FREGE's LoGic

« The most important event in the history of philosophy in the nineteenth century was the invention of mathematical logic.This was not only a refoundation of the science of logic itself, but had important consequences for the philosophy ofmathematics, the philosophy of language, and ultimately for philosophers' understanding of the nature of philosophy itself.The principal founder of mathematical logic was Gottlob Frege.

Born on the Baltic coast of Germany in 1848, Frege(1848–1925) took his doctorate in philosophy at Göttingen, and taught at the University of Jena from 1874 until hisretirement in 1918.

Apart from his intellectual activity his life was uneventful and secluded; his work was little read in hislifetime, and even after his death his influence was exercised originally through the writings of other philosophers.

Butgradually he came to be recognized as the greatest of all philosophers of mathematics, and as a philosopher of logic fit tobe compared to Aristotle.

His invention of mathematical logic was one of the major contributions to the devel-opments inmany disciplines which resulted in the invention of computers.

Thus Frege affected the lives of all of us.Frege's productive career began in 1879 with the publication of a pamphlet with the title Begriffschrift, or Concept Script.The concept script which gave the book its title was a new symbolism designed to bring out with clarity logical relationshipswhich were concealed in ordinary language.

Frege's own script, which was logically elegant but typographicallycumbersome, is no longer used in symbolic logic; but the calculus which it formulated has ever since formed the basis ofmodern logic.Instead of the Aristotelian syllogistic, Frege placed at the front of logic the propositional calculus first explored by theStoics: that is to say, the branch of logic that deals with those inferences which depend on the force of negation, conjunction, disjunction etc.

when applied to sentences as wholes.

Its fundamental principle – which again goes back tothe Stoics – is to treat the truth-value (i.e.

the truth or falsehood) of sentences which contain connectives such as ‘and',‘if', ‘or', as being determined solely by the truth-values of the component sentences which are linked by the connectives –in the way in which the truth-value of ‘John is fat and Mary is slim' depends on the truth-values of ‘John is fat' and ‘Mary isslim'.

Composite sentences, in the logicians' technical term, are treated as truth-functions of the simple sentences of whichthey are put together.

Frege's Begriffschrift contains the first systematic formulation of the propositional calculus; it ispresented in an axiomatic manner in which all laws of logic are derived, by specified rules of inference, from a number ofprimitive principles.Frege's greatest contribution to logic was his invention of quantification theory: that is to say, a method of symbolizing andrigorously displaying those inferences that depend for their validity on expressions such as ‘all' or ‘some', ‘any' or ‘every',‘no' or ‘none'.

This new method enabled him, among other things, to reformulate traditional syllogistic.There is an analogy between the inferenceAll men are mortal Socrates is a man So Socrates is mortaland the inferenceIf Socrates is a man, then Socrates is mortal Socrates is a manSo Socrates is mortal.The second inference is a valid inference in the propositional calculus (if p then q; but p, therefore q).

But it cannot beregarded as a translation of the first, since its first premiss seems to state something about Socrates in particular,whereas if ‘All men are mortal' is true, thenif x is a man, then x is mortalwill be true no matter whose name is substituted for the variable ‘x'.

Indeed, it will remain true even if we substitute thename of a non-man for x, since in that case the antecedent will be false, and the whole sentence, in accordance with thetruth-functional rules for ‘if'-sentences, will turn out true.

So we can express the traditional proposition:All men are mortalin this wayFor all x, if x is a man, x is mortal. This reformulation forms the basis of Frege's quantification theory: to see how, we have to explain how he conceived eachof the items which go together to make up the complex sentence.Frege introduced into logic the terminology of algebra.

An algebraic expression such as ‘x/2 + 1' may be said to represent afunction of x: the value of the number represented by the whole expression will depend on what we substitute for thevariable ‘x', or, in the technical term, what we take as the argument of the function.

Thus 3 will be the value of the functionfor the argument 4, and 4 will be the value of the function for the argument 6.

Frege applied the terminology of argument,function, and value to expressions of ordinary language as well as to expressions in mathematical notation.

He replacedthe grammatical notions of subject and predicate with the mathematical notions of argument and function, and heintroduced truth-values as well as numbers as possible values for expressions.

Thus ‘x is a man' represents a functionwhich for the argument Socrates takes the value true, and for the argument Venus takes the value false.

The expression‘for all x', which introduces the sentence above, says, in Fregean terms, that what follows (‘if x is a man, x is mortal') is afunction which is true for every argument.

Such an expression is called a quantifier.Besides ‘for all x', the universal quantifier, there is also the particular quantifier ‘for some x' which says that what follows istrue for at least one argument.

Thus, ‘some swans are black' can be represented in a Fregean dialect as ‘For some x, x is aswan and x is black'.

This sentence can be taken as equivalent to ‘there are such things as black swans'; and indeed Fregemade general use of the particular quantifier in order to represent existence.

Thus, ‘God exists' or ‘there is a God' isrepresented in his system as ‘For some x, x is God'.Using his novel notation for quantification, Frege was able to present a calculus which formalized the theory of inference ina way more rigorous and more general than the traditional Aristotelian syllogistic which up to the time of Kant had beenlooked on as the be-all and end-all of logic.

After Frege, for the first time, formal logic could handle arguments whichinvolved sentences with multiple quantification, sentences which are as it were quantified at both ends, such as ‘Nobodyknows everybody' and ‘any schoolchild can master any language'.. »