Proof Theory
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Author(s):  
Pablo Cobreros ◽  
Elio La Rosa ◽  
Luca Tranchini

AbstractBuilding on early work by Girard (1987) and using closely related techniques from the proof theory of many-valued logics, we propose a sequent calculus capturing a hierarchy of notions of satisfaction based on the Strong Kleene matrices introduced by Barrio et al. (Journal of Philosophical Logic 49:93–120, 2020) and others. The calculus allows one to establish and generalize in a very natural manner several recent results, such as the coincidence of some of these notions with their classical counterparts, and the possibility of expressing some notions of satisfaction for higher-level inferences using notions of satisfaction for inferences of lower level. We also show that at each level all notions of satisfaction considered are pairwise distinct and we address some remarks on the possible significance of this (huge) number of notions of consequence.


Author(s):  
DALE MILLER

Abstract Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this article, we will survey the use of structural proof theory as an alternative foundation. Researchers have been using this foundation for the past 35 years to elevate logic programming from its roots in first-order classical logic into higher-order versions of intuitionistic and linear logic. These more expressive logic programming languages allow for capturing stateful computations and rich forms of abstractions, including higher-order programming, modularity, and abstract data types. Term-level bindings are another kind of abstraction, and these are given an elegant and direct treatment within both proof theory and these extended logic programming languages. Logic programming has also inspired new results in proof theory, such as those involving polarity and focused proofs. These recent results provide a high-level means for presenting the differences between forward-chaining and backward-chaining style inferences. Anchoring logic programming in proof theory has also helped identify its connections and differences with functional programming, deductive databases, and model checking.


2021 ◽  
Author(s):  
◽  
David Friggens

<p>The abstract mathematical structures known as coalgebras are of increasing interest in computer science for their use in modelling certain types of data structures and programs. Traditional algebraic methods describe objects in terms of their construction, whilst coalgebraic methods describe objects in terms of their decomposition, or observational behaviour. The latter techniques are particularly useful for modelling infinite data structures and providing semantics for object-oriented programming languages, such as Java. There have been many different logics developed for reasoning about coalgebras of particular functors, most involving modal logic. We define a modal logic for coalgebras of polynomial functors, extending Rößiger’s logic [33], whose proof theory was limited to using finite constant sets, by adding an operator from Goldblatt [11]. From the semantics we define a canonical coalgebra that provides a natural construction of a final coalgebra for the relevant functor. We then give an infinitary axiomatization and syntactic proof relation that is sound and complete for functors constructed from countable constant sets.</p>


2021 ◽  
Author(s):  
◽  
David Friggens

<p>The abstract mathematical structures known as coalgebras are of increasing interest in computer science for their use in modelling certain types of data structures and programs. Traditional algebraic methods describe objects in terms of their construction, whilst coalgebraic methods describe objects in terms of their decomposition, or observational behaviour. The latter techniques are particularly useful for modelling infinite data structures and providing semantics for object-oriented programming languages, such as Java. There have been many different logics developed for reasoning about coalgebras of particular functors, most involving modal logic. We define a modal logic for coalgebras of polynomial functors, extending Rößiger’s logic [33], whose proof theory was limited to using finite constant sets, by adding an operator from Goldblatt [11]. From the semantics we define a canonical coalgebra that provides a natural construction of a final coalgebra for the relevant functor. We then give an infinitary axiomatization and syntactic proof relation that is sound and complete for functors constructed from countable constant sets.</p>


2021 ◽  
Vol 17 (4) ◽  
pp. 1693-1757
Author(s):  
Samuel Buss ◽  
Rosalie Iemhoff ◽  
Ulrich Kohlenbach ◽  
Michael Rathjen

ARHE ◽  
2021 ◽  
Vol 27 (34) ◽  
pp. 61-83
Author(s):  
KATARINA MAKSIMOVIĆ

The goal of this paper is to introduce the reader to the distinction between intensional and extensional as a distinction between different approaches to meaning. We will argue that despite the common belief, intensional aspects of mathematical notions can be, and in fact have been successfully described in mathematics. One that is for us particularly interesting is the notion of deduction as depicted in general proof theory. Our considerations result in defending a) the importance of a rule-based semantical approach and b) the position according to which non-reductive and somewhat circular explanations play an essential role in describing intensionality in mathematics.


2021 ◽  
Vol 333 ◽  
pp. 230-246
Author(s):  
Tarmo Uustalu ◽  
Niccolò Veltri ◽  
Noam Zeilberger

Author(s):  
Cezary Cieśliński

AbstractWe present a construction of a truth class (an interpretation of a compositional truth predicate) in an arbitrary countable recursively saturated model of first-order arithmetic. The construction is fully classical in that it employs nothing more than the classical techniques of formal proof theory.


Author(s):  
Paul Égré ◽  
Lorenzo Rossi ◽  
Jan Sprenger

AbstractIn Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti (1936) and Reichenbach (1935, 1944) on the one hand, and by Cooper (Inquiry, 11, 295–320, 1968) and Cantwell (Notre Dame Journal of Formal Logic, 49, 245–260, 2008) on the other. Here we provide the proof theory for the resulting logics and , using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: allows for algebraic completeness, but not for the construction of a canonical model, while fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects.


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