The Oxford Companion to Philosophy Part 56 doc

propositional connectives, which include accounts of *material implication, *strict implication, and relevant implication. The Megarians and the Stoics also investigated various logical *antinomies, including the *liar paradox. The leading logician of this school was Chrysippus, credited with over 100 works in logic. There were few developments in logic in succeeding periods, other than a number of handbooks, summaries, translations, and commentaries, usually in a simplified and combined form. The more influential authors include Cicero, Porphyry, and Boethius in the later Roman Empire; the Byzantine scholiast Philoponus; and al- Fa¯ra¯bı¯, Avicenna, and Averroës in the Arab world. The next major logician known to us is an innovator of the first rank: Peter Abelard, who worked in the early twelfth century. He composed an independent treatise on logic, the Dialectica, and wrote extensive commentaries. There are discussions of conversion, opposition, quantity, quality, tense logic, a reduction of de dicto to *de re modality, and much else. Abelard also clearly formulates several semantic principles, including the Tarski biconditional for the theory of truth, which he rejects. Perhaps most important, Abelard is responsible for the clear formulation of a pair of relevance criteria for logical consequences. (*Rele- vance logic.) The failure of his criteria led later logicians to reject relevance implication and to endorse material implication. Spurred by Abelard’s teachings and problems he proposed, and by further translations, other logicians began to grasp the details of Aristotle’s texts. The result, coming to fruition in the middle of the thirteenth century, was the first phase of *supposition theory, an elaborate doctrine about the reference of terms in various propositional contexts. Its development is preserved in handbooks by Peter of Spain, Lambert of Auxerre, and William of Sherwood. The theory of *obligationes, a part of non-formal logic, was also invented at this time. Other topics, such as the relation between time and modality, the conventionality of semantics, and the theory of *truth, were investigated. The fourteenth century is the apex of medieval logical theory, containing an explosion of creative work. Suppo- sition theory is developed extensively in its second phase by logicians such as William of Ockham, Jean Buridan, Gregory of Rimini, and Albert of Saxony. Buridan also elaborates a full theory of consequences, a cross between entailments and inference rules. From explicit semantic principles, Buridan constructs a detailed and extensive investigation of syllogistic, and offers completeness proofs. Nor is Buridan an isolated figure. Three new liter- ary genres emerged: treatises on syncategoremata (logical particles), which attempted to codify their behaviour and the inferences they license; treatises on sentences, called ‘sophisms’, that are puzzling or challenging given background assumptions about logic and language; and treatises on insolubles, such as the liar paradox. The creative energy that drove the logical inquiries of the fourteenth century was not sustained. By the middle of the fifteenth century little if any new work was being done. There were instead many simplified handbooks and manuals of logic. The descendants of these textbooks came to be used in the universities, and the great innova- tions of medieval logicians were forgotten. Probably the best of these works is the *Port Royal Logic, by Antoine Arnauld and Pierre Nicole, which was published in 1662. When writers refer to ‘traditional logic’, they usually have this degenerate textbook tradition in mind. (*Logic, traditional.) Since the beginning of the modern era most of the con- tributions to logic have been made by mathematicians. Leibniz envisioned the development of a universal language to be specified with mathematical precision. The syntax of the words is to correspond to the metaphysical make-up of the designated entities. The goal, in effect, was to reduce scientific and philosophical speculation to com- putation. Although this grandiose project was not developed very far, and it did not enjoy much direct influence, the Universal Characteristic is a precursor to much of the subsequent work in mathematical logic. In the early nineteenth century Bolzano developed a number of notions central to logic. Some of these, like analyticity and logical consequence, are seen to be relative to a collection of ‘variable’ concepts. For example, a proposition C is a consequence of a collection P of propositions relative to a group G of variable items, if every appropriate uniform substitution for the members of G that makes every member of P true also makes C true. This may be the first attempt to characterize consequence in non-modal terms, and it is the start of a long tradition of characterizing logical notions in semantic terms, using a distinction between logical and non-logical terminology. Toward the end of the nineteenth century one can dis- tinguish three overlapping traditions in the development of logic. One of them originates with Boole and includes, among others, Peirce, Jevons, Schröder, and Venn. This ‘algebraic school’ focused on the relationship between reg- ularities in correct reasoning and operations like addition and multiplication. A primary aim was to develop calculi common to the reasoning in different areas, such as propositions, classes, and probabilities. The orientation is that of abstract algebra. One begins with one or more systems of related operations and articulates a common, abstract structure. A set of axioms is then formulated which is satisfied by each of the system. The system that Boole developed is quite similar to what is now called Boolean algebra. Other members of the school developed rudimen- tary *quantifiers, which were sometimes taken to be extended, even infinitary, conjunctions and disjunctions. The aim of the second tradition, the ‘logicist school’, was to codify the underlying logic of all rational, scientific discourse into a single system. For them, logic is not the result of abstractions from the reasoning in particular dis- ciplines and contexts. Rather, logic concerns the most general features of actual precise discourse, features independent of subject-matter. The major logicists were Russell, the early Wittgen- stein perhaps, and the greatest logician since Aristotle, 530 logic, history of Gottlob Frege. In his Begriffsschrift (translated in van Hei- jenoort (ed.), From Frege to Gödel), Frege developed a rich formal language with full mathematical rigour. Despite the two-dimensional notation, it is easily recognized as a contemporary *Higher-order logic. Quantifiers are understood as they are in current logic textbooks, not as extended conjunctions and disjunctions. Unlike the algebraists, Frege did not envision various domains of discourse, each of which can serve as an interpretation of the language. Rather, each (first-order) variable is to range over all objects whatsoever. Moreover, in contemporary terms, the systems of the logicists had no non-logical terminology. Frege made brilliant use of his logical insights when developing his philosophical programmes concerning mathematics and language. He held that arithmetic and analysis are parts of logic (*logicism; mathematics, history of the philosophy of ), and made great strides in casting number theory within the system of the Begriffsschrift. To capture mathematical induction, minimal closures, and a host of other mathematical notions, he developed and exploited the *ancestral relation, in purely logical terms. Unfortunately, the system Frege eventually developed was shown to be inconsistent. It entails the existence of a concept R which holds of all and only those extensions that do not contain themselves. A contradiction, known as *Russell’s paradox, follows. A major response was the multi-volume Principia Mathematica, by Russell and Whitehead, which attempts to recapture the logicist programme by developing an elaborate theory of *types. (*Higher- order logic.) Antino- mies are avoided by enforcing a *‘vicious-circle principle’ that no item may be defined by reference to a totality that contains the item to be defined. Despite its complexity, Principia Mathematica enjoyed a wide influence among logicians and philosophers. An elegant version of the theory, called simple type theory, was introduced by Ramsey. It violates the vicious-circle principle, but still avoids formal paradox. The third tradition dates back to at least Euclid and, in this period, includes Dedekind, Peano, Hilbert, Pasch, Veblen, Huntington, Heyting, and Zermelo. The aim of this ‘mathematical school’ is the axiomatization of particular branches of mathematics, like geometry, arithmetic, analysis, and set theory. Zermelo, for example, produced an axiomatization of set theory in 1908, drawing on insights of Cantor and others. The theory now known as Zermelo–Fraenkel set theory is the result of some modifi- cations and clarifications, due to Skolem, Fraenkel, and von Neumann, among others. Unlike Euclid, some members of the mathematical school thought it important to include an explicit formulation of the rules of inference—the logic—in the axiomatic development. In some cases, such as Hilbert and his followers, this was part of a formalist philosophical agenda, sometimes called the Hilbert programme. (*Formalism.) Others, like Heyting, produced axiomatic versions of the logic of *intuitionism and intuitionistic mathematics, in order to contrast and highlight their revi- sionist programmes (see Brouwer). A variation on the mathematical theme took place in Poland under Łukasiewicz and others. Logic itself became the branch of mathematics to be brought within axiomatic methodology. Systems of propositional logic, modal logic, tense logic, Boolean algebra, and *mereology were designed and analysed. A crucial development occurred when attention was focused on the languages and the axiomatizations themselves as objects for direct mathematical study. Drawing on the advent of non-Euclidean geometry, mathematicians in this school considered alternative interpretations of their languages and, at the same time, began to consider metalogical questions about their systems, including issues of *independence, *consistency, *categoricity, and *completeness. Both the Polish school and those pursuing the Hilbert programme developed an extensive programme for such ‘metamathematical’ investigation. (*Metalanguage; *metalogic.) Eventually, notions about syntax and proof, such as consistency and derivability, were carefully distinguished from semantic, or model- theoretic counterparts, such as satisfiability and logical consequence. This metamathematical perspective is foreign to the logicist school. For them, the relevant languages were already fully interpreted, and were not to be limited to any particular subject-matter. Because the languages are completely general, there is no interesting perspective ‘out- side’ the system from which to study it. The orientation of the logicists has been called ‘logic as language’, and that of the mathematicians and algebraists ‘logic as calculus’. Despite problems of communication, there was signifi- cant interaction between the schools. Contemporary logic is a blending of them. In 1915 Löwenheim carefully delineated what would later be recognized as the first-order part of a logical system, and showed that if a first-order formula is satisfiable at all, then it is satisfiable in a countable (or finite) domain. He was firmly rooted in the algebraic school, using techniques developed there. Skolem went on to generalize that result in several ways, and to produce more enlight- ening proofs of them. The results are known as the Löwenheim–Skolem theorems. (*Skolem’s paradox.) The intensive work on metamathematical problems culminated in the achievements of Kurt Gödel, a logician whose significance ranks with Aristotle and Frege. In his 1929 doctoral dissertation, Gödel showed that a given first-order sentence is deducible in common deductive systems for logic if and only if it is logically true in the sense that it is satisfied by all interpretations. This is known as Gödel’s completeness theorem. A year later, he proved that for common axiomatizations of a sufficiently rich version of arithmetic, there is a sentence which is nei- ther provable nor refutable therein. This is called Gödel’s incompleteness theorem, or simply *Gödel’s theorem. The techniques of Gödel’s theorem appear to be general, applying to any reasonable axiomatization that logic, history of 531 includes a sufficient amount of arithmetic. But what is ‘reasonable’? Intuitively, an axiomatization should be effec- tive: there should be an *algorithm to determine whether a given string is a formula, an axiom, etc. But what is an ‘algorithm’? Questions like this were part of the motiva- tion for logicians to turn their attention to the notions of computability and effectiveness in the middle of the 1930s. There were a number of characterizations of computability, developed more or less independently, by logicians like Gödel (recursiveness), Post, Church (lambda-definability), Kleene, Turing (the *Turing machine), and Markov (the Markov algorithm). Many of these were by- products of other research in mathematical logic. It was shown that all of the characterizations are coextensive, indicating that an important class had been identified. Today, it is widely held that an arithmetic function is com- putable if and only if it is recursive, Turing machine com- putable, etc. This is known as *Church’s thesis. Later in the decade Gödel developed the notion of set theoretic constructibility, as part of his proof that the axiom of *choice and Cantor’s *continuum hypothesis are consistent with Zermelo–Fraenkel set theory (formulated without the axiom of choice). In 1963 Paul Cohen showed that these statements are independent of Zermelo– Fraenkel set theory, introducing the powerful technique of forcing. (*Independence.) There was (and is) a spirited inquiry among set theorists, logicians, and philosophers, including Gödel himself, into whether assertions like the continuum hypothesis have determinate truth-values. (*Continuum problem; *mathematics, problems of the philosophy of.) Alfred Tarski, a pupil of Łukasiewicz, was one of the most creative and productive logicians of this, or any other, period. His influence spreads among a wide range of philosophical and mathematical schools and locations. Among philosophers, he is best known for his definitions of *truth and logical consequence, which introduce the fruitful semantic notion of *satisfaction. This, however, is but a small fraction of his work, which illuminates the methodology of deductive systems, and such central notions as completeness, decidability, consistency, satisfiability, and definability. His results are the foundation of several ongoing research programmes. Alonzo Church was another major influence in both mathematical and philosophical logic. He and students such as Kleene and Henkin have developed a wide range of areas in philosophical and mathematical logic, including completeness, definability, computability, and a number of Fregean themes, such as second-order logic and sense and reference. Church’s theorem is that the collection of first-order logical truths is not recursive. It follows from this and Church’s thesis that there is no algorithm for determining whether a given first-order formula is a logical truth. Church was a founder of the Association for Symbolic Logic and long-time guiding editor of the Journal of Symbolic Logic, which began publication in 1936. Vol- umes 1 and 3 contain an extensive bibliography of work in symbolic logic since antiquity. The development of logic in the first few decades of this century is one of the most remarkable events in intellec- tual history, bringing together many brilliant minds work- ing on closely related concepts. Mathematical logic has come to be a central tool of contemporary analytic philosophy, forming the backbone of the work of major figures like Quine, Kripke, Davidson, and Dummett. Since about the 1950s special topics of interest to contemporary philosophers, such as modal logic, tense logic, *many-valued logic (used in the study of *vagueness), *deontic logic, relevance logic, and nonstan- dard logic, have been vigorously studied. The field still attracts talented mathematicians and philosophers, and there is no sign of abatement. p.k. s.s. *logic, traditional; logical laws. I. M. Bochen´ski, A History of Formal Logic, tr. and ed. Ivo Thomas (New York, 1956). Alonzo Church, Introduction to Mathematical Logic (Princeton, NJ, 1956). Martin Davis (ed.), The Undecidable (New York, 1965). Jean van Heijenoort (ed.), From Frege to Gödel (Cambridge, Mass., 1967). William Kneale and Martha Kneale, The Development of Logic (Oxford, 1962). Alfred Tarski, Logic, Semantics and Metamathematics, 2nd edn., tr. J. H. Woodger, ed. John Corcoran (Indianapolis, 1983). logic, informal. Informal logic examines the nature and function of arguments in natural language, stressing the craft rather than the formal theory of reasoning. It supple- ments the account of simple and compound statements offered by *formal logic and, reflecting the character of arguments in natural language, widens the scope to include inductive as well as deductive patterns of inference. Informal logic’s own account of arguments begins with assertions—the premisses and conclusions—whose rich meaning in natural language is largely ignored by formal logic. Assertions have meaning as statements as well as actions and often reveal something about the person who makes them. Not least, they are the main ingredient in patterns of inference. Apart from the crucial action of claiming statements to be true, the assertions found in an argument may play other performative roles, such as war- ranting a statement’s truth (on one’s own authority or that of another), conceding its truth, contesting it, or—instead of asserting it at all—assuming the statement as a hypothesis. Assertions also have an epistemic dimension. It is a convention of natural language (though hardly a universal truth) that speakers believe what they assert. Appraising the full meaning of a premiss or conclusion therefore involves gauging whether the statement was asserted merely as a belief or, in addition, as an objective fact or even as an item of knowledge. Finally, assertions have an emotive side. Few arguments of natural language are utterly impersonal. Attitudes and feelings seep from the language of argument and can easily influence what direction a sequence of reasoning may take. Because 532 logic, history of informal logic sees assertions and arguments as woven into the fabric of discourse, the threads it traces are extremely varied: embedded but possibly incomplete patterns of deductive and non-deductive inference, hidden assumptions, conversational implications, vagueness, rhetorical techniques of persuasion, and, of course, fallac- ies. Such topics, though important for understanding arguments in natural language, lead it far from the concerns of formal logic. That informal logic lacks the precision and elegance of a formal theory is hardly surprising, therefore, but it probably comes as close as any enterprise ever will to being a science of argumentation. r.e.t. I. Copi, Informal Logic (New York, 1986). F. W. Dauer, Critical Thinking: An Introduction to Reasoning (Oxford, 1989). logic, intuitionist: see intuitionist logic. logic, many-valued: see many-valued logic. logic, modal: see modal logic. logic, modern. Logic, whether modern or traditional, is about sound reasoning and the rules which govern it. In the mid-nineteenth century (say from 1847, the date of Boole’s book The Mathematical Analysis of Logic), logic began to be developed as a rigorous mathematical system. Its development was soon speeded along by controversies about the foundations of mathematics. The resulting discoveries are now used constantly by mathematicians, philosophers, lin- guists, computer scientists, electronic engineers, and less regularly by many others (for example, music composers and psychologists). Gödel’s incompleteness theorem of 1931 was a high point not only for logic but also for twentieth-century culture. Gödel’s argument showed that there are absolute limits to what we can achieve by reasoning within a formal system; but it also showed how powerful mechanical calculation can be, and so it led almost directly to the invention of digital computers. Many arguments are valid because of their form; any other argument of the same form would be valid too. For example: Fifty-pence pieces are large seven-sided coins. This machine won’t take large coins. Therefore this machine won’t take fifty-pence pieces. An auk is a short-necked diving bird. What Smith saw was not a short-necked bird. Therefore what Smith saw was not an auk. Both of these arguments can be paraphrased into the form: (1) Every X is a Y and a Z. No Y is a W. Therefore no X is a W. (Thus for the first, X = fifty-pence piece, Y = large coin, Z = seven-sided object, W = thing that this machine will take.) This form (1) is an argument schema; it has schematic letters in it, and it becomes an argument when we translate the letters into phrases. Moreover, every argument got from the schema in this way is valid: the conclusion (after ‘Therefore’) does follow from the premisses (the sentences before ‘Therefore’). So we call (1) a valid argument schema. Likewise some statements are true purely by virtue of their form and hence are logically valid. We can write down a statement schema to show the form, for example: If p and q then p. Here the schematic letters p, q have to be translated into clauses; but whatever clauses we use, the resulting sentence must be true. Such a schema is logically valid; we can regard it as a valid argument schema with no premisses. What does it mean to say that a particular argument, expressed in English, has a particular argument schema as its form? Unfortunately this question has no exact answer. As we saw in the examples above, the words in an argument can be rearranged or paraphrased to bring out the form. Words can be replaced by synonyms too; an argument doesn’t become invalid because it says ‘gramo- phone’ at one point and ‘record-player’ at another. For the last 100 years or more, it has been usual to split logic into an exact part which deals with precisely defined argument schemas, and a looser part which has to do with translat- ing arguments into their logical *form. This looser part has been very influential in philosophy. One doctrine—we may call it the logical form doctrine— states that every proposition or sentence has a logical form, and the logical forms of arguments consist of the logical forms of the sentences occurring in them. In the early years of the century Russell and Wittgenstein put forward this doctrine in a way which led to the programme of *analytic philosophy: analysing a proposition was regarded as uncovering its logical form. Chomsky has argued that each sentence of a natural language has a structure which can be analysed at several levels, and one of these levels is called LF for logical form—roughly speak- ing, this level carries the meaning of the sentence. How- ever, Chomsky’s reasons for this linguistic analysis have nothing to do with the forms of valid arguments, though his analysis does use devices from logic, such as quantifiers and variables. One can hope for a general linguistic theory which gives each natural-language sentence a logical form that explains its meaning and also satisfies the logical form doctrine; logicians such as Montague and his student Kamp have made important suggestions in this direction, but the goal is still a long way off. Let us turn to the more exact part of logic. Experience shows that in valid argument schemas we constantly meet words such as ‘and’, ‘or’, ‘if’; moreover, the sentences can be paraphrased so that these words are used to connect clauses, not single words. For example, the sentence logic, modern 533 Fifty-pence pieces are large seven-sided coins can be paraphrased as Fifty-pence pieces are large coins and fifty-pence pieces are seven-sided. We can introduce symbols to replace these words, for example for ‘and’, ∨ for ‘or’, ¬ for ‘it is not true that . . . ’ and → for ‘if . . . then’. Unlike the schematic letters, these new symbols have a fixed meaning and they can be translated into English. They are known as *logical constants. Round about 1880 Frege and Peirce independently suggested another kind of expression for use in argument schemas. We write ∀ x …x… to mean that ‘…x…’ is true however x is interpreted. The expression ∀ x can be read as ‘For all x’. For example, the sentence Fifty-pence pieces are large seven-sided coins can be rewritten as ∀ x (if x is a fifty-pence piece then x is a large seven- sided coin), or, using the logical constants, (2) ∀x (x is a fifty-pence piece → (x is a large coin x is seven-sided) ). This last sentence says that whatever thing we consider (as an interpretation for x), if it’s a fifty-pence piece then it’s a large coin and it’s seven-sided. The symbol x is not a schematic letter in (2), because the expression ∀x becomes nonsense if we give x an interpretation. Instead it is a new kind of symbol which we call a bound variable. The expression ∀x has a twin, ∃x, which is read as ‘For some x’. These two expressions are the main examples of logical *quantifiers. Quantifiers are somewhere between logical constants and schematic letters. Like logical constants, they do have a fixed meaning. But this meaning needs to be filled out by the context, because we need to known what range of interpretations of the bound variable is allowed. This range is called the domain of quantification. (Frege assumed that the domain of quantification is always the class of all objects. But in practice when we say ‘everybody’ we usually mean everybody in the room, or all adults of sound mind, or some other restricted class of people.) With the help of the symbols described above, we can translate English sentences into a *formal language. For example we can translate (2) into ∀x (A(x) → (B(x) C(x))). Here A, B, and C are schematic letters which need to be interpreted as clauses containing x, such as ‘x is a fifty- pence piece’; this is what the (x) in A(x) indicates. The grammar of this formal language can be written down in a mathematical form. By choosing a particular set of symbols and saying exactly what range of interpretations is allowed for the schematic letters and the quantifiers, we single out a precise formal language, and we can start to ask mathematical questions about the valid argument schemas which are expressible in that language. For example a first-order language is a formal language built up from the symbols described above, where all quantifiers are interpreted as having the same domain of quantification but this domain can be any non-empty set. First-order logic is logic based on argument schemas written in a first-order language. What is the dividing-line between valid and invalid argument schemas? There are two main approaches to this question. In the first approach, which we may call the rule-based or syntactic one, we suppose that we can intuitively tell when a simple argument is valid, just by looking at it; we count a complicated argument as valid if it can be broken down into simple steps which we immediately recognize as valid. This approach naturally leads us to write down a set of simple valid argument schemas and some rules for fitting them together. The result will be a logical *calculus, i.e. a mathematical device for generating valid argument schemas. The array of symbols written down in the course of generating an argument schema by the rules is called a formal proof of the schema. Once we have a logical calculus up and running, the mathematicians may suggest ways of revamping it to make it easier to teach to undergraduates, or faster to run on a computer. There is a great variety of logical calculuses for first-order logic, all of them giving the same class of valid argument schemas. Two well-known examples are the *natural deduction calculus (Gentzen, 1934), which breaks down complex arguments into intuitively ‘natural’ pieces, and the tableau or truth-tree calculus (Beth, 1955) which is very easy to learn and can be thought of as a systematic search for counter-examples (see the next paragraph). There is another approach to defining validity, the semantic approach. In this approach we count an argument schema as valid precisely if every interpretation which makes the premisses true makes the conclusion true too. To phrase this a little differently, a counter- example to an argument schema is an interpretation which turns the premisses into true sentences and the conclusion into a false sentence; the semantic definition says that an argument schema is valid if and only if it has no counter- examples. At first sight this is a very paradoxical definition; it makes the following highly implausible argument schema valid just because the conclusion is true whatever we put for X: The Emperor Caligula’s favourite colour was X. Therefore Omsk today is a town in Siberia with a popu- lation of over a million and a large petroleum industry, and X = X. Nevertheless, one can argue that the semantic approach works if the language of our logic doesn’t contain any words (such as ‘Omsk’ or ‘today’) that tie us down to specific features of our world. This is an untidy view, because the notion of a specific feature of our world is not sharp; ∨ ∨ ∨ 534 logic, modern should it include the physical laws of the universe, or the mathematical properties of sets? One has to answer questions like these in order to draw a line between logical necessity and other kinds of necessity (physical or mathematical), and probably there will always be philosophical debate about how best to do this. For first-order logic the problem happily doesn’t arise. One can prove that every first-order argument schema which is justified by any of the standard logical calculuses is valid in the semantic sense. This is a mathematical theorem, the soundness theorem for first-order logic. Conversely if an argument schema is not proved valid by the logical calculuses, then we can show that there is an interpretation of the schema which makes the premisses true and the conclusion false. This again is a mathematical theorem, the *completeness theorem for first-order logic (Gödel, 1930; this is quite different from his incompleteness theorem of 1931). The completeness theorem justifies both the rule-based approach and the semantic one, in the following way. The chief danger with the rule-based approach was that we might have overlooked some rule that was needed. The completeness theorem assures us that any schema not justified by our logical calculus would have a counter-example, so it certainly wouldn’t be valid. And conversely the chief danger with the semantic approach was that it might make some argument schema valid for spurious reasons (like the example with Omsk above). The completeness theorem shows that if an argument has no counter-example, then it is justified by the logical calculus. In this way the valid first-order argument schemas are trapped securely on both sides, so we can be very con- fident that we have the dividing-line in the right place. For other logics the position is less clear. For example, in monadic second-order logic we have some quantifiers whose domain of quantification is required to be the fam- ily of subsets of a particular set. Because of this restriction, some truths of set theory can be expressed as valid schemas in this logic, and one consequence is that the logic doesn’t admit a completeness theorem. In temporal logics there are logical constants such as ‘until’ or ‘it will sometime be true that . . . ’; to define validity in these logics, we need to decide what background assumptions we can make about time, for example whether it is continu- ous or discrete. For these and other logics, the normal practice today is to give a precise mathematical definition of the allowed interpretations, and then use the semantic definition of validity. The result is an exact notion, even if some people are unhappy to call it logical validity. This is the place to mention a muddle in some recent psychological literature. The question at issue is how human beings carry out logical reasoning. One often reads that there are two possible answers: (1) by rules as in a logical calculus, or (2) by models (which are interpretations stripped down to the relevant essentials) as in the semantic approach. This is a confusion. There is no distinction between rule-based and semantic ways of reasoning. The rule-based and semantic approaches are different explana- tions of what we achieve when we do perform a proof: on the rule-based view, we correctly follow the rules, whereas on the semantic view we eliminate counter- examples. Can we mechanically test whether a given argument schema is logically valid, and if so, how? For first-order logic, half of the answer is positive. Given any standard logical calculus, we can use it to list in a mechanical way all possible valid argument schemas; so if an argument schema is valid, we can prove this by waiting until it appears in the list. In fact most logical calculi do much bet- ter than this; we can use them to test the schema system- atically, and if it is valid they will eventually say ‘Yes’. The bad news is that there is no possible computer pro- gram which will tell us when a given first-order argument schema is invalid. This was proved by Church in 1936, adapting Gödel’s incompleteness theorem. (Strictly it also needs Turing’s 1936 analysis of what can be done in principle by a computer.) This does not mean that there are some first-order argument schemas which are undecidable, in the sense that it’s impossible for us to tell whether they are valid or not—that might be true, but it would need further arguments about the nature of human cre- ativity. Church’s theorem does mean that there is no purely mechanical test which will give the right answer in all cases. A similar argument, again based on Gödel’s incompleteness theorem, shows that for many other logics including monadic second-order logic, it is not even possible to list mechanically the valid argument schemas. On the other hand there are many less adventurous logics— for example, the logic of Aristotle’s *syllogisms—for which we have a decision procedure, meaning that we can mechanically test any argument schema for validity. A final question: Is there a particular logical calculus which can be used to justify all valid reasoning (say, in science or mathematics)? For the intuitionist school of Brouwer, it is an article of faith that the answer is ‘No’. On the other side, Frege believed that he had given a logical calculus which was adequate at least for arithmetic; but *Russell’s paradox showed that Frege’s system was inconsistent. For the moment, the heat has gone out of this question. In modern mathematics we assume that every argument can be translated into the first-order language appropriate for set theory, and that the steps in the argument can all be justified using a first-order logical calculus together with the axioms of Zermelo–Fraenkel *set theory. This has become a criterion of sound mathematical reasoning, though nobody ever carries out the translation in practice (it would be horrendously tedious). Versions of this translation are used to check the correctness of computer soft- ware, for example where lives may depend on it. There is a more radical reading of our question. In many situations we carry out reasoning along quite different lines from the logical calculuses mentioned above. For example, when someone pays us money, we normally take for granted that it is legal tender and not a forgery, and so when it adds up correctly we infer that we have logic, modern 535 been given the correct change. Strictly this is not logical reasoning, because even when the premisses are true, the conclusion could be false (and occasionally is). But it is reasoning of a kind, and it does follow some rules. Logicians generally disregarded this kind of reasoning until they found they needed it to guide intelligent databases. For this purpose a number of non-monotonic logics have been proposed; the name refers to the fact that in this kind of reasoning a valid conclusion may cease to be valid when a new premiss is added (for example, that the five pound note has no metal strip). Several other alternative logics have been suggested, each for its own purposes. Linear logic tries to formalize the idea that there is a cost incurred each time we use a premiss, and perhaps we can only afford to use it once. An older example is intuitionist logic (Heyting, 1930), which incorporates a *verifiability principle: we can’t claim to have proved that there is an A until we can show how to produce an example of an A. Each of these logics must be justified on its own terms. There is no reason to think that the list of useful logics is complete yet. w.a.h. *logic, traditional; quantification. J. C. Beall and Bas C. von Fraassen, Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic (Oxford, 2003). H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 2nd edn. (New York, 1996). D. Gabbay, Elementary Logics: A Procedural Perspective (London, 1998). ——and F. Guenthner (eds.), Handbook of Philosophical Logic, 2nd edn. in 18 vols. (Dordrecht, 2001– ). Wilfrid Hodges, Logic, 2nd edn. (London, 2001). W. H. Newton-Smith, Logic: An Introductory Course (London, 1985). W. V. Quine, Philosophy of Logic, 2nd edn. (Cambridge, Mass., 1986). A Tarski, Introduction to Logic and to the Methodology of Deductive Sciences, 4th edn. (New York, 1994). logic, paraconsistent. A logical system is paraconsistent if it does not sanction the principle that anything follows from a contradiction. The rejected inference is called ex falso quodlibet, and is expressed in symbols thus: p, ¬pq. Paraconsistent logics have application to the logic of belief, and other propositional attitudes, especially if one wants to develop something analogous to *possible worlds semantics. A person who has contradictory beliefs is not thereby committed to every proposition whatsoever. A ‘world’ that is ‘compatible’ with one’s beliefs need not be consistent, but it should not trivially make every proposition true. Other applications of paraconsistent logic concern reasoning with faulty data and *dialetheism, the view that some contradictions are true. Dialetheism is one attempt to deal with paradoxes like the Liar. Most systems of *relevance logic are paraconsistent. s.s. Graham Priest, ‘Paraconsistent Logic’, in Dov M. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, vi, 2nd edn. (Dordrecht, 2002). logic, philosophical: see philosophical logic. logic, relevance: see relevance logic. logic, second-order. Consider ‘Socrates is wise’. In a first- order logic the name ‘Socrates’ may be replaced by a bound variable to yield ‘something is wise’. It is a further question whether the predicate in this sentence may also be replaced by a bound variable. A formal logic that per- mits this replacement is called ‘second-order’. In the standard semantics for second-order logic, first-order variables range over a domain of individuals, whereas second-order variables range over sets, properties, relations, or func- tions on the range of the first-order variables. So understood, second-order logic is extremely powerful. It is *incomplete (there can be no finite deductive system in which every second-order logical truth is deducible), *categorical (any two models that satisfy a set S of sentences are isomorphic), and not compact (even if every finite sub- set of S has a model, S itself may lack a model). In a non- standard (Henkin) semantics the second-order variables range over a separate domain of individuals. So understood, second-order logic is complete, categorical, and compact. f.m acb. *higher-order logic; categoricity; incompleteness. Stewart Shapiro, Foundations without Foundationalism: A Case for Second-Order Logic (Oxford, 1991). logic, traditional. The rough-and-ready title given by later logicians to the methods and doctrines which once dominated the universities, but which were supplanted in the twentieth century by the ‘modern’ or ‘mathematical’ logic with which the names of Frege and Russell are especially associated. Sometimes called ‘Aristotelian’—or ‘syllogistic’, or the ‘logic of terms’—it originated with Aristotle in the fourth century bc, though it acquired a great many accretions in the intervening 2,000 years. The older logic was limited, it is customary to say, by the uncritical assumption that propositions are of the subject–predicate form. This contention, however, is mis- leading; not least because the subject–predicate distinction is actually quite at odds with the formal system which is supposed to be based on it. Most traditional logicians certainly accepted that non- compound propositions invariably contain *subjects and predicates. At its vaguest, the idea was perhaps that to make any judgement at all is to say something about something. It is easy to drift from this to the more specific doctrine that every proposition contains two distinct elements: an element which names or refers to something (a ‘subject-term’), and an element (the ‘predicate-term’) which expresses what is said about it. Thus, in ‘Socrates is bald’, the name ‘Socrates’ refers to a person, and the expression ‘is bald’ says something about this person. The subject of a proposition in this sense—what it is about—is not part of the proposition but something to which part of it refers, not the name ‘Socrates’ but the T 536 logic, modern person who bears it. If some traditional logicians failed to stress the difference, this may have reflected uncertainty about the status of the predicate. The difference between ‘Socrates’ and Socrates is perfectly clear; not quite so clear is the difference between ‘is bald’ and is bald. This asymmetry is one aspect of what is really a very considerable difference: subjects and predicates belong to quite distinct categories. Granted that an expression like ‘. . . is bald’ plays a predicative role, a subject is anything of which this may be said. A subject-term is therefore a word or expression which fulfils two conditions: it constitutes a grammatical answer to a question like ‘You said that something (someone) is bald: of what (whom) did you say this?’ and it must produce good English when it is substi- tuted for x in ‘x is bald’. Proper names, referring expressions like ‘Plato’s teacher’, and a variety of other items, satisfy these conditions; but it is obvious that predicative expressions cannot themselves be subject-terms, because ‘is bald is bald’ makes no sense at all. The subject–predicate distinction, then, revolves around the difference between naming or referring to something and saying something about it. But no such distinction can sensibly be applied to the traditional system. The crowning glory of that system, it is agreed on all sides, is the doctrine of the syllogism. But this doctrine, as we shall see, requires—as indeed does the rest of the system— that what is the predicate of one proposition can be the subject of another. Traditional logic was for the most part concerned with the logical properties of four forms of proposition. More often than not these were said to be All S is P. No S is P. Some S is P. Some S is not P. ‘All S is P’ was called the ‘universal affirmative’ or ‘A’ form, ‘No S is P’ the ‘universal negative’ or ‘E’ form, ‘Some Sis P’ the ‘particular affirmative’ or ‘I’ form, and ‘Some S is not P’ the ‘particular negative’ or ‘O’ form. That a proposition is universal or particular was called its quantity, and that it is affirmative or negative was called its quality. A moment’s reflection shows that ‘All S is P’ cannot properly belong in the same list as the rest, because ‘No Greek is bald’ is good English, while ‘All Greek is bald’ is merely good gibberish. This drawback, though, could be remedied simply by taking ‘Every S is P’ to be the correct form. A more serious problem concerns the innuendo in the symbolism, which is in any case frankly espoused by those who use it, that S and P stand for subjects and predicates. If ‘is bald’ is a predicative expression, P clearly cannot be a predicate in ‘No S is P’, since ‘No Greek is is bald’ looks like a mere typing error. The stuttering ‘is’ could be removed in one of at least two ways. One would be to give up the idea that the predicate is ‘is bald’ in favour of saying that it is merely ‘bald’. This is no doubt the ulterior motive behind the half-baked suggestion that pro-positions contain a third element, over and above the subject and the predicate, namely the copula (i.e. ‘is’). Another way would be to give up the practice of writing, for example, ‘No S is P’ in favour of ‘No S P’. But the difficulties do not end there. We have seen that a subject-term is anything that takes the place of x in an expression like ‘x is bald’. According to this criterion, ‘Every man’, ‘No man’, and ‘Some man’ are perfectly good subject-terms. But substituting them in the standard forms again produces meaningless repetition: ‘Every every man is bald’, and so on. Again there are two ways of coping: one is to say that not ‘Every S is P’ but the simple S P is the correct form, the other that not ‘Every man’ but merely ‘man’ is the subject-term. These different ways of coping led our symbolism in quite different directions. One leaves us with only two elements (subject and predicate); the other first with three elements (subject, predicate, copula), then with four (subject, predicate, copula, and a sign of quantity). All these distinct, and mutually inconsistent, ways of analysing propositions are at least hinted at in the traditional textbooks. As we saw at the outset, the subject–predicate distinction arises in the context of singular propositions like ‘Socrates is bald’. In the traditional textbooks, singulars are treated as universals, on the feeble pretext that in ‘Socrates is bald’ the name ‘Socrates’ refers to everything it can. This notion was generally expressed in technical terminology: the name was said to be ‘distributed’ or to ‘refer to its whole extension’. These obscurities presum- ably reflect a disinclination to say something that is obviously absurd (that one is talking about the whole of Socrates), something that is obviously false (that only one person can be called Socrates), or something that is obviously vacuous (that the name is here meant to name everyone it is here meant to name). Be that as it may, it is worth noticing that the singular propositions which are paradigmatic in the exposition of the subject–predicate distinction become quite peripheral in the exposition of the syllogism. What this indicates is that the subject–predicate distinction is merely a nuisance so far as the formal system of traditional logic is concerned. How then should the propositions discussed in traditional logic be symbolized? The only analysis which is truly consistent with the traditional system is one in which propositions are treated as containing two distinct sorts of elements, but these are not subjects and predicates; they are logical *constants and *terms. The constants, four in number, are: ‘All . . . are . . . ’ (A) ‘No . . . are . . . ’ (E) ‘Some . . . are . . . ’ (I) ‘Some . . . are not . . . ’ (O) These are two-place term-operators, which is to say, expressions which operate on any two terms to generate propositions. What are terms? Given our operators and the require- ment that a term must be capable of filling either place in logic, traditional 537 them, this question answers itself. A term is typically a plural noun—like ‘baldies’—or any expression—like ‘persons challenged in the hair department’—that does the work of an actual or possible plural noun (‘possible’ because any particular language may or may not have a single word with the same meaning as a complex expression). Small letters from the beginning of the alphabet will be used to stand for terms, i.e. as term-variables, and these will be written after the term-operator. Thus ‘Anyone who disagrees with me is a complete fool’ is of the form Aab, where a =‘persons who disagree with me’ and b =‘ complete fools’. The traditional system relied upon two kinds of *negation. The distinction between ‘Not everything which glisters is gold’ (negating a proposition) and ‘Everything which glisters is not gold’ (negating a term) is worth fight- ing for, despite the common practice of using the second to mean the first. Propositional-negation will be represented by N (meaning ‘It is not that . . . ’); term-negation by n (meaning ‘non-’). Term-negation may preface either or both terms. Thus ‘Everything which doesn’t glister is gold’ is Anab, ‘Everything which glisters isn’t gold’) is Aanb, and ‘Everything which doesn’t glister isn’t gold’ is Ananb. We need in our symbolism also ways of representing connections between propositions. Aab & Abc will signify the conjunction of these two propositions. Aab → Anbna will signify the (in this case true) assertion that the second proposition follows from the first, and Aab ≡ Aba the (in this case false) assertion that these two propositions are equivalent, i.e. that each follows from the other. The laws of the traditional system may be classified under two headings: those which apply to only two propositions, and those which apply to three or more. The square of opposition and immediate inference fall under the first heading, syllogisms and polysyllogisms under the second. The *square of opposition depicts various kinds of ‘opposition’ between the four propositional forms. A and E are contraries, meaning that, if a and b stand for the same terms in Aab and Eab, these two propositions cannot both be true but may both be false; hence Aab → NEab and Eab → NAab. I and O are subcontraries, meaning that they cannot both be false but may both be true; hence NIab → Oab and NOab → Iab. A and O are contradictories, as are E and I, meaning that one of each pair must be true, the other false; hence Aab ≡ NOaband Eab ≡ NIab. I is subaltern to A, as O is to E, meaning that in each instance the second implies the first; hence Aab → Iab and Eab → Oab. *Immediate inference, which consists in drawing a conclusion from a single premiss, encompasses conversion, obversion, contraposition, and inversion. Conversion consists in reversing the order of terms. It is valid for E and I, invalid for A and O; hence Eab → Ebaand Iab → Iba. The valid inferences Eab → Oba and Aab → Iba are called conversion per accidens. Obversion consists in negating the second term of a proposition and changing its quality. It is valid for all four forms; hence Eab → Aanb, Aab → Eanb, Oab → Ianb, and Iab → Oanb. Contraposition consists in negating both terms and reversing their order. It is valid for A and O; hence Aab → Anbna and Oab → Onbna. Inversion consists in inferring from a given proposition another having for its subject the negation of the original subject. It is valid in the following cases: Eab → Inab, Eab → Onanb, Aab → Onab, and Aab → Inanb. *Syllogisms draw a conclusion from two premisses. They contain three terms: one (the middle term) is common to the premisses, another is common to the conclusion and one of the premisses, and the third is common to the conclusion and the other premiss. We will use b to signify the middle term, a and c to signify what are called the extreme terms. Perhaps the best-known syllogism (it was called Barbara) may be illustrated by the following simple example: Any workers who voted for that party were voting for their own unemployment. Those who vote for their own unemployment are fools to themselves. Any workers who voted for that party are fools to themselves. Traditionally syllogisms were set out this way, with the conclusion under the premisses like the lines of a sum. In our symbolism, this example is of the form (Aab &Abc) → Aac. Polysyllogisms have more than two premisses but may be reduced to a series of conventional syllogisms: Some university teachers profess to believe in acade- mic freedom but do nothing to defend it. Those who profess such a thing but do nothing about it are not practising what they preach. Teachers who fail to practise what they preach are a disgrace to their profession. Some university teachers are a disgrace to their profession. This has the form (Iab &Abc &Acd) → Iad, but it may be regarded as the summation of two conventional syllogisms, namely (Iab &Abc) → Iac and (Iac & Acd) → Iad. It is customary to say that there are 256 forms of syllogism. This number results from a convention concerning how syllogisms are depicted: the order of terms in the conclusion is fixed, but that in the premisses is reversable. The conclusion is thus restricted to taking one of four forms: Eac, Aac, Oac, or Iac. Each premiss, however, may take any one of eight forms: one is Eab, Eba, Aab, Aba, Iab, Iba, Oab, or Oba, and the other is Ebc, Ecb, Abc, Acb, Ibc, Icb, Obc, or Ocb. The number 256 is simply 4 × 8 × 8. Syllogisms were classified in the traditional textbooks according to their mood and figure. The mood of a syllogism is essentially the sequence of term-operators it contains. The mood of Barbara, for example, is AAA (hence the name). The various moods, 64 in all, are therefore EEE, EEA, EEO, EEI, and so on. The figure of a syllogism is determined by the arrangement of terms in its premisses. Aristotle distinguished three figures; later 538 logic, traditional logicians, whose conception of figure differed significantly from his, decreed that there are four: (1) ab, bc. (2) ab, cb. (3) ba, bc. (4) ba, cb. The identity of a syllogism is completely specified by its mood and figure, so the number 256 is the product of 4 (figures) and 64 (moods). Of these 256, 24 are said to be valid (some authors, for reasons that will be indicated in a moment, say 19, or even 15). Omitting brackets, the 24, arranged in their figures, are: (1) Aab &Abc → Aac Aab &Abc → Iac Iab &Abc → Iac Aab &Ebc → Eac Aab &Ebc → Oac Iab &Ebc → Oac (2) Aab &Ecb → Eac Aab &Ecb → Oac Eab &Acb → Eac Eab &Acb → Oac Iab &Ecb → Oac Oab &Acb→Oac (3) Aba &Abc → Iac Aba &Ibc → Iac Iba &Abc → Iac Aba &Ebc → Oac Aba &Obc → Oac Iba &Ebc → Oac (4) Aba &Acb → Iac Eba &Acb → Eac Eba &Acb → Oac Aba &Ecb → Oac Aba &Icb → Iac Iba &Ecb → Oac Of these, five are ‘weakened’, meaning that they draw particular conclusions from premisses that merit a universal one. If these are omitted, the number of valid forms is 19. Among these 19, 15 either draw a universal conclusion from universal premisses or a particular conclusion from one universal and one particular premiss: these were sometimes called ‘fundamental’. But the convention behind the numbers given in the traditional textbooks is wholly improper. The effect of reversing the order of terms in E and I propositions is to produce mere equivalents, while in A and O non- equivalents are produced. The textbook account therefore includes duplication. It excludes from the syllogism, moreover, the varieties of negation that are permitted in immediate inferences, and is as a consequence incomplete. The traditional system encompassed what were really eight logically distinct propositional forms: Eab (Eba, etc.). Enab (Aba, etc.). Eanb (Aab, etc.). Enanb (Anab, etc.). NEab (Iab, etc.). NEnab (Oba, etc.). NEanb (Oab, etc.). NEnanb (Onab, etc.). Any one of these eight forms is expressible in eight ways. Eab, for example, is equivalent to Eba, Aanb, Abna, NIab, NIba, NOanb and NObna. A proper account of the syllogism, then, would cover 64 forms of proposition: the correct number of syllogisms is therefore 262,144 (i.e. 64 × 64 × 64). c.w. P. T. Geach, ‘History of the Corruptions of Logic’, in Logic Matters (Oxford, 1972). J. N. Keynes, Formal Logic, 4th edn. (London, 1906). J. Łukasiewicz, Aristotle’s Syllogistic, 2nd edn. (Oxford, 1957). A. N. Prior, Formal Logic, 2nd edn. (Oxford, 1962), pt. 2, ch. 6. C. Williamson, ‘Traditional Logic as a Logic of Distribution- Values’, Logique et analyse (1971). —— ‘How Many Syllogisms Are There?’, History and Philosophy of Logic (1988). logic of discovery. *Deduction in the testing of scientific theories. For example, the exhibiting of logical relations between the sentences of a theory (such as equivalence, derivability, consistency, inconsistency) or between a theory and estabilished theories; the logical inferring of predictions from a theory. *Popper argues against the view that scientific theories are conclusively inductively verifiable but argues for their deductive and empirical falsifiability. A claim of the form ( ∀a) (Fa), ‘Every a is F’, cannot be confirmed by any finite number of observations of a’s that are F, because there could always in principle exist an undiscovered a that is not F, but ( ∀a)(Fa) can be refused by the discovery of just one a that is not F. *Popper has an evolutionary epistemology of scientific discovery. The formulation of theories is analogous to genetic mutation in evolutionary theory. Theories and mutations arise randomly as putative solutions to environmental problems, and only those conducive to the survival of the species in that environment themselves survive through trial and error. Popper adopts a Platonist view of logic, on the grounds that proofs are (sometimes surprising) discoveries not unsurprising inventions. s.p. Karl R. Popper, The Logic of Scientific Discovery (London, 1980), ch. 1, sect. 3. logical atomism: see atomism, logical. logical constants. An argument’s logical form is shown by analysing its constituent propositions into constant and variable parts, constants representing what is common to proportions, variables their differing content. The constants peculiar to syllogistic logic are ‘All … are …’, ‘No … are …’, ‘Some … are …’, Some … are not …’; those of propositional calculus are truth-functional connectives like implication, conjunction, and disjunction; those of predicate calculus add the quantifiers ‘For all x …’ and ‘There is an x such that …’. Constants concerning identity, tense, modality, etc. may be introduced in more complex systems. c.w. logical determinism: see determinism, logical. logical empiricism: see empiricism, logical. logical form: see form, logical. logical form 539 . premisses, another is common to the conclusion and one of the premisses, and the third is common to the conclusion and the other premiss. We will use b to signify the middle term, a and c to signify. example whether it is continu- ous or discrete. For these and other logics, the normal practice today is to give a precise mathematical definition of the allowed interpretations, and then use the semantic definition. proposed, and by further translations, other logicians began to grasp the details of Aristotle’s texts. The result, coming to fruition in the middle of the thirteenth century, was the first phase of

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