The Oxford Companion to Philosophy Part 57 pptx

logical laws. Propositions true on logical grounds alone; logical truths. For example, the laws of non-contradiction, identity, excluded middle, and double negation. In propo- sitional calculus the law of non-contradiction is: –(p & –p), ‘It is not the case that both p and not p’ in predicate calculus: ( ∀x) –(Fx. –Fx) ‘For any x, it is not the case that x isF and x is not F’ In propositional calculus the law of identity is: (p → p), ‘If p then p’ in predicate calculus: ( ∀x) (Fx → Fx), ‘For any x, if x is F then x is F’ in predicate calculus with identity: ( ∀x) (x = x), ‘For any x, x is x’ in modal predicate calculus with identity: ٗ(∀x) (x = x), ‘Necessarily, for any x, x is x’ In propositional calculus the law of excluded middle is: p v –p, ‘Either p or not p’ in predicate calculus: ( ∀x) (Fx v –Fx), ‘For any x, either x is F or x is not F’ In propositional calculus the laws of double negation are: ––p → p, ‘If not not p then p’, and p → ––p, ‘If p then not not p’ and in predicate calculus: ( ∀x) (––Fx → Fx) ‘For any x, if x is not not F then x is F’ and ( ∀x) (Fx →––Fx), ‘For any x, if x is F then x is not not F’. Aristotle does not distinguish sharply between logical laws, laws of thought, and laws of being, so the consistent, the *conceivable, and what could exist coincide, and the inconsistent, the inconceivable, and what could not exist coincide. Aristotle’s informal statements of the law of non-contradiction include: ‘For the same thing to hold good and not to hold good simultaneously of the same thing and in the same respect is impossible’ (Metaphysics Γ 1005 b ): (∀x) –(Fx . –Fx) or arguably: (∀x) –◊ (Fx . –Fx), and ‘Nor [. . .] is it possible that there should be anything in the middle of a contra- diction’ (1011 b ): –◊ (p . –p). His statement of the law of excluded middle is ‘but it is necessary either to assert or deny any one thing of one thing’ (1011 b ), (∀x) (Fx v –Fx) or arguably; ٗ(∀x) (Fx v –Fx). Aristotle says it shows a lack of education to demand a proof of logical laws. He does, however, bring a self-refutation argument against their putative denial by his Pre-Socratic predecessors, Protago- ras, who thinks that every claim is true but there is no truth over and above belief by or appearance to persons, and Heraclitus, who thinks that everything is changing in every respect so there is no truth. Aristotle points out that saying anything meaningful or true—for example, making Protagorean or Heraclitean claims—presupposes logical laws. Mill maintains that logical laws are not a priori or necessary, but empirical generalizations confirmed by all experience but, so far, refuted by none. He thinks all deduction is really induction. Quine has suggested revision of the law of excluded middle to simplify quantum mechanics. Plantinga has commented that this is rather like revising a law of arith- metic to simplify the doctrine of the Holy Trinity. It is widely taken as axiomatic that if the description of a putative phenomenon entails a violation of a logical law, then that phenomenon cannot exist. However, if we are persuaded, for example, that Zeno has found contradic- tions in the concept of motion (for example: If x moves, then x is at a place at a time and x is not at that place at that time), we do not thereby conclude that nothing moves; ‘Foolish, foolish us! We thought things moved. But no. That philosopher Zeno has shown that the concept of motion entails a contradiction. Clearly we should give up this widespread, perceptually compelling but incoherent assumption! Motion is logically impossible.’ Rather, we retain the view that things move and look for a consistent theory of motion. The implications for philosophy, sci- ence, and theology are wide. Perhaps time-travel is not logically impossible, it is just that we so far lack a consis- tent theory of it. Arguably, something is possible if and only if there is at least one consistent description of it. Perhaps nothing is logically impossible, because contra- dictions do not pick out any putative states of affairs. If not, they do not pick out any impossible putative states of affairs. ‘Ah yes, “Both (p. –p)”, it is the putative state of affairs picked out by that sentence that could not come about!’ But what state of affairs could not come about? s.p. Aristotle’s Metaphysics, Book Γ, tr. with notes by Christopher Kirwan (Oxford, 1971). E. J. Lemmon, Beginning Logic (London, 1967). John Stuart Mill, A System of Logic, 2 vols. (London, 1879). Alvin Plantinga, The Nature of Necessity (Oxford, 1974). W. V. O. Quine, ‘Two Dogmas of Empiricism’, in his From a Log- ical Point of View (Cambridge, Mass., 1953), 20–46. logically perfect language. Natural *languages may be thought in various ways to be ‘logically imperfect’. Certain grammatical forms may mislead us about logical form; thus, ‘It is raining’ looks as if it refers to something (‘it’). More radically, certain concepts may even involve us in contradiction or incoherence. For example, Tarski argued that the ordinary concept ‘true’ did this, since it generated such paradoxes as the *liar. A logically perfect language would be one lacking these faults, as well, perhaps, as some other ‘defects’, such as ambiguity and redundancy. Frege attempted to create such a language (the Begriffss- chrift), in which to couch the truths of logic and mathematics. Rather later, the *Logical Positivists were interested in the idea of a logically perfect language with which to express the whole of natural science. r.p.l.t. G. Frege, Begriffsschrift, in Translations from the Philosophical Writ- ings of Gottlob Frege, tr. P. T. Geach and M. Black, 2nd edn. (Oxford, 1960), ch. 1. 540 logical laws logically proper names. The term Bertrand Russell uses for names that are logically guaranteed to have a bearer. For Russell the meaning of a logically proper name is the object it stands for. If there is no object that the name stands for, it is literally meaningless. To know the mean- ing of a logically proper name is to know the object it stands for, where this is a matter of being directly acquainted with the object. Since Russell supposed that the only objects we were directly acquainted with were private items of sensory experience or memory, only these items could be picked out by logically proper names. Conversely, if a name could be used in a sentence mean- ingfully even if it did not stand for an existing entity, for example ‘Santa Claus’, then that name could not be a logically proper name, but was instead an abbreviation of a definite description. For Russell ordinary proper names did not count as logically proper names. The only genuine examples of logically proper names in English were expressions such ‘this’, ‘that’, and ‘I’, standing for items with which the thinker was immediately acquainted. Wittgenstein thought Russell had matters the wrong way round. Instead of starting with a logical test of a genuine name, only to discover that hardly any of the expressions we ordinarily called names passed the test, a proper account of names should start by characterizing the expressions we called names. Others maintain that Russell is right about names, but wrong to restrict the entities we can name and know to items in sensory experience. To mean something by a name, we must know who, or what, we are referring to, but such knowledge can take many forms, and is not limited to direct acquaintance with the object itself. b.c.s. B. Russell, ‘The Philosophy of Logical Atomism’, in R. C. Marsh (ed.), Logic and Knowledge (London, 1984). logical notations: see notations, logical. Logical Positivism. This twentieth-century movement is sometimes also called logical (or linguistic) empiricism. In a narrower sense it also carries the name of the *Vienna Circle since such thinkers in this tradition as Rudolph Car- nap, Herbert Feigl, Otto Neurath, Moritz Schlick, and Friedrich Waismann formed an influential study group in Vienna in the early 1920s to articulate and propagate the group’s positivist ideas. In the broader sense, however, Logical Positivism includes such non-Viennese thinkers as A. J. Ayer, C. W. Morris, Arne Naess, and Ernest Nagel. Central to the movement’s doctrines is the principle of verifiability, often called the *verification principle, the notion that individual sentences gain their meaning by some specification of the actual steps we take for deter- mining their truth or falsity. As expressed by Ayer, sen- tences (statements, propositions) are meaningful if they can be assessed either by an appeal directly (or indirectly) to some foundational form of sense-experience or by an appeal to the meaning of the words and the grammatical structure that constitute them. In the former case, sen- tences are said to be synthetically true or false; in the latter, analytically true or false. If the sentences under examina- tion fail to meet the verifiability test, they are labelled meaningless. Such sentences are said to be neither true nor false. Famously, some say infamously, many posi- tivists classed metaphysical, religious, aesthetic, and ethi- cal claims as meaningless. For them, as an example, an ethical claim would have meaning only in so far as it pur- ported to say something empirical. If part of what was meant by ‘x is good’ is roughly ‘I like it’, then ‘x is good’ is meaningful because it makes a claim that could be verified by studying the behaviour of the speaker. If the speaker always avoided x, we could verify that ‘x is good’ is false. But the positivists typically deny that ‘x is good’ and simi- lar claims can be assessed as true or false beyond this sort of report. Instead, they claim that the primary ‘meaning’ of such sentences is *emotive or evocative. Thus, ‘x is good’ (as a meaningless utterance) is comparable to ‘Hooray!’ In effect, this sort of analysis shows the posi- tivists’ commitment to the fact–value distinction. Given the role that the verifiability principle plays in their thinking, it is not surprising that the Logical Positivists were admirers of science. One might say they were science- intoxicated. For them it was almost as if philosophy were synonymous with the philosophy of science, which in turn was synonymous with the study of the logic (language) of science. Typically, their philosophy of science treated sense-experience (or sense-data) as foundational and thus tended to be ‘bottom up’ in nature. That is, it tended to con- sider the foundational claims of science as being more directly verifiable (and thus more trustworthy) than the more abstract law and theoretic claims that science issues. Their philosophy of science also tended to be ‘atomistic’ rather than holistic in nature. Each foundational claim was thought to have its own truth-value in isolation from other claims. After the Second World War these doctrines of positivism, as well as the verifiability principle, atomism, and the fact–value distinction, were put under attack by such thinkers as Nelson Goodman, W. V. Quine, J. L. Austin, Peter Strawson, and, later, by Hilary Putnam and Richard Rorty. By the late 1960s it became obvious that the movement had pretty much run its course. n.f. *verificationism. A. J. Ayer, Language, Truth and Logic (New York, 1946). Herbert Feigl and May Brodbeck (eds.), Readings in the Philosophy of Science (New York, 1953). Jørgen Jørgensen, The Development of Logical Positivism (Chicago, 1951). logical symbols: see Appendix on Logical Symbols; notations, logical. logical theory. Like all parts of philosophy, logical theory is best seen as a vaguely delimited and shifting group of problems. A rough characterization would be that they concern (1) how to understand the activities of logicians and the nature of the systems that logicians construct (phil- osophy of logic), and (2) how to apply the systems to what has always been logic’s primary purpose, the appraisal of logical theory 541 *arguments. In its heyday, the twentieth century, the subject has also had important ramifications (3) 1. It is possible to see a logical system as something abstract, formal, and uninterpreted (unexplicated). The logician takes a vocabulary of words or symbols (ele- ments), and devises rules of two kinds: rules for concat- enating the elements into strings (well-formed formulae), and rules for selecting and manipulating formulae or sequences of them so as to produce other formulae or sequences (derivation rules). Doing logic consists in following these rules; logical results, or theorems, are to the effect ‘Such-and-such an output can be got by the rules’. So conceived, the activity has no use at all: it is part of pure mathematics. It is no surprise that, historically, the pure-mathematical approach came late: in its origins, logic was supposed to serve a purpose. If it is to do so, the rules must be designed to detect some property or relation, and if the purpose is to count as logical in the currently accepted sense, that property or relation must be defined in terms of truth (or of some allied notion such as satisfaction, or war- ranted assertibility). The way this works out is as follows: first we define ‘Formula φ is valid’ (a kind of *logical truth) to mean ‘φ is true on all interpretations’, and ‘Formula φ is a consequence of the set of formulae Γ’ to mean ‘φ is true on all interpretations on which all the members of Γ are true’; and then we understand ‘Such-and-such an output can be got by the rules’ as asserting that the output is a valid formula or a consequence-related sequence of for- mulae, provided that the input is (or unconditionally, if there is no input to a particular rule). This procedure interprets (explicates) the originally abstract claim that some result comes out by the rules; it gives us interpreted logic. But at once it imposes two new obligations on the logician: he must tell us what he means by ‘interpretation’ in his definitions of ‘valid’ and ‘conse- quence’, and he must show us that the rules do establish what we are now to understand their users as asserting. The first of these obligations can, in fact, be discharged in more than one way, but roughly speaking an ‘interpret- ation’ (or instance) of a formula is a sentence that results from it by replacing all its schematic letters uniformly by ordinary words. The second obligation requires the logi- cian to prove that his system of rules is sound, i.e. does what he (now) says it does. Proof of soundness depends on ways of telling when an ‘interpretation’ of a formula is true—or rather, what turns out to be enough, on ways of telling when it’s bound to be the case that every ‘interpretation’ of a given formula is true (or of a given sequence of formulae is ‘truth- preserving’). That means that we need truth-conditions for the constant elements in each formula, the elements which are unchanged through all its various ‘interpretations’. So soundness depends on truth-conditions of constants. This is something that has come to consciousness in twentieth- century logical theory, but was implicit all along. Besides soundness, logical theory is concerned with other properties of logical systems, among them com- pleteness, which is the ability of a system to generate every- thing that is, according to a given set of truth-conditions, valid or a consequence. 2. If you want to apply logic to appraising an argument, two steps are needed: fitting the argument’s premisses and con- clusion to a sequence of logical formulae, and evaluating the sequence. Evaluation goes by the rules of the logical system, provided they are sound, and is sometimes wholly mechan- ical. Logical theory must then argue (or assume) that only valid arguments fit the favourably evaluated sequences— the ones for which the consequence relation holds. Fitting is a quite different kind of operation, not mechanical and often difficult: it is symbolizing or formal- izing or ‘translating from’ ordinary words into a ‘logical language’. Pitfalls have long been known: for example, why is this not a valid argument? Man is a species. Socrates is a man. So Socrates is species. The twentieth century saw a strong revival of interest in these pitfalls, whose existence is a large part of the reason why in the first half of the century logic seemed to analytic philosophers to lie at the centre of their subject. Here are a few more examples. The President of New York is or is not black. Is that true, given that there is no such person? If not, does it falsify the law of *excluded middle? If not true, is it false? If it is false, is that because the definite *description ‘the President of New York’ is, as Russell thought, not its logical subject but an *incomplete symbol like ‘some president’? If you swallow an aspirin, you will feel better. So if you dip an aspirin in cyanide and swallow it, you will feel better. If ‘if’ worked in the same way as its surrogate ‘ → ’ in propositional logic, the argument would be valid. If the argument is invalid, as it certainly appears to be, how does ‘if’ work? Some things don’t exist (Gandalf, for example). According to Kant ‘existence is not a predicate’, and this developed into Frege’s doctrine that ‘exist’ ‘really’ has the syntactic role of a *quantifier equivalent to ‘some existing thing’, making a sentence when attached not to a subject but to a predicate. If so, the last proposition above is non- sense, mere bad grammar. Even if we readmit ‘exists’ as a genuine predicate and symbolize the last proposition in the way of predicate logic as ‘ ∃x ¬ (x exists)’, that has the unintended feature of being false, or even self- contradictory. One solution is to rejig the truth-conditions of predicate logic so that ‘ ∃xφ(x)’ means ‘Something is φ’, where that is to be distinguished from ‘Some existing thing is φ’ (free logic). Everyone who voted could have been a teller. So there could have been voting tellers. 542 logical theory One trouble is that the premiss is three-ways ambiguous. Does it mean ‘There’s a possible situation in which all those who would then have voted would then have been tellers’ or ‘There is a possible situation in which all those who actually voted would have been tellers’, or ‘For any one of those who actually voted, there is a possible situa- tion in which that one would have been a teller’? Only the first meaning licenses the inference, and then only if its ‘all’ implies ‘some’. A second difficulty is that classical predicate logic rejects that implication: ‘all’, ‘every’, etc. do not always work in the same way as their logical surrogate ‘ ∀’. Examples of similar problems could be multiplied. 3. During the twentieth century logical theory infiltrated three other disciplines: linguistics, mathematics, and metaphysics. The influence on linguistics came partly from logicians’ interest in well-formedness—what were called above the rules of concatenation. In linguistic study such rules are a part of syntax, which is a part of grammar, and although the grammar of real languages is immensely more complex, and never stable, some linguists have found the logicians’ model a helpful one. Also, as logicians came to see that the logical powers of sentences, their interrelations of *entailment and consistency and the like, depend on truth-conditions, so the thought natu- rally arose that truth-conditions determine meaning. Frege’s distinction of sense and tone had already moder- ated that enthusiasm, but the theory of meaning (seman- tics) has remained beholden to logicians’ ideas, and philosophy of *language is still not quite an independent domain. Logic was assured of an influence on mathematics by the circumstance that its nineteenth-century revival was due to mathematicians. At first they wanted foundations for arithmetic and geometry (Frege, Russell). By the 1930s conceptions (e.g. ω-consistency) and theorems (e.g. Gödel’s *incompleteness theorems) had emerged which belong to pure logic but which only a mathematical mind could compass. The infiltration into metaphysics was due mainly to Wittgenstein and Russell, and proved short-lived. In 1919 both those philosophers thought that the outline of the way things are is to be discovered by attention to how one must speak if one’s speech is to be formalizable into predi- cate, or even propositional, logic. ‘Practically all tradi- tional metaphysics’, said Russell, ‘is filled with mistakes due to bad grammar’ (‘The Philosophy of Logical Atom- ism’, 269). Kant’s idea that metaphysics explores the bounds of sense came, at the hands of Ryle and also of the *Logical Positivists, to be combined briefly with the hope that logic could chart those bounds. A bright afterglow remains in the work of Strawson, Quine, D. K. Lewis, Davidson, and very many others. c.a.k. *logic, modern; logic, traditional; metalogic. Aristotle, De interpretatione, tr. J. L. Ackrill, in Aristotle’s Cate- gories and De interpretatione (Oxford, 1963). G. Frege, ‘Über Sinn und Bedeutung’, Zeitschrift für Philosophie und philosophische Kritik (1892), tr. as ‘On Sense and Reference’, in Translations from the Philosophical Writings of Gottlob Frege, ed. P. T. Geach and M. Black (Oxford, 1952). C. A. Kirwan, Logic and Argument (London, 1978). B. A. W. Russell, ‘On Denoting’, Mind (1905), repr. in Logic and Knowledge, ed. R. C. Marsh (London, 1956), and elsewhere. —— ‘The Philosophy of Logical Atomism’, in Logic and Knowl- edge, ed. R. C. Marsh (London, 1956). P. F. Strawson, Individuals (London, 1959). logical truth. The expression has various meanings, all connected to the idea of a logical system. Logical systems have always shared two features: they are at least partly symbolic, using letters or similar devices, and they assert, or preferably prove, results about their symbolic expressions (in the modern jargon, the ‘formu- lae’ of their ‘logical language’), results such as: any argument of the form ‘No Bs are Cs, some As are Bs, so some As are not Cs’ is valid; ‘¬P’ is a consequence of ‘(P → ¬P)’. 1. One current meaning of ‘logical truth’ is ‘result in some sound logical system’ (‘sound’ is not redundant here: it excludes faulty logical systems in which not all the results are true). A true result will usually be a proved result, therefore a theorem, for example (as above): ‘¬P’ is a consequence of ‘(P → ¬P)’. 2. Sometimes certain symbolic expressions are them- selves described as logical truths, for example: If some As are Bs, then some Bs are As. ((P → ¬P) → ¬P). Here explanation is needed, since strictly speaking these expressions are not truths at all (they do not say anything). What is meant is that all their instances are true, where an instance is what you can express by uniformly replacing certain schematic or—in a loose sense—‘variable’ sym- bols (the letters A and B in the first example, the letter P in the second) by syntactically permissible words from an adequately rich vocabulary; or, alternatively, that they are true under all interpretations, where an interpretation assigns meanings uniformly to those same ‘variables’ from a syntactically limited but adequately rich range of meanings. In this usage, truth and falsity do not exhaust the field: in between logical truths, all of whose instances are true, and logical falsehoods, all of whose instances are false, are symbolic expressions such as ‘P or not Q’, having some true and some false instances. 3. Finally, and perhaps most commonly, ‘logical truth’ may mean ‘truth that is true in virtue of some result in a sound logical system’. The basic kind of case is a truth that is an instance (or interpretation) of a symbolic expression all of whose instances (or interpretations) are true, i.e. an instance of a type 2 logical truth, for example: If some men are Greeks, then some Greeks are men. If a condition for your believing erroneously that you exist is that the belief is not erroneous, then it is not erroneous. logical truth 543 The range of type 3 logical truths is indeterminate, since it depends on which sorts of system you are willing to count as logical. Propositional logic, predicate logic, and syllogistic are accredited systems, but not all philoso- phers are so happy about, say, *modal logic, epistemic logic, *tense logic, *deontic logic, *set theory, *mereol- ogy. On the other hand it is disputable whether any boundary conditions can rationally be set; certainly none are agreed. Type 3 logical truths can be defined in other roughly equivalent ways: ‘true in virtue of its (logical) form’, that is, in virtue of being an instance of some type 2 logical truth; ‘true in virtue of the meanings of its logical words’, that is, of the words in it that can be represented by con- stants in some logical system; or ‘true under all reinterpre- tations of its non-logical words’, similarly. Basic type 3 logical truths are often described as ‘logi- cally necessary’, as if their origin in logic guarantees their necessity. Part (only part) of the guarantee comes from using intuitively satisfying methods to prove the logical results, the type 1 truths, methods which may be semantic, resting on the truth-conditions of the system’s constants, or logistic, resting on self-commending manipulation of (‘derivation from’) self-commending primitive expres- sions (‘axioms’). Other truths can be deduced from the basic logical truths by means of definitions; for example, ‘A mastax is a pharynx’ from ‘The pharynx of a rotifer is a pharynx’ by the definition of ‘mastax’. But usually these aren’t counted as logical truths, though they are counted as logically nec- essary. There’s a warning in all the above: it would be mistake to suppose that you can always tell at a glance whether some proposition is a type 3 logical truth. You must know your type 1 truths, the theorems of sound systems, many of which are far from obvious; you must judge whether the systems they belong to deserve to be called logical; you must take care over the notions of ‘instance’ and ‘interpretation’ (for example, ‘If she’s wrong, she’s wrong’ will not be an instance of the type 2 logical truth ‘If P, P’, unless the ‘she’s’ refer to the same person); and defini- tions—if the use of them is allowed—are often hazy (for example, is water liquid by definition?). c.a.k. W. V. Quine, ‘Carnap and Logical Truth’, in B. H. Kazemier and D. Vuysje (eds.), Logic and Language (Dordrecht, 1962); repr. in P. A. Schilpp (ed.), The Philosophy of Rudolf Carnap (La Salle, Ill., 1963), and in The Ways of Paradox (New York, 1966). —— Philosophy of Logic (Englewood Cliffs, NJ, 1970), ch. 4. P. F. Strawson, ‘Propositions, Concepts, and Logical Truths’, Philosophical Quarterly (1957); repr. in Logico-linguistic Papers (London, 1971). logicism. The slogan of the programme is ‘Mathematics is logic’. The goal is to provide solutions to problems in the philosophy of *mathematics, by reducing mathematics, or some of its branches, to logic. There are several aspects of, and variations on, this theme. On the semantic front, logicism can be a thesis about the meaning of some mathematical statements, in which case mathematical truth would be a species of logical truth and mathematical knowledge would be logical knowledge. Mathematics, or some of its branches, might be seen as either having no ontology at all or else having only the ontology of logic (whatever that might be). In any case, the value of the enterprise depends on what logic is. The traditional logicist programme consists of system- atic translations of statements of mathematics into a language of pure logic. For Frege, statements about nat- ural numbers are statements about the extensions of cer- tain concepts. The number three, for example, is the extension of the concept that applies to all and only those concepts that apply to exactly three objects. Frege was not out to eliminate mathematical ontology, since he held that logic itself has an ontology, containing concepts and their extensions. Frege’s complete theory of extensions was shown to be inconsistent, due to the original *Rus- sell’s paradox. For Russell, statements of arithmetic are statements of ramified *type theory, or *higher-order logic. Here, too, logic has an ontology, consisting of prop- erties, propositional functions, and, possibly, classes. To complete the reduction of arithmetic, however, Russell had to postulate an axiom of *infinity; and he conceded that this is not known on logical grounds alone. So state- ments of mathematics are statements of logic, but mathe- matical knowledge goes beyond logical knowledge. On the other hand, a principle of infinity is a consequence of the (consistent) arithmetic fragment of Frege’s system. Apparently, there was no consensus on the contents and boundaries of logic, a situation that remains with us today. There are a number of views in the philosophy of math- ematics which resemble parts of logicism. It was held by some positivists that mathematical statements are *ana- lytic, true or false in virtue of the meanings of the terms. Some contemporary philosophers hold that the essence of mathematics is the determination of logical consequences of more or less arbitrary sets of axioms or postulates. As far as mathematics is concerned, the axioms might as well be meaningless. To know a theorem of arithmetic, for exam- ple, is to know that the statement is a consequence of the axioms of arithmetic. On such views, mathematical knowledge is logical knowledge. Today, a number of philosophers think of logic as the study of first-order languages, and it is widely held that logic should have no ontology. Higher-order systems are either regarded as too obscure to merit attention or are consigned to set theory, part of mathematics proper. From this perspective, logicism is an absurd undertaking. Nothing that merits the title of ‘logic’ is rich enough to do complete justice to mathematics. It is often said that the logicists accomplished (only) a reduction of some branches of mathematics to set theory. On the other hand, a number of logicians do regard higher-order logic, and the like, as part of logic, and there is extensive mathemati- cal study of such logical systems. It is not much of an exag- geration to state that logic is now part of mathematics, rather than the other way round. s.s. 544 logical truth *Logical Positivism. Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathe- matics, 2nd edn. (Cambridge, 1983). Gottlob Frege, Die Grundlagen der Arithmetik (Breslau, 1884). Alfred North Whitehead and Bertrand Russell, Principia Mathe- matica (Cambridge, 1910). logistic method. A postulational method of constructing formalized logical systems by specifying one’s symbols, recursively defining the well-formed formulae, and laying down an economical set of axioms and inference rules for proving theorems. Such a procedure is axiomatic, which historically was the norm. The currently more popular variant, *natural deduction, uses only rules of inference, for proving theorems as well as the validity of derivations. Generally, the notion of proof or of valid derivation is given a strict formal definition. This approach is moti- vated by a desire for rigour and interpretative versatility. k.w. Alonzo Church, Introduction to Mathematical Logic (Princeton, NJ, 1956), i, Intro., sect. 7. logocentrism. Term deployed most frequently by Jacques Derrida and the proponents of *deconstruction in philosophy and literary theory. In this usage a logocentric discourse is one that subscribes to the traditional order of priorities as regards language, meaning, and truth. Thus it is taken for granted first that language (spoken language) is a more or less adequate expression of ideas already in the mind, and second that writing inhabits a realm of deriva- tive, supplementary signs, a realm twice removed from the ‘living presence’ of the logos whose truth can only be revealed through the medium of authentic (self-present) speech. c.n. *différance. Jacques Derrida, Of Grammatology, tr. G. C. Spivak (Baltimore, 1976). logos. A Greek word, of great breadth of meaning, pri- marily signifying in the context of philosophical discus- sion the rational, intelligible principle, structure, or order which pervades something, or the source of that order, or giving an account of that order. The cognate verb legein means ‘say’, ‘tell’, ‘count’. Hence the ‘word’ which was ‘in the beginning’ as recounted at the start of St John’s Gospel is also logos. The root occurs in many English compounds such as biology, epistemology, and so on. Aristotle, in his Nicomachean Ethics, makes use of a distinction between the part of the soul which originates a logos (our *reason) and the part which obeys or is guided by a logos (our *emo- tions). The idea of a generative intelligence (logos sper- matikos) is a profound metaphysical notion in Neoplatonic and Christian discussion. n.j.h.d. As good a place as any to see the notion of logos at work in general is in Stoic metaphysics; see J. M. Rist, Stoic Philosophy (Cambridge, 1969). London philosophy. For a long time after the foundation of University College London in 1828 the main centres of philosophy in Britain were still Oxford, Cambridge, and the universities of Scotland. There was nothing in London like the circle of philosophers round Mersenne in seven- teenth-century Paris or the salons where the *philosophes met in the eighteenth century until the philosophical rad- icals came together in the early nineteenth century, presided over by Bentham and united, for a time, by the Westminster Review. The first element of what was to become the University of London was brought into exis- tence by this group of Benthamites. Their firmly secular intentions were at first frustrated in philosophy by the appointment of a clerical nonentity as the first professor of the subject. The official exponents of philosophy in London Uni- versity, although often worthy and competent, did not have much impact. Croom Robertson, the first editor of Mind, James Sully, principally a psychologist, Carveth Read, a follower of Mill who attached evolutionary specu- lations to his empiricist inheritance, H. Wildon Carr, a businessman who dabbled in Bergson and Croce, and the more professional and durable (he was professor at Uni- versity College from 1904 to 1928) George Dawes Hicks, a critical realist hostile to the prevailing sense-datum the- ory, can have set no one’s pulses racing. Between the wars there were some more colourful figures in various parts of the university. At Bedford was L. Susan Stebbing, aggres- sive critic of the metaphysical speculations of such scien- tists as Jeans and Eddington; at Birkbeck C. E. M. Joad, ardent and useful popularizer after his initial invest- ment in Bergson had proved unrewarding; at University College, John Macmurray, a gifted lecturer and writer, an exponent of British *idealism in its Scottish and more reli- gious form. But they were intellectually lightweight. There was, however, an altogether more interesting set of thinkers, concerned with philosophy and of high philo- sophical capacity, teaching mathematics and science in London: the logician Augustus de Morgan (who impressed his pupil Walter Bagehot), the brilliant, short- lived W. K. Clifford (whose severe ethics of belief was rejected by William James), and his follower, Karl Pear- son. Clifford and Pearson, both admirable writers, elabo- rated a phenomenalistic *positivism closely similar to that of Mach. (London, it may be noted, was the centre of the increasingly sectarian and eccentric English branch of Comtian positivism, a different and philosophically more questionable undertaking.) Other London professors of philosophical interest whose chairs were not in philosophy were L. T. Hob- house, the sociologist, and Edward Westermarck, the anthropologist, theorists, respectively, of the evolution and of the relativity of morals. The great reviews of the Victorian age were hospitable to such gifted metropolitan philosophical amateurs as G. H. Lewes, Leslie Stephen, Samuel Butler, and Fredric Harrison. Philosophy in London came into its own after 1945 and the arrival of K. R. Popper and A. J. Ayer, in their different London philosophy 545 ways continuing the tradition of Clifford and Pearson, Popper as a philosopher of science, Ayer as a scientistic philosopher. With their respective circles of active follow- ers they greatly enhanced the philosophical vitality of the capital. It came to be a third force, opposed to the amor- phous Wittgensteinianism of Cambridge and the minute lexicography of Austin’s Oxford. Ayer’s seminars of the post-war years were notable for their hard-hitting argu- mentativeness. His readiness to appear in public, on television and in the press, and the liveliness with which he did so, made him the exemplar of a philosopher for the general public. He conveyed his argumentative energy to a number of influential philosophers, just as Popper passed on his commitment to clarity to others. Ayer was succeeded by the very different Stuart Hamp- shire, shortly after the latter’s Thought and Action came out in 1959, a book whose systematic aim and fine mandarin prose were both unusual for an Oxford philosopher of the time. Also in London throughout the 1950s and 1960s was Michael Oakeshott, the even more stylish reanimator of conservative political theory. Through much the same period J. N. Findlay was at King’s College, a former Wittgensteinian who proclaimed to a surprised philo- sophical community in 1955 the merits of Hegelianism. But these imaginative, rather literary philosophers did not succeed in undermining the science-favouring tendency of London philosophy. a.q. *Cambridge philosophy; Oxford philosophy. lore, social: see social science, philosophy of. lottery paradox. Suppose I buy one ticket in a lottery with a million tickets and one prize. It would be irrational to believe my ticket will win. Some philosophers have thought that because we are so prone to error, we are bound to believe what is no more than highly probable, hence, as here, to believe that my ticket won’t win. But the same holds for each ticket, so we are bound to believe that no ticket will win. But one ticket is, ex hypothesi, certain to win: hence the paradox. What the paradox shows is that there is a difference between believing that something is— to however great a degree—probable and believing it. m.c. L. J. Cohen, The Probable and the Provable (Oxford, 1977). Lotze, Rudolf Hermann (1817–81). German physiologist and philosopher, who tried to reconcile the idealist tradi- tion, running from Leibniz to Fichte and Hegel, with nat- ural science. He argued, especially in Mikrokosmos (1856–64), that nature, including life, can be explained mechanistically, but the unity of consciousness (our abil- ity to compare two presentations and judge them (un)like) resists mechanical explanation. The causal interactions of nature presuppose that it is an organic unity of relatively permanent entities. Such entities can only be understood as finite spirits, analogues of our consciousness, and their unity is grounded in an infinite spirit or (personal) God. Natural laws are the mode of God’s activity, which aims at the realization of moral value and is to be understood by analysis of the concept of the good. ‘His work is character- istic of the woolly and emotional nebulosities which in Germany followed the collapse of the idealist school’ (Collingwood). m.j.i. H. Schnädelbach, Philosophy in Germany 1831–1933 (Cambridge, 1984). love. Affection or attachment, especially sexual, and in this sense studied by philosophers since Plato, who viewed love as a desire for beauty, which should transcend the physical and even the personal, culminating in *phi- losophy—the love of wisdom itself. In reaction to such lofty views, love has been thought of as reducible either to the sex drive (e.g. Schopenhauer) or to a struggle for power—‘in its means, war: at bottom, the deadly hatred of the sexes’ (Nietzsche). The latter view is close to that of much *feminist philosophy, which regards love as part of a male ideology for securing the subordination of women. Yet reductionism of these sorts encounters the objection that true love must be something over and above these things in virtue of the high value we set on it (as on *friendship). p.g. Irving Singer, The Nature of Love (Chicago, 1989). love-feast: see agape¯. Lovejoy, Arthur O. (1873–1962). American philosopher and historian of ideas at Johns Hopkins University who advocated *Critical Realism, temporalistic realism, and a method of tracing ideas through history. A dualist in epis- temology, he held that there are ‘changes in certain physi- cal structures which generate existents that are not physical . . . and these non-physical particulars are indis- pensable means to any knowledge of physical realities’. ‘[T]emporalism’, he said, ‘is the metaphysical theory which maintains . . . the essentially transitive and unfin- ished and self-augmentative character of reality’. In his conception of intellectual history unit-ideas are assump- tions or habits which become ‘dialectical motives’ when, vague and general as they are, they ‘influence the course of men’s reflections on almost any subject’. The historian traces each unit-idea ‘through . . . the provinces of history in which it figures in any important degree, whether those provinces are called philosophy, science, literature, art, religion or politics’. Lovejoy was also an influential and courageous advocate of academic freedom. p.h.h. Daniel J. Wilson, Arthur O. Lovejoy and the Quest for Intelligibility (Chapel Hill, NC, 1980). loyalty. A disposition, normally regarded as admirable, by which a person remains faithful and committed to a per- son or cause, despite danger and difficulty attendant on that allegiance, and often despite evidences that that per- son or cause may not be quite as meritorious or creditable 546 London philosophy as they seem. The fact that loyalty can be blind to or unmoved by such evidences gives rise to problems about its value, as the phrases misguided, misplaced, or unques- tioning loyalty suggest. None the less, we are apt to see the capacity for selfless commitment contained in loyalty as presumptively good (if it does not become fanaticism). Loyalty need not be to universal or impartial causes; it is often very limited and exclusive in its scope. In this way, too, it can give rise to injustice. Only rarely has it been seen as a cardinal *virtue. n.j.h.d. *trust. J. Royce, The Philosophy of Loyalty (New York, 1908) contains an exhaustive discussion. Lucretius (c.95–52 bc). He was a Roman poet whose work De rerum natura (On the Nature of Things) is both a major source for Epicurean philosophy and one of the master- pieces of Latin literature. He wrote the poem to transmit into Latin culture the message from Greek *Epicureanism that nothing infringes our autonomy in securing happi- ness. The centre-piece of the poem is an extended argu- ment that human beings are purely material things and so they cannot survive the destruction of their physical bod- ies; religion which seeks to teach otherwise, is damaging superstition. To support his case he had to mount exten- sive investigations of physical and psychological phenom- ena, which are described with great literary power. His attempt to prove that people are irrational to be worried about their future non-existence is often cited in contem- porary moral philosophy. j.d.g.e. C. Bailey, Titi Lucreti Cari: De rerum natura (Oxford, 1947), i. 1–171. D. Sedley, Lucretius and the Transformation of Greek Wisdom (Cam- bridge, 1998). Lukács, Georg (1885–1971). The most prominent Marxist philosopher in the Hegelian tradition, Lukács is best known for his book History and Class Consciousness (1923), which attempts a philosophical justification of the Bolshe- vik enterprise. He stressed the distinction between actual class consciousness and ‘ascribed’ class consciousness— the attitudes that the proletariat would have if they were aware of all the facts. Lukács here emphasized *dialectics over *materialism, and made concepts such as *alienation and reification central to his theory well before the publi- cation of some of Marx’s key earlier writings vindicated this interpretation. Later in his long life, which he divided between his native Hungary and the Soviet Union, Lukács became the leading Marxist theoretician of literature, before producing a monumental work on social ontology in his last decade. d.m cl. *Marxist philosophy. G. Parkinson (ed.), Georg Lukács: The Man, his Work, and his Ideas (London, 1970). Lukasiewicz, Jan (1878–1956). Logician who is the author of many innovative ideas in logic, including *many-valued logic, bracketless or *‘Polish’ notation, a formal axiomati- zation of *syllogisms including modal syllogistic, and the historical recognition of Stoic logic as the original form of modern propositional logic. Łukasiewicz intended three- valued logic to reflect Aristotle’s ideas about future con- tingent propositions in De interpretatione. If ‘There will be a sea battle tomorrow’ is true today then the sea battle’s occurrence seems predetermined or inevitable; if false then its non-occurrence seems inevitable. But by the prin- ciple of bivalence every proposition is either true or false. To ensure the contingency of future events Łukasiewicz proposed that future-tense propositions be considered neither true nor false, but instead take a third truth-value ‘indefinite’ or ‘possible’. Where 1 is ‘true’, 0 ‘false’, and ½ ‘indefinite’, Łukasiewicz’s three-valued logic is defined by the following matrices: ⊃ 1 ½ 0~ 11½ 00 ½ 11½½ 01111 s.mcc. *modal logic; many-valued logic. J. Łukasiewicz, Aristotle’s Syllogistic from The Standpoint of Modern Formal Logic (Oxford, 1957). —— Selected Works (Amsterdam, 1970). Lumber of the Schools. ’Tis you must put us in the Way; Let us (for shame) no more be fed With antique Reliques of the Dead, The Gleanings of Philosophy, Philosophy! the Lumber of the Schools . . . (Jonathan Swift, ‘Ode to Sir William Temple’, line 20) Virtue, says Swift in this over-long ode, was broken at the Fall, and ancient wisdom will never reconstitute it. To ‘dig the leaden Mines of deep Philosophy’ only produces life- less leavings—a perverse confirmation, apparently, of Plato’s theory of recollection. The poem’s almost existen- tialist excoriation of academia is perhaps connected with Swift’s having obtained his degree only by ‘special grace’ three years before writing it. Its dedicatee, Sir William Temple, who was kind enough to employ him, is declared to be the one person fit to discover ‘Virtue’s Terra Incog- nita’. j.o’g. Luther, Martin (1483–1546). German theologian, Profes- sor of Philosophy and then of Theology at Wittenberg, leader of the Protestant Reformation. Luther is notorious among philosophers for speaking of *reason as ‘the Devil’s Whore’, which must be sacrificed as the enemy of God. He sees reason as having being corrupted by original sin, and therefore incapable of coming to a true estimate of the relation between God and man. The Mosaic law, which crushes men but which would at the same time bind God to a human contract, is the fruit of reason. Salvation can only come through the divine gift of grace and revelation. While in human affairs reason ought to be followed, in the Luther, Martin 547 theological realm it must stand aside for the rebirth afforded by grace, confining its efforts to the elucidation of what God reveals through Scripture. Historically and the- ologically Luther is a pivotal point in the tradition leading from Paul’s doctrine of justification through faith and Augustine’s two cities through to the anti-rationalism of Karl Barth. a.o’h. B. A. Gerrish, Grace and Reason: A Study in the Theology of Luther (Oxford, 1962). Lycan, William G. (1945– ). Lycan develops a *truth- conditions theory of sentence-meaning in Logical Form and Natural Language, and assays the standard kinds of objec- tions to truth-conditions semantics. These arise from facts about vagueness, indexicality, tense, and other features of language in use, e.g. presupposition and conversational implications of what one says. Lycan’s truth-theoretic semantic theory is applied to fundamental questions in psycholinguistics and in an account of linguistic and cognitive abilities. In Consciousness he develops a functionalist theory of the nature of mind, ‘homuncular functionalism’. This view emphasizes the levels at which psychological and cognitive accounts of thought and action find application, from the surface level of common sense to the level at which representations are attributed to cognitive systems housed in the brain and thence to subcognitive systems which carry out semi-intelligent roles the execution of which constitute our psychological lives. Judgement and Justification contains an application of this form of *func- tionalism to the nature and role of belief. Here and else- where Lycan defends the representational theory of mind. d.g. William G. Lycan, Consciousness (Cambridge, Mass., 1987). —— Judgement and Justification (Cambridge, 1988). lying. Some church fathers held that lying, almost always prohibited, is occasionally right, as when only thus can the community be protected from invasive inquiries by perse- cutors. Augustine argued that lying is always prohibited and Aquinas agreed. Later moral philosophers divide similarly. Kant judged that a lie violates a duty to oneself and to others, because rational beings owe each other truthfulness in communication. Mill severely condemned almost all lying as injurious to human trust and therefore to the social fabric, but judged it right on rare occasions, as when only thus can some great and unmerited evil be averted. An adequate treatment of lying would have to consider whether and how it violates the norms govern- ing speech-acts of assertion and what kind of injury it involves to the trust which constitutes central human rela- tionships. a.m aci. *absolutism, moral; self-deception; noble lie. Sissela Bok, Lying (New York, 1978). Lyotard, Jean-François (1924– ). An exponent of so-called *‘post-modernism’, lately much in vogue among cultural and literary theorists. His arguments may be summarized briefly as follows. Our epoch has witnessed the collapse of all those grand ‘metanarrative’ schemas (Kantian, Hegelian, Marxist, or whatever) that once promised truth or justice at the end of inquiry. What we are left with is an open multiplicity of ‘heterogeneous’ or strictly incommen- surable *language-games, each disposing of its own imma- nent criteria. This requires that we should not presume to judge any one such discourse according to the standards, values, or truth-conditions of any other, but should instead seek to maximize the current range of ‘first-order natural pragmatic’ narratives. Moreover, anyone who rejects these premisses—who seeks (like Jürgen Habermas) to uphold the values of enlightenment, critique, and rational consen- sus as against Lyotard’s ill-defined notion of ‘dissensus’ as the touchstone of democratic freedom—must ipso facto be arguing from a ‘totalitarian’ or rigidly doctrinaire stand- point. What this amounts to, in short, is a mélange of Wittgensteinian, post-structuralist, and kindred ideas pre- sented in an oracular style that raises bafflement to a high point of principle. c.n. Jean-François Lyotard, The Postmodern Condition: A Report on Knowledge, tr. Geoff Bennington and Brian Massumi (Min- neapolis, 1983). 548 Luther, Martin Mach, Ernst (1838–1916). Austrian scientist, several times nominated for the Nobel Prize. He made important con- tributions to optics (Doppler effect), acoustics (shock waves), physiology (Mach bands), and the history and phil- osophy of *science. Writing in a vivid style he recom- mended the ‘bold intellectual move’, emphasized that sensations and physical objects were ‘as . . . preliminary as the elements of alchemy’, and criticized the scientists of his time (the defenders of the theory of relativity included) for neglecting this aspect. Making physics a measure of reality, they blocked the unification of physical, biological, and psychological phenomena. Most of Mach’s demands have by now become commonplace (*evolutionary epis- temology, *constructivism, *complementarity), though not always in a way Mach would have enjoyed. p.k.f. Bibliography, literature, and evaluations in R. S. Cohen and R. J. Seeger (eds.), Ernst Mach, Physicist and Philosopher (Dordrecht, 1970); P. K. Feyerabend, Studies in the History of the Philosophy of Science (1984); J. T. Blackmore, Ernst Mach (Los Angeles, 1972). Machiavelli, Niccolò (1469–1527). Italian statesman and political theorist who turned political thought in a new direction. Whereas traditional political theorists were concerned with morally evaluating the state in terms of fulfilling its function of promoting the common good and preserving justice, Machiavelli was more interested in empirically investigating how the state could most effect- ively use its *power to maintain law and order (political science). His famous claim that the end justifies the means also seems to advocate the use of immoral means to acquire and maintain political power. However, what he seems to mean by this is that sometimes in order to main- tain law and order it is necessary for a ruler to do things that, considered in themselves, are not right, but which, considered in their context, are right because necessary to prevent great evils. r.d.m. *ends and means; dirty hands. N. Machiavelli, The Discourses (1513). —— The Prince (1513). MacIntyre, Alasdair C. (1929– ). MacIntyre is best known for the work he has produced since 1980, although there was significant output before then. His work is primarily concerned with morality, especially with the historical changes which have shaped moral belief and practice, and also shaped theorizing about morality. Starting with his early A Short History of Ethics (London, 1966), MacIntyre has eschewed the close, often narrow, analytical and lin- guistic work which characterized much academic moral philosophy, preferring to explore the significance of moral ideas (and shifts in moral vocabulary) against the wider background of historical, cultural, sociological, religious, and other influences forming society and the individual. This has given his work an unusual breadth of reference, and has made it more accessible to non-professional per- sons interested in understanding our moral predicament. It is central to MacIntyre’s more recent work, as set out in three substantial books After Virtue (London, 1981), Whose Justice? Which Rationality? (London, 1988), and Three Rival Versions of Moral Enquiry (London, 1990, the Gifford Lectures given at the University of Edinburgh in 1988), that what many recent moral philosophers have presented as timeless truths about the nature of moral dis- course or the foundations of moral judgement are nothing of the kind. The representation of the individual as a sov- ereign chooser who by his or her own decision determines the values to live by is, in fact, the obscure manifestation of massive dislocations in society, and the dissolution of social ties and modes of life which alone can give dignity and meaning to human activity. MacIntyre has argued for an attempt to recover an Aristotelian way of viewing the purposes and activities central to human realization and fulfilment. Born in Scotland and largely educated in England, MacIntyre has worked in America since 1970. n.j.h.d. *narrative; histories of moral philosophy. Mackie, John L. (1917–81). Born in Australia, lived and taught in Australia and New Zealand before moving to England, teaching finally at Oxford University. He was the author of six books and numerous papers on a wide range of topics, especially in metaphysics, ethics, philosophy of religion, and the history of philosophy. Mackie was influ- ential for his ‘error theory’ of moral values—the view that there are no objective moral values, yet ordinary moral judgements include an implicit claim to objectivity, and hence are all false. The objectivity-claim is at least partly M . (1 957) ; repr. in Logico-linguistic Papers (London, 1971). logicism. The slogan of the programme is ‘Mathematics is logic’. The goal is to provide solutions to problems in the philosophy of *mathematics,. of interest in these pitfalls, whose existence is a large part of the reason why in the first half of the century logic seemed to analytic philosophers to lie at the centre of their subject. Here. or by an appeal to the meaning of the words and the grammatical structure that constitute them. In the former case, sen- tences are said to be synthetically true or false; in the latter, analytically