Let ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let $M$ be a countable transitive model of ZF. The method of forcing extends $M$ to another model $M\lbrack G\rbrack$ of ZF (a "generic extension"). If the axiom of choice holds in $M$ it also holds in $M\lbrack G\rbrack$, that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive (...) an application to Boolean toposes. (shrink)

This is a collection of the key articles written by renowned Wittgenstein scholar, G.P. Baker, on Wittgenstein’s later philosophy, published posthumously. Following Baker’s death in 2002, the volume has been edited by collaborator and partner, Katherine Morris. Contains articles previously only available in other languages, and one previously unpublished paper. Completely distinct from the widely-known work Baker did with P.M.S. Hacker in the Analytical Commentary on the Philosophical Investigations.

In ‘Wittgenstein on Language and Rules’, Professor N. Malcolm took us to task for misinterpreting Wittgenstein's arguments on the relationship between the concept of following a rule and the concept of community agreement on what counts as following a given rule. Not that we denied that there are any grammatical connections between these concepts. On the contrary, we emphasized that a rule and an act in accord with it make contact in language. Moreover we argued that agreement in judgments and (...) in definitions is indeed necessary for a shared language. But we denied that the concept of a language is so tightly interwoven with the concept of a community of speakers as to preclude its applicabilty to someone whose use of signs is not shared by others. Malcolm holds that ‘This is an unwitting reduction of Wittgenstein's originality. That human agreement is necessary for “shared” language is not so striking a thought as that it is essential for language simpliciter.’ Though less striking, we believe that it has the merit of being a true thought. We shall once more try to show both that it is correct, and that it is a correct account of Wittgenstein's arguments. (shrink)

This chapter contains sections titled: The Tractatus and rules of logical syntax From logical syntax to philosophical grammar Rules and rule‐formulations Philosophy and grammar The scope of grammar Some morals.

This chapter contains sections titled: Following a rule Practices and techniques Doing the right thing and doing the same thing Privacy and the community view On not digging below bedrock.

This chapter contains sections titled: Setting the stage Leitmotifs External guidelines Necessary propositions and norms of representation Concerning the truth and falsehood of necessary propositions What necessary truths are about Illusions of correspondence: ideal objects, kinds of reality and ultra‐physics The psychology and epistemology of the a priori Propositions of logic and laws of thought Alternative forms of representation The arbitrariness of grammar A kinship to the non‐arbitrary Proof in mathematics Conventionalism.

Among the most outstanding discoveries of the last century is one that is not quite as momentous as the theory of relativity or cybernetics. It may even still be enigmatic. It has no one single author, it is not expressed in a single formula, conception, or invention. Nonetheless it is worth all the others combined.

The title of this article will perhaps seem excessively contentious to the reader. What solitude of Vladimir Ivanovich Vernadskii, he might ask, if he has even the slightest familiarity with the scientist's biography?

This chapter contains sections titled: Is a language necessarily shared with a community of speakers? Innate knowledge of a language Robinson Crusoe sails again Solitary cavemen and monologuists Private languages and ‘private languages’ Overview.

This chapter contains sections titled: The scaffolding of facts The role of our nature Forms of life Agreement: consensus of human beings and their actions.

This chapter contains sections titled: Initial compass bearings Accord and the harmony between language and reality Rules of inference and logical machinery Formulations and explanations of rules by examples Interpretations, fitting and grammar Further misunderstandings.

This chapter contains sections titled: The continuation of the Early Draft into philosophy of mathematics Hidden isomorphism A common methodology The flatness of philosophical grammar.