12,570 research outputs found
Some modal and temporal translations of generalized basic logic
We introduce a family of modal expansions of Ćukasiewicz logic that are designed to accommodate modal translations of generalized basic logic (as formulated with exchange, weakening, and falsum). We further exhibit algebraic semantics for each logic in this family, in particular showing that all of them are algebraizable in the sense of Blok and Pigozzi. Using this algebraization result and an analysis of congruences in the pertinent varieties, we establish that each of the introduced modal Ćukasiewicz logics has a local deduction-detachment theorem. By applying Jipsen and Montagnaâs poset product construction, we give two translations of generalized basic logic with exchange, weakening, and falsum in the style of the celebrated Gödel-McKinsey-Tarski translation. The first of these interprets generalized basic logic in a modal Ćukasiewicz logic in the spirit of the classical modal logic S4, whereas the second interprets generalized basic logic in a temporal variant of the latter
Contingent composition as identity
When the necessity of identity is combined with composition as identity, the contingency of composition is at risk. In the extant literature, either NI is seen as the basis for a refutation of CAI or CAI is associated with a theory of modality, such that: either NI is renounced ; or CC is renounced. In this paper, we investigate the prospects of a new variety of CAI, which aims to preserve both NI and CC. This new variety of CAI is the quite natural product of the attempt to make sense of CAI on the background of a broadly Kripkean view of modality, such that one and the same entity is allowed to exist at more than one possible world. CCAI introduces a world-relative kind of identity, which is different from standard identity, and claims that composition is this kind of world-relative identity. CCAI manages to preserve NI and CC. We compare CCAI with Gibbardâs and Galloisâ doctrines of contingent identity and we show that CCAI can be sensibly interpreted as a form of Weak CAI, that is of the thesis that composition is not standard identity, yet is significantly similar to it
On FO2 quantifier alternation over words
We show that each level of the quantifier alternation hierarchy within
FO^2[<] -- the 2-variable fragment of the first order logic of order on words
-- is a variety of languages. We then use the notion of condensed rankers, a
refinement of the rankers defined by Weis and Immerman, to produce a decidable
hierarchy of varieties which is interwoven with the quantifier alternation
hierarchy -- and conjecturally equal to it. It follows that the latter
hierarchy is decidable within one unit: given a formula alpha in FO^2[<], one
can effectively compute an integer m such that alpha is equivalent to a formula
with at most m+1 alternating blocks of quantifiers, but not to a formula with
only m-1 blocks. This is a much more precise result than what is known about
the quantifier alternation hierarchy within FO[<], where no decidability result
is known beyond the very first levels
A Note on Parameterised Knowledge Operations in Temporal Logic
We consider modeling the conception of knowledge in terms of temporal logic.
The study of knowledge logical operations is originated around 1962 by
representation of knowledge and belief using modalities. Nowadays, it is very
good established area. However, we would like to look to it from a bit another
point of view, our paper models knowledge in terms of linear temporal logic
with {\em past}. We consider various versions of logical knowledge operations
which may be defined in this framework. Technically, semantics, language and
temporal knowledge logics based on our approach are constructed. Deciding
algorithms are suggested, unification in terms of this approach is commented.
This paper does not offer strong new technical outputs, instead we suggest new
approach to conception of knowledge (in terms of time).Comment: 10 page
Now there will be trouble
The paper considers sentences in which ânowâ occurs in initial position and shows that the meaning they convey differs from the meaning of sentences that are otherwise identical except for ânowâ occurring in final position. We argue that the occurrence of ânowâ in initial position triggers a particular kind of modal reading for the sentence to which the adverb is prefixed. A general notion of modal forcing is proposed to provide a uniform account of this kind of reading. Armed with this account, we offer a solution to two tense-modal puzzles, which have to do with fatalism and the possibility of a changing past
Modal Logics that Bound the Circumference of Transitive Frames
For each natural number we study the modal logic determined by the class
of transitive Kripke frames in which there are no cycles of length greater than
and no strictly ascending chains. The case is the G\"odel-L\"ob
provability logic. Each logic is axiomatised by adding a single axiom to K4,
and is shown to have the finite model property and be decidable.
We then consider a number of extensions of these logics, including
restricting to reflexive frames to obtain a corresponding sequence of
extensions of S4. When , this gives the famous logic of Grzegorczyk, known
as S4Grz, which is the strongest modal companion to intuitionistic
propositional logic. A topological semantic analysis shows that the -th
member of the sequence of extensions of S4 is the logic of hereditarily
-irresolvable spaces when the modality is interpreted as the
topological closure operation. We also study the definability of this class of
spaces under the interpretation of as the derived set (of limit
points) operation.
The variety of modal algebras validating the -th logic is shown to be
generated by the powerset algebras of the finite frames with cycle length
bounded by . Moreover each algebra in the variety is a model of the
universal theory of the finite ones, and so is embeddable into an ultraproduct
of them
Moving up and down in the generic multiverse
We give a brief account of the modal logic of the generic multiverse, which
is a bimodal logic with operators corresponding to the relations "is a forcing
extension of" and "is a ground model of". The fragment of the first relation is
called the modal logic of forcing and was studied by us in earlier work. The
fragment of the second relation is called the modal logic of grounds and will
be studied here for the first time. In addition, we discuss which combinations
of modal logics are possible for the two fragments.Comment: 10 pages. Extended abstract. Questions and commentary concerning this
article can be made at
http://jdh.hamkins.org/up-and-down-in-the-generic-multiverse
Admissibility via Natural Dualities
It is shown that admissible clauses and quasi-identities of quasivarieties
generated by a single finite algebra, or equivalently, the quasiequational and
universal theories of their free algebras on countably infinitely many
generators, may be characterized using natural dualities. In particular,
axiomatizations are obtained for the admissible clauses and quasi-identities of
bounded distributive lattices, Stone algebras, Kleene algebras and lattices,
and De Morgan algebras and lattices.Comment: 22 pages; 3 figure
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