Iterated lower bound formulas: a diagonalization-based approach to proof complexity

We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS and in the context of propositional proof systems more generally. We use the approach to give an explicit sequence of CNF formulas {φn} such that VNP ≠ VP iff there are no polynomial-size IPS proofs for the formulas φn. This provides a natural equivalence between proof complexity lower bounds and standard algebraic complexity lower bounds. Our proof of this fact uses the implication from IPS lower bounds to algebraic complexity lower bounds due to Grochow and Pitassi together with a diagonalization argument: the formulas φn themselves assert the non-existence of short IPS proofs for formulas encoding VNP ≠ VP at a different input length. Our result also has meta-mathematical implications: it gives evidence for the difficulty of proving strong lower bounds for IPS within IPS. For any strong enough propositional proof system R, we define the *iterated R-lower bound formulas*, which inductively assert the non-existence of short R proofs for formulas encoding the same statement at a different input length, and propose them as explicit hard candidates for the proof system R. We observe that this hypothesis holds for Resolution following recent results of Atserias and Muller and of Garlik, and give evidence in favour of it for other proof systems

Frege's Basic Law V and Cantor's Theorem

The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These ontological issues are interesting in their own right. And if and only if in case ontological considerations make a strong case for something like (BLV) we have to trouble us with inconsistency and paraconsistency. These ontological issues also lead to a renewed methodological reflection what to assume or recognize as an axiom

Fourteen Arguments in Favour of a Formalist Philosophy of Real Mathematics

The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attention to some arguments in favour of game formalism as an appropriate philosophy of real mathematics. For the most part, these arguments have not yet been used or were neglected in past discussions

Topics in Programming Languages, a Philosophical Analysis through the case of Prolog

[EN]Programming languages seldom find proper anchorage in philosophy of logic, language and science. is more, philosophy of language seems to be restricted to natural languages and linguistics, and even philosophy of logic is rarely framed into programming languages topics. The logic programming paradigm and Prolog are, thus, the most adequate paradigm and programming language to work on this subject, combining natural language processing and linguistics, logic programming and constriction methodology on both algorithms and procedures, on an overall philosophizing declarative status. Not only this, but the dimension of the Fifth Generation Computer system related to strong Al wherein Prolog took a major role. and its historical frame in the very crucial dialectic between procedural and declarative paradigms, structuralist and empiricist biases, serves, in exemplar form, to treat straight ahead philosophy of logic, language and science in the contemporaneous age as well. In recounting Prolog's philosophical, mechanical and algorithmic harbingers, the opportunity is open to various routes. We herein shall exemplify some: - the mechanical-computational background explored by Pascal, Leibniz, Boole, Jacquard, Babbage, Konrad Zuse, until reaching to the ACE (Alan Turing) and EDVAC (von Neumann), offering the backbone in computer architecture, and the work of Turing, Church, Gödel, Kleene, von Neumann, Shannon, and others on computability, in parallel lines, throughly studied in detail, permit us to interpret ahead the evolving realm of programming languages. The proper line from lambda-calculus, to the Algol-family, the declarative and procedural split with the C language and Prolog, and the ensuing branching and programming languages explosion and further delimitation, are thereupon inspected as to relate them with the proper syntax, semantics and philosophical élan of logic programming and Prolog

Reference and Indexicality

Tese arquivada ao abrigo da Portaria nº 227/2017 de 25 de Julho-Registo de Grau EstrangeiroThis thesis is a general defence of a context-dependent description theory of reference with special regards to indexical reference on the basis of a truth-conditional theory of meaning. It consists of two parts. In the first part, the roots of the Frege-Russell view are laid out and contrasted with various aspects of direct reference theory and the New Theory of Reference. Two description-based accounts of the reference of proper names, nominal and external description theory, are defended against various known counter-arguments such as Kripke’s circularity objection and the Church-Langford translation test. It is shown how the resulting analysis of de dicto belief ascriptions can be made compositional, but also argued that compositionality is not mandatory. The second part deals with forms of indexical and non-indexical contextdependence. Taking into account a range of typological data, referential features of indexical expressions like their egocentricity, token-reflexivity, and the vagueness of spatial and temporal indexicals are laid out. Kaplan’s Logic of Demonstratives is then reformulated, but following Cresswell (1990) it is argued that full quantification over modal indices is needed. Various indicators and demonstratives are analyzed on the basis of a description theory of reference in a variant of first-order predicate logic with non-traditional predication theory and two sorts of reified contexts. Examples analyzed include: I, now, here, actually, we, the former president, the left entrance, context-shifting indexicals, and demonstratives like Japanese are. Finally, essential indexicality is addressed and it is conceded that description theory cannot deal with attitudes de se. In defense of indirect reference it is argued that the cognitive phenomena underlying essential indexicality, as for example I-thoughts, aren’t aspects of the public meaning of natural language expressions and that speaking of a ‘language of thinking’ or ‘reference in thinking’ are unfitting metaphors for general semiotic reasons