language, in the spirit of natural deduction. An argument is derivable if there is a deduction from some or all of its premises to its conclusion. Section 4 provides a model-theoretic semantics. An argument is valid if there is no interpretation (in the semantics) in which its premises are all true and its conclusion false. This reflects the longstanding view that a valid argument is truth-preserving. In Section 5, we turn to relationships between the deductive system and the semantics, and in particular, the relationship between derivability and validity. We show that an argument is derivable only if it is valid. This pleasant feature, called soundness, entails that no deduction takes one from true premises to a false conclusion. Thus, deductions preserve truth. Then we establish a converse, called completeness, that an argument is valid only if it is derivable. This establishes that the deductive system is rich enough to provide a deduction for every valid argument. So there are enough deductions: all and only valid arguments are derivable. We briefly indicate other features of the logic, some of which are corollaries to soundness and completeness. The final section, Section 6, is devoted to the a brief examination of the philosophical position that classical logic is "the one right logic". 1. Introduction 2. Language 2.1 Building blocks 2.2 Atomic formulas 2.3 Compound formulas 2.4 Features of the syntax 3. Deduction 4. Semantics 5. Meta-theory 6. The One Right Logic? Bibliography Further Reading Academic Tools Other Internet Resources Related Entries Typically, ordinary deductive reasoning takes place in a natural language, or perhaps a natural language augmented with some mathematical symbols. So our question begins with the relationship between a natural language and a formal language. Without attempting to be comprehensive, it may help to sketch several options on this matter. One view is that the formal languages accurately exhibit actual features of certain fragments of a natural language. Some philosophers claim that declarative sentences of natural language have underlying logical forms and that these forms are displayed by formulas of a formal language. Other writers hold that (successful) declarative sentences express propositions; and formulas of formal languages somehow display the forms of these propositions. On views like this, the components of a logic provide the underlying deep structure of correct reasoning. A chunk of reasoning in natural language is correct if the forms underlying the sentences constitute a valid or deducible argument. See for example, Montague [1974], Davidson [1984], Lycan [1984] (and the entry on logical form). Another view, held at least in part by Gottlob Frege and Wilhelm Leibniz, is that because natural languages are fraught with vagueness and ambiguity, they should be replaced by formal languages. A similar view, held by W. V. O. Quine (e.g., [1960], [1986]), is that a natural language should be regimented, cleaned up for serious scientific and metaphysical work. One desideratum of the enterprise is that the logical structures in the regimented language should be transparent. It should be easy to "read off" the logical properties of each sentence. A regimented language is similar to a formal language regarding, for example, the explicitly presented rigor of its syntax and its truth conditions. On a view like this, deducibility and validity represent idealizations of correct reasoning in natural language. A chunk of reasoning is correct to the extent that it corresponds to, or can be regimented by, a valid or deducible argument in a formal language. When mathematicians and many philosophers engage in deductive reasoning, they occasionally invoke formulas in a formal language to help disambiguate, or otherwise clarify what they mean. In other words, sometimes formulas in a formal language are used in ordinary reasoning. This suggests that one might think of a formal language as an addendum to a natural language. Then our present question concerns the relationship between this addendum and the original language. What do deducibility and validity, as sharply defined on the addendum, tell us about correct deductive reasoning in general? Another view is that a formal language is a mathematical model of a natural language in roughly the same sense as, say, a collection of point masses is a model of a system of physical objects, and the Bohr construction is a model of an atom. In other words, a formal language displays certain features of natural languages, or idealizations thereof, while ignoring or simplifying other features. The purpose of mathematical models is to shed light on what they are models of, without claiming that the model is accurate in all respects or that the model should replace what it is a model of. On a view like this, deducibility and validity represent mathematical models of (perhaps different aspects of) correct reasoning in natural languages. Correct chunks of deductive reasoning correspond, more or less, to valid or deducible arguments; incorrect chunks of reasoning roughly correspond to invalid or nondeducible arguments. See, for example, Corcoran [1973], Shapiro [1998], and Cook [2002]. There is no need to adjudicate this matter here. Perhaps the truth lies in a combination of the above options, or maybe some other option is the correct, or most illuminating one. We raise the matter only to lend some philosophical perspective to the formal treatment that follows. 2. Language Here we develop the basics of a formal language, or to be precise, a class of formal languages. Again, a formal language is a recursively defined set of strings on a fixed alphabet. Some aspects of the formal languages correspond to, or have counterparts in, natural languages like English. Technically, this "counterpart relation" is not part of the formal development, but we will mention it from time to time, to motivate some of the features and results. 2.1 Building blocks We begin with analogues of singular terms, linguistic items whose function is to denote a person or object. We call these terms. We assume a stock of individual constants. These are lowercase letters, near the beginning of the Roman alphabet, with or without numerical subscripts: We envisage a potential infinity of individual constants. In the present system each constant is a single character, and so individual constants do not have an internal syntax. Thus we have an infinite alphabet. This could be avoided by taking a constant like , for example, to consist of three characters, a lowercase " " followed by a pair of subscript "2"s. We also assume a stock of individual variables. These are lowercase letters, near the end of the alphabet, with or without numerical subscripts: In ordinary mathematical reasoning, there are two functions terms need to fulfill. We need to be able to denote specific, but unspecified (or arbitrary) objects, and sometimes we need to express generality. In our system, we use some constants in the role of unspecified reference and variables to express generality. Both uses are recapitulated in the formal treatment below. Some logicians employ different symbols for unspecified objects (sometimes called "individual parameters") and variables used to express generality. Constants and variables are the only terms in our formal language, so all of our terms are simple, corresponding to proper names and some uses of pronouns. We call a term closed if it contains no variables. In general, we use to represent variables, and to represent a closed term. Some authors also introduce function letters, which allow complex terms corresponding to: " " and "the wife of Bill Clinton", or complex terms containing variables, like "the father of " and " ". Logic books aimed at mathematicians are likely to contain function letters, probably due to the centrality of functions in mathematical discourse. Books aimed at a more general audience (or at philosophy students), may leave out function letters, since it simplifies the syntax and theory. We follow the latter route here. This is an instance of a general tradeoff between presenting a system with greater expressive resources, at the cost of making its formal treatment more complex.