An Intuitionistic Formula Hierarchy Based on High-School Identities

We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms corresponding to the high-school identities, we show that one can obtain a more compact variant of a proof system, consisting of non-invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalisation procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non-invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first-order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism

Classical BI: Its Semantics and Proof Theory

We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O'Hearn and Pym's logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBI-formulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the proof-theoretic level, a very natural formalism for CBI is provided by a display calculus \`a la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics.Comment: 42 pages, 8 figure

Tools for reformulating logical forms into zero-one mixed integer programs (MIPS)

A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Programming (ILP) formulation or a Mixed Integer Programming (MIP) formulation is presented. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. A prototype user interface by which logical forms can be reformulated and the corresponding MIP constructed and analysed within an existing Mathematical Programming modelling system is illustrated. Finally, the steps to formulate a discrete optimisation model in this way are demonstrated by means of an example

The Broadest Necessity

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given

Non-Fregean Logics of Analytic Equivalence (I)

The identity connective is usually interpreted in non-Fregean logic as an operator representing the identity of situations. This interpretation is related to the modal criterion of the identity of sentence correlates, characteristic of the WT system and some stronger systems. However, this connective can also be interpreted in a different way – as an operator representing the identity of propositions. The “propositional” interpretation is in turn associated with the modal-contents criterion of the identity of sentence correlates. This begs the question of whether there is a system of non-Fregean logic, providing an adequate formalization of this criterion. The aim of the paper is to systematize the metalogical and philosophical context of the issue and to point to a system that provides its solution.The research reported in this paper is a part of the project financed from the funds supplied by the National Science Centre, Poland (decision no. DEC-2011/03/N/HS1/04580)

Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ where $x,y,z$ range over Muchnik reals, $w$ ranges over functions from Muchnik reals to Muchnik reals, and $A(x,y)$ is a formula not containing $w$ or $z$. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page