The construction of logical space (Q5891078)

One of the main tenets in this book is that our conception of logical space is shaped by the acceptance or rejection of a certain sort of statements labeled by the author as just-is statements, i.e., statements of the form `For \(A\) to be \(P\) just is for it to be \(Q\)' or the form `For \(A\) to \(R\) just is for it to \(Q\)', or analogous forms. A change in the acceptance of a just-is statement would involve, according to the author, a change in the conception of logical space. Accordingly, different conceptions of logical space are possible by association to different sets of just-is statements. The author discusses the kind of considerations grounding the acceptance or rejection of that sort of statements. He also offers a formal description of the just-is operator and establishes connections between such an operator and four other notions, viz.: the notions of possibility, inconsistency, sameness of truth conditions and of what he calls the Why-closure of a given statement \(p\) (i.e., the unableness to make sense of the question `Why is it the case that \(p\)'). In addition, he addresses the question of what the truth and falsity of a just-is statement consists in together with the question whether there is an objectively correct conception of logical space. There is no definite answer in the book to the latter question. Now, also in connection with just-is statements, the author proposes a certain philosophy of language that he calls compositionalism. This philosophical theory makes room for certain sorts of just-is statements that, according to the author, should not be rejected on linguistic or metaphysical grounds. Compositionalism involves two claims, one concerning the nature of singular terms and the other of reference. On the basis of this philosophy of language as well as his view of just-is statements and their ties to logical space, the author develops a defense of a variant of mathematical Platonism that he calls trivialist Platonism, i.e., the view that the truths of pure mathematics have trivial truth conditions and the falsities of pure mathematics have trivial falsity conditions. His argument first involves a compositional truth conditions semantic theory for arithmetic, in which every true (false) sentence of pure arithmetic is assigned trivial truth conditions (falsity conditions), and then an extension of such a semantic theory to the case of set theory with urelements. Furthermore, he argues that the trivialist could develop a conception of set theory based on the iterative conception of sets. Finally, the author provides an account of the epistemology of mathematics by offering an explanation of cognitive accomplishment in logic and mathematics. He argues that such accomplishments could be understood in terms of information transfer abilities. Other topics discussed in the book include the connection the author thinks there is between just-is statements and metaphysical possibility and the paraphrase methods for the language of arithmetic from the perspective of a trivialist Platonist. A compositionalist account of linguistic stipulation in mathematics and a device for simulating quantification over possible objects from the perspective of a modal actualist are also presented in the book.