Actuality, Necessity, and Logical Truth

The traditional view that all logical truths are metaphysically necessary has come under attack in recent years. The contrary claim is prominent in David Kaplan’s work on demonstratives, and Edward Zalta has argued that logical truths that are not necessary appear in modal languages supplemented only with some device for making reference to the actual world (and thus independently of whether demonstratives like ‘I’, ‘here’, and ‘now’ are present). If this latter claim can be sustained, it strikes close to the heart of the traditional view. I begin this paper by discussing and refuting Zalta’s argument in the context of a language for propositional modal logic with an actuality connective (section 1). This involves showing that his argument in favor of real world validity his preferred explication of logical truth, is fallacious. Next (section 2) I argue for an alternative explication of logical truth called general validity. Since the rule of necessitation preserves general validity, the argument of section 2 provides a reason for affirming the traditional view. Finally (section 3) I show that the intuitive idea behind the discredited notion of real world validity finds legitimate expression in an object language connective for deep necessity. Earlier versions of this paper were read at the universities of Graz, Maribor, and Salzburg, and at a workshop on the philosophy of logic at the National Autonomous University of Mexico (UNAM) in Mexico City. My thanks to those present at these events for many helpful suggestions. Thanks are also due to an anonymous referee for Philosophical Studies.     

Formal Logic vs. Philosophical Argument

The wider topic to which the content of this paper belongs is that of the relationship between formal logic and real argumentation. Of particular potential interest in this connection are held to be substantive arguments constructed by philosophers reputed equally as authorities in logical theory. A number of characteristics are tentatively indicated by the author as likely to be encountered in such arguments. The discussion centers afterwards, by way of specification, on a remarkable piece of argument quoted in Cicero’s dialog On Divination and ascribed to Stoic thinkers. The Stoics’ formal theory of inference is summarily referred to in this context, with special emphasis on their basic deductive schemata (‘indemonstrables’), some of them recognizable as links in the overall structure of the quoted argument. The main lines of Cicero’s criticism of the Stoic argument are next commented upon, with emphasis on his implied view as to the requirements of a good argument. Towards the end of the paper, a few considerations are added on the changes in the prevailing style of argumentation conspicuous in the three famous Roman Stoics.     

Inductive Incompleteness

Nelson Goodman cast the ‘problem of induction’ as the task of articulating the principles and standards by which to distinguish valid from invalid inductive inferences. This paper explores some logical bounds on the ability of a rational reasoner to accomplish this task. By a simple argument, either an inductive inference method cannot admit its own fallibility, or there exists some non-inferable hypothesis whose non-inferability the method cannot infer (violating the principle of ‘negative introspection’). The paper discusses some implications of this limited self-knowledge for the justifiability of inductive inferences, auto-epistemic logic, and the epistemic foundations of game theory.     

First-Order da Costa Logic

Priest (2009) formulates a propositional logic which, by employing the worldsemantics for intuitionist logic, has the same positive part but dualises the negation, to produce a paraconsistent logic which it calls ‘Da Costa Logic’. This paper extends matters to the first-order case. The paper establishes various connections between first order da Costa logic, da Costa’s own C_ω, and classical logic. Tableau and natural deductions systems are provided and proved sound and complete.     

Aquinas and quantifier mistakes

In his “Third Way” Aquinas appears to argue in a way that relies upon shifting quantifiers in a fallacious way. Some have tried to save this and other parts of the “Third Way” by introducing sophisticated logical and metaphysical machinery. Alternatively, Aquinas’ apparently fallacious quantifier shift can be seen to be part of a valid argument if we supply a simple premise which an Aristotelian natural philosopher would surely hold. In this short paper, I consider candidates for this premise, defend a specific premise, and from that discussion draw a moral about quantifier predicate logic. I conclude that Aristotelian natural philosophy is more than an historical backdrop to Aquinas’ arguments.     

Applied Logic without Psychologism

Logic is a celebrated representation language because of its formal generality. But there are two senses in which a logic may be considered general, one that concerns a technical ability to discriminate between different types of individuals, and another that concerns constitutive norms for reasoning as such. This essay embraces the former, permutation-invariance conception of logic and rejects the latter, Fregean conception of logic. The question of how to apply logic under this pure invariantist view is addressed, and a methodology is given. The pure invariantist view is contrasted with logical pluralism, and a methodology for applied logic is demonstrated in remarks on a variety of issues concerning non-monotonic logic and non-monotonic inference, including Charles Morgan’s impossibility results for non-monotonic logic, David Makinson’s normative constraints for non-monotonic inference, and Igor Douven and Timothy Williamson’s proposed formal constraints on rational acceptance.     

Some Embedding Theorems for Conditional Logic

We prove some embedding theorems for classical conditional logic, covering ‘finitely cumulative’ logics, ‘preferential’ logics and what we call ‘semi-monotonic’ logics. Technical tools called ‘partial frames’ and ‘frame morphisms’ in the context of neighborhood semantics are used in the proof.     

From Modal Discourse to Possible Worlds

The possible-worlds semantics for modality says that a sentence is possibly true if it is true in some possible world. Given classical prepositional logic, one can easily prove that every consistent set of propositions can be embedded in a ‘maximal consistent set’, which in a sense represents a possible world. However the construction depends on the fact that standard modal logics are finitary, and it seems false that an infinite collection of sets of sentences each finite subset of which is intuitively ‘possible’ in natural language has the property that the whole set is possible. The argument of the paper is that the principles needed to shew that natural language possibility sentences involve quantification over worlds are analogous to those used in infinitary modal logic.     

Space Cannot Be Cut—Why Self-Identity Naturally Includes Neighbourhood

Psychology is not alone in its struggle with conceptualizing the dynamic relationship between space and individual or collective identity. This general epistemological issue haunts biology where it has a specific focus in evolutionary arguments. It arises because of the incompatibility between definitive logical systems of ‘contradiction or unity’, which can only apply to inert material systems, and natural evolutionary processes of cumulative energetic transformation. This incompatibility makes any attempt to apply definitive logic to evolutionary change unrealistic and paradoxical. It is important to recognise, because discrete perceptions of self and group, based on the supposition that any distinguishable identity can be completely cut free, as an ‘independent singleness’, from the space it inescapably includes and is included in, are a profound but unnecessary source of psychological, social and environmental conflict. These perceptions underlie Darwin’s definition of ‘natural selection’ as ‘the preservation of favoured races in the struggle for life’. They result in precedence being given to striving for homogeneous supremacy, through the competitive suppression of others, instead of seeking sustainable, co-creative evolutionary relationship in spatially and temporally heterogeneous communities. Here, I show how ‘natural inclusion’, a new, post-dialectic understanding of evolutionary process, becomes possible through recognising space as a limitless, indivisible, receptive (non-resistive) ‘intangible presence’ vital for movement and communication, not as empty distance between one tangible thing and another. The fluid boundary logic of natural inclusion as the co-creative, fluid dynamic transformation of all through all in receptive spatial context, allows all form to be understood as flow-form, distinctive but dynamically continuous, not singularly discrete. This simple move from regarding space and boundaries as sources of discontinuity and discrete definition to sources of continuity and dynamic distinction correspondingly enables self-identity to be understood as a dynamic inclusion of neighbourhood, through the inclusion of space throughout and beyond all natural figural forms as configurations of energy. Fully to appreciate and communicate the significance of this move, it is necessary to widen the linguistic, mathematical and imaginative remit of conventional scientific argument and explication so as to include more poetic, fluid and artistic forms of expression.     

What is <Emphasis Type="Italic">the</Emphasis> Logic of Inference?

The topic of this paper is the question whether there is a logic which could be justly called the logic of inference. It may seem that at least since Prawitz, Dummett and others demonstrated the proof-theoretical prominency of intuitionistic logic, the forthcoming answer is that it is this logic that is the obvious choice for the accolade. Though there is little doubt that this choice is correct (provided that inference is construed as inherently single-conclusion and complying with the Gentzenian structural rules), I do not think that the usual justification of it is satisfactory. Therefore, I will first try to clarify what exactly is meant by the question, and then sketch a conceptual framework in which it can be reasonably handled. I will introduce the concept of ‘inferentially native’ logical operators (those which explicate inferential properties) and I will show that the axiomatization of these operators leads to the axiomatic system of intuitionistic logic. Finally, I will discuss what modifications of this answer enter the picture when more general notions of inference are considered. Presented by Hannes Leitgeb     

Temporal Reference in Linear Tense Logic

The paper introduces a first-order theory in the language of predicate tense logic which contains a single simple axiom. It is shewn that this theory enables times to be referred to and sentences involving ‘now’ and ‘then’ to be formalised. The paper then compares this way of increasing the expressive capacity of predicate tense logic with other mechanisms, and indicates how to generalise the results to other modal and tense systems.     

The Logic of Analogy in the Law

This paper deals with two issues in the field of reasoning by analogy in the law. The one issue is whether there exists such a thing as analogous rule application, or whether there is only the ‘normal’ application of a broadened rule. It is argued that if rules, as the entities made by a legislator, are distinguished from generalised solutions for cases, the idea of analogous application of rules makes sense. It is also shown how the so-called ‘reason-based model of rule application’, in contrast to the traditional modus ponens or subsumption model, makes it easy to give a logical account of analogous rule application. The other issue is how to argue logically about whether two cases are sufficiently similar to adopt the outcome of the one case for the other case. Section 3 provides a general logical model to establish this. The model is based on the comparison of the reasons for and against a particular solution in the two cases.     

Reassessing logical hylomorphism and the demarcation of logical constants

The paper investigates the propriety of applying the form versus matter distinction to arguments and to logic in general. Its main point is that many of the currently pervasive views on form and matter with respect to logic rest on several substantive and even contentious assumptions which are nevertheless uncritically accepted. Indeed, many of the issues raised by the application of this distinction to arguments seem to be related to a questionable combination of different presuppositions and expectations; this holds in particular of the vexed issue of demarcating the class of logical constants. I begin with a characterization of currently widespread views on form and matter in logic, which I refer to as ‘logical hylomorphism as we know it’—LHAWKI, for short—and argue that the hylomorphism underlying LHAWKI is mereological. Next, I sketch an overview of the historical developments leading from Aristotelian, non-mereological metaphysical hylomorphism to mereological logical hylomorphism (LHAWKI). I conclude with a reassessment of the prospects for the combination of hylomorphism and logic, arguing in particular that LHAWKI is not the only and certainly not the most suitable version of logical hylomorphism. In particular, this implies that the project of demarcating the class of logical constants as a means to define the scope and nature of logic rests on highly problematic assumptions.     

Design and its discontents

The design argument was rebutted by David Hume. He argued that the world and its contents (such as organisms) were not analogous to human artifacts. Hume further suggested that there were equally plausible alternatives to design to explain the organized complexity of the cosmos, such as random processes in multiple universes, or that matter could have inherent properties to self-organize, absent any external crafting. William Paley, writing after Hume, argued that the functional complexity of living beings, however, defied naturalistic explanations. In effect he dared anyone to come up with an alternative to his inference to design, and hence a designer, outside of nature. Charles Darwin explained the apparent design of functional complexity by his theory of natural selection. Asa Gray, however, in essays as well as in correspondence with Darwin argued that natural selection allowed for a type of ‘evolutionary teleology’ in which design at most could be considered the result of universal principles. F.E. Hicks updated Hume by specifically objecting to the use of design arguments by Paley. Hicks argued that the apparent design seen in nature reflected order at a deep level in nature. The design argument was briefly revived by Lawrence Henderson early in the twentieth century but he ultimately concluded that design and teleology were not necessarily mutually entailing and he retracted his design argument in favor of one that he termed ‘natural teleology’. The current claims of ‘intelligent design’ have the same logical problems that have beset previous design arguments. If design is divorced from teleology and its discontents put behind us, then there is a possibility that the latter can have a place in the development of theories to explain the phenomena of emergent complexity.     

Logical and Epistemic Foundationalism About Grounding: The Triviality of Facts and Principles

In this paper, I seek to undermine G.A. Cohen’s polemical use of a metaethical claim he makes in his article, ‘Facts and Principles’, by arguing that that use requires an unsustainable equivocation between epistemic and logical grounding. I begin by distinguishing three theses that Cohen has offered during the course of his critique of Rawls and contractualism more generally, the foundationalism about grounding thesis, the justice as non-regulative thesis, and the justice as all-encompassing thesis, and briefly argue that they are analytically independent of each other. I then offer an outline of the foundationalism about grounding thesis, characterising it, as Cohen does, as a demand of logic. That thesis claims that whenever a normative principle is dependent on a fact, it is so dependent in virtue of some other principle. I then argue that although this is true as a matter of logic, it, as Cohen admits, cannot be true of actual justifications, since logic cannot tell us anything about the truth as opposed to the validity of arguments. Facts about a justification cannot then be decisive for whether or not a given argument violates the foundationalism about grounding thesis. As long as, independently of actual justifications, theorists can point to plausible logically grounding principles, as I argue contractualists can, Cohen’s thesis lacks critical bite.     

Yet Another Run around the Circle

In a recent article, D. A. Truncellito (2004, ‘Running in Circles about Begging the Question’, Argumentation 18, 325–329) argues that the discussion between Robinson (1971, ‘Begging the Question’, Analysis 31, 113–117), Sorensen (1996, ‘Unbeggable Questions’, Analysis 56, 51–55) and Teng (1997, ‘Sorensen on Begging the Question’, Analysis 57, 220–222) shows that we need to distinguish between logical fallacies, which are mistakes in the form of the argument, and rhetorical fallacies, which are mistakes committed by the arguer. While I basically agree with Truncellito’s line of thinking, I believe this distinction is not tenable and offer a different view. In addition, I will argue that the conclusion to draw from the abovementioned discussion is that validity is not a sufficient criterion of begging the question, and that we should be wary of the containment-metaphor of a deductive argument.     

A defense of contingent logical truths

A formula is a contingent logical truth when it is true in every model M but, for some model M, false at some world of M. We argue that there are such truths, given the logic of actuality. Our argument turns on defending Tarski’s definition of truth and logical truth, extended so as to apply to modal languages with an actuality operator. We argue that this extension is the philosophically proper account of validity. We counter recent arguments to the contrary presented in Hanson’s ‘Actuality, Necessity, and Logical Truth’ (Philos Stud 130:437–459, 2006).     

Dynamic Deontic Logic and its Paradoxes

In Meyer’s promising account [7] deontic logic is reduced to a dynamic logic. Meyer claims that with his account “we get rid of most (if not all) of the nasty paradoxes that have plagued traditional deontic logic.” But as was shown by van der Meyden in [4], Meyer’s logic also contains a paradoxical formula. In this paper we will show that another paradox can be proven, one which also effects Meyer’s “solution” to contrary to duty obligations and his logic in general. Presented by Hannes Leitgeb     

Rule-following, ideal conditions and finkish dispositions

This paper employs some outcomes (for the most part due to David Lewis) of the contemporary debate on the metaphysics of dispositions to evaluate those dispositional analyses of meaning that make use of the concept of a disposition in ideal conditions. The first section of the paper explains why one may find appealing the notion of an ideal-condition dispositional analysis of meaning and argues that Saul Kripke’s well-known argument against such analyses is wanting. The second section focuses on Lewis’ work in the metaphysics of dispositions in order to call attention to some intuitions about the nature of dispositions that we all seem to share. In particular, I stress the role of what I call ‘Actuality Constraint’. The third section of the paper maintains that the Actuality Constraint can be used to show that the dispositions with which ideal-condition dispositional analyses identify my meaning addition by ‘+’ do not exist (in so doing, I develop a suggestion put forward by Paul Boghossian). This immediately implies that ideal-condition dispositional analyses of meaning cannot work. The last section discusses a possible objection to my argument. The point of the objection is that the argument depends on an illicit assumption. I show (1) that, in fact, the assumption in question is far from illicit and (2) that even without this assumption it is possible to argue that the dispositions with which ideal-condition dispositional analyses identify my meaning addition by ‘+’ do not exist.