First-Order Reasoning and Primitive Recursive Natural Number Notations

If the collection of models for the axioms \mathfrakA{\mathfrak{A}} of elementary number theory (Peano arithmetic) is enlarged to include not just the “natural numbers” or their non-standard infinitistic extensions but also what are here called “primitive recursive notations”, questions arise about the reliability of first-order derivations from \mathfrakA{\mathfrak{A}}. In this enlarged set of “models” some derivations usually accepted as “reliable” may be problematic. This paper criticizes two of these derivations which claim, respectively, to establish the totality of exponentiation and to prove Euclid’s theorem about the infinity of primes.     

Learning and transfer of category knowledge in an indirect categorization task

Knowledge representations acquired during category learning experiments are ‘tuned’ to the task goal. A useful paradigm to study category representations is indirect category learning. In the present article, we propose a new indirect categorization task called the “same”–“different” categorization task. The same–different categorization task is a regular same–different task, but the question asked to the participants is about the stimulus category membership instead of stimulus identity. Experiment 1 explores the possibility of indirectly learning rule-based and information-integration category structures using the new paradigm. The results suggest that there is little learning about the category structures resulting from an indirect categorization task unless the categories can be separated by a one-dimensional rule. Experiment 2 explores whether a category representation learned indirectly can be used in a direct classification task (and vice versa). The results suggest that previous categorical knowledge acquired during a direct classification task can be expressed in the same–different categorization task only when the categories can be separated by a rule that is easily verbalized. Implications of these results for categorization research are discussed.     

Varieties of Linear Calculi

A uniform calculus for linear logic is presented. The calculus has the form of a natural deduction system in sequent calculus style with general introduction and elimination rules. General elimination rules are motivated through an inversion principle, the dual form of which gives the general introduction rules. By restricting all the rules to their single-succedent versions, a uniform calculus for intuitionistic linear logic is obtained. The calculus encompasses both natural deduction and sequent calculus that are obtained as special instances from the uniform calculus. Other instances give all the invertibilities and partial invertibilities for the sequent calculus rules of linear logic. The calculus is normalizing and satisfies the subformula property for normal derivations.     

Expressive Power and Incompleteness of Propositional Logics

Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules (however formulated) underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the standard readings, for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this.     

Cut-Elimination and a Permutation-Free Sequent Calculus for Intuitionistic Logic

We describe a sequent calculus, based on work of Herbelin, of which the cut-free derivations are in 1-1 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cut-elimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.     

The enduring scandal of deduction

Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed.     

Proof-Theoretic Functional Completeness for the Hybrid Logics of Everywhere and Elsewhere

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding so-called satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results.     

Rough Sets and 3-Valued Logics

In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is treated as designated, the above logic is a “common denominator” for Kleene and Łukasiewicz 3-valued logics, which represent its two different “determinizations”. In turn, the weak semantics—where both t and u are treated as designated—represents such a “common denominator” for two major 3-valued paraconsistent logics. We give sound and complete, cut-free sequent calculi for both versions of the logic generated by the rough set Nmatrix. Then we derive from these calculi sequent calculi with the same properties for the various “determinizations” of those two versions of the logic (including Łukasiewicz 3-valued logic). Finally, we show how to embed the four above-mentioned determinizations in extensions of the basic rough set logics obtained by adding to those logics a special two-valued “definedness” or “crispness” operator.     

Commuting Conversions vs. the Standard Conversions of the “Good” Connectives

Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that ‘one tends to think that natural deduction should be modified to correct such atrocities.’ We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions. Presented by Heinrich Wansing and Jacek Malinowski     

Normality and naturalness: A comparison of the meanings of concepts used within veterinary medicine and human medicine

This article analyses the different connotations of “normality” and “being natural,” bringing together the theoretical discussion from both human medicine and veterinary medicine. We show how the interpretations of the concepts in the different areas could be mutually fruitful. It appears that the conceptions of “natural” are more elaborate in veterinary medicine, and can be of value to human medicine. In particular they can nuance and correct conceptions of nature in human medicine that may be too idealistic. Correspondingly, the wide ranging conceptions of “normal” in human medicine may enrich conceptions in veterinary medicine, where the discussions seem to be sparse. We do not argue that conceptions from veterinary medicine should be used in human medicine and vice versa, but only that it could be done and that it may well be fruitful. Moreover, there are overlaps between some notions of normal and natural, and further conceptual analysis on this overlap is needed.     

Natural Deduction Calculi and Sequent Calculi for Counterfactual Logics

In this paper we present labelled sequent calculi and labelled natural deduction calculi for the counterfactual logics CK + {ID, MP}. As for the sequent calculi we prove, in a semantic manner, that the cut-rule is admissible. As for the natural deduction calculi we prove, in a purely syntactic way, the normalization theorem. Finally, we demonstrate that both calculi are sound and complete with respect to Nute semantics [12] and that the natural deduction calculi can be effectively transformed into the sequent calculi.     

A New Normalization Strategy for the Implicational Fragment of Classical Propositional Logic

The introduction and elimination rules for material implication in natural deduction are not complete with respect to the implicational fragment of classical logic. A natural way to complete the system is through the addition of a new natural deduction rule corresponding to Peirce’s formula (((A → B) → A) → A). E. Zimmermann [6] has shown how to extend Prawitz’ normalization strategy to Peirce’s rule: applications of Peirce’s rule can be restricted to atomic conclusions. The aim of the present paper is to extend Seldin’s normalization strategy to Peirce’s rule by showing that every derivation Π in the implicational fragment can be transformed into a derivation Π′ such that no application of Peirce’s rule in Π′ occurs above applications of →-introduction and →-elimination. As a corollary of Seldin’s normalization strategy we obtain a form of Glivenko’s theorem for the classical {→}-fragment.     

"本质"的双重含义:自然科学与人文科学——黑格尔、狄尔泰、胡塞尔之间的一点链接

Early in Aristotle’s terminology, and ever since, “essence” has been conceived as having two meanings, namely “universality” and “individuality”. According to the tradition of thought that has dominated throughout the history of Western philosophy, “essence” unequivocally refers to “universality”. As a matter of fact, however, “universality” cannot cover Aristotle’s definition and formulation of “essence”: Essence is what makes a thing “happen to be this thing.” “Individuality” should be the deep meaning of “essence”. By means of an analysis of some relevant Western thoughts and a review of cultural realities, it can be concluded that the difference between the attitudes toward things of the natural sciences and the humane sciences mainly lies in the fact that the former focus on the pursuit of universal regularity, whereas the latter go after the value and significance of human life. The movement from natural things to cultural things is a process in which essence shifts from universality to individuality. It is the author’s contention that what should be stressed in the fields of human culture and society is the construction of an ideal society that is “harmonious yet not identical”, on the basis of respecting and developing individual peculiarity and otherness. Translated by Zhang Lin from Beijing daxue xuebao 北京大学学报 (Journal of Peking University), 2007, (11): 23–29     

The Aesthetic Import of the Act of Knowledge and its European Roots in Merab Mamardašvili

What Mamardašvili meant by “process of knowledge” is not an all-embracing vision of reality accomplished “once-and-for-all”; it is not a step by step procedure of deduction; rather it is an anti-dialectical reconstruction of a constellation of signs put together over and over again by the subject by an act of non-premeditated genius. It is a kind of aesthetic act that makes the sense appear, like a vertical cut in the sequential line of space and time.     

What determines biological fitness? The problem of the reference environment

Organisms’ environments are thought to play a fundamental role in determining their fitness and hence in natural selection. Existing intuitive conceptions of environment are sufficient for biological practice. I argue, however, that attempts to produce a general characterization of fitness and natural selection are incomplete without the help of general conceptions of what conditions are included in the environment. Thus there is a “problem of the reference environment”—more particularly, problems of specifying principles which pick out those environmental conditions which determine fitness. I distinguish various reference environment problems and propose solutions to some of them. While there has been a limited amount of work on problems concerning what I call “subenvironments”, there appears to be no earlier work on problems of what I call the “whole environment”. The first solution I propose for a whole environment problem specifies the overall environment for natural selection on a set of biological types present in a population over a specified period of time. The second specifies an environment relevant to extinction of types in a population; this kind of environment is especially relevant to certain kinds of long-term evolution.     

Can the World Help Us in Fixing the Reference of Natural Kind Terms?

According to Putnam the reference of natural kind terms is fixed by the world, at least partly; whether two things belong to the same kind depends on whether they obey the same objective laws. We show that Putnam's criterion of substance identity only “works” if we read “objective laws” as “OBJECTIVE LAWS”. Moreover, at least some of the laws of some of the special sciences have to be included. But what we consider to be good special sciences and what not depends upon our values. Hence, “objective laws” cannot be read as “OBJECTIVE LAWS”. It follows that the reference of natural kind terms cannot be fixed by the world, not even partly. The final conclusion applies to a variety of realisms. This revised version was published online in August 2006 with corrections to the Cover Date.     

Assigning biological functions: making sense of causal chains

A meaningful distinction can be made between functions and mere effects in biological systems without resorting to teleological arguments: (i) biological systems must cope with a multitude of problems or they will cease to exist; (ii) the solutions to these problems invariably depend on circular causal chains (“feedback loops”); and (iii) biological functions are attributes of elements in biological systems that have an effect which, by contributing to the correcting behavior of a feedback control system, assists in solving a biological problem. The analysis is applied to several biological systems. The proposed solution is discussed primarily in its relation to two popular approaches to the concept of biological function, i.e., the “causal role accounts” and the “selected effect accounts”.     

The temporality of Merleau-Ponty’s intertwining

In his last work, The Visible and the Invisible, Merleau-Ponty explored the fact that we believe that perception occurs in our heads (“in the recesses of a body”) and, hence, assert that the perceptual world is “in” us, while also believing that we are “in” the world we perceive.  In this article, I examine how this intertwining of self and world justifies the faith we have in perception. I shall do so by considering a number of examples.  In each case, the object “in itself” will turn out to be neither within us nor outside of us, but rather at the intersection set by the intertwining. I will then turn to what this disclosure of this object reveals about human temporality and, indeed, about human being as a place (or “clearing”) that permits disclosure.     

郭象"独化"范畴释义

Seemingly, “independent genesis” refers to the independent existence and changes of each thing, but it is clear that there cannot be any truly “independent” things at all. Each thing in the world has to stay in connection or relationship with other things outside itself if it wants to represent its own “independence” and “genesis” in terms of form; and inevitably such connection or relationship itself has to be embodied in the internal nature of each thing. In the metaphysical thought of Guo Xiang, the former was known as the quality of “interdependence”; and the latter the characteristics of “quality” or “quality image.” Such characteristics of “quality” or “quality image” were interdependent, which constituted the essence of each thing itself as “beingness” and “beinglessness,” and thus resulted in the independent manifestation and change of things in terms of their external forms. The grasping of essence of things as “beingness” and “beinglessness” depended upon comprehension or rational intuition, and that was the realm of “profundity” in Guo Xiang’s terms. Translated by Huang Deyuan from Zhexue Yanjiu 哲哲哲哲 (Philosophical Researches), 2007, (11): 37–43