Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
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an adequate syntactic analysis of our language has priority<br />
<strong>in</strong> philosophical explanation. If we grant <strong>the</strong> fur<strong>the</strong>r <strong>the</strong>sis<br />
that Frege’s new language of quantifiers <strong>and</strong> variables<br />
represents <strong>the</strong> most perspicuous means of represent<strong>in</strong>g<br />
natural language, we can apparently <strong>in</strong> good conscience<br />
justify <strong>the</strong> privileged role of logical analysis <strong>in</strong> analytic philosophy.<br />
To privilege <strong>the</strong> role of Frege’s predicate logic is<br />
not to understate <strong>the</strong> importance for <strong>the</strong> semantic tradition<br />
of ei<strong>the</strong>r <strong>the</strong> attack on psychologism, which Dummett calls<br />
‘<strong>the</strong> extrusion of thoughts from <strong>the</strong> m<strong>in</strong>d’, or <strong>the</strong> context<br />
pr<strong>in</strong>ciple. This is because <strong>the</strong>se two tenets of analytical<br />
philosophy <strong>in</strong> its classical phase are coeval with <strong>the</strong> <strong>in</strong>troduction<br />
of Frege’s new logical symbolism. Frege’s notions<br />
of concept <strong>and</strong> object are correlative to <strong>the</strong> symbolic notions<br />
of function <strong>and</strong> argument; by tak<strong>in</strong>g concept as a<br />
function of an argument, we can underst<strong>and</strong> <strong>the</strong> process of<br />
concept formation without appeal to extraneous psychological<br />
considerations. And <strong>the</strong> context pr<strong>in</strong>ciple is, as<br />
Frege states explicitly, <strong>in</strong>spired by <strong>the</strong> rigorisation of <strong>the</strong><br />
calculus, whereby <strong>in</strong>f<strong>in</strong>itesimals are banished through an<br />
explanation of <strong>the</strong> mean<strong>in</strong>g of ‘contexts’ conta<strong>in</strong><strong>in</strong>g expressions<br />
such as df(x) or dx ra<strong>the</strong>r than seek<strong>in</strong>g to expla<strong>in</strong><br />
<strong>the</strong>m <strong>in</strong> isolation.<br />
It is generally acknowledged that <strong>the</strong> <strong>in</strong>troduction of<br />
quantifier notation <strong>and</strong> bound variables was <strong>the</strong> s<strong>in</strong>gle<br />
most important advance <strong>in</strong> logic s<strong>in</strong>ce Aristotle. Frege’s<br />
way of pars<strong>in</strong>g sentences <strong>in</strong>volv<strong>in</strong>g quantifiers offers a<br />
tremendous <strong>in</strong>crease <strong>in</strong> expressive power <strong>in</strong>sofar as it can<br />
adequately represent <strong>the</strong> statements of multiple generality<br />
that had troubled traditional syllogistic. Although <strong>the</strong> significance<br />
of Frege’s revolution <strong>in</strong> logic is well-known, however,<br />
<strong>the</strong> orig<strong>in</strong>al <strong>in</strong>tention <strong>in</strong>form<strong>in</strong>g his development of his<br />
new conceptual notation is easily understated <strong>in</strong> <strong>the</strong> contemporary<br />
context. Dummett’s statement that ‘<strong>the</strong> orig<strong>in</strong>al<br />
task which Frege set himself to accomplish, at <strong>the</strong> outset<br />
of his career, was to br<strong>in</strong>g to ma<strong>the</strong>matics <strong>the</strong> means to<br />
achieve absolute rigor <strong>in</strong> <strong>the</strong> process of proof’ (1973) is<br />
obviously accurate, but, <strong>in</strong>formed by an awareness of <strong>the</strong><br />
<strong>in</strong>completeness of second-order proof procedures, also<br />
understates <strong>the</strong> extent of Frege’s ambition.<br />
An historically unprejudiced read<strong>in</strong>g of <strong>the</strong> preface<br />
to Begriffsschrift cannot avoid <strong>the</strong> conclusion that Frege<br />
conceived of his new formula language as a vital contribution<br />
to <strong>the</strong> realization of <strong>the</strong> Enlightenment project of a<br />
ma<strong>the</strong>sis universalis, a universal methodical procedure<br />
capable of provid<strong>in</strong>g answers to all possible problems.<br />
While conced<strong>in</strong>g <strong>the</strong> slow advance <strong>in</strong> <strong>the</strong> development of<br />
formalized languages, he notes recent successes <strong>in</strong> <strong>the</strong><br />
particular sciences of arithmetic, geometry <strong>and</strong> chemistry<br />
(1879, XI), <strong>and</strong> also suggests that his own symbolism<br />
represents a particularly significant step forward <strong>in</strong>sofar as<br />
logic has a central place with respect to all o<strong>the</strong>r symbolic<br />
languages <strong>and</strong> can be used to fill <strong>in</strong> <strong>the</strong> gaps <strong>in</strong> <strong>the</strong>ir exist<strong>in</strong>g<br />
proof procedures (1879, XII). On account of its seem<strong>in</strong>gly<br />
limitless generality, <strong>the</strong> new predicate calculus, with<br />
its expressive power to represent functions <strong>and</strong> relations of<br />
higher level, is conceived by Frege as <strong>the</strong> most significant<br />
advance yet made on <strong>the</strong> way towards Leibniz’s gr<strong>and</strong>iose<br />
goal of a universal characteristic.<br />
Dummett on <strong>the</strong> Orig<strong>in</strong>s of Analytical <strong>Philosophy</strong> — George Duke<br />
3. Transformative Analysis <strong>and</strong> Semantic<br />
Logicism<br />
Recent work by Michael Beaney (2007) <strong>and</strong> Robert Br<strong>and</strong>om<br />
(2006) fur<strong>the</strong>r clarifies <strong>the</strong> dist<strong>in</strong>ctive philosophical<br />
perspective of <strong>the</strong> semantic tradition. Br<strong>and</strong>om’s characterization<br />
of <strong>the</strong> notion of ‘semantic logicism’ is particularly<br />
reveal<strong>in</strong>g, <strong>in</strong> that it provides a way of br<strong>in</strong>g<strong>in</strong>g toge<strong>the</strong>r<br />
philosophers for whom logical analysis of language <strong>and</strong><br />
mean<strong>in</strong>g is <strong>the</strong> core concern <strong>and</strong> naturalistic <strong>and</strong> empiricist<br />
approaches which are less easily accommodated by<br />
Dummett’s fundamental axiom.<br />
Beaney explicates three conceptions of analysis <strong>in</strong><br />
<strong>the</strong> Western philosophical tradition, claim<strong>in</strong>g that <strong>the</strong> third<br />
of <strong>the</strong>se - transformative analysis - is characteristic of analytical<br />
philosophy <strong>in</strong> its classical phase as embodied by<br />
Frege, Russell, <strong>the</strong> early Wittgenste<strong>in</strong> <strong>and</strong> Carnap. The<br />
first form of analysis is <strong>the</strong> decompositional - <strong>the</strong> break<strong>in</strong>g<br />
of a concept down <strong>in</strong>to its more simple parts. The decompositional<br />
approach is prevalent <strong>in</strong> early modern philosophy<br />
<strong>and</strong> encapsulated <strong>in</strong> Descartes’ 13th rule for <strong>the</strong> direction<br />
of <strong>the</strong> m<strong>in</strong>d that if we are to underst<strong>and</strong> a problem we<br />
must abstract from it every superfluous conception <strong>and</strong> by<br />
means of enumeration, divide it up <strong>in</strong>to its smallest possible<br />
parts. The second k<strong>in</strong>d of analysis is regressive analysis,<br />
accord<strong>in</strong>g to which one works back towards first pr<strong>in</strong>ciples<br />
by means of which someth<strong>in</strong>g can be demonstrated.<br />
This conception is predom<strong>in</strong>ant <strong>in</strong> classical Greek thought,<br />
for example <strong>in</strong> Euclidean geometry. Transformative analysis<br />
works on <strong>the</strong> assumption that statements need to be<br />
translated <strong>in</strong>to <strong>the</strong>ir ‘correct’ logical form before decomposition<br />
<strong>and</strong> regression can take place. Classic examples are<br />
Frege’s attempt to reduce ma<strong>the</strong>matics to logic <strong>and</strong> Russell’s<br />
<strong>the</strong>ory of def<strong>in</strong>ite descriptions. The epistemological<br />
<strong>and</strong> ontological explanatory power of Frege’s predicate<br />
logic would thus appear to be <strong>the</strong> major assumption of<br />
analytical philosophy <strong>in</strong> its classical phase.<br />
Robert Br<strong>and</strong>om <strong>in</strong>troduces <strong>the</strong> notion of ‘semantic<br />
logicism’ to characterize ‘classical’ analytical philosophy.<br />
Accord<strong>in</strong>g to Br<strong>and</strong>om, analytical philosophy <strong>in</strong> its classical<br />
phase is concerned with <strong>the</strong> relations between vocabularies<br />
– ‘its characteristic form of question is whe<strong>the</strong>r <strong>and</strong> <strong>in</strong><br />
what way one can make sense of <strong>the</strong> mean<strong>in</strong>gs expressed<br />
by one k<strong>in</strong>d of locution <strong>in</strong> terms of <strong>the</strong> mean<strong>in</strong>gs expressed<br />
by ano<strong>the</strong>r k<strong>in</strong>d of locution’ (2006, 1). So, what is dist<strong>in</strong>ctive<br />
of analytical philosophy is that ‘logical vocabulary is<br />
accorded a privileged role’ (2006, 2) <strong>in</strong> specify<strong>in</strong>g semantic<br />
relations that are thought to make <strong>the</strong> true epistemological<br />
<strong>and</strong> ontological commitments of <strong>the</strong> former explicit.<br />
In explicat<strong>in</strong>g <strong>the</strong> classical project of analysis as<br />
‘semantic logicism’, Br<strong>and</strong>om notes that it <strong>in</strong>volves, to employ<br />
Dummettian phraseology, <strong>the</strong> translation of epistemological<br />
<strong>and</strong> ontological questions <strong>in</strong>to a semantic key.<br />
Br<strong>and</strong>om describes how two core programs of classical<br />
analytical philosophy, empiricism <strong>and</strong> naturalism, were<br />
transformed <strong>in</strong> <strong>the</strong> twentieth century ‘by <strong>the</strong> application of<br />
<strong>the</strong> newly available logical vocabulary to <strong>the</strong> selfconsciously<br />
semantic programs <strong>the</strong>y <strong>the</strong>n became’ (2006,<br />
2). The generic challenge posed by such projects is to<br />
demonstrate how target vocabularies, for example, statements<br />
about <strong>the</strong> external world, can be reconstructed from<br />
‘what is expressed by <strong>the</strong> base vocabulary when it is<br />
elaborated by <strong>the</strong> use of logical vocabulary’ (2006, 3).<br />
Br<strong>and</strong>om’s characterization of semantic logicism is<br />
more <strong>in</strong>clusive than Dummett’s fundamental axiom, but<br />
none<strong>the</strong>less does not completely cover <strong>the</strong> range of philosophers<br />
who would commonly be considered analytic.<br />
Apart from th<strong>in</strong>kers like Moore <strong>and</strong> Ryle, to whom it does<br />
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