Online research in philosophy

I make an attempt at the description of the delicate role of the standard model of arithmetic for the syntax of formal systems. I try to assess whether the possible instability in the notion of finiteness deriving from the nonstandard interpretability of arithmetic affects the very notions of syntactic metatheory and of formal system. I maintain that the crucial point of the whole question lies in the evaluation of the phenomenon of formalization. The ideas of Skolem, Zermelo, Beth and Carnap (...) (among others) on the problem are discussed. ‘A tries to explain to B the meaning of negation. Finally A gives up, saying: “You don’t understand what I mean, and I am not going to explain any longer,” to which B replies: “Yes, I see what you mean, and I am glad you are willing to continue your explanations”’. G. Mannoury, reported by E. W. Beth (Beth, 1963, 489). (shrink)

I discuss and try to evaluate the argument about constructible sets made by Putnam in ‘ ”Models and Reality”, and some of the counterarguments directed against it in the literature. I shall conclude that Putnam’s argument, while correct in substance, nevertheless has no direct bearing on the philosophical question of unintended models of set theory.

We try to explain Tarski's conception of logical notions, as it emerges from alecture of his, delivered in 1966 and published posthumously in 1986 (Historyand Philosophy of Logic 7, 143–154), a conception based on the idea ofinvariance. The evaluation of Tarski's proposal leads us to consider an interesting(and neglected) reply to Skolem in which Tarski hints at his own point of view onthe foundations of set theory. Then, comparing the lecture of 1966 with Tarski'slast work and with an earlier paper (...) written with Lindenbaum, it is shown thatTarski's conception of logical notions, with its essentially type-theoretic character,did not undergo any significant modifications throughout his life. A remark onTarski's prudential attitude on the topic in the famous paper on the concept oflogical consequence (and elsewhere) concludes our paper. (shrink)