Logic in the Tractatus

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects

Encoding logical theories of programs

Nowadays, in many critical situations (such as on-board software), it is manda-tory to certify programs and systems, that is, to prove formally that they meet their specifications. To this end, many logics and formal systems have been proposed for rea-soning rigorously on properties of programs and systems. Their usage on non-trivial cases, however, is often cumbersome and error-prone; hence, a computerized proof assistant is required. This thesis is a contribution to the field of computer-aided formal reasoning. In recent years, Logical Frameworks (LF's) have been proposed as general metalan-guages for the description (encoding) of formal systems. LF's streamline the implementa-tion of proof systems on a machine; moreover, they allow for conceptual clarification of the object logics. The encoding methodology of LF's (based on the judgement as types, proofs as \u3bb-terms paradigm) has been successfully applied to many logics; however, the encoding of the many peculiarities presented by formal systems for program logics is problematic. In this thesis we propose a general methodology for adequately encoding formal systems for reasoning on programs. We consider Structured and Natural Operational Semantics, Modal Logics, Dynamic Logics, and the \ub5-calculus. Each of these systems presents distinc-tive problematic features; in each case, we propose, discuss and prove correct, alternative solutions. In many cases, we introduce new presentations of these systems, in Natural Deduction style, which are suggested by the metalogical analysis induced by the method-ology. At the metalogical level, we generalize and combine the concept of consequence relation by Avron and Aczel, in order to handle schematic and multiple consequences. We focus on a particular Logical Framework, namely the Calculus of Inductive Con-structions, originated by Coquand and Huet, and its implementation, Coq. Our inves-tigation shows that this framework is particularly flexible and suited for reasoning on properties of programs and systems. Our work could serve as a guide and reference to future users of Logical Frameworks

Informal proof, formal proof, formalism

Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened

What's Right With a Syntactic Approach to Theories and Models?

Syntactic approaches in the philosophy of science, which are based on formalizations in predicate logic, are often considered in principle inferior to semantic approaches, which are based on formalizations with the help of structures. To compare the two kinds of approach, I identify some ambiguities in common semantic accounts and explicate the concept of a structure in a way that avoids hidden references to a specific vocabulary. From there, I argue that contrary to common opinion (i) unintended models do not pose a significant problem for syntactic approaches to scientific theories, (ii) syntactic approaches can be at least as language independent as semantic ones, and (iii) in syntactic approaches, scientific theories can be as well connected to the world as in semantic ones. Based on these results, I argue that syntactic and semantic approaches fare equally well when it comes to (iv) ease of application, (iv) accommodating the use of models in the sciences, and (vi) capturing the theory-observation relation