Related papers: Prawitz's Conjecture is False; So What?
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…
Our understanding about things is conceptual. By stating that we reason about objects, it is in fact not the objects but concepts referring to them that we manipulate. Now, so long just as we acknowledge infinitely extending notions such as…
General acceptance of a mathematical proposition $P$ as a theorem requires convincing evidence that a proof of $P$ exists. But what constitutes "convincing evidence?" I will argue that, given the types of evidence that are currently…
Probability theory, epistemically interpreted, provides an excellent, if not the best available account of inductive reasoning. This is so because there are general and definite rules for the change of subjective probabilities through…
In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a…
This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett's and Dag Prawitz' philosophical motivations and…
I discuss two approaches to monotonic proof-theoretic semantics. In the first one, which I call SVA, consequence is understood in terms of existence of valid arguments. The latter involve the notions of argument structure and justification…
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
The analysis of theory-confirmation generally takes the deductive form: show that a theory in conjunction with physical data and auxiliary hypotheses yield a prediction about phenomena; verify the prediction; provide a quantitative measure…
In this introductory note, I describe my particular view of the notion of ontological commitments as honest and pragmatic working hypotheses that assume the existence (out there) of certain entities represented by the symbols in our theory.…
Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on the rule-following…
Logically constrained term rewriting is a relatively new formalism where rules are equipped with constraints over some arbitrary theory. Although there are many recent advances with respect to rewriting induction, completion, complexity…
In two recent papers Khrennikov uses what he calls Ozawa intersubjectivity theorem to claim that intersubjectivity is necessarily verified in quantum mechanics and to criticize QBism and more generally all interpretations that are…
The field of proof-theoretic semantics (P-tS) offers an alternative approach to meaning in logic that is based on inference and argument (rather than truth in a model). It has been successfully developed for various logics; in particular,…
It is known that intuitionistic Kripke semantics can be generalized so that it can treat arbitrary propositional connectives characterized by truth functions. We extend this generalized Kripke semantics to first-order logic, and study how…
Ordinary and transfinite recursion and induction and ZF set theory are used to construct from a fully interpreted object language and from an extra formula a new language. It is fully interpreted under a suitably defined interpretation.…
The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$.…
How does the mathematical community accept that a given proof is correct? Is objective verification based on explicit axioms feasible, or must the reviewer's experiences and prejudices necessarily come into play? Can automated provers avoid…
I deal with two approaches to proof-theoretic semantics: one based on argument structures and justifications, which I call reducibility semantics, and one based on consequence among (sets of) formulas over atomic bases, called base…
For a long time, Collatz Conjecture has been assumed to be true, although a formal proof has eluded all efforts to date. In this article, evidence is presented that suggests such an assumption is incorrect. By analysing the stopping times…