126 research outputs found
Local Sentences and Mahlo Cardinals
Local sentences were introduced by J.-P. Ressayre who proved certain
remarkable stretching theorems establishing the equivalence between the
existence of finite models for these sentences and the existence of some
infinite well ordered models. Two of these stretching theorems were only proved
under certain large cardinal axioms but the question of their exact
(consistency) strength was left open in [O. Finkel and J.-P. Ressayre,
Stretchings, Journal of Symbolic Logic, Volume 61 (2), 1996, p. 563-585 ].
Here, we solve this problem, using a combinatorial result of J. H. Schmerl. In
fact, we show that the stretching principles are equivalent to the existence of
n-Mahlo cardinals for appropriate integers n. This is done by proving first
that for each integer n, there is a local sentence phi_n which has well ordered
models of order type alpha, for every infinite ordinal alpha > omega which is
not an n-Mahlo cardinal
Initial Semantics for Strengthened Signatures
We give a new general definition of arity, yielding the companion notions of
signature and associated syntax. This setting is modular in the sense requested
by Ghani and Uustalu: merging two extensions of syntax corresponds to building
an amalgamated sum. These signatures are too general in the sense that we are
not able to prove the existence of an associated syntax in this general
context. So we have to select arities and signatures for which there exists the
desired initial monad. For this, we follow a track opened by Matthes and
Uustalu: we introduce a notion of strengthened arity and prove that the
corresponding signatures have initial semantics (i.e. associated syntax). Our
strengthened arities admit colimits, which allows the treatment of the
\lambda-calculus with explicit substitution.Comment: In Proceedings FICS 2012, arXiv:1202.317
Automatic Generation of Proof Tactics for Finite-Valued Logics
A number of flexible tactic-based logical frameworks are nowadays available
that can implement a wide range of mathematical theories using a common
higher-order metalanguage. Used as proof assistants, one of the advantages of
such powerful systems resides in their responsiveness to extensibility of their
reasoning capabilities, being designed over rule-based programming languages
that allow the user to build her own `programs to construct proofs' - the
so-called proof tactics.
The present contribution discusses the implementation of an algorithm that
generates sound and complete tableau systems for a very inclusive class of
sufficiently expressive finite-valued propositional logics, and then
illustrates some of the challenges and difficulties related to the algorithmic
formation of automated theorem proving tactics for such logics. The procedure
on whose implementation we will report is based on a generalized notion of
analyticity of proof systems that is intended to guarantee termination of the
corresponding automated tactics on what concerns theoremhood in our targeted
logics
Uncertainty About Evidence
We develop a logical framework for reasoning about knowledge and evidence in
which the agent may be uncertain about how to interpret their evidence. Rather
than representing an evidential state as a fixed subset of the state space, our
models allow the set of possible worlds that a piece of evidence corresponds to
to vary from one possible world to another, and therefore itself be the subject
of uncertainty. Such structures can be viewed as (epistemically motivated)
generalizations of topological spaces. In this context, there arises a natural
distinction between what is actually entailed by the evidence and what the
agent knows is entailed by the evidence -- with the latter, in general, being
much weaker. We provide a sound and complete axiomatization of the
corresponding bi-modal logic of knowledge and evidence entailment, and
investigate some natural extensions of this core system, including the addition
of a belief modality and its interaction with evidence interpretation and
entailment, and the addition of a "knowability" modality interpreted via a
(generalized) interior operator.Comment: In Proceedings TARK 2019, arXiv:1907.0833
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