A framework for the transfer of proofs,lemmas and strategies from classical to non classical logics

There exist valuable methods for theorem proving in non classical logics based on translation from these logics into first-order classical logic (abbreviated henceforth FOL). The key notion in these approaches istranslation from aSource Logic (henceforth abbreviated SL) to aTarget Logic (henceforth abbreviated TL). These methods are concerned with the problem offinding a proof in TL by translating a formula in SL, but they do not address the very important problem ofpresenting proofs in SL via a backward translation. We propose a framework for presenting proofs in SL based on a partial backward translation of proofs obtained in a familiar TL: Order-Sorted Predicate Logic. The proposed backward translation transfers some formulasF _TL belonging to the proof in TL into formulasF _SL , such that the formulasF _SL either (a) belong to a corresponding deduction in SL (in the best case) or, (b) are semantically related in some precise way, to formulas in the corresponding deduction in SL (in the worst case). The formulasF _TL andF _SL can obviously be considered aslemmas of their respective proofs. Therefore the transfer of lemmas of TL gives at least a skeleton of the corresponding proof in SL. Since the formulas of a proof keep trace of the strategy used to obtain the proof, clearly the framework can also help in solving another fundamental and difficult problem:the transfer of strategies from classical to non classical logics. We show how to apply the proposed framework, at least to S5, S4(p), K, T, K4. Two conjectures are stated and we propose sufficient (and in general satisfactory) conditions in order to obtain formulas in the proof in SL. Two particular cases of the conjectures are proved to be theorems. Three examples are treated in full detail. The main lines of future research are given.     

An abstract setting for Henkin proofs

A general result is proved about the existence of maximally consistent theories satisfying prescribed closure conditions. The principle is then used to give streamlined proofs of completeness and omitting-types theorems, in which inductive Henkin-style constructions are replaced by a demonstration that a certain theory respects a certain class of inference rules.     

Knowledge of proofs

If proofs are nothing more than truth makers, then there is no force in the standard argument against classical logic (there is no guarantee that there is either a proof forA or a proof fornot A). The standard intuitionistic conception of a mathematical proof is stronger: there are epistemic constraints on proofs. But the idea that proofs must be recognizable as such by us, with our actual capacities, is incompatible with the standard intuitionistic explanations of the meanings of the logical constants. Proofs are to be recognizable in principle, not necessarily in practice, as shown in section 1. Section 2 considers unknowable propositions of the kind involved in Fitch's paradox:p and it will never be known thatp. It is argued that the intuitionist faces a dilemma: give up strongly entrenched common sense intuitions about such unknowable propositions, or give up verificationism. The third section considers one attempt to save intuitionism while partly giving up verificationism: keep the idea that a proposition is true iff there is a proof (verification) of it, and reject the idea that proofs must be recognizable in principle. It is argued that this move will have the effect that some standard reasons against classical semantics will be effective also against intuitionism. This is the case with Dummett's meaning theoretical argument. At the same time the basic reason for regarding proofs as more than mere truth makers is lost.I am much indebted for comments to Lars Bergström, Per Martin-Löf, Wlodek Rabinowicz, Fredrik Stjernberg, Dag Westerståhl and Tim Williamson. I owe even more to the many seminars about truth and meaning, led by Dag Prawitz, at the philosophy department of Stockholm University. These were especially intense in the mideighties, when I was a graduate student.     

An existential interpretation of success neurosis

The fear of success is typically interpreted as a neurosis in which the fear of change plays a central role. In spite of agreeing with the existence of this syndrome in clinical practice. this author holds that a similar ambivalence about change lurks within ourselves as well. We are especially like patients in that, in spite of their often unique conflicts, they also run away, like us, from true success. In defining the latter, the author interprets it as synonymous with living an ethical life filled with reverence and respect for all.Grateful acknowledgement is made to Stephen Getsinger, Ph.D., for having read and made numerous cogent remarks on this paper.     

释"孚"

本文分析了"孚"字的甲骨文形体,其形体义为父或母用手抱起幼儿之状。父或母用手抱起幼儿完全出于真诚而不虚伪的爱心,从这样的慈爱之举中引申出"诚信"义来。     

Tree proofs for syllogistic

This paper presents a tree method for testing the validity of inferences, including syllogisms, in a simple term logic. The method is given in the form of an algorithm and is shown to be sound and complete with respect to the obvious denotational semantics. The primitive logical constants of the system, which is indebted to the logical works of Jevons, Brentano and Lewis Carroll, are term negation, polyadic term conjunction, and functors affirming and denying existence, and use is also made of a metalinguistic concept of formal synonymy. It is indicated briefly how the method may be extended to other systems.To the memory of Jerzy Supecki     

Interactional value theory: An interpretation

Summary The imperatives of organization, technology, and planning operate similarly in both capitalist and Marxist systems. Differences in behavioural outputs (such as organizational productivity, industrial relations behaviour, or the outcomes of different health services systems) can be explained by adopting a framework of an interactional value theory which accepts convergence of different value systems and points out and analyzes differences in outputs in light of interactions between available climatic techniques and preferred values. The interactional approach links together ethical theory and various issues in social philosophy, especially in the context of the resolution of conflict among values. The dynamics of economic and social change stem from differing values held by key actors in society - government, business and the public. Values are not categorical but integrative, and values held by various parties converge. Further research should be focussed to clarify value-beliefs of government, business and other key players in identifying the interactional characteristics of public policy issues.     

释《周易·渐卦》

通过对渐卦卦象的分析,并考《说文》段注“陂陀曰渐”,得出渐卦取名为渐,当取坡渐之意,进而引伸为渐进之意。又从卦爻象出发,对渐卦的卦爻辞作出分析,认为渐卦的卦爻辞是根据其卦爻象而系出,而且其整卦的卦爻辞都是围绕着一个中心主题而展开,且其思想与《诗经》中的《鸿雁》等篇不无关系。这或可为进一步解读《周易》古经及其产生时代问题提供某种有益的启示。     

Recognition of proofs in conditional reasoning

Relatively little is known about those who consistently produce the valid response to Modus Tollens (MT) problems. In two studies, people who responded correctly to MT problems indicated how “convinced” they were by proofs of conditional reasoning conclusions. The first experiment showed that MT competent reasoners found accurate proofs of MT reasoning more convincing than similar “proofs” of invalid reasoning. Similarly, there was a tendency for MT competent reasoners to find an initial counterfactual supposition more convincing than did people who were less competent in MT. The second experiment showed that when individuals produced the correct MT response, and found correct MT proofs to be more convincing than “bogus” proofs, they were also less likely to find the conclusions to Denying the Antecedent, or Affirming the Consequent problems valid, compared to individuals who could not discriminate between valid and bogus MT proofs. These findings are discussed in terms of both their implications for the mental logic and mental models positions, and individual differences in System 1 and System 2 reasoning.