Papers by Michał Godziszewski
Logic and Its Applications
Yablo's paradox results in a set of formulas which (with local disquotation in the background) tu... more Yablo's paradox results in a set of formulas which (with local disquotation in the background) turns out consistent, but ω-inconsistent. Adding either uniform disquotation or the ω-rule results in inconsistency. One might think that it doesn't arise in finitary contexts. We study whether it does. It turns out that the issue turns on how the finitistic approach is formalized.
We answer the question of computational reasons for epistemic hardness of certain class of philos... more We answer the question of computational reasons for epistemic hardness of certain class of philosophically interesting mathematical concepts. We justify the statement that mathematical knowability may be identified with algorithmic learnability. We present framework of experimental logics equivalent to the notion of learnability. Then we prove the main result. By adjoining the minimal possible set of undecidable sentences to recursive axiomatization of arithmetics and closing it under logical consequence, we obtain a non-learnable theory. This gives an explanation to the fact that undecidable arithmetical sentences are cognitively difficult. We conclude that cognitively accessible mathematical concepts are exactly within the scope of learnablity. 1 Knowability as algorithmic learnability Emergence and development of recursion theory and computer science enable us to rigorously address the question of characterising the class of mathematical concepts that are cognitively accessible t...
We study approval-based committee elections for the case where the voters’ preferences come from ... more We study approval-based committee elections for the case where the voters’ preferences come from a 2D-Euclidean model. We consider two main issues: First, we ask for the complexity of computing election results. Second, we evaluate election outcomes experimentally, following the visualization technique of Elkind et al. (2017). Regarding the first issue, we find that many NP-hard rules remain intractable for 2D-Euclidean elections. For the second one, we observe that the behavior and nature of many rules strongly depend on the exact protocol for choosing the approved candidates.
We answer the question of computational reasons for epistemic hardness of certain class of philos... more We answer the question of computational reasons for epistemic hardness of certain class of philosophically interesting mathematical concepts. We justify the statement that mathematical knowability may be identified with algorithmic learnability. We present framework of experimental logics equivalent to the notion of learnability. Then we prove the main result. By adjoining the minimal possible set of undecidable sentences to recursive axiomatization of arithmetics and closing it under logical consequence, we obtain a non-learnable theory. This gives an explanation to the fact that undecidable arithmetical sentences are cognitively difficult. We conclude that cognitively accessible mathematical concepts are exactly within the scope of learnablity. 1 Knowability as algorithmic learnability Emergence and development of recursion theory and computer science enable us to rigorously address the question of characterising the class of mathematical concepts that are cognitively accessible t...
We rigorously specify the class of nonprobabilistic agents which are, we argue, immune to the cla... more We rigorously specify the class of nonprobabilistic agents which are, we argue, immune to the classical Dutch Book argument. We also discuss the notion of expected value used in the argument as well as sketch future research connecting our results to those concerning incoherence measures.
The relation between indicative conditionals in natural language and material implication wasn’t ... more The relation between indicative conditionals in natural language and material implication wasn’t a major topic in the Lvov-Warsaw school. However, a major defense of the claim that the truth conditions of these two are the same has been developed by Ajdukiewicz (Studia Logica IV:117–134, 1956). The first major goal of this paper is to present, assess, and improve his strategy. It turns out that it is quite similar to the approach developed by Grice (Studies in the Way of Words, Harvard University Press, Cambridge, MA, 1991), so our second goal is to compare these two and to argue that the accuracy of Ajdukiewicz’s explanation is less dependent on controversial properties of a systematic but convoluted general theory of cooperative communicative behavior. In Lvov-Warsaw school the relation between material implication and indicative conditionals was also discussed by Gol ąb (Matematyka 3(5):27–29, 1949) and Slupecki (Matematyka 4(6):32–35, 1949), so the third part of our paper is devoted to their discussion and relating it to Ajdukiewicz’s views.
We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient prese... more We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient presentations of models of arithmetic and set theory. Specifically, no nonstandard model of arithmetic has a computable quotient presentation by a c.e. equivalence relation. No \(\Sigma _1\)-sound nonstandard model of arithmetic has a computable quotient presentation by a co-c.e. equivalence relation. No nonstandard model of arithmetic in the language \(\{+,\cdot ,\le \}\) has a computably enumerable quotient presentation by any equivalence relation of any complexity. No model of ZFC or even much weaker set theories has a computable quotient presentation by any equivalence relation of any complexity. And similarly no nonstandard model of finite set theory has a computable quotient presentation.
We consider the notion of intuitive learnability and its relation to intuitive computability. We ... more We consider the notion of intuitive learnability and its relation to intuitive computability. We briefly discuss the Church's Thesis. We formulate the Learnability Thesis. Further we analyse the proof of the Church's Thesis presented by M. Mostowski. We indicate which assumptions of the Mostowski's argument implicitly include that the Church's Thesis holds. The impossibility of this kind of argument is strengthened by showing that the Learnability Thesis does not imply the Church's Thesis. Specifically, we show a natural interpretation of intuitive computability under which intuitively learnable sets are exactly algorithmically learnable but intuitively computable sets form a proper superset of recursive sets.
The paper describes properties of Yablo sequences over growing domains of finite arithmetical mod... more The paper describes properties of Yablo sequences over growing domains of finite arithmetical models and over partial models of Kripke truth theory. We show that for any partial fixed-point model and for the Strong Kleene, Weak Kleene and Supervaluation valuation schema, all Yablo sentences Y (n) are neither true nor false under these schema or equivalently: the truth-value of all Yablo sentences Y (n) in fixed-point partial models under any of the above valuation scheme, is indeterminate. Furthermore, we show that under the logic of sufficiently large finite models (logic of potential infinity) all the Yablo sentences are false in the limit. The main philosophical conclusion is that a finitist, not accepting the notion of actual infinity may adopt the sl-theory of arithmetics as an adequate explication of the concept of potential infinity and simply give an answer to the Yablo paradox that in the limit (i.e. under the sl-semantics on the FM-domain of a given arithmetical model) all the Yablo sentences are false.
This is a note in which we respond to a conjecture concerning generalized and classical probabili... more This is a note in which we respond to a conjecture concerning generalized and classical probability spaces.
This note on a recent award-winning paper (Feintzeig (2014)) contains three results: 1) that, con... more This note on a recent award-winning paper (Feintzeig (2014)) contains three results: 1) that, contrary to Feintzeig's conjecture, subadditivity does not guarantee the existence of a classical extension of a given generalized probability space; 2) that the claim made in the paper that subadditivity of a generalized probability space guarantees its nonsusceptability to a Dutch Book (in Feintzeig's sense) is also false; 3) and that Feintzeig's proposed program of finding generalized probability spaces which would be both nonsusceptible to a Dutch Book and would not possess a classical extension is doomed to fail (at least among finite probability spaces): a general theorem is shown according to which a space has a classical extension if and only if it is not susceptible to a Dutch Book.
We study natural language constructions which are deemed to express the existence of certain kind... more We study natural language constructions which are deemed to express the existence of certain kinds of similarities between partial orderings. Specifically, we give examples of natural language sentences and their plausible logical forms that express the existence of homomorphism, embedding and variations of those. Semantically, we interpret the constructions as polyadic generalized quantifiers. We examine some of the quantifiers in question with respect to their F O-definability over finite models. Since they are definable in the existential fragment of SO, we investigate their completeness in the class N P . We prove that among the quantifiers under investigation, only the homomorphism quantifier is tractable. We stress methodological importance of our results for linguistics. Finally, we discuss some interconnections between computational and evolutionary semantics.
We answer the question of computational reasons for epistemic hardness of certain class of philos... more We answer the question of computational reasons for epistemic hardness of certain class of philosophically interesting mathematical concepts. We justify the statement that mathematical knowability may be identified with algorithmic learnability. We present framework of experimental logics equivalent to the notion of learnability. Then we prove the main result. By adjoining the minimal possible set of undecidable sentences to recursive axiomatization of arithmetics and closing it under logical consequence, we obtain a non-learnable theory. This gives an explanation to the fact that undecidable arithmetical sentences are cognitively difficult. We conclude that cognitively accessible mathematical concepts are exactly within the scope of learnablity.
Oświadczam, że niniejsza praca została przygotowana pod moim kierunkiem i stwierdzam, że spełnia ... more Oświadczam, że niniejsza praca została przygotowana pod moim kierunkiem i stwierdzam, że spełnia ona warunki do przedstawienia jej w postępowaniu o nadanie tytułu zawodowego.
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Papers by Michał Godziszewski