Logical Constants and the Sorites Paradox

Logical form is thought to be discovered by keeping fixed the logical constants and allowing the non-logical content in the sentence to vary. The problem of logical constants is the problem of defining what counts as a logical constant. In this paper, I will argue that the concept ’logical constant’ is vague. I demonstrate the vagueness of logical constancy by providing a sorites argument, thereby showing the sorites-susceptibility of the concept. Many prior papers in the literature on logical constants hint at this vagueness, but do not explore how theories of vagueness apply to logical constants. In the second half of this paper, I do just this. I consider approaches to logical constants that resemble nihilism about vagueness and more recent theories that relativize truth to precisifications. Finally, I argue that approaches that accept the potential indeterminate status of putative logical constants are preferable to nihilism or relativism about logical constancy

An Argument for Minimal Logic

The problem of negative truth is the problem of how, if everything in the world is positive, we can speak truly about the world using negative propositions. A prominent solution is to explain negation in terms of a primitive notion of metaphysical incompatibility. I argue that if this account is correct, then minimal logic is the correct logic. The negation of a proposition A is characterised as the minimal incompatible of A composed of it and the logical constant ¬. A rule based account of the meanings of logical constants that appeals to the notion of incompatibility in the introduction rule for negation ensures the existence and uniqueness of the negation of every proposition. But it endows the negation operator with no more formal properties than those it has in minimal logic

Second Order Logic and Logical Form

This thesis explores several related issues surrounding second order logic. The central problem running throughout is whether second order logic should provide the underlying logic for formalizations of natural language. A prior problem is determining the significance of this choice. Such controversies over the adoption of a logic usually involve assessing the merits of challengers to first order logic. In some of these rival systems various first order logical truths do not hold. The failure of the Law of the Excluded Middle in intuitionistic systems is the most common example. The other alternatives to first order logic accept it as a part of the truth, but extend it by adding new logical constants. Some modal systems of logic are formed by adding to first order logic a symbol intended to be read as \u27it is logically necessary that.\u27 The first order semantics is extended to provide truth conditions for sentences containing this new symbol. In such cases the debate is whether we are justified in expanding the list of logical constants provided by first order logic. We accept the first order logical constants and are deciding whether, e.g., \u27it is logically necessary that\u27 should be added to the list

Mapping-equivalence and oid-equivalence of single-function object-creating conjunctive queries

Conjunctive database queries have been extended with a mechanism for object creation to capture important applications such as data exchange, data integration, and ontology-based data access. Object creation generates new object identifiers in the result, that do not belong to the set of constants in the source database. The new object identifiers can be also seen as Skolem terms. Hence, object-creating conjunctive queries can also be regarded as restricted second-order tuple-generating dependencies (SO tgds), considered in the data exchange literature. In this paper, we focus on the class of single-function object-creating conjunctive queries, or sifo CQs for short. We give a new characterization for oid-equivalence of sifo CQs that is simpler than the one given by Hull and Yoshikawa and places the problem in the complexity class NP. Our characterization is based on Cohen's equivalence notions for conjunctive queries with multiplicities. We also solve the logical entailment problem for sifo CQs, showing that also this problem belongs to NP. Results by Pichler et al. have shown that logical equivalence for more general classes of SO tgds is either undecidable or decidable with as yet unknown complexity upper bounds.Comment: This revised version has been accepted on 11 January 2016 for publication in The VLDB Journa

Proof-theoretic semantics, a problem with negation and prospects for modality

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings of modal operators in terms of rules of inference

Combining decision procedures for the reals

We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which "local" decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let Tadd[Q] be the first-order theory of the real numbers in the language of ordered groups, with negation, a constant 1, and function symbols for multiplication by rational constants. Let Tmult[Q] be the analogous theory for the multiplicative structure, and let T[Q] be the union of the two. We show that although T[Q] is undecidable, the universal fragment of T[Q] is decidable. We also show that terms of T[Q]can fruitfully be put in a normal form. We prove analogous results for theories in which Q is replaced, more generally, by suitable subfields F of the reals. Finally, we consider practical methods of establishing quantifier-free validities that approximate our (impractical) decidability results.Comment: Will appear in Logical Methods in Computer Scienc

Model-checking Quantitative Alternating-time Temporal Logic on One-counter Game Models

We consider quantitative extensions of the alternating-time temporal logics ATL/ATLs called quantitative alternating-time temporal logics (QATL/QATLs) in which the value of a counter can be compared to constants using equality, inequality and modulo constraints. We interpret these logics in one-counter game models which are infinite duration games played on finite control graphs where each transition can increase or decrease the value of an unbounded counter. That is, the state-space of these games are, generally, infinite. We consider the model-checking problem of the logics QATL and QATLs on one-counter game models with VASS semantics for which we develop algorithms and provide matching lower bounds. Our algorithms are based on reductions of the model-checking problems to model-checking games. This approach makes it quite simple for us to deal with extensions of the logical languages as well as the infinite state spaces. The framework generalizes on one hand qualitative problems such as ATL/ATLs model-checking of finite-state systems, model-checking of the branching-time temporal logics CTL and CTLs on one-counter processes and the realizability problem of LTL specifications. On the other hand the model-checking problem for QATL/QATLs generalizes quantitative problems such as the fixed-initial credit problem for energy games (in the case of QATL) and energy parity games (in the case of QATLs). Our results are positive as we show that the generalizations are not too costly with respect to complexity. As a byproduct we obtain new results on the complexity of model-checking CTLs in one-counter processes and show that deciding the winner in one-counter games with LTL objectives is 2ExpSpace-complete.Comment: 22 pages, 12 figure

Carnap's problem for intuitionistic propositional logic

We show that intuitionistic propositional logic is \emph{Carnap categorical}: the only interpretation of the connectives consistent with the intuitionistic consequence relation is the standard interpretation. This holds relative to the most well-known semantics with respect to which intuitionistic logic is sound and complete; among them Kripke semantics, Beth semantics, Dragalin semantics, and topological semantics. It also holds for algebraic semantics, although categoricity in that case is different in kind from categoricity relative to possible worlds style semantics.Comment: Keywords: intuitionistic logic, Carnap's problem, nuclear semantics, algebraic semantics, logical constants, consequence relations, categoricity. Versions: 3rd version has minor additions, and correction of an error in 2nd version (not in 1st version