Three experiments are reported which show that in certain contexts subjects reject instances of the valid modus ponens and modus tollens inference form in conditional arguments. For example, when a conditional premise, such as: If she meets her friend then she will go to a play, is accompanied by a conditional containing an additional requirement: If she has enough money then she will go to a play, subjects reject the inference from the categorical premise: She meets her (...) friend, to the conclusion: She will go to a play. Other contexts suppress the conditional fallacies. The first experiment demonstrates the effects of context on conditional reasoning. The second experiment shows that the inference suppression disappears when the categorical premise refers to both of the antecedents, such as: She meets her friend and she has enough money. In this case, subjects make both the valid inferences and the fallacies, regardless of the contextual information. The third experiment establishes that when subjects are given general information about the duration of a situation in which a conditional inducement was uttered, such as: If you shout then I will shoot you, they reject both the valid inferences and the fallacies. The results suggest that the interpretation of premises plays an even more central role in reasoning than has previously been admitted. Trois expériences montrent que dans certains contextes les sujets rejettent des instances valides de modus ponens et de modus tollens dans des énoncés conditionnels. Par exemple, quand une prémisse conditionnelle comme: Si elle rencontre son ami, alors elle ira jouer, est accompagnée par une phrase contenant une condition supplémentaire: Si elle a assez d'argent, alors elle ira jouer, les sujets rejettent l'inférence liant la prémisse catégorielle: Elle rencontre son ami, à la conclusion: Elle ira jouer. D'autres contextes éliminent également les inférences conditionnelles fallacieuses. La premiére expérience démontre l'effet du contexte sur la raisonnement conditionnel. La deuxiéme expérience montre que la suppression de l'inférence disparait quand la prémisse catégorielle fait référence aux deux antécédants, comme: Elle rencontre son ami et elle a assez d'argent. Dans ce cas, les sujets font aussi bien les inférences valides et fallacieuses, quelle que soit l'information contextuelle. La troisiéme expérience établit que quand l'on donne aux sujets des informations générales à propos de la durée d'une situation dans laquelle une conditionnelle a été prononcée, comme Si tu cries, alors je te tire dessus, ils rejettent autant les inférences valides que les inférences fallacieuses. Ces résultats suggèrent que l'interprétation des prémisses joue un rŏle encore plus central dans le raisonnement que l'on admettait auparavant. (shrink)

Participants were asked to draw inferences about correlation from single x,y observations. In Experiment 1 statistically sophisticated participants were given the univariate characteristics of distributions of x and y and asked to infer whether a single x, y observation came from a correlated or an uncorrelated population. In Experiment 2, students with a variety of statistical backgrounds assigned posterior probabilities to five possible populations based on single x, y observations, again given knowledge of the univariate statistics. In Experiment 3, statistically (...) naïve participants were given a problem analogous to that given in Experiment 1, framed verbally. Experiment 4 replicated Experiment 3 but added an “impossible to determine” response option. Models that rely on computing sample correlations make no predictions about these investigations. From a Bayesian perspective, participants’ inferences in all four experiments tended to make probabilistically valid inferences as long as the single datum was directional. The results are discussed in light of the Brunswikian notion of vicarious functioning. (shrink)

True beliefs and truth-preserving inferences are, in some sense, good beliefs and good inferences. When an inference is valid though, it is not merely truth-preserving, but truth-preserving in all cases. This motivates my question: I consider a Modus Ponens inference, and I ask what its validity in particular contributes to the explanation of why the inference is, in any sense, a good inference. I consider the question under three different definitions of ‘case’, and hence of (...) ‘validity’: the orthodox definition given in terms of interpretations or models, a metaphysical definition given in terms of possible worlds, and a substitutional definition defended by Quine. I argue that the orthodox notion is poorly suited to explain what's good about a Modus Ponens inference. I argue that there is something good that is explained by a certain kind of truth across possible worlds, but the explanation is not provided by metaphysical validity in particular; nothing of value is explained by truth across all possible worlds. Finally, I argue that the substitutional notion of validity allows us to correctly explain what is good about a valid inference. (shrink)

The problem analysed in this paper is whether we can gain knowledge by using valid inferences, and how we can explain this process from a model-theoretic perspective. According to the paradox of inference (Cohen & Nagel 1936/1998, 173), it is logically impossible for an inference to be both valid and its conclusion to possess novelty with respect to the premises. I argue in this paper that valid inference has an epistemic significance, i.e., it can (...) be used by an agent to enlarge his knowledge, and this significance can be accounted in model-theoretic terms. I will argue first that the paradox is based on an equivocation, namely, it arises because logical containment, i.e., logical implication, is identified with epistemological containment, i.e., the knowledge of the premises entails the knowledge of the conclusion. Second, I will argue that a truth-conditional theory of meaning has the necessary resources to explain the epistemic significance of valid inferences. I will explain this epistemic significance starting from Carnap’s semantic theory of meaning and Tarski’s notion of satisfaction. In this way I will counter (Prawitz 2012b)’s claim that a truth-conditional theory of meaning is not able to account the legitimacy of valid inferences, i.e., their epistemic significance. (shrink)

The traditional picture of logic takes it for granted that "valid arguments have a fundamental epistemic significance", but neither model theory nor traditional proof theory dealing with formal system has been able to give an account of this significance. Since valid arguments as usually understood do not in general have any epistemic significance, the problem is to explain how and why we can nevertheless use them sometimes to acquire knowledge. It is suggested that we should distinguish between arguments (...) and acts of inferences and that we have to reconsider the latter notion to arrive at the desired explanation. More precisely, the notions should be developed so that the following relationship holds: one gets in possession of a ground for a conclusion by inferring it from premisses for which one already has grounds, provided that the inference in question is valid. The paper proposes explications of the concepts of ground and deductively valid inference so that this relationship holds as a conceptual truth. Logical validity of inference is seen as a special case of deductive validity, but does not add anything as far as epistemic significance is concerned—it resides already in the deductively valid inferences. (shrink)

Byrne has demonstrated that although subjects can make deductively valid inferences of the modus ponens and modus tollens forms, these valid inferences can be suppressed by presenting an appropriate additional premise “If R then Q” with the original conditional “If P then Q”. This suppression effect challenges the assumption of all syntactic theories of conditional reasoning that formal rules of inference such as modus ponens is part of mental logic. This paper argues that both the syntactic and (...) the mental model accounts of the suppression effect are inadequate because they fail to give a principled account of the critical interpretive component involved in reasoning. In contrast, the relative salience model proposed in this study emphasized the centrality of the interpretative processes with the critical component being the relative salience of premises as judged by subjects on the basis of their prior knowledge activated in particular problem situations. Using 120 undergraduates and 120 policemen as subjects, predictions from the model were tested and confirmed in a suppression paradigm and evidence of convergent validity for the construct of salience were obtained. The results cannot be reconciled with either the syntactic view or the mental model view that have dominated theories of conditional reasoning. (shrink)

This chapter contains sections titled: Validity and Necessity The Validity Paradox Gödel Arithmetizing the Validity Paradox The Validity Paradox in S5 Validity, Necessity, and Deductive Inference.

The so-called materially valid inferences have come to new prominence through the work of Robert Brandom. This paper introduces a fragment of a logic of concepts that does not reduce concepts to their extensions. Concept logic and ist semantics allow us to represent the conceptual knowledge used in material inferences and thus suggests a way to deal with them.

The paper gives a tentative reconstruction of the classical theory of so called fallacious arguments. Its title refers to the following observations.One of the fallacies listed in traditional logic ispetitio principii. It seem natural to add to the list another, similar fallacy. An argumentationArg + considered as a part of a theoretical contextC commits this fallacy relatively toC, if it contains an inference such that the principle of this inference has not been proved inC. By principle of a (...) given inference the conditional is meant whose antecedent and consequent are the conjuction of all the premisses and the conclusion of the inference respectively.If the principle of a given inference has been proved in a given contextC, the inference is valid relatively toC and the premisses are implying the conclusion relatively toC. Both these concepts, of valid inference and of implication do involve the concept of an effectively performed proof; hence they are pragmatic concepts. (shrink)

We may try to explain proofs as chains of valid inference, but the concept of validity needed in such an explanation cannot be the traditional one. For an inference to be legitimate in a proof it must have sufficient epistemic power, so that the proof really justifies its final conclusion. However, the epistemic concepts used to account for this power are in their turn usually explained in terms of the concept of proof. To get out of this (...) circle we may consider an idea within intuitionism about what it is to justify the assertion of a proposition. It depends on Heyting’s view of the meaning of a proposition, but does not presuppose the concept of inference or of proof as chains of inferences. I discuss this idea and what is required in order to use it for an adequate notion of valid inference. (shrink)

The main idea that we want to defend in this paper is that the question of what a logic is should be addressed differently when structural properties enter the game. In particular, we want to support the idea according to which it is not enough to identify the set of valid inferences to characterize a logic. In other words, we will argue that two logical theories could identify the same set of validities, but not be the same logic.

By a valid inference is here understood an inference that succeeds in its aim to justify its conclusion given that its premisses are already justified. For an inference to be valid it is thus not enough that the sentence asserted in the conclusion is a logical consequence of the sentences asserted in the premisses. A proof is understood as a succession of valid inferences that is closed (i.e. all its assumptions are discharged and all (...) its free variables are bound by some of the inferences). On the other hand, in mathematics the justification of an assertion has traditionally been taken to consist in a proof of it, which gives rise to an explanatory circle.In an attempt to avoid this circle, I have in previous works tried to explain the notion of ground for an assertion without referring to valid inferences. However, it turns out that the resulting notion of ground has unwanted consequences like the notion of proof in the so-called BHK-interpretation of intuitionistic logic. For instance, one is guaranteed a ground or a proof of an implication A → B as soon as there is in fact no ground or proof of A, regardless of whether one really knows this fact, and hence without having a real justification of the assertion of the implication.In the absence of a non-circular explanation of the notions of valid inference and proof, I suggest that we should try to formulate a number of principles of how these two epistemic notions depend on each other. A more careful explanation of the validity of an inference results in what I call a heuristic idea about the validity of inference, which is used as a guide to formulate precise principles. One such principle generalizes Gentzen’s idea that his introduction rules define the meaning of the logical constants. Another principle generalizes a previously noted property of Gentzen’s elimination rules that I have called that of satisfying the inversion principle. It results in a notion that I call non-creativity, which implies validity. (shrink)

Epistemologists have known for two-and-a-half centuries that there are serious difficulties surroundingnon-demonstrativeinference. The best-known problem,theproblem of induction, was first diagnosed by Hume in theTreatise. In our own century, several more problems were added, e.g., by Hempel —the paradox of the ravens—and by Goodman —the “new,” or exacerbated, problem of induction. But an even greater blow lay ahead: within the decade after Goodman's problem appeared, Gettier was to publish his famous challenge to the traditional analysis of knowledge which, again, underscored how (...) problematic inductive inferences are. (shrink)

It has been common in contemporary logic and philosophy of logic to identify the validity of an inference with its conclusion being a (logical) consequence of its premisses.

It would seem possible in principle … to reconstruct the ordinary sentential calculus in terms of phrastics only, and then apply it to indicatives and imperatives alike simply by adding the appropriate neustics.… It might be asked how we are to know, given two premisses in different moods, in what mood the conclusion is to be. The problem of the effect upon inferences of the moods of premisses and conclusion has been ignored by logicians who have not looked beyond the (...) indicative mood, … But if, as I do, we regard the entailment-relations of ordinary logic as relations between the phrastics of sentences, the problem becomes pressing. (R. M. Hare, The Language of Morals, pp. 26–28.). (shrink)

The purpose of teaching logic in philosophy is to enable us to evaluate arguments with respect to (formal) validity. Standard logics refer to a concept of validity which allows for the relation of implication to hold between premises and conclusion even in cases where there is no “relevant” connection between the premises and the conclusion. A prominent example for this is the rule “Ex-Falso-Quodlibet” (EFQ), which allows us to infer an arbitrary proposition from a contradiction. The tolerance of irrelevance endorsed (...) by standard logics unfortunately engenders that they cannot adequately fulfill their intended task of analyzing and evaluating philosophical, scientific and everyday-life arguments – instead, their application even gives rise to a multitude of artificial philosophical pseudoproblems (like the problem of the disposition predicates or the problem of counterfactuals). As alternatives to standard logics, there exist non-standard systems called “relevance logics” or “relevant logics” meant to avoid irrelevance. The problem with these systems, however, is that the mainstream relational semantics (“worlds semantics”) available for them is to be considered unintuitive and complex to a degree which is apt to render relevant logics unattractive to the majority of philosophers who are on the lookout not only for adequate, but also simple and efficient technical means for evaluating arguments. Therefore, the main aim of this treatise is to provide an alternative semantics (“rules semantics”) which is comparatively easy to grasp and simple in application. A second aim of the book is to extend the semantics as least as far as it takes to cover more or less all the logical notions philosophers need in their “everyday analyzing”. This includes first order predicate logic, higher order logic (for analyzing talk about “properties” etc.), identity, definite descriptions, abstraction principles and modal logic. This book can be read without having any more background than a good introductory course in classical logic provides. (shrink)

An inference is valid if it guarantees the transferability of knowledge from the premisses to the conclusion. If knowledge is here understood as demonstrative knowledge, and demonstration is explained as a chain of valid inferences, we are caught in an explanatory circle. In recent lectures, Per Martin-Löf has sought to avoid the circle by specifying the notion of knowledge appealed to in the explanation of the validity of inference as knowledge of a kind weaker than demonstrative (...) knowledge. The resulting explanation is the main topic of this article. (shrink)

Inferences from the absence of evidence to something are common in ordinary speech, but when used in scientific argumentations are usually considered deficient or outright false. Yet, as demonstrated here with the help of various examples, archaeologists frequently use inferences and reasoning from absence, often allowing it a status on par with inferences from tangible evidence. This discrepancy has not been examined so far. The article analyses it drawing on philosophical discussions concerning the validity of inference from absence, using (...) probabilistic models that were originally developed to show that such inferences are weak and inconclusive. The analysis reveals that inference from absence can indeed be justified in many important situations of archaeological research, such as excavations carried out to explore the past existence and time-span of sedentary human habitation. The justification is closely related to the fact that archaeology explores the human past via its material remains. The same analysis points to instances where inference from absence can have comparable validity in other historical sciences, and to research questions in which archaeological inference from absence will be problematic or totally unwarranted. (shrink)

The purpose of this paper is to explore the issue of how the validity of the parallel inference is possible in view of its deep semantic-syntactic structure. I fi...

Thomas Hofweber takes the semantic paradoxes to motivate a radical reconceptualization of logical validity, rejecting the idea that an inference rule is valid just in case every instance thereof is necessarily truth-preserving. Rather than this “strict validity”, we should identify validity with “generic validity”, where a rule is generically valid just in case its instances are truth preserving, and where this last sentence is a generic, like “Bears are dangerous”. While sympathetic to Hofweber’s view that strict validity (...) should be replaced by something allowing for exceptions, I argue that this should instead be truth-conduciveness, a matter of a large majority of instances being truth-preserving. The fact that inference rules have uncountably many instances, I argue, can be handled by defining truth-conduciveness as relative to finite sets of instances. Moreover, I argue that Hofweber’s position, while seemingly radical, may be less so under closer scrutiny. For the mere claim that classical rules and unrestricted truth rules are generically valid (or truth-conducive) is not in itself controversial. Also, if there is an answer to the question of which of these are also strictly valid, then the claim that generic validity is the central notion in logic is at most an interest-relative matter, since there is then also a fact of the matter as to which inference rules are (not) strictly valid. If no solution operating with strict validity to the paradoxes can be had, however, then the claim that validity should be identified with a weaker notion is better motivated. I argue that it is reasonable, in view of past failures, to conjecture that no ordinary solution will score high enough relative to standard (uncontroversial) desiderata to merit justified belief. Hence, it is unknowable which solution is correct. I further argue that this is best explained by its being metaphysically indeterminate which solution is correct. If this conjecture is true, then we ought instead to think of validity as allowing exceptions, and then take the “true logic” to be one that takes both the classical rules and the truth rules to be valid, i.e., truth-conducive. This logic scores very high on the desiderata on logics, and is therefore preferable to any logic operating with strict validity. (shrink)

본 연구는 “교육학적 추론을 위한 베이지언 모델”의 형식적 타당성 및 이 모델이 갖는 교육학적 함의와 인식론적 함의에 대해 비판적으로 검토한다.BR 베이즈주의 학습이론가들에 따르면, 교육학적 목표를 가장 잘 성취하기 위해서는 “정확한 가설”(h)에 대한 학습자의 믿음을 최대화하는 “데이터”(d)를 교사가 선택해야 한다. 달리 말하면, 학생이 추측하는 문제의 가설(개념)이 교사가 목표로 하는 바로 그 가설(개념)에 최대로 가까워지게 하는 예시를 교사가 학생에게 제공해야 한다. 이를 위해서는 교사가 생산하는 “데이터의 분포”(p teacher (d｜h))가 “가설(h)에 대한 학습자의 사후 믿음”(p learner (h｜d))을 최대화하는 데이터들을 중심으로 균등하게 분포되어야 할 것이다. (...) 학생은 처음부터 문제의 개념을 소유하고 있지 않다. 학생은 교사가 제공하는 예시를 통해서 비로소 개념을 구성해야 하는 위치에 있다. 이를 위해서 우선 교사는 학생의 인지 상태를 알고 있어야 한다. 즉 학생이 어떤 데이터를 접하게 될 때, 어떤 개념을 상상할 것이라고 교사는 예측해야 한다. 결국 교사는 “p teacher (d｜h)” 와 “p learner (h｜d)”, 양자 모두를 알아야 한다. 이러한 맥락에서 교사가 활용할 수 있는 모델이 아래와 같은 방정식이다.BR 이것이 베이지언 학습이론가들이 지지하는 “교육학적 추론 모델”이다. 그들에 따르면, 이 방정식의 해는 문제의 개념(가설)을 가장 잘 전달하게 할 예시 선택을 가능하게 하는 대안이다. 그런데 이 모델이 구체적 교육 현장에서 어떻게 적용될 수 있는가를 보기 위해서, 또한 이 모델이 가지는 논리-형식적 타당성 및 교육-인식론적 함의를 이해하기 위해서, 필자는 베이지언 이론가들이 제시하는 많은 예시들 중에서 가장 단순한 예시인 “선 게임 이론”(theory of line game)을 분석하고자 한다.BR. (shrink)

This paper explores an idea of Stoic descent that is largely neglected nowadays, the idea that an argument is valid when the conditional formed by the conjunction of its premises as antecedent and its conclusion as consequent is true. As it will be argued, once some basic features of our naıve understanding of validity are properly spelled out, and a suitable account of conditionals is adopted, the equivalence between valid arguments and true conditionals makes perfect sense. The account (...) of validity outlined here, which displays one coherent way to articulate the Stoic intuition, accords with standard formal treatments of deductive validity and encompasses an independently grounded characterization of inductive validity. (shrink)

Inference versus consequence , an invited lecture at the LOGICA 1997 conference at Castle Liblice, was part of a series of articles for which I did research during a Stockholm sabbatical in the autumn of 1995. The article seems to have been fairly effective in getting its point across and addresses a topic highly germane to the Uppsala workshop. Owing to its appearance in the LOGICA Yearbook 1997 , Filosofia Publishers, Prague, 1998, it has been rather inaccessible. Accordingly it (...) is republished here with only bibliographical changes and an afterword. (shrink)

An important field of probability logic is the investigation of inference rules that propagate point probabilities or, more generally, interval probabilities from premises to conclusions. Conditional probability logic (CPL) interprets the common sense expressions of the form “if . . . , then . . . ” by conditional probabilities and not by the probability of the material implication. An inference rule is probabilistically informative if the coherent probability interval of its conclusion is not necessarily equal to the (...) unit interval [0, 1]. Not all logically valid inference rules are probabilistically informative and vice versa. The relationship between logically valid and probabilistically informative inference rules is discussed and illustrated by examples such as the modus ponens or the affirming the consequent. We propose a method to evaluate the strength of CPL inference. (shrink)

Mathematical rigor is commonly formulated by mathematicians and philosophers using the notion of proof gap: a mathematical proof is rig­orous when there is no gaps in the mathematical reasoning of the proof. Any philosophical approach to mathematical rigor along this line requires then an account of what a proof gap is. However, the notion of proof gap makes sense only relatively to a given conception of valid mathematical reasoning, i.e., to a given conception of the validity of mathematical (...) class='Hi'>inference. A proof gap can in particular be conceived as a failure in drawing a valid mathematical inference. The aim of this paper is to discuss two possible views of the validity of math­ematical inference with respect to their capacity to yield a plausible account of the intuitive notion(s) of proof gap present in mathematical practice. The first view is the one provided by the contemporary standards of mathematical rigor: a mathematical inference is valid if and only if its conclusion can be formally derived from its premises. We will argue that this conception does not lead to a plausible account of the intuitive notion(s) of proof gap. The second view is based on a new account of the validity of inference proposed by Prawitz: an inference is valid if and only if it consists in an operation that provides a ground for its conclusion given (previously obtained) grounds for its premises. We will first specify Prawitz's account to mathematical inference and we will then argue that the resulting ground-based account is able to capture various intuitive notions of proof gap as different types of failure in drawing valid mathematical inferences. We conclude that the ground-based account ap­pears of particular interest for the philosophy of mathematical practice, and we finally raise several challenges facing a full development of a ground-based account of the notions of mathematical rigor, proof gap and the validity of mathematical inference. (shrink)

ABSTRACT: A traditional objection to inferentialism states that not all inferences can be meaning-constitutive and therefore inferentialism has to comprise an analytic-synthetic distinction. As a response, Peregrin argues that meaning is a matter of inferential rules and only the subset of all the valid inferences for which there is a widely shared corrective behaviour corresponds to rules and so determines meaning. Unfortunately, Peregrin does not discuss what counts as “widely shared”. In the paper, I argue for an empirical plausibility (...) of Peregrin’s proposal. The aim of the paper is to show that we can find examples of meaning-constitutive linguistic action, which sustain Peregrin’s response. The idea is supported by examples of meaning modulation. If Peregrin is right, then we should be able to find specific meaning modulations in which a new meaning is publicly available and modulated in such a way that it has a potential to be widely shared. I believe that binding modulations – a specific type of meaning modulations – satisfy this condition. (shrink)

In this paper the informativeness account of assertion (Pagin in Assertion. Oxford University Press, Oxford, 2011) is extended to account for inference. I characterize the conclusion of an inference as asserted conditionally on the assertion of the premises. This gives a notion of conditional assertion (distinct from the standard notion related to the affirmation of conditionals). Validity and logical validity of an inference is characterized in terms of the application of method that preserves informativeness, and contrasted with (...) consequence and logical consequence, that is defined in terms of truth preservation. The proposed account is compared with that of Prawitz (Logica yearbook 2008, pp. 175-192. College Publications, London, 2009). (shrink)

Some arguments include imperative clauses. For example: ‘Buy me a drink; you can’t buy me that drink unless you go to the bar; so, go to the bar!’ How should we build a logic that predicts which of these arguments are good? Because imperatives aren’t truth apt and so don’t stand in relations of truth preservation, this technical question gives rise to a foundational one: What would be the subject matter of this logic? I argue that declaratives are used to (...) produce beliefs, imperatives are used to produce intentions, and beliefs and intentions are subject to rational requirements. An argument will strike us as valid when anyone whose mental state satisfies the premises is rationally required to satisfy the conclusion. For example, the above argument reflects the principle that it is irrational not to intend what one takes to be the necessary means to one’s intended ends. I argue that all intuitively good patterns of imperative inference can be explained using off-the-shelf formulations of our rational requirements. I then develop a formal-semantic theory embodying this view that predicts a range of data, including free-choice effects and Ross’s paradox. The resulting theory shows one way that our aspirations to rational agency can be discerned in the patterns of our speech, and is a case study in how the philosophy of language and the philosophy of action can be mutually illuminating. (shrink)

Many classically valid meta-inferences fail in a standard supervaluationist framework. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. (...) Thus, supervaluationists can provide an account of good deductive reasoning. Our proof system moreover brings to light how one can revise the standard supervaluationist framework to make room for higher-order vagueness. We prove that the resulting logic is sound and complete with respect to the consequence relation that preserves truth in a model of the non-normal modal logic _NT_. Finally, we extend our approach to a first-order setting and show that supervaluationism can treat vagueness in the same way at every order. The failure of conditional proof and other meta-inferences is a crucial ingredient in this treatment and hence should be embraced, not lamented. (shrink)

In reply to Geach's objection against expressivism, some have claimed that there is a plurality of truth predicates. I raise a difficulty for this claim: valid inferences can involve sentences assessable by any truth predicate, corresponding to 'lightweight' truth as well as to 'heavyweight' truth. To account for this, some unique truth predicate must apply to all sentences that can appear in inferences. Mixed inferences remind us of a central platitude about truth: truth is what is preserved in (...) class='Hi'>valid inferences. The question is why we should postulate truth predicates that do not satisfy this platitude. (shrink)

I first propose to distinguish between two kinds of external validity inferences, predictive and explanatory. I then argue that we have a satisfactory answer to the question of the conditions under which predictive external validity inferences are good. If this claim is correct, then it has two immediate consequences: First, some external validity inferences are deductive, contrary to what is commonly assumed. Second, Steel’s requirement that an account of external validity inference break what he calls the ‘Extrapolator’s Circle’ is (...) misplaced, at least when it comes to predictive external validity inferences. (shrink)

The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on: admissible or permissible inference rules the derivability of the admissible inference rules the structural completeness of logics the bases for admissible and valid inference rules. There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered. The book (...) is basically self-contained and special attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included. The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook. (shrink)