Abstraction in Algorithmic Logic

We develop a functional abstraction principle for the type-free algorithmic logic introduced in our earlier work. Our approach is based on the standard combinators but is supplemented by the novel use of evaluation trees. Then we show that the abstraction principle leads to a Curry fixed point, a statement C that asserts C ⇒ A where A is any given statement. When A is false, such a C yields a paradoxical situation. As discussed in our earlier work, this situation leaves one no choice but to restrict the use of a certain class of implicational rules including modus ponens.     

Logics with the Qualitative Probability Operator

The paper presents several strongly complete axiomatizationsof qualitative probability within the framework of probabilisticlogic. We show that in the proposed semantics qualitative probabilitiesare characterized by probability functions, so they also arecomparative probabilities.     

浅谈医学科研思维的逻辑性

医学科研思维过程中最重要的就是医学假说的提出、验证,推理和遵守逻辑思维的过程。它以抽象的概念、判断、推理为思维形式,通过分析、综合、比较、分类等多种逻辑思维方法进行操作,以达到它的最终目的。逻辑思维能力对于医学科研工作者是至关重要的,医学科研工作者应该不断加强逻辑修养,不断提升科研思维水准。     

A Note on Priest's Finite Inconsistent Arithmetics

We give a complete characterization of Priest's Finite Inconsistent Arithmetics observing that his original putative characterization included arithmetics which cannot in fact be realized.     

Non-Classical Negation in the Works of Helena Rasiowa and Their Impact on the Theory of Negation

The paper is devoted to the contributions of Helena Rasiowa to the theory of non-classical negation. The main results of Rasiowa in this area concerns–constructive logic with strong (Nelson) negation,–intuitionistic negation and some of its generalizations: minimal negation of Johansson and semi-negation.We discuss also the impact of Rasiowa works on the theory of non-classical negation.A lecture presented at the International Conference Trends in Logic III : A. Mostowski, H. Rasiowa and C. Rauszer in memoriam, Warsaw, Ruciane-Nida September 23-26, 2005.     

An Impossibility Theorem on Beliefs in Games

A paradox of self-reference in beliefs in games is identified, which yields a game-theoretic impossibility theorem akin to Russell’s Paradox. An informal version of the paradox is that the following configuration of beliefs is impossible:Ann believes that Bob assumes thatAnn believes that Bob’s assumption is wrongThis is formalized to show that any belief model of a certain kind must have a ‘hole.’ An interpretation of the result is that if the analyst’s tools are available to the players in a game, then there are statements that the players can think about but cannot assume. Connections are made to some questions in the foundations of game theory.Special Issue Ways of Worlds II. On Possible Worlds and Related Notions Edited by Vincent F. Hendricks and Stig Andur Pedersen     

Propositions, Sets, and Worlds

If we agree with Michael Jubien that propositions do not exist, while accepting the existence of abstract sets in a realist mathematical ontology, then the combined effect of these ontological commitments has surprising implications for the metaphysics of modal logic, the ontology of logically possible worlds, and the controversy over modal realism versus actualism. Logically possible worlds as maximally consistent proposition sets exist if sets generally exist, but are equivalently expressed as maximally consistent conjunctions of the same propositions in corresponding sets. A conjunction of propositions, even if infinite in extent, is nevertheless itself a proposition. If sets and hence proposition sets exist but propositions do not exist, then whether or not modal realism is true depends on which of two apparently equivalent methods of identifying, representing, or characterizing logically possible worlds we choose to adopt. I consider a number of reactions to the problem, concluding that the best solution may be to reject the conventional model set theoretical concept of logically possible worlds as maximally consistent proposition sets, and distinguishing between the actual world alone as maximally consistent and interpreting all nonactual merely logically possible worlds as submaximal. I am grateful to the Netherlands Institute for Advanced Study in the Humanities and Social Sciences (NIAS), Royal Netherlands Academy of Arts and Sciences (KNAW), for supporting this among related research projects in philosophical logic and philosophy of mathematics during my Resident Research Fellowship in 2005-2006.     

MV-Algebras and Quantum Computation

We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a “flat” quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra over the complex numbers. Presented by Heinrich Wansing     

Monotonicity and Reasoning with Exceptions

A proposal by Ferguson [2003, Argumentation 17, 335–346] for a fully monotonic argument form allowing for the expression of defeasible generalizations is critically examined and rejected as a general solution. It is argued that (i) his proposal reaches less than the default-logician’s solution allows, e.g., the monotonously derived conclusion is one-sided and itself not defeasible. (ii) when applied to a suitable example, his proposal derives the wrong conclusion. Unsuccessful remedies are discussed.     

Is ZF a hack?: Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics

This paper presents Automath encodings (which are also valid in LF/λP) of various kinds of foundations of mathematics. Then it compares these encodings according to their size, to find out which foundation is the simplest.

The systems analyzed in this way are two kinds of set theory (ZFC and NF), two systems based on Church's higher order logic (Isabelle/Pure and HOL), three kinds of type theory (the calculus of constructions, Luo's extended calculus of constructions, and Martin-Löf's predicative type theory) and one foundation based on category theory.

The conclusions of this paper are that the simplest system is type theory (the calculus of constructions), but that type theories that know about serious mathematics are not simple at all. In that case the set theories are the simplest. If one looks at the number of concepts needed to explain such a system, then higher order logic is the simplest, with twenty-five concepts. On the other side of the scale, category theory is relatively complex, as is Martin-Löf's type theory.

(The full Automath sources of the contexts described in this paper are one the web at http://www.cs.ru.nl/~freek/zfc-etc/.)