FORMAL LOGIC

the abstract study of propositions, statements, or assertively used sentences and of deductive arguments. The discipline abstracts from the content of these elements the structures or logical forms that they embody. The logician customarily uses a symbolic notation to express such structures clearly and unambiguously and to enable manipulations and tests of validity to be more easily applied. Although the following discussion freely employs the technical notation of modern symbolic logic, its symbols are introduced gradually and with accompanying explanations so that the serious and attentive general reader should be able to follow the development of ideas. Formal logic is an a priori, and not an empirical, study. In this respect it contrasts with the natural sciences and with all other disciplines that depend on observation for their data. Its nearest analogy is with pure mathematics; indeed, many logicians and pure mathematicians would regard their respective subjects as indistinguishable, or as merely two stages of the same unified discipline. Formal logic, therefore, is not to be confused with the empirical study of the processes of reasoning, which belongs to psychology. It must also be distinguished from the art of correct reasoning, which is the practical skill of applying logical principles to particular cases; and, even more sharply, it must be distinguished from the art of persuasion, in which invalid arguments are sometimes more effective than valid ones. Additional reading Michael Dummett, Elements of Intuitionism (1977), offers a clear presentation of the philosophic approach that demands constructibility in logical proofs. G.e. Hughes and M.J. Cresswell, An Introduction to Modal Logic (1968, reprinted 1989), treats operators acting on sentences in first-order logic (or predicate calculus) so that, instead of being interpreted as statements of fact, they become necessarily or possibly true or true at all or some times in the past, or they denote obligatory or permissible actions, and so on. Jon Barwise et al. (eds.), Handbook of Mathematical Logic (1977), provides a technical survey of work in the foundations of mathematics (set theory) and in proof theory (theories with infinitely long expressions). Elliott Mendelson, Introduction to Mathematical Logic, 3rd ed. (1987), is the standard text; and G. Kreisel and J.L. Krivine, Elements of Mathematical Logic: Model Theory (1967, reprinted 1971; originally published in French, 1967), covers all standard topics at an advanced level. A.S. Troelstra, Choice Sequences: A Chapter of Intuitionistic Mathematics (1977), offers an advanced analysis of the philosophical position regarding what are legitimate proofs and logical truths; and A.S. Troelstra and D. van Dalen, Constructivism in Mathematics, 2 vol. (1988), applies intuitionist strictures to the problem of the foundations of mathematics. The predicate calculus Modal logic True propositions can be divided into thoselike 2 + 2 = 4that are true by logical necessity (necessary propositions), and thoselike France is a republicthat are not (contingently true propositions). Similarly, false propositions can be divided into thoselike 2 + 2 = 5that are false by logical necessity (impossible propositions), and thoselike France is a monarchythat are not (contingently false propositions). Contingently true and contingently false propositions are known collectively as contingent propositions. A proposition that is not impossible (i.e., one that is either necessary or contingent) is said to be a possible proposition. Intuitively, the notions of necessity and possibility are connected in the following way: to say that a proposition is necessary is to say that it is not possible for it to be false; and to say that a proposition is possible is to say that it is not necessarily false. If it is logically impossible for a certain proposition, p, to be true without a certain proposition, q, being also true (i.e., if the conjunction of p and not-q is logically impossible), then it is said that p strictly implies q. An alternative, equivalent way of explaining the notion of strict implication is by saying that p strictly implies q if and only if it is necessary that p materially implies q. John's tie is scarlet, for example, strictly implies John's tie is red, because it is impossible for John's tie to be scarlet without being red (or: it is necessarily true that if John's tie is scarlet it is red); and in general, if p is the conjunction of the premises, and q the conclusion, of a deductively valid inference, p will strictly imply q. The notions just referred tonecessity, possibility, impossibility, contingency, strict implicationand certain other closely related ones are known as modal notions, and a logic designed to express principles involving them is called a modal logic. The most straightforward way of constructing such a logic is to add to some standard nonmodal system a new primitive operator intended to represent one of the modal notions mentioned above, to define other modal operators in terms of it, and to add certain special axioms or transformation rules or both. A great many systems of modal logic have been constructed, but attention will be restricted here to a few closely related ones in which the underlying nonmodal system is ordinary PC. Alternative systems of modal logic All the systems to be considered here have the same wffs but differ in their axioms. The wffs can be specified by adding to the symbols of PC a primitive monadic operator L and to the formation rules of PC the rule that if a is a wff, so is La. L is intended to be interpreted as It is necessary that, so that Lp will be true if and only if p is a necessary proposition. The monadic operator M and the dyadic operator (to be interpreted as It is possible that and strictly implies, respectively) can then be introduced by the following definitions, which reflect in an obvious way the informal accounts given above of the connections between necessity, possibility, and strict implication: if a is any wff, then Ma is to be an abbreviation of ~L~a; and if a and b are any wffs, then a b is to be an abbreviation of L(a b) [or alternatively of ~M(a ~b)]. The modal system known as T has as axioms some set of axioms adequate for PC (such as those of PM), and in addition Axiom 1 expresses the principle that whatever is necessarily true is true, and 2 the principle that, if q logically follows from p, then, if p is a necessary truth, so is q (i.e., that whatever follows from a necessary truth is itself a necessary truth). These two principles seem to have a high degree of intuitive plausibility, and 1 and 2 are theorems in almost all modal systems. The transformation rules of T are uniform substitution, modus ponens, and a rule to the effect that if a is a theorem so is La (the rule of necessitation). The intuitive rationale of this rule is that, in a sound axiomatic system, it is expected that every instance of a theorem a will be not merely true but necessarily trueand in that case every instance of La will be true. Among the simpler theorems of T are and There are many modal formulas that are not theorems of T but that have a certain claim to express truths about necessity and possibility. Among them are Lp LLp, Mp LMp, and p LMp. The first of these means that if a proposition is necessary, its being necessary is itself a necessary truth; the second means that if a proposition is possible, its being possible is a necessary truth; and the third means that if a proposition is true, then not merely is it possible but its being possible is a necessary truth. These are all various elements in the general thesis that a proposition's having the modal characteristics it has (such as necessity, possibility) is not a contingent matter but is determined by logical considerations. Although this thesis may be philosophically controversial, it is at least plausible, and its consequences are worth exploring. One way of exploring them is to construct modal systems in which the formulas listed above are theorems. None of these formulas, as was said, is a theorem of T; but each could be consistently added to T as an extra axiom to produce a new and more extensive system. The system obtained by adding Lp LLp to T is known as S4; that obtained by adding Mp LMp to T is known as S5; and the addition of p LMp to T gives the Brouwerian system, here called B for short. The relations between these four systems are as follows: S4 is stronger than T i.e., it contains all the theorems of T and others besides. B is also stronger than T. S5 is stronger than S4 and also stronger than B. S4 and B, however, are independent of each other in the sense that each contains some theorems that the other does not have. It is of particular importance that if Mp LMp is added to T then Lp LLp can be derived as a theorem but that if one merely adds the latter to T the former cannot then be derived. Examples of theorems of S4 that are not theorems of T are Mp MMp, MLMp Mp, and (p q) (Lp Lq). Examples of theorems of S5 that are not theorems of S4 are Lp MLp, L(p Mq) (Lp Mq), M(p Lq) (Mp Lq), and (Lp Lq) (Lq Lp). One important feature of S5 but not of the other systems mentioned is that any wff that contains an unbroken sequence of monadic modal operators (Ls or Ms or both) is probably equivalent to the same wff with all these operators deleted except the last. Considerations of space preclude an account of the many other axiomatic systems of modal logic that have been investigated. Some of these are weaker than T; such systems normally contain the axioms of T either as axioms or as theorems but have only a restricted form of the rule of necessitation. Another group comprises systems that are stronger than S4 but weaker than S5; some of these have proved fruitful in developing a logic of temporal relations. Yet another group includes systems that are stronger than S4 but independent of S5 in the sense explained above. Modal predicate logics can be formed also by making analogous additions to LPC instead of to PC.