894 research outputs found
From Many-Valued Consequence to Many-Valued Connectives
Given a consequence relation in many-valued logic, what connectives can be
defined? For instance, does there always exist a conditional operator
internalizing the consequence relation, and which form should it take? In this
paper, we pose this question in a multi-premise multi-conclusion setting for
the class of so-called intersective mixed consequence relations, which extends
the class of Tarskian relations. Using computer-aided methods, we answer
extensively for 3-valued and 4-valued logics, focusing not only on conditional
operators, but on what we call Gentzen-regular connectives (including negation,
conjunction, and disjunction). For arbitrary N-valued logics, we state
necessary and sufficient conditions for the existence of such connectives in a
multi-premise multi-conclusion setting. The results show that mixed consequence
relations admit all classical connectives, and among them pure consequence
relations are those that admit no other Gentzen-regular connectives.
Conditionals can also be found for a broader class of intersective mixed
consequence relations, but with the exclusion of order-theoretic consequence
relations.Comment: Updated version [corrections of an incorrect claim in first version;
two bib entries added
Generic Modal Cut Elimination Applied to Conditional Logics
We develop a general criterion for cut elimination in sequent calculi for
propositional modal logics, which rests on absorption of cut, contraction,
weakening and inversion by the purely modal part of the rule system. Our
criterion applies also to a wide variety of logics outside the realm of normal
modal logic. We give extensive example instantiations of our framework to
various conditional logics. For these, we obtain fully internalised calculi
which are substantially simpler than those known in the literature, along with
leaner proofs of cut elimination and complexity. In one case, conditional logic
with modus ponens and conditional excluded middle, cut elimination and
complexity were explicitly stated as open in the literature
LP, K3, and FDE as Substructural Logics
Building on recent work, I present sequent systems for the non-classical logics LP, K3, and FDE with two main virtues. First, derivations closely resemble those in standard Gentzen-style systems. Second, the systems can be obtained by reformulating a classical system using nonstandard sequent structure and simply removing certain structural rules (relatives of exchange and contraction). I clarify two senses in which these logics count as “substructural.
Harmony and modality
It is argued that the meaning of the modal connectives must be given inferentially, by the rules for the assertion of formulae containing them, and not semantically by reference to possible worlds. Further, harmony confers transparency on the inferentialist account of meaning, when the introduction-rule specifies both necessary and sufficient conditions for assertion, and the elimination-rule does no more than exhibit the consequences of the meaning so conferred. Hence, harmony is not to be identified with normalization, since the standard modal natural deduction rules, though normalizable, are not in this sense harmonious. Harmonious rules for modality have lately been formulated, using labelled deductive systems
A Nonmonotonic Sequent Calculus for Inferentialist Expressivists
I am presenting a sequent calculus that extends a nonmonotonic consequence relation over an atomic language to a logically complex language. The system is in line with two guiding philosophical ideas: (i) logical inferentialism and (ii) logical expressivism. The extension defined by the sequent rules is conservative. The conditional tracks the consequence relation and negation tracks incoherence. Besides the ordinary propositional connectives, the sequent calculus introduces a new kind of modal operator that marks implications that hold monotonically. Transitivity fails, but for good reasons. Intuitionism and classical logic can easily be recovered from the system
Hilbert's "Verunglueckter Beweis," the first epsilon theorem, and consistency proofs
In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's Programme,
were working on consistency proofs for arithmetical systems. One proposed
method of giving such proofs is Hilbert's epsilon-substitution method. There
was, however, a second approach which was not reflected in the publications of
the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's
first epsilon theorem and a certain 'general consistency result' due to
Bernays. An analysis of the form of this so-called 'failed proof' sheds further
light on an interpretation of Hilbert's Programme as an instrumentalist
enterprise with the aim of showing that whenever a `real' proposition can be
proved by 'ideal' means, it can also be proved by 'real', finitary means.Comment: 18 pages, final versio
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