6,319 research outputs found
On formal aspects of the epistemic approach to paraconsistency
This paper reviews the central points and presents some recent developments of the epistemic approach to paraconsistency in terms of the preservation of evidence. Two formal systems are surveyed, the basic logic of evidence (BLE) and the logic of evidence and truth (LET J ), designed to deal, respectively, with evidence and with evidence and truth. While BLE is equivalent to Nelson’s logic N4, it has been conceived for a different purpose. Adequate valuation semantics that provide decidability are given for both BLE and LET J . The meanings of the connectives of BLE and LET J , from the point of view of preservation of evidence, is explained with the aid of an inferential semantics. A formalization of the notion of evidence for BLE as proposed by M. Fitting is also reviewed here. As a novel result, the paper shows that LET J is semantically characterized through the so-called Fidel structures. Some opportunities for further research are also discussed
Belief Semantics of Authorization Logic
Authorization logics have been used in the theory of computer security to
reason about access control decisions. In this work, a formal belief semantics
for authorization logics is given. The belief semantics is proved to subsume a
standard Kripke semantics. The belief semantics yields a direct representation
of principals' beliefs, without resorting to the technical machinery used in
Kripke semantics. A proof system is given for the logic; that system is proved
sound with respect to the belief and Kripke semantics. The soundness proof for
the belief semantics, and for a variant of the Kripke semantics, is mechanized
in Coq
From truth to computability I
The recently initiated approach called computability logic is a formal theory
of interactive computation. See a comprehensive online source on the subject at
http://www.cis.upenn.edu/~giorgi/cl.html . The present paper contains a
soundness and completeness proof for the deductive system CL3 which axiomatizes
the most basic first-order fragment of computability logic called the
finite-depth, elementary-base fragment. Among the potential application areas
for this result are the theory of interactive computation, constructive applied
theories, knowledgebase systems, systems for resource-bound planning and
action. This paper is self-contained as it reintroduces all relevant
definitions as well as main motivations.Comment: To appear in Theoretical Computer Scienc
De Jongh's Theorem for Intuitionistic Zermelo-Fraenkel Set Theory
We prove that the propositional logic of intuitionistic set theory IZF is
intuitionistic propositional logic IPC. More generally, we show that IZF has
the de Jongh property with respect to every intermediate logic that is complete
with respect to a class of finite trees. The same results follow for CZF.Comment: 12 page
A Galois connection between classical and intuitionistic logics. II: Semantics
Three classes of models of QHC, the joint logic of problems and propositions,
are constructed, including a class of subset/sheaf-valued models that is
related to solutions of some actual problems (such as solutions of algebraic
equations) and combines the familiar Leibniz-Euler-Venn semantics of classical
logic with a BHK-type semantics of intuitionistic logic.
To test the models, we consider a number of principles and rules, which
empirically appear to cover all "sufficiently simple" natural conjectures about
the behaviour of the operators ! and ?, and include two hypotheses put forward
by Hilbert and Kolmogorov, as formalized in the language of QHC. Each of these
turns out to be either derivable in QHC or equivalent to one of only 13
principles and 1 rule, of which 10 principles and 1 rule are conservative over
classical and intuitionistic logics. The three classes of models together
suffice to confirm the independence of these 10 principles and 1 rule, and to
determine the full lattice of implications between them, apart from one
potential implication.Comment: 35 pages. v4: Section 4.6 "Summary" is added at the end of the paper.
v3: Major revision of a half of v2. The results are improved and rewritten in
terms of the meta-logic. The other half of v2 (Euclid's Elements as a theory
over QHC) is expected to make part III after a revisio
Intuitionistic computability logic
Computability logic (CL) is a systematic formal theory of computational tasks
and resources, which, in a sense, can be seen as a semantics-based alternative
to (the syntactically introduced) linear logic. With its expressive and
flexible language, where formulas represent computational problems and "truth"
is understood as algorithmic solvability, CL potentially offers a comprehensive
logical basis for constructive applied theories and computing systems
inherently requiring constructive and computationally meaningful underlying
logics.
Among the best known constructivistic logics is Heyting's intuitionistic
calculus INT, whose language can be seen as a special fragment of that of CL.
The constructivistic philosophy of INT, however, has never really found an
intuitively convincing and mathematically strict semantical justification. CL
has good claims to provide such a justification and hence a materialization of
Kolmogorov's known thesis "INT = logic of problems". The present paper contains
a soundness proof for INT with respect to the CL semantics. A comprehensive
online source on CL is available at http://www.cis.upenn.edu/~giorgi/cl.htm
Automated Synthesis of Tableau Calculi
This paper presents a method for synthesising sound and complete tableau
calculi. Given a specification of the formal semantics of a logic, the method
generates a set of tableau inference rules that can then be used to reason
within the logic. The method guarantees that the generated rules form a
calculus which is sound and constructively complete. If the logic can be shown
to admit finite filtration with respect to a well-defined first-order semantics
then adding a general blocking mechanism provides a terminating tableau
calculus. The process of generating tableau rules can be completely automated
and produces, together with the blocking mechanism, an automated procedure for
generating tableau decision procedures. For illustration we show the
workability of the approach for a description logic with transitive roles and
propositional intuitionistic logic.Comment: 32 page
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