Rather later, the *Logical Positivists were interested in the idea of a logically perfect language with which to express the whole of natural science.. A rough characterization would be
Trang 1logical 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 is F 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’ (MetaphysicsΓ 1005b): (∀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’ (1011b): –◊ (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;(∀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
Trang 2logically 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
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
Trang 3*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 ‘∃x φ(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
Trang 4One 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,
*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
Trang 5The 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 mathemsystem-atics 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 statestate-ments 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 logimathemati-cal systems It is not much of an exag-geration to state that logic is now part of mathematics,
544 logical truth
Trang 6*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)
*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
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
Trang 7ways 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
*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’
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
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
Trang 8as 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
*trust
J Royce, The Philosophy of Loyalty (New York, 1908) contains an
exhaustive discussion
Lucretius(c.95–52bc) 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
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
*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:
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
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
Trang 9theological 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
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-condiobjec-tions 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
*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 ‘dissenconsen-sus’ 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
Jean-François Lyotard, The Postmodern Condition: A Report on Knowledge, tr Geoff Bennington and Brian Massumi
(Min-neapolis, 1983)
548 Luther, Martin
Trang 10Mach, 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
*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