Life without definitions

The question ‘what is life?’ has long been a source of philosophical debate and in recent years has taken on increasing scientific importance. The most popular approach among both philosophers and scientists for answering this question is to provide a “definition” of life. In this article I explore a variety of different definitional approaches, both traditional and non-traditional, that have been used to “define” life. I argue that all of them are deeply flawed. It is my contention that a scientifically compelling understanding of the nature of life presupposes an empirically adequate scientific theory (vs. definition) of life; as I argue, scientific theories are not the sort of thing that can be encapsulated in definitions. Unfortunately, as I also discuss, scientists are currently in no position to formulate even a tentative version of such a theory. Recent discoveries in biology and biochemistry have revealed that familiar Earth life represents a single example that may not be representative of life. If this is the case, life on Earth today provides an empirically inadequate foundation for theorizing about life considered generally. I sketch a strategy for procuring the needed additional examples of life without the guidance of a definition or theory of life, and close with an application to NASA’s fledgling search for extraterrestrial life.     

Towards a Semantics Based on the Notion of Justification

Suppose we want to take seriously the neoverificationist idea that an intuitionistic theory of meaning can be generalized in such a way as to be applicable not only to mathematical but also to empirical sentences. The paper explores some consequences of this attitude and takes some steps towards the realization of this program. The general idea is to develop a meaning theory, and consequently a formal semantics, based on the idea that knowing the meaning of a sentence is tantamount to having a criterion for establishing what is a justification for it. Section 1 motivates a requirement of epistemic transparency imposed onto justifications conceived as mental states. In Section 2, the formal notion of justification for an atomic formula is defined, in terms of the notion of cognitive state. In Section 3, the definition is extended to logically complex formulas. In Section 4, the notion of truth-ground is introduced and is used to give a definition of logical validity.     

On Generalization of Definitional Equivalence to Non-Disjoint Languages

For simplicity, most of the literature introduces the concept of definitional equivalence only for disjoint languages. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to non-disjoint languages and they show that their generalization is not equivalent to intertranslatability in general. In this paper, we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce another formalization of definitional equivalence due to Andréka and Németi which is equivalent to the Barrett–Halvorson generalization in the case of disjoint languages. We show that the Andréka–Németi generalization is the smallest equivalence relation containing the Barrett–Halvorson generalization and it is equivalent to intertranslatability, which is another definition for definitional equivalence, even for non-disjoint languages. Finally, we investigate which definitions for definitional equivalences remain equivalent when we generalize them for theories in non-disjoint languages.

     

The Notion of Creativity Revisited: A Philosophical Perspective on Creativity Research

This article is a critical, yet constructive, review of some recent attempts to define and understand creativity, informed by the methods and debates of contemporary philosophy. I argue that the definitional project is not essential to creativity research, but important nevertheless. The standard definition of creativity as the production of something that is both novel and appropriate is on the right track, but needs further qualification and tends to be elaborated in ways that make it either too narrow or too broad. I argue that the product, and not the person or process, should be viewed as the primary bearer of creativity and criticize some influential theorists for making creativity too strongly dependent on social acceptance, while also recognizing that the realist alternative tends to widen, and thus threatens to trivialize, the central notion of an appropriate product. The notion of response‐dependence might be of some help to find the proper balance between the two extremes, and some comparisons with evolutionary theory also help to shed further light on the problem. Finally, I try to spell out the practical consequences of my investigation for creativity research.     

The categorical and the hypothetical: a critique of some fundamental assumptions of standard semantics

The hypothetical notion of consequence is normally understood as the transmission of a categorical notion from premisses to conclusion. In model-theoretic semantics this categorical notion is ??truth??, in standard proof-theoretic semantics it is ??canonical provability??. Three underlying dogmas, (I) the priority of the categorical over the hypothetical, (II) the transmission view of consequence, and (III) the identification of consequence and correctness of inference are criticized from an alternative view of proof-theoretic semantics. It is argued that consequence is a basic semantical concept which is directly governed by elementary reasoning principles such as definitional closure and definitional reflection, and not reduced to a categorical concept. This understanding of consequence allows in particular to deal with non-wellfounded phenomena as they arise from circular definitions.     

Nouns and Verbs: A Comparison of Definitional Style

The present investigation is a study of the definitional style of nouns and verbs in typically developing school-age children. A total of 30 children in upper-elementary grades provided verbal definitions for 10 common high-frequency nouns (e.g., apple, boat, baby) and 10 common high-frequency verbs (e.g., climb, sing, throw). All definitions were coded and scored for semantic content (meaning) and grammatical form (syntax). Results revealed no significant difference between noun and verb definitions for content scores. For form, however, noun definition scores were significantly higher than verb definition scores. A supplementary analysis was conducted to explore development of noun and verb definitions in upper-elementary grades. Input factors, word frequency, as well as theory of the organization of the mental lexicon are discussed in relation to definitional skill.     

The Justification of the Logical Laws Revisited

The proof-theoretic analysis of logical semantics undermines the received view of proof theory as being concerned with symbols devoid of meaning, and of model theory as the sole branch of logical theory entitled to access the realm of semantics. The basic tenet of proof-theoretic semantics is that meaning is given by some rules of proofs, in terms of which all logical laws can be justified and the notion of logical consequence explained. In this paper an attempt will be made to unravel some aspects of the issue and to show that this justification as it stands is untenable, for it relies on a formalistic conception of meaning and fails to recognise the fundamental distinction between semantic definitions and rules of inference. It is also briefly suggested that the profound connection between meaning and proofs should be approached by first reconsidering our very notion of proof.     

Proof and Canonical Proof

Certain anti-realisms about mathematics are distinguished by their taking proof rather than truth as the central concept in the account of the meaning of mathematical statements. This notion of proof which is meaning determining or canonical must be distinguished from a notion of demonstration as more generally conceived. This paper raises a set of objections to Dummett's characterisation of the notion via the notion of a normalised natural deduction proof. The main complaint is that Dummett's use of normalised natural deduction proofs relies on formalisation playing a role for which it is unfit. Instead I offer an alternative account which does not rely on formalisation and go on to examine the relation of proof to canonical proof, arguing that rather than requiring an explicit characterisation of canonical proofs we need to be more aware of the complexities of that relation.     

Models of Deduction*

In standard model theory, deductions are not the things one models. But in general proof theory, in particular in categorial proof theory, one finds models of deductions, and the purpose here is to motivate a simple example of such models. This will be a model of deductions performed within an abstract context, where we do not have any particular logical constant, but something underlying all logical constants. In this context, deductions are represented by arrows in categories involved in a general adjoint situation. To motivate the notion of adjointness, one of the central notions of category theory, and of mathematics in general, it is first considered how some features of it occur in set-theoretical axioms and in the axioms of the lambda calculus. Next, it is explained how this notion arises in the context of deduction, where it characterizes logical constants. It is shown also how the categorial point of view suggests an analysis of propositional identity. The problem of propositional identity, i.e., the problem of identity of meaning for propositions, is no doubt a philosophical problem, but the spirit of the analysis proposed here will be rather mathematical. Finally, it is considered whether models of deductions can pretend to be a semantics. This question, which as so many questions having to do with meaning brings us to that wall that blocked linguists and philosophers during the whole of the twentieth century, is merely posed. At the very end, there is the example of a geometrical model of adjunction. Without pretending that it is a semantics, it is hoped that this model may prove illuminating and useful. *Since the text of this talk was written in 1999, the author has published several papers about related matters (see ‘Identity of proofs based on normalization and generality’, The Bulletin of Symbolic Logic 9 (2003), 477–503, corrected version available at: http://arXiv.org/math.LO/0208094; other titles are available in the same archive).     

The Dual Foundation of Qualitative Truth Approximation

The main formal notion involved in qualitative truth approximation by the HD-method, viz. ‘more truthlike’, is shown to not only have, by its definition, an intuitively appealing ‘model foundation’, but also, at least partially, a conceptually plausible ‘consequence foundation’. Moreover, combining the relevant parts of both leads to a very appealing ‘dual foundation’, the more so since the relevant methodological notions, viz. ‘more successful’ and its ingredients provided by the HD-method, can be given a similar dual foundation. According to the resulting dual foundation of ‘naive truth approximation’, the HD-method provides successes (established true consequences) and counterexamples (established wrongly missing models) of theories. Such HD-results may support the tentative conclusion that one theory seems to remain more successful than another in the naive sense of having more successes and fewer counterexamples. If so, this provides good reasons for believing that the more successful theory is also more truthlike in the naive sense of having more correct models and more true consequences. In the dual foundation of ‘refined truth approximation’, HD-results remain of the same two kinds, but ‘more successful’ is taken in the refined sense of accommodating counterexamples while saving relevant successes, in which case ‘more truthlike’ can be taken in the refined sense of improving relevant models while saving relevant consequences. In this way one gets a realistic dual account of qualitative truth approximation by the HD-method. The model foundation can also be extended to the methodological notions, but not in a very plausible way. The consequence foundation only seems specifiable for naive truth approximation, in which case it is plausible. In sum, the dual foundation is superior to both. This revised version was published online in July 2006 with corrections to the Cover Date.     

Frege on the psychological significance of definitions

Conclusion In contemporary work, the distinction between the proposition expressed by a sentence and its psychological significance is usually motivated by a familiar kind of counterfactual argument; and the discussion of these issues usually centers around the role of external factors in determining the meaning of our words. My primary goal in this paper has been to show that a similar, though not identical, distinction between two aspects of meaning can be developed entirely on the basis of considerations internal to language users — their cognitive limitations. To make this point, I have focused on symbols introduced through stipulative definitions. In a language containing such symbols, certain expressions and their definitional reductions will seem to differ in psychological significance for creatures with limited intellects, and so in any aspect of meaning that is supposed to correlate with psychological significance; but it seems also that there is some important aspect of meaning that they share.I have argued that a distinction in meaning like this — between sense and psychological significance — should be drawn even in the kind of languages of most concern to Frege, and that his failure to do so led to tensions in his thought. Of course, this observation only touches on the many issues involved in interpreting Frege's theory of definition more generally. I have not tried to describe here, for example, the ways in which the weak interpretation of fruitfulness might interact with the more robust interpretation mentioned earlier; I have only mentioned Frege's view on explicative definitions and the paradox of analysis, and failed even to mention either his treatment of contextual definition, or his peculiar objections to conditional definitions. I do want to emphasize, however, that the distinction drawn here is not simply a matter of Frege scholarship, but that it has some contemporary relevance as well. As we have seen, Frege's semantic goals often coincide with our own; and a number of contemporary writers are explicitly concerned, like Frege, to construct a semantic theory that is able to account for differences in meaning among logically equivalent expressions. Any such theorist should recognize a distinction like that drawn here between sense and psychological significance, and should avoid subjecting an account of one notion to constraints appropriate only for the other.     

Socrates’ wisdom in definition

Why does Socrates favour definitional speech discussion of virtue’s instances and attributes? Why does he take such a dim view of applied ethics? In this article, I criticise the received answers to these questions and offer a different view. I argue that Socrates favours definitional dialogue because it actualises knowledge that the logic of his argument shows to be essential to virtue. By leading the interlocutor to a paradoxical definition of virtue as knowledge, Socrates engenders this knowledge in his soul.     

Routes for Roots: A Mapping Shorthand Symbolism with Reference to Nelson Goodman’s Hidden Ars Combinatoria

A shorthand symbolism for the relational mapping of categories (MSS) is introduced and developed on the basis of Nelson Goodman's structural methodology. Through a reconstruction of extensional isomorphism that Goodman introduces as a criterion for definitional accuracy, and a brief reminder of the argument structure behind his ‘new riddle of induction’, Goodman's radical ontological relativism is turned into a protological principle of what I call ‘domain constituting philosophy’. MSS is demonstrated with reference to Goodman's symbol theory, particularly his notion of exemplification, as well as in a final structured comparison of Frege's ‘sense’ with Goodman's straightforward construction of ‘secondary extension’.     

Meaning Approached Via Proofs

According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development of this notion. Another main part of the paper is concerned with a modification of this notion as it occurs in Michael Dummett’s book The Logical Basis of Metaphysics. The two notions are compared and there is a discussion of how they fare as a foundation for a theory of meaning. It is noted that Dummett’s notion has a simpler structure, but it is argued that it is less appropriate for the foundation of a theory of meaning, because the possession of a valid argument for a sentence in Dummett’s sense is not enough to be warranted to assert the sentence.     

On the Priority of Agent-Based Argumentative Norms

This paper argues against the priority of pure, virtue-based accounts of argumentative norms [VA]. Such accounts are agent-based and committed to the priority thesis: good arguments and arguing well are explained in terms of some prior notion of the virtuous arguer arguing virtuously. Two problems with the priority thesis are identified. First, the definitional problem: virtuous arguers arguing virtuously are neither sufficient nor necessary for good arguments. Second, the priority problem: the goodness of arguments is not explained virtuistically. Instead, being excellences, virtues are instrumental in relation to other, non-aretaic goods—in this case, reason and rationality. Virtues neither constitute reasons nor explain their goodness. Two options remain for VA: either provide some account of reason and rationality in virtuistic terms, or accept them as given but non-aretaic goods. The latter option, though more viable, demands the concession that VA cannot provide the core norms of argumentation theory.     

Luck,Knowledge, and “Mere” Coincidence

There are good reasons for pursuing a theory of knowledge by way of understanding the connection between knowledge and luck. Not surprisingly, then, there has been a burgeoning of interest in “luck theories” of knowledge as well as in theories of luck in general. Unfortunately, “luck” proves to be as recalcitrant an analysandum as “knows.” While it is well worth pursuing a general theory of luck despite these difficulties, our theory of knowledge might be made more manageable if we could find a more restricted notion that captured the core phenomena of luck that are relevant to whether or not someone knows. This essay makes the attempt to delineate such a notion, called “mere coincidence.”     

Decidability ofstit theory with a single agent andRefref Equivalence

The purpose of this paper is to prove the decidability ofstit theory (a logic of “seeing to it that”) with a single agent andRefref Equivalence. This result is obtained through an axiomatization of the theory and a proof that it has thefinite model property. A notion ofcompanions to stit formulas is introduced and extensively used in the proof.     

"In Search of Token Women in Academia": Some Definitions and Clarifications

This article makes an important distinction between two definitions of "token woman." In the first definition, a token woman is one of few women in a predominantly male setting. The second meaning of "Token Woman" identifies that subset of such women who have made the distinctive psychological adaptation described by Laws (1975). The methodological decisions in Young, MacKenzie, and Sherif's (1980) research are justified as based on that definitional distinction. Constantinople's critique is shown to be appropriate as an alternative to Laws' theory, but not as a criticism of our research. Alternative generational explanations for previous findings about Token Women are not supported by existing data.     

Displaying and Deciding Substructural Logics 1: Logics with Contraposition

Many logics in the relevant family can be given a proof theory in the style of Belnap's display logic (Belnap, 1982). However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modified proof theory which more closely models relevant logics. In addition, we use this proof theory to show decidability for a large range of substructural logics.     

Überlegungen zur theoretischen Fundierung der Disziplin Wirtschaftsinformatik, ausgehend von der allgemeinen Systemtheorie

Using general system theory as a theoretical foundation of business information systems science. The science of business information systems is looking for its theoretical foundations. In this article general system theory is adapted to serve as such a theoretical foundation. The adapted general system theory is reconstructed using the structuralistic approach to reveal the empirical claims of this theory and to analyse scientific progress in the science of business information systems. This revised version was published online in August 2006 with corrections to the Cover Date.