Harmony and Autonomy in Classical Logic

Michael Dummett and Dag Prawitz have argued that a constructivist theory of meaning depends on explicating the meaning of logical constants in terms of the theory of valid inference, imposing a constraint of harmony on acceptable connectives. They argue further that classical logic, in particular, classical negation, breaks these constraints, so that classical negation, if a cogent notion at all, has a meaning going beyond what can be exhibited in its inferential use.I argue that Dummett gives a mistaken elaboration of the notion of harmony, an idea stemming from a remark of Gerhard Gentzen"s. The introduction-rules are autonomous if they are taken fully to specify the meaning of the logical constants, and the rules are harmonious if the elimination-rule draws its conclusion from just the grounds stated in the introduction-rule. The key to harmony in classical logic then lies in strengthening the theory of the conditional so that the positive logic contains the full classical theory of the conditional. This is achieved by allowing parametric formulae in the natural deduction proofs, a form of multiple-conclusion logic.     

Carroll’s Regress and the Epistemology of Logic

On an internalist account of logical inference, we are warranted in drawing conclusions from accepted premises on the basis of our knowledge of logical laws. Lewis Carroll’s regress challenges internalism by purporting to show that this kind of warrant cannot ground the move from premises to conclusion. Carroll’s regress vindicates a repudiation of internalism and leads to the espousal of a standpoint that regards our inferential practice as not being grounded on our knowledge of logical laws. Such a standpoint can take two forms. One can adopt either a broadly externalist model of inference or a sceptical stance. I will attempt, in what follows, to defend a version of internalism which is not affected by the regress. The main strategy will be to show that externalism and scepticism are not satisfying standpoints to adopt with regard to our inferential practice, and then to suggest an internalist alternative.     

逻辑常项与保守性：以tonk为例

根据逻辑推理论,逻辑常项的意义是由它的引入和消去规则确定的。普莱尔（Arthur Prior）提出的tonk对推理论构成了严重挑战。库克（Roy Cook）最近构造了一个禁止传递性的相干的逻辑系统,即Tonk-逻辑,并借助四值语义学重新定义了Tonk-后承概念,在这种概念之下,tonk的引入规则和消去规则都是有效的,同时系统还不是平凡的。本文探讨了保守性与常项的引入和消去规则的协调性之间的联系,并定义了两种较强的协调性概念,即HCU-协调性和HML-协调性概念。借助这两个概念,本文论证,tonk不是HCU-协调的也不是HML-协调的,因而它不是合法的逻辑常项,Tonk-逻辑也不是一种合法的逻辑系统。     

Logical knowledge and ordinary reasoning

This paper argues that the prominent accounts of logical knowledge have the consequence that they conflict with ordinary reasoning. On these accounts knowing a logical principle, for instance, is having a disposition to infer according to it. These accounts in particular conflict with so-called ‘reasoned change in view’, where someone does not infer according to a logical principle but revise their views instead. The paper also outlines a propositional account of logical knowledge which does not conflict with ordinary reasoning.     

Brandom on the Normativity of Meaning

Brandom's "inferentialism"–his theory that an expression's or state's contentfulness consists in its use or occurrence being governed by inferential norms–proves dubiously compatible with his own deflationary approach to underwriting the objectivity of intentional content (an approach that is one of the theory's essential presuppositions). This is because a deflationist argument, adapted from the case of truth to that of correct inference , undermines the key criterion of adequacy Brandom employs in motivating inferentialism. Once that constraint is abandoned, furthermore, Brandom is left vulnerable to the charge that his inferential norms are unavailable to play the meaning-constituting role he claims for them. Yet Brandom's account of meaning tacitly intertwines inferentialism with a separate explanatory project, one that in explaining the pragmatic significance of meaning-attributions does yield a convincing construal of the claim that the concept of meaning is a normative one.     

Moral perception,inference, and intuition

Sarah McGrath argues that moral perception has an advantage over its rivals in its ability to explain ordinary moral knowledge. I disagree. After clarifying what the moral perceptualist is and is not committed to, I argue that rival views are both more numerous and more plausible than McGrath suggests: specifically, I argue that (a) inferentialism can be defended against McGrath’s objections; (b) if her arguments against inferentialism succeed, we should accept a different rival that she neglects, intuitionism; and (c), reductive epistemologists can appeal to non-naturalist commitments to avoid McGrath’s counterexamples.

     

Acceptance, inference, and the multiple-conclusion sequent

This paper offers an interpretation of multiple-conclusion sequents as a kind of meta-inference rule: just as single-conclusion sequents represent inferences from sentences to sentences, so multiple-conclusion sequents represent a certain kind of inference from single-conclusion sequents to single-conclusion sequents. The semantics renders sound and complete the standard structural rules of reflexivity, monotonicity (or thinning), and transitivity (or cut). The paper is not the first one to attempt to account for multiple-conclusion sequents without invoking notions of truth or falsity??but unlike earlier such efforts, which have typically helped themselves to primitive notions of both acceptance and rejection, the present one makes do with the former alone. For technical reasons, the treatment is limited to sequents with non-empty succedents.     

Building bridges with words: an inferential account of ethical univocity

Abstract

Explaining genuine moral disagreement is a challenge for metaethical theories. For expressivists, this challenge comes from the plausibility of agents making seemingly univocal claims while expressing incongruent conative attitudes. I argue that metaethical inferentialism – a deflationary cousin to expressivism, which locates meaning in the inferential import of our moral assertions rather than the attitudes they express – offers a unique solution to this problem. Because inferentialism doesn’t locate the source of moral disagreements in a clash between attitudes, but instead in conflicts between the inferential import of ethical assertions, the traditional problem for expressivism can be avoided. After considering two forms of inferentialism that lead to revenge versions of the problem, I conclude by recommending that we understand the semantics of moral disagreements pragmatically: the source of univocity does not come from moral or semantic facts waiting to be described, but instead from the needs that ethical and semantic discourses answer – a solution to the problems of what we are to do and how we are to talk about it.     

The Case for Psychologism in Default and Inheritance Reasoning

Default reasoning occurs whenever the truth of the evidence available to the reasoner does not guarantee the truth of the conclusion being drawn. Despite this, one is entitled to draw the conclusion “by default” on the grounds that we have no information which would make us doubt that the inference should be drawn. It is the type of conclusion we draw in the ordinary world and ordinary situations in which we find ourselves. Formally speaking, ‘nonmonotonic reasoning’ refers to argumentation in which one uses certain information to reach a conclusion, but where it is possible that adding some further information to those very same premises could make one want to retract the original conclusion. It is easily seen that the informal notion of default reasoning manifests a type of nonmonotonic reasoning. Generally speaking, default statements are said to be true about the class of objects they describe, despite the acknowledged existence of “exceptional instances” of the class. In the absence of explicit information that an object is one of the exceptions we are enjoined to apply the default statement to the object. But further information may later tell us that the object is in fact one of the exceptions. So this is one of the points where nonmonotonicity resides in default reasoning. The informal notion has been seen as central to a number of areas of scholarly investigation, and we canvass some of them before turning our attention to its role in AI. It is because ordinary people so cleverly and effortlessly use default reasoning to solve interesting cognitive tasks that nonmonotonic formalisms were introduced into AI, and we argue that this is a form of psychologism, despite the fact that it is not usually recognized as such in AI. We close by mentioning some of the results from our empirical investigations that we believe should be incorporated into nonmonotonic formalisms.     

Inferentializing Semantics

The entire development of modern logic is characterized by various forms of confrontation of what has come to be called proof theory with what has earned the label of model theory. For a long time the widely accepted view was that while model theory captures directly what logical formalisms are about, proof theory is merely our technical means of getting some incomplete grip on this; but in recent decades the situation has altered. Not only did proof theory expand into new realms, generalizing the concept of proof in various directions; many philosophers also realized that meaning may be seen as primarily consisting in certain rules rather than in language-world links. However, the possibility of construing meaning as an inferential role is often seen as essentially compromised by the limits of proof-theoretical means. The aim of this paper is to sort out the cluster of problems besetting logical inferentialism by disentangling and clarifying one of them, namely determining the power of various inferential frameworks as measured by that of explicitly semantic ones.     

Inferentialism and Communicative Action: Robust Conceptions of Intersubjectivity

Brandom's inferentialism provides a semantics that complements Habermas's theory of communicative action without sacrificing its intersubjectivist insights. Pace Habermas, Brandom's conception of communication is robustly intersubjective. At the pragmatic level, interlocutors inherit each other's commitments and entitlements and must justify their claims when challenged; at the semantic level, anaphora show how the web of meaning is knit together, connecting expressions of the language as well as interlocutors. Finally, Habermas's thesis that there are three irreducible types of validity claim is preserved by linking claims to truth and rightness with mutually irreducible patterns of inference.     

The reference class problem is your problem too

The reference class problem arises when we want to assign a probability to a proposition (or sentence, or event) X, which may be classified in various ways, yet its probability can change depending on how it is classified. The problem is usually regarded as one specifically for the frequentist interpretation of probability and is often considered fatal to it. I argue that versions of the classical, logical, propensity and subjectivist interpretations also fall prey to their own variants of the reference class problem. Other versions of these interpretations apparently evade the problem. But I contend that they are all “no-theory” theories of probability - accounts that leave quite obscure why probability should function as a guide to life, a suitable basis for rational inference and action. The reference class problem besets those theories that are genuinely informative and that plausibly constrain our inductive reasonings and decisions. I distinguish a “metaphysical” and an “epistemological” reference class problem. I submit that we can dissolve the former problem by recognizing that probability is fundamentally a two-place notion: conditional probability is the proper primitive of probability theory. However, I concede that the epistemological problem remains.     

The fate of the direct argument and the case for incompatibilism

In this paper, I distinguish causal from logical versions of the direct argument for incompatibilism. I argue that, contrary to appearances, causal versions are better equipped to withstand an important recent challenge to the direct-argument strategy. The challenge involves arguing that support for the argument’s pivotal inference principle falls short just when it is needed most, namely when a deterministic series runs through an agent’s unimpaired deliberations. I then argue that, while there are limits to what causal versions can accomplish, they can be used to buttress the ultimacy argument, another important argument for incompatibilism.     

Traditional Logic,Modern Logic and Natural Language

In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. For traditionals the checking is local, i.e. separately for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’.     

Hard and Blind: On Wittgenstein’s Genealogical View of Logical Necessity

My main aim is to sketch a certain reading (‘genealogical’) of later Wittgenstein’s views on logical necessity. Along the way, I engage with the inferentialism currently debated in the literature on the epistemology of deductive logic.     

Ten reasons why ANOVA theory and research fail to explain attribution processes: 1. Conceptual problems

Ten reasons are proposed to explain why Kelley’s attribution theory and related questionnaire research fail to account for the cognitive processes underlying ordinary explanations of behaviour. In this first paper, five salient conceptual problems of the ANOVA model are reviewed: the inadequate analysis of behaviours and their contexts, and the inferences to which they give rise; deployment of the vague and ambiguous internal-external distinction; the normative view of people as poor social scientists; the crude analogy between ordinary reasoning and statistical procedures; and, fundamentally, the naive conceptualization of cognitive processes. Schank’s computational theory of social inference is identified as a sophisticated process model which avoids the conceptual problems inherent in the ANOVA model.     

Introducing Identity

The best-known syntactic account of the logical constants is inferentialism . Following Wittgenstein’s thought that meaning is use, inferentialists argue that meanings of expressions are given by introduction and elimination rules. This is especially plausible for the logical constants, where standard presentations divide inference rules in just this way. But not just any rules will do, as we’ve learnt from Prior’s famous example of tonk, and the usual extra constraint is harmony. Where does this leave identity? It’s usually taken as a logical constant but it doesn’t seem harmonious: standardly, the introduction rule (reflexivity) only concerns a subset of the formulas canvassed by the elimination rule (Leibniz’s law). In response, Read [5, 8] and Klev [3] amend the standard approach. We argue that both attempts fail, in part because of a misconception regarding inferentialism and identity that we aim to identify and clear up.

     

Paradoxes and Failures of Cut

This paper presents and motivates a new philosophical and logical approach to truth and semantic paradox. It begins from an inferentialist, and particularly bilateralist, theory of meaning—one which takes meaning to be constituted by assertibility and deniability conditions—and shows how the usual multiple-conclusion sequent calculus for classical logic can be given an inferentialist motivation, leaving classical model theory as of only derivative importance. The paper then uses this theory of meaning to present and motivate a logical system—ST—that conservatively extends classical logic with a fully transparent truth predicate. This system is shown to allow for classical reasoning over the full (truth-involving) vocabulary, but to be non-transitive. Some special cases where transitivity does hold are outlined. ST is also shown to give rise to a familiar sort of model for non-classical logics: Kripke fixed points on the Strong Kleene valuation scheme. Finally, to give a theory of paradoxical sentences, a distinction is drawn between two varieties of assertion and two varieties of denial. On one variety, paradoxical sentences cannot be either asserted or denied; on the other, they must be both asserted and denied. The target theory is compared favourably to more familiar related systems, and some objections are considered and responded to.     

Toward a Noninferentialist,Nonreliabilist Account of Perceptual Justification

While it may be a datum of common sense that perceptual experiences can justify beliefs, there is no clear consensus about how they can do so. According to what I call “inferentialism,” perceptual experiences can justify beliefs because perceptual experiences have propositional contents and thus can serve as reasons for belief. A critical commitment of inferentialism is that justification requires the obtaining of a nonarbitrary or nonaccidental semantic relation between justifier and justified, a requirement that I call semantic appropriateness (SA). By contrast, reliabilists reject SA and argue that perceptual experiences can justify beliefs because perceptual experiences are part of a reliable belief‐forming process. In this paper, I explore whether a commitment to SA inevitably leads to a commitment to inferentialism. This exploration is largely motivated by doubts over whether perceptual experiences have propositional contents. If those doubts prove to be well‐founded, then it seems either that perceptual experiences cannot justify beliefs or that some form of reliabilism is true. I argue that although we should take the doubts seriously, there is a way to make sense of SA that does not require inferentialism.