scholarly journals Existential second-order logic and modal logic with quantified accessibility relations

2016 ◽  
Vol 247 ◽  
pp. 217-234
Author(s):  
Lauri Hella ◽  
Antti Kuusisto
Keyword(s):  
Modal Logic ◽  
Second Order ◽  
Order Logic ◽  
Second Order Logic

Representability in second-order propositional poly-modal logic

2002 ◽  
Vol 67 (3) ◽  
pp. 1039-1054 ◽  
Author(s):  
G. Aldo Antonelli ◽  
Richmond H. Thomason
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Second Order ◽  
Order Logic ◽  
Multiple Modalities ◽  
Modal Operators ◽  
Second Order Systems ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Propositional System

AbstractA propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.In this paper we generalize this framework by allowing multiple modalities. While this does not affect the undecidability of K, B, T, K4 and S4, poly-modal second-order S5 is dramatically more expressive than its mono-modal counterpart. As an example, we establish the definability of the transitive closure of finitely many modal operators. We also take up the decidability issue, and, using a novel encoding of sets of unordered pairs by partitions of the leaves of certain graphs, we show that the second-order propositional logic of two S5 modalitities is also equivalent to full second-order logic.

Noncompactness in propositional modal logic

1972 ◽  
Vol 37 (4) ◽  
pp. 716-720 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Predicate Logic ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
Relational Models ◽  
Propositional Modal Logic ◽  
Bimodal Logic ◽  
Deductive Power ◽  
Second Order Logic

We have come to believe that propositional modal logic (with the usual relational semantics) must be understood as a rather strong fragment of classical second-order predicate logic. (The interpretation of propositional modal logic in second-order predicate logic is well known; see e.g. [2, §1].) “Strong” refers of course to the expressive power of the languages, not to the deductive power of formal systems. By “rather strong” we mean sufficiently strong that theorems about first-order logic which fail for second-order logic usually fail even for propositional modal logic. Some evidence for this belief is contained in [2] and [3]. In the former is exhibited a finitely axiomatized consistent tense logic having no relational models, and the latter presents a finitely axiomatized modal logic between T and S4, such that □p → □2p is valid in all relational models of the logic but is not a thesis of the logic. The result of [2] is strong evidence that bimodal logic is essentially second-order, but that of [3] does not eliminate the possibility that unimodal logic only appears to be incomplete because we have not adopted sufficiently powerful rules of inference. In the present paper we present stronger evidence of the essentially second-order nature of unimodal logic.

Categoricity and the sets

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Set Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order ◽  
The Status ◽  
Second Order Logic ◽  
Philosophy Of Set Theory ◽  
Order Set

In this chapter, the focus shifts from numbers to sets. Again, no first-order set theory can hope to get anywhere near categoricity, but Zermelo famously proved the quasi-categoricity of second-order set theory. As in the previous chapter, we must ask who is entitled to invoke full second-order logic. That question is as subtle as before, and raises the same problem for moderate modelists. However, the quasi-categorical nature of Zermelo's Theorem gives rise to some specific questions concerning the aims of axiomatic set theories. Given the status of Zermelo's Theorem in the philosophy of set theory, we include a stand-alone proof of this theorem. We also prove a similar quasi-categoricity for Scott-Potter set theory, a theory which axiomatises the idea of an arbitrary stage of the iterative hierarchy.

Algorithmic Logic with Stacks and Its Model-Theoretical Properties

1984 ◽  
Vol 7 (4) ◽  
pp. 391-428
Author(s):  
Wiktor Dańko
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Algorithmic Logic ◽  
Second Order Logic

In this paper we propose to transform the Algorithmic Theory of Stacks (cf. Salwicki [30]) into a logic for expressing and proving properties of programs with stacks. We compare this logic to the Weak Second Order Logic (cf. [11, 15]) and prove theorems concerning axiomatizability without quantifiers (an analogon of Łoś-Tarski theorem) and χ 0 - categoricity (an analogon of Ryll-Nardzewski’s theorem).

Monadic second-order logic on finite sequences

2017 ◽  
Vol 52 (1) ◽  
pp. 232-245
Author(s):  
Loris D'Antoni ◽  
Margus Veanes
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Finite Sequences ◽  
Second Order Logic ◽  
Monadic Second Order Logic
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