78,387 research outputs found
Hybrid type theory: a quartet in four movements
This paper sings a song -a song created by bringing together the work of four great names in the history of logic: Hans Reichenbach, Arthur Prior, Richard Montague, and Leon Henkin. Although the work of the first three of these authors have previously been combined, adding the ideas of Leon Henkin is the addition required to make the combination work at the logical level. But the present paper does not focus on the underlying technicalities (these can be found in Areces, Blackburn, Huertas, and Manzano [to appear]) rather it focusses on the underlying instruments, and the way they work together. We hope the reader will be tempted to sing along
On the Correspondence between Display Postulates and Deep Inference in Nested Sequent Calculi for Tense Logics
We consider two styles of proof calculi for a family of tense logics,
presented in a formalism based on nested sequents. A nested sequent can be seen
as a tree of traditional single-sided sequents. Our first style of calculi is
what we call "shallow calculi", where inference rules are only applied at the
root node in a nested sequent. Our shallow calculi are extensions of Kashima's
calculus for tense logic and share an essential characteristic with display
calculi, namely, the presence of structural rules called "display postulates".
Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable
for proof search due to the presence of display postulates and other structural
rules. The second style of calculi uses deep-inference, whereby inference rules
can be applied at any node in a nested sequent. We show that, for a range of
extensions of tense logic, the two styles of calculi are equivalent, and there
is a natural proof theoretic correspondence between display postulates and deep
inference. The deep inference calculi enjoy the subformula property and have no
display postulates or other structural rules, making them a better framework
for proof search
Defining Original Presentism
It is surprisingly hard to define presentism. Traditional definitions
of the view, in terms of tensed existence statements, have
turned out not to to be capable of convincingly distinguishing
presentism from eternalism. Picking up on a recent proposal
by Tallant, I suggest that we need to locate the break between
eternalism and presentism on a much more fundamental level.
The problem is that presentists have tried to express their
view within a framework that is inherently eternalist. I call
that framework the Fregean nexus, as it is defined by Frege’s
atemporal understanding of predication. In particular, I show
that the tense-logical understanding of tense which is treated
as common ground in the debate rests on this very same
Fregean nexus, and is thus inadequate for a proper definition
of presentism. I contrast the Fregean nexus with what I call
the original temporal nexus, which is based on an alternative,
inherently temporal form of predication. Finally, I propose
to define presentism in terms of the original temporal nexus,
yielding original presentism. According to original presentism,
temporal propositions are distinguished from atemporal ones not
by aspects of their content, as they are on views based on the
Fregean nexus, but by their form—in particular, by their form of
predication
Complete Additivity and Modal Incompleteness
In this paper, we tell a story about incompleteness in modal logic. The story
weaves together a paper of van Benthem, `Syntactic aspects of modal
incompleteness theorems,' and a longstanding open question: whether every
normal modal logic can be characterized by a class of completely additive modal
algebras, or as we call them, V-BAOs. Using a first-order reformulation of the
property of complete additivity, we prove that the modal logic that starred in
van Benthem's paper resolves the open question in the negative. In addition,
for the case of bimodal logic, we show that there is a naturally occurring
logic that is incomplete with respect to V-BAOs, namely the provability logic
GLB. We also show that even logics that are unsound with respect to such
algebras do not have to be more complex than the classical propositional
calculus. On the other hand, we observe that it is undecidable whether a
syntactically defined logic is V-complete. After these results, we generalize
the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van
Benthem's theme of syntactic aspects of modal incompleteness
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