87,284 research outputs found
Buying Logical Principles with Ontological Coin: The Metaphysical Lessons of Adding epsilon to Intuitionistic Logic
We discuss the philosophical implications of formal results showing the con-
sequences of adding the epsilon operator to intuitionistic predicate logic. These
results are related to Diaconescu’s theorem, a result originating in topos theory
that, translated to constructive set theory, says that the axiom of choice (an
“existence principle”) implies the law of excluded middle (which purports to be
a logical principle). As a logical choice principle, epsilon allows us to translate
that result to a logical setting, where one can get an analogue of Diaconescu’s
result, but also can disentangle the roles of certain other assumptions that are
hidden in mathematical presentations. It is our view that these results have not
received the attention they deserve: logicians are unlikely to read a discussion
because the results considered are “already well known,” while the results are
simultaneously unknown to philosophers who do not specialize in what most
philosophers will regard as esoteric logics. This is a problem, since these results
have important implications for and promise signif i cant illumination of contem-
porary debates in metaphysics. The point of this paper is to make the nature
of the results clear in a way accessible to philosophers who do not specialize in
logic, and in a way that makes clear their implications for contemporary philo-
sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism.
Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett,
realism, antirealis
Proof Complexity of Systems of (Non-Deterministic) Decision Trees and Branching Programs
This paper studies propositional proof systems in which lines are sequents of decision trees or branching programs, deterministic or non-deterministic. Decision trees (DTs) are represented by a natural term syntax, inducing the system LDT, and non-determinism is modelled by including disjunction, ?, as primitive (system LNDT). Branching programs generalise DTs to dag-like structures and are duly handled by extension variables in our setting, as is common in proof complexity (systems eLDT and eLNDT).
Deterministic and non-deterministic branching programs are natural nonuniform analogues of log-space (L) and nondeterministic log-space (NL), respectively. Thus eLDT and eLNDT serve as natural systems of reasoning corresponding to L and NL, respectively.
The main results of the paper are simulation and non-simulation results for tree-like and dag-like proofs in LDT, LNDT, eLDT and eLNDT. We also compare them with Frege systems, constant-depth Frege systems and extended Frege systems
Sense and Substance in Wittgenstein"s Tractatus
In the early pages of his Tractatus, Wittgenstein says that
the substance of the world consists of unalterable, simple
objects (Gegenstände) (2.021, 2.027). Substance is connected
to the Sinn of a proposition in the following cryptic
way:
"If the world had no substance, then whether a proposition
had sense [Sinn] would depend on whether another
proposition was true.� (2.0211)�
"In that case we could not sketch any picture of the
world (true or false).� (2.0212)
How can the sense of a proposition depend on the existence
of simple, unalterable objects? No connection between
Sinn and substance is evident on the basis of these
short remarks. Furthermore, the remarks sound strange
because elsewhere in the Tractatus Wittgenstein makes it
clear that we can understand the sense of a proposition
without knowing anything about how things actually stand
in the world (see e.g. 4.024)
LTLf and LDLf Monitoring: A Technical Report
Runtime monitoring is one of the central tasks to provide operational
decision support to running business processes, and check on-the-fly whether
they comply with constraints and rules. We study runtime monitoring of
properties expressed in LTL on finite traces (LTLf) and in its extension LDLf.
LDLf is a powerful logic that captures all monadic second order logic on finite
traces, which is obtained by combining regular expressions and LTLf, adopting
the syntax of propositional dynamic logic (PDL). Interestingly, in spite of its
greater expressivity, LDLf has exactly the same computational complexity of
LTLf. We show that LDLf is able to capture, in the logic itself, not only the
constraints to be monitored, but also the de-facto standard RV-LTL monitors.
This makes it possible to declaratively capture monitoring metaconstraints, and
check them by relying on usual logical services instead of ad-hoc algorithms.
This, in turn, enables to flexibly monitor constraints depending on the
monitoring state of other constraints, e.g., "compensation" constraints that
are only checked when others are detected to be violated. In addition, we
devise a direct translation of LDLf formulas into nondeterministic automata,
avoiding to detour to Buechi automata or alternating automata, and we use it to
implement a monitoring plug-in for the PROM suite
Categoricity, Open-Ended Schemas and Peano Arithmetic
One of the philosophical uses of Dedekind’s categoricity theorem for Peano Arithmetic is to provide support for semantic realism. To this end, the logical framework in which the proof of the theorem is conducted becomes highly significant. I examine different proposals regarding these logical frameworks and focus on the philosophical benefits of adopting open-ended schemas in contrast to second order logic as the logical medium of the proof. I investigate Pederson and Rossberg’s critique of the ontological advantages of open-ended arithmetic when it comes to establishing the categoricity of Peano Arithmetic and show that the critique is highly problematic. I argue that Pederson and Rossberg’s ontological criterion deliver the bizarre result that certain first order subsystems of Peano Arithmetic have a second order ontology. As a consequence, the application of the ontological criterion proposed by Pederson and Rossberg assigns a certain type of ontology to a theory, and a different, richer, ontology to one of its subtheories
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
Non-normal modalities in variants of Linear Logic
This article presents modal versions of resource-conscious logics. We
concentrate on extensions of variants of Linear Logic with one minimal
non-normal modality. In earlier work, where we investigated agency in
multi-agent systems, we have shown that the results scale up to logics with
multiple non-minimal modalities. Here, we start with the language of
propositional intuitionistic Linear Logic without the additive disjunction, to
which we add a modality. We provide an interpretation of this language on a
class of Kripke resource models extended with a neighbourhood function: modal
Kripke resource models. We propose a Hilbert-style axiomatization and a
Gentzen-style sequent calculus. We show that the proof theories are sound and
complete with respect to the class of modal Kripke resource models. We show
that the sequent calculus admits cut elimination and that proof-search is in
PSPACE. We then show how to extend the results when non-commutative connectives
are added to the language. Finally, we put the logical framework to use by
instantiating it as logics of agency. In particular, we propose a logic to
reason about the resource-sensitive use of artefacts and illustrate it with a
variety of examples
- …