The Oxford Companion to Philosophy Part 57

540 logical laws 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 pand not p’ in predicate calculus: (∀x) –(Fx. –Fx) ‘For any x, it is not the case that xisFand xis not F’ In propositional calculus the law of identity is: (p®p), ‘If pthenp’ in predicate calculus: (∀x) (Fx ® Fx), ‘For any x, if xisFthenxisF’ in predicate calculus with identity: (∀x) (x=x), ‘For any x,xisx’ in modal predicate calculus with identity: u(∀x) (x=x), ‘Necessarily, for any x,xisx’ In propositional calculus the law of excluded middle is: pv –p, ‘Either por not p’ in predicate calculus: (∀x) (Fxv –Fx), ‘For any x, either xisForxis not F’ In propositional calculus the laws of double negation are: ––p ® p, ‘If not not pthenp’, and p® ––p,‘Ifpthen not not p’ and in predicate calculus: (∀x) (––Fx® Fx) ‘For any x, if xis not not Fthen x is F’ and (∀x) (Fx ®––Fx), ‘For any x, if xisFthenxis 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 1005b): (∀x) –(Fx. –Fx) or arguably: (∀x) –à (Fx. –Fx), and ‘Nor[. . .] is it possible that there should be anything in the middle of a contra-diction’ (1011b): –à (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’ (1011b), (∀x) (Fx v –Fx) or arguably;u(∀x) (Fxv –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, thenxis at a place at a time and xis 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. logical theory 541 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: seenotations, 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 ‘xis good’ is roughly ‘I like it’, then ‘xis 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 ‘xis 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 542 logical theory *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 useat 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 φis valid’ (a kind of *logical truth) to mean ‘φis true on all interpretations’, and ‘Formula φis a consequence of the set of formulae Γ’ to mean ‘φis 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 hemeans 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. logical truth 543 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 mathematicsby 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-goriesandDe interpretatione (Oxford, 1963). G. Frege, ‘Über Sinn und Bedeutung’, Zeitschrift für Philosophie und philosophische Kritik(1892), tr. as ‘On Sense and Reference’, inTranslations 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 AandBin the first example, the letter Pin 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 ‘Por 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. 544 logical truth 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. ... - tailieumienphi.vn