The standard first-order reading of modality does not bind individual variables, i.e., if x is fr... more The standard first-order reading of modality does not bind individual variables, i.e., if x is free in F (x), then x remains free in ✷F (x). Accordingly, if ✷ stands for ‘provable in arithmetic, ’ ∀x✷F (x) states that F (n) is provable for any given value of n = 0, 1, 2,...; this corresponds to a de re reading of modality. The other, de dicto meaning of ✷F (x), suggesting that F (x) is derivable as a formula with a free variable x, is not directly represented by a modality, though, semantically, it could be approximated by compound constructions, e.g., ✷∀xF (x). We introduce the first-order logic FOS4 ∗ in which modalities can bind individual variables and, in particular, can directly represent both de re and de dicto modalities. FOS4 ∗ extends first-order S4 and is the natural forgetful projection of the first-order logic of proofs FOLP. The same method of introducing binding modalities obviously works for other modal logics as well. 1
In this case study we describe an approach to a general logical framework for tracking evidence w... more In this case study we describe an approach to a general logical framework for tracking evidence within epistemic contexts. We consider as basic an example which features two justifications for a true statement, one which is correct and one which is not. We formalize this example in a system of Justification Logic with two knowers: the object agent and the observer, and we show that whereas the object agent does not logically distinguish between factive and non-factive justifications, such distinctions can be attained at the observer level by analyzing the structure of evidence terms. Basic logic properties of the corresponding two-agent Justification Logic system have been established, which include Kripke-Fitting completeness. We also argue that a similar evidence-tracking approach can be applied to analyzing paraconsistent systems.
The well-known Church-Fitch paradox shows that the verificationist knowability principle all trut... more The well-known Church-Fitch paradox shows that the verificationist knowability principle all truths are knowable, yields an unacceptable omniscience property. Our semantic analysis establishes that the knowability principle fails because it misses the stability assumption ‘the proposition in question does not change from true to false in the process of discovery,’ hidden in the verificationist approach. Once stability is made explicit, the resulting stable knowability principle accurately represents verificationist knowability, does not yield the omniscience property, and can be offered as a resolution of the knowability paradox. Two more principles are considered: total knowability stating that it is possible to know whether a proposition holds or not, and monotonic knowability stemming from the intrinsically intuitionistic reading of knowability. The study of these four principles yields a “knowability diamond” describing their logical strength. These results are obtained within a...
In his dissertation of 1950, Nash based his concept of the solution to a game on the assumption t... more In his dissertation of 1950, Nash based his concept of the solution to a game on the assumption that “a rational prediction should be unique, that the players should be able to deduce and make use of it.” We study when such definitive solutions exist for strategic games with ordinal payoffs. We offer a new, syntactic approach: instead of reasoning about the specific model of a game, we deduce properties of interest directly from the description of the game itself. This captures Nash’s deductive assumptions and helps to bridge a well-known gap between syntactic game descriptions and specific models which could require unwarranted additional epistemic assumptions, e.g., common knowledge of a model. We show that games without Nash equilibria do not have definitive solutions under any notion of rationality, but each Nash equilibrium can be a definitive solution for an appropriate refinement of Aumann rationality. With respect to Aumann rationality itself, games with multiple Nash equili...
The standard first-order reading of modality does not bind individual variables, i.e., if x is fr... more The standard first-order reading of modality does not bind individual variables, i.e., if x is free in F (x), then x remains free in ✷F (x). Accordingly, if ✷ stands for ‘provable in arithmetic, ’ ∀x✷F (x) states that F (n) is provable for any given value of n = 0, 1, 2,...; this corresponds to a de re reading of modality. The other, de dicto meaning of ✷F (x), suggesting that F (x) is derivable as a formula with a free variable x, is not directly represented by a modality, though, semantically, it could be approximated by compound constructions, e.g., ✷∀xF (x). We introduce the first-order logic FOS4 ∗ in which modalities can bind individual variables and, in particular, can directly represent both de re and de dicto modalities. FOS4 ∗ extends first-order S4 and is the natural forgetful projection of the first-order logic of proofs FOLP. The same method of introducing binding modalities obviously works for other modal logics as well. 1
In this case study we describe an approach to a general logical framework for tracking evidence w... more In this case study we describe an approach to a general logical framework for tracking evidence within epistemic contexts. We consider as basic an example which features two justifications for a true statement, one which is correct and one which is not. We formalize this example in a system of Justification Logic with two knowers: the object agent and the observer, and we show that whereas the object agent does not logically distinguish between factive and non-factive justifications, such distinctions can be attained at the observer level by analyzing the structure of evidence terms. Basic logic properties of the corresponding two-agent Justification Logic system have been established, which include Kripke-Fitting completeness. We also argue that a similar evidence-tracking approach can be applied to analyzing paraconsistent systems.
The well-known Church-Fitch paradox shows that the verificationist knowability principle all trut... more The well-known Church-Fitch paradox shows that the verificationist knowability principle all truths are knowable, yields an unacceptable omniscience property. Our semantic analysis establishes that the knowability principle fails because it misses the stability assumption ‘the proposition in question does not change from true to false in the process of discovery,’ hidden in the verificationist approach. Once stability is made explicit, the resulting stable knowability principle accurately represents verificationist knowability, does not yield the omniscience property, and can be offered as a resolution of the knowability paradox. Two more principles are considered: total knowability stating that it is possible to know whether a proposition holds or not, and monotonic knowability stemming from the intrinsically intuitionistic reading of knowability. The study of these four principles yields a “knowability diamond” describing their logical strength. These results are obtained within a...
In his dissertation of 1950, Nash based his concept of the solution to a game on the assumption t... more In his dissertation of 1950, Nash based his concept of the solution to a game on the assumption that “a rational prediction should be unique, that the players should be able to deduce and make use of it.” We study when such definitive solutions exist for strategic games with ordinal payoffs. We offer a new, syntactic approach: instead of reasoning about the specific model of a game, we deduce properties of interest directly from the description of the game itself. This captures Nash’s deductive assumptions and helps to bridge a well-known gap between syntactic game descriptions and specific models which could require unwarranted additional epistemic assumptions, e.g., common knowledge of a model. We show that games without Nash equilibria do not have definitive solutions under any notion of rationality, but each Nash equilibrium can be a definitive solution for an appropriate refinement of Aumann rationality. With respect to Aumann rationality itself, games with multiple Nash equili...
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Papers by Sergei Artemov