Various formally valid counterexamples have been adduced against the Humean dictum that one cannot derive an “ought” from an “is.” There are formal rebuttals—some very sophisticated now (e.g., Charles R. Pigden’s and Gerhard Schurz’s)—to such counterexamples. But what follows is an intuitive and informal argument against them. I maintain that it is better than these sophisticated formal defenses of the Humean dictum and that it also helps us see why it implausible to think that we can be as (...) decisive about the truth or falsity of the dictum as both the formal counterexamples or formal barriers to them purport to be. (shrink)

The paper provides a reconstruction of proof by contradiction in Kant’s pure general logic. A seemingly less-explored point of view on this topic is how apagogical proof can account for the formal truth of a judgement. Integrating the argument held by Kjosavik (2019), I intend to highlight how one can use proof by contradiction, conceived as a modus tollens, to establish the logical actuality (logical or formal truth) of a cognition. Although one might agree on the capacity of (...) the proof to prove formal falsity, the logical actuality of a judgement is assessable based on a logically grounded judgement and, as for transcendental logic, this cognitive operation has to presuppose the real possibility of an object. (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)

Formalizations in first-order logic are standardly used to represent logical forms of sentences and to show the validity of ordinary-language arguments. Since every sentence admits of a variety of formalizations, a challenge arises: why should one valid formalization suffice to show validity even if there are other, invalid, formalizations? This paper suggests an explanation with reference to criteria of adequacy which ensure that formalizations are related in a hierarchy of more or less specific formalizations. This proposal is then (...) compared with stronger criteria and assumptions, especially the idea that sentences essentially have just one logical form. (shrink)

Olympiodorus led the Platonist school of philosophy at Alexandria for several decades in the sixth century, and both Platonic and Aristotelian commentaries ascribed to him survive. During this time the school’s attitude to the teaching of Aristotelian syllogistic, originally owing something to Ammonius, changed markedly, with an early tendency to reinforce the teaching of syllogistic even in Platonist lectures giving way to a greater awareness of its limitations. The vocabulary for arguments and their construction becomes far commoner than the language (...) of syllogistic and syllogistic figures, and also of demonstration. I discuss the value of these changes for the dating of certain works, especially where the text lectured on does not demand different emphases. The commitment to argument rather than to authority continues, but a greater emphasis eventually falls on the establishment of the premises than on formal validity. (shrink)

This book features essays that investigate the nature of legal validity from the point of view of different traditions and disciplines. Validity is a fascinating and elusive characteristic of law that in itself deserves to be explored, but further investigation is made more acute and necessary by the production, nowadays, of soft law products of regulation, such as declarations, self-regulatory codes, and standardization norms. These types of rules may not exhibit the characteristics of formal law, and may (...) lack full formal validity but yet may have a very real impact on people's lives. The essays focus on the structural properties of hard and soft legal phenomena and the basis of their validity. Some propose to redefine validity: to allow for multiple concepts instead of one and/or to allow for a gradual concept of validity. Others seek to analyze the new situation by linking it to familiar historical debates and well-established theories of law. In addition, coverage looks at the functions of validity itself. The discussion considers both international law as well as domestic law arrangements. What does it mean to say that something is valid? Should we discard validity as the determining aspect of law? If so, what does this mean for our concept of law? Should we differentiate between kinds of validity? Or, can we say that rules can be "more" or "less" valid? After reading this book, practitioners, scholars and students will have a nuanced understanding of these questions and more. Chapter 6 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com. (shrink)

A natural language argument may be valid in at least two nonequivalent senses: it may be interpretationally or representationally valid (Etchemendy in The concept of logical consequence. Harvard University Press, Cambridge, 1990). Interpretational and representational validity can both be formally exhibited by classical first-order logic. However, as these two notions of informal validity differ extensionally and first-order logic fixes one determinate extension for the notion of formal validity (or consequence), some arguments must be formalized by unrelated (...) nonequivalent formalizations in order to formally account for their interpretational or representational validity, respectively. As a consequence, arguments must be formalized subject to different criteria of adequate formalization depending on which variant of informal validity is to be revealed. This paper develops different criteria that formalizations of an argument have to satisfy in order to exhibit the latter’s interpretational or representational validity. (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)

A formula is (materially) valid iff all its instances are true sentences; and an axiomatic system is called (materially) sound and complete iff it proves all and only valid formulas. These are 'natural' concepts of validity and completeness, which were, however, in the course of the history of modern logic, stealthily replaced by their formal descendants: formal validity and completeness. A formula is formally valid iff it is true under all interpretations in all universes; and an (...) axiomatic system is called formally sound and complete iff it proves all and only formulas valid in this sense. Though the step from material to formal validity and completeness may seem to be merely an unproblematic case of explication, I argue that it is not; and that mistaking the latter concepts for the former ones may lead to serious conceptual confusions. (shrink)

This thesis consists of a collection of five papers on naturalized formal epistemology of uncertain reasoning. In all papers I apply coherence based probability logic to make fundamental epistemological questions precise and propose new solutions to old problems. I investigate the rational evaluation of uncertain arguments, develop a new measure of argument strength, and explore the semantics of uncertain indicative conditionals. Specifically, I study formally and empirically the semantics of negated apparently selfcontradictory conditionals (Aristotle’s theses), resolve a number of (...) paradoxes of the material conditional in a purely semantical way without employing pragmatics and investigate the psychological plausibility of the proposed semantics. Moreover, I defend the formalization of defeasible inferences within a probabilistic framework of nonmonotonic reasoning and empirically justify the formalizations by a series of psychological experiments. I investigate general properties of uncertain argument forms and the interrelations among logical validity, Adams’ p-validity and probabilistic informativeness. (shrink)

This article introduces a novel approach for the analysis of the dynamics of reasoning processes and explores its applicability for the reasoning pattern called reasoning by assumption. More specifically, for a case study in the domain of a Master Mind game, it is shown how empirical human reasoning traces can be formalized and automatically analyzed against dynamic properties they fulfill. To this end, for the pattern of reasoning by assumption a variety of dynamic properties have been specified, some of which (...) are considered characteristic for the reasoning pattern, whereas some other properties can be used to discriminate among different approaches to the reasoning. These properties have been automatically checked for the traces acquired in experiments undertaken. The approach turned out to be beneficial from two perspectives. First, checking characteristic properties contributes to the empirical validation of a theory on reasoning by assumption. Second, checking discriminating properties allows the analyst to identify different classes of human reasoners. (shrink)

There is a long-standing debate whether propositions, sentences, statements or utterances provide an answer to the question of what objects logical formulas stand for. Based on the traditional understanding of logic as a science of valid arguments, this question is firstly framed more exactly, making explicit that it calls not only for identifying some class of objects, but also for explaining their relationship to ordinary language utterances. It is then argued that there are strong arguments against the proposals commonly put (...) forward in the debate. The core of the problem is that an informative account of the objects formulas stand for presupposes a theory of formalization; that is, a theory that explains what formulas may adequately substitute for an inference in proofs of validity. Although such theories are still subject to research, some consequences can be drawn from an analysis of the reasons why the common accounts featuring sentences, propositions or utterances fail. Theories of formalization cannot refer to utterances qua expressions of propositions; instead they may refer to sentences and rely on additional information about linguistic structure and pragmatic context. (shrink)

Since the crisis of Fordism, capitalism has been characterised by the ever more central role of knowledge and the rise of the cognitive dimensions of labour. This is not to say that the centrality of knowledge to capitalism is new per se. Rather, the question we must ask is to what extent we can speak of a new role for knowledge and, more importantly, its relationship with transformations in the capital/labour relation. From this perspective, the paper highlights the continuing (...) class='Hi'>validity of Marx's analysis of the knowledge/power relation in the development of the division of labour. More precisely, we are concerned with the theoretical and heuristic value of the concepts of formal subsumption, real subsumption and general intellect for any interpretation of the present change of the capital/labour relation in cognitive capitalism. In this way, we show the originality of the general intellect hypothesis as a sublation of real subsumption. Finally, the article summarises key contradictions and new forms of antagonism in cognitive capitalism. (shrink)

This article identifies problems with regard to providing criteria that regulate the matching of logical formulae and natural language. We then take on to solve these problems by defining a necessary and sufficient criterion of adequate formalization. On the basis of this criterion we argue that logic should not be seen as an ars iudicandi capable of evaluating the validity or invalidity of informal arguments, but as an ars explicandi that renders transparent the formal structure of informal reasoning.

"Logic", one of the central words in Western intellectual history, compre hends in its meaning such diverse things as the Aristotelian syllogistic, the scholastic art of disputation, the transcendental logic of the Kantian critique, the dialectical logic of Hegel, and the mathematical logic of the Principia Mathematica of Whitehead and Russell. The term "Formal Logic", following Kant is generally used to distinguish formal logical reasonings, precisely as formal, from the remaining universal truths based on reason. (Cf. SCHOLZ, (...) 1931). A text-book example of a formal-logical inference which from "Some men are philosophers" and "All philosophers are wise" concludes that "Some men are wise" is called formal, because the validity of this inference depends only on the form ofthe given sentences -in particular it does not depend on the truth or falsity of these sentences. (On the dependence of logic on natural language, English, for example, compare Section 1 and 8). The form of a sentence like "Some men are philosophers", is that which remains preserved when the given predicates, here "men" and "philosophers" are replaced by arbitrary ones. The form itself can thus be represented by replacing the given predicates by variables. Variables are signs devoid of meaning, which may serve merely to indicate the place where meaningful constants (here the predicates) are to be inserted. As variables we shall use - as did Aristotle - letters, say P, Q and R, as variables for predicates. (shrink)

This article argues for the formal validity of and the truth of the premises and conclusion of a version of Aquinas' "Third Way" that says: If each of the parts of nature is contingent, the whole of nature is contingent. Each of the parts of nature is contingent. Therefore, the whole of nature is contingent--where "contingent" means having a cause and not existing self-sufficiently.

This dissertation constitutes an inquiry into the formal aspects of a particular form of conceptual intentional realism: Holistic Realist Ramified Conceptualism. Several axiomatic systems, which have this theory as their philosophical background are developed and/or studied from a syntactical and semantical point of view. ;Among systems studied are Cocchiarella's RRC$\sbsp{\lambda}{\*}$ and HRRC$\sbsp{\lambda}{\*}$. A set theoretical semantics for these systems is developed. Also, completeness theorems with respect to certain extensions of RRC$\sbsp{\lambda}{\*}$ and HRRC$\sbsp{\lambda}{\*}$ and certain notions of validity related (...) to that semantics are proved. A relative consistency proof of RRC$\sbsp{\lambda}{\*}$, HRRC$\sbsp{\lambda}{\*}$ and other systems is given. ;An axiomatic system for identity free realist ramified conceptualism involving indiscernibility relative to constructive concepts is developed. A set theoretical semantics for this system is constructed. A completeness theorem with respect to certain extensions of the identity-free system and a notion of validity related to that semantics is proved. ;Epistemic considerations are brought into this inquiry by introducing an epistemic operator into the logical syntax. Such an operator corresponds intuitively to the idea of constructive knowability. A philosophical discussion of this idea is presented and on the basis of this discussion an epistemic logical theory is formulated. It is shown that this system has HRRC$\sbsp{\lambda}{\*}$ as a proper subsystem. On the basis of the epistemic logical theory, an axiomatic system is introduced from which Peano's arithmetic is deduced. In this way, it is shown that arithmetic can be developed axiomatically on conceptualist foundations. (shrink)

While the philosophical literature has extensively studied how decisions relate to arguments, reasons and justifications, decision theory almost entirely ignores the latter notions. In this article, we elaborate a formal framework to introduce in decision theory the stance that decision-makers take towards arguments and counter-arguments. We start from a decision situation, where an individual requests decision support. We formally define, as a commendable basis for decision-aid, this individual’s deliberated judgment, a notion inspired by Rawls’ contributions to the philosophical literature, (...) and embodying the requirement that the decision-maker should carefully examine arguments and counter-arguments. We explain how models of deliberated judgment can be validated empirically. We then identify conditions upon which the existence of a valid model can be taken for granted, and analyze how these conditions can be relaxed. We then explore the significance of our framework for the practice of decision analysis. Our framework opens avenues for future research involving both philosophy and decision theory, as well as empirical implementations. (shrink)

This dissertation is focused on the validity of “the data” that are collected in psychotherapy research for the purpose of evidencing treatment efficacy. In the ‘Evidence Based Treatment’ paradigm, researchers rely on the so-called ‘gold standard methodology’ to gather sound and trustworthy evidence, which increasingly influences the organization of mental health care worldwide. In the gold standard, data are collected by quantified self-report measures, to assess the presence and severity of symptoms before and after treatment. When the pre-post difference (...) is larger for a group of people that received the treatment of interest and a group of people who received no or an alternative treatment, the treatment of interest is called effective. In this methodological procedure, researchers tend to assume that when this gold standard methodology is conducted properly, “the data” will speak for themselves. However, when evidence is based on data, that evidence is principally limited by that data. In other words: output depends on input. That implies that when input is flawed, output will be flawed, even when the very best methods of analysis would be used. So what if “the data” yield validity issues despite being collected by validated measures? When “the data” do not straightforwardly evidence treatment effects, will the subsequent steps in the analysis of these data be enough to secure that the evidence in the end does evidence treatment efficacy? In this dissertation, the focus is turned to validity of “the data” that are concretely provided by patient-participants by scoring their own experienced symptoms in self-report questionnaires, in function of evidencing treatment efficacy. A series of empirical case studies were conducted to scrutinize how patient-participants in a randomized controlled psychotherapy study experienced the process of data collection, and how these experiences affected the data that they provided. Each of the studied patient-participants experienced a substantial effect of the questionnaire administration on their complaints. This impacted the level of complaints, but also changed the way in which the complaints were understood at all, and how patient-participants perceived themselves. Thus, rather than neutrally measuring symptoms, questionnaire administration changed the experienced complaints ‘performatively’, which turned measurement into a clinical intervention of its own. Consequently, what is measured cannot straightforwardly be called ‘treatment efficacy’, as it may be entangled with, enabled by or even obstructed by effects of measurement and research. Therefore, the act of measurement itself can pose a vital threat to the validity of “the data”. This way, the empirical case studies exhibited that data can yield validity problems despite the use of validated measures. Consequently, the validity of a measure as such is no guarantee for the validity of data. Nonetheless, in gold standard research, the data are straightforwardly taken as input for analyses of general treatment efficacy. The question is what happens with those validity issues on the level of individual data, when they pursue their journey towards becoming evidence. In this dissertation, it was argued that the validity issues are not sufficiently solved in the methodological steps after data collection, so when data are invalid in the beginning, these validity issues will simply become part of the data set that forms the input for analysis of the final evidence. This urges that validity issues should be solved on the level of individually provided data, as the validity issues will otherwise become inherent to “the data”. Put formally: valid data is a precondition for evidence in EBT. In conclusion, it is crucial for a sound evidence-base to scrutinize the validity of data in function of the overall goal and utility of the research. For this, it is important not to take “the data” as speaking for themselves, but to regard them as clinical narratives, which are framed in a specific format to be communicated between researcher and respondent in a research context. This emphasizes that the choice for a certain format determines what can be evidenced, so it is vital that these choices indeed allow for obtaining evidence that is useful and valid to serve the clinical goal of EBT. (shrink)

Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space and time. This new meta-methodological concept (...) is designed to cover not only “traditional” formal proofs but all kinds of proofs and inference steps, including incomplete or purported proofs. Our approach attempts to make proof-events more comprehensive to express the complete trajectory of a mathematical proof-event until the ultimate validation of the proving outcome. Thus, we advance an extended version of proof-event calculus which is built on argumentation theories designed to capture the internal and external structure of collaborative mathematical practice and highlight the relationship between proof, human reasoning, and cognitive processes. In addition, another area in which argumentation can make a significant contribution is dealing with the defeasible knowledge of the Web which is a product of its open and ubiquitous nature. This approach seems to be sufficient for the presentation of Web-based proving processes as manifested in the case of the Mini-Polymath 4 project. (shrink)

We investigate the philosophical significance of the existence of different semantic systems with respect to which a given deductive system is sound and complete. Our case study will be Corcoran's deductive system D for Aristotelian syllogistic and some of the different semantic systems for syllogistic that have been proposed in the literature. We shall prove that they are not equivalent, in spite of D being sound and complete with respect to each of them. Beyond the specific case of syllogistic, the (...) goal is to offer a general discussion of the relations between informal notions—in this case, an informal notion of deductive validity—and logical apparatuses such as deductive systems and (model-theoretic or other) semantic systems that aim at offering technical, formal accounts of informal notions. Specifically, we will be interested in Kreisel's famous 'squeezing argument'; we shall ask ourselves what a plurality of semantic systems (understood as classes of mathematical structures) may entail for the cogency of specific applications of the squeezing argument. More generally, the analysis brings to the fore the need for criteria of adequacy for semantic systems based on mathematical structures. Without such criteria, the idea that the gap between informal and technical accounts of validity can be bridged is put under pressure. (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)

Formal logic provides us with a powerful set of techniques for criticizing some arguments and showing others to be valid. These techniques are relevant to all of us with an interest in being skilful and accurate reasoners. In this highly accessible book, Peter Smith presents a guide to the fundamental aims and basic elements of formal logic. He introduces the reader to the languages of propositional and predicate logic, and then develops formal systems for evaluating arguments translated (...) into these languages, concentrating on the easily comprehensible 'tree' method. His discussion is richly illustrated with worked examples and exercises. A distinctive feature is that, alongside the formal work, there is illuminating philosophical commentary. This book will make an ideal text for a first logic course, and will provide a firm basis for further work in formal and philosophical logic. (shrink)

For whatever reason, scholars have recently reapproached the moral philosophy of Thomas Hobbes with a renewed interest in establishing its validity. Two influential interpretations have emerged, a theistic interpretation and a concep- tualistic interpretation, the former by Howard Warrender in The Political Philosophy of Hobbes, and the latter by David Gauthier in tfhe fcogic of leviathan.Both Warrender and Gauthier maintain that Hobbes's egoistic psychology invalidates his moral theory, and undertake to rescue its formal validity by regrounding the (...) theory on his theology, on the one hand, and on his methodological (rather than metaphysical) materialism, on the other. The result in both instances is a piecemeal analysis that dissolves the political realism for which Hobbes was famous, and ignores altogether the comprehensive intentions which he so carefully expressed. Hobbes takes on the appearance of something that might be best described as a pre-Kantian Kant. (shrink)

Given the frequency of human error, it seems rational to believe that some of our own rational beliefs are false. This is the axiom of epistemic modesty. Unfortunately, using standard propositional quantification, and the usual relational semantics, this axiom is semantically inconsistent with a common logic for rational belief, namely KD45. Here we explore two alternative semantics for KD45 and the axiom of epistemic modesty. The first uses the usual relational semantics and bisimulation quantifiers. The second uses a topological semantics (...) and standard propositional quantification. We show the two different semantics validate many of the same formulas, though we do not know whether they validate exactly the same formulas. Along the way we address various philosophical concerns. (shrink)

Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: an argument is defined to be logically valid if and only if there is no substitution instance with true premises and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or (...) not. It is shown that the resulting notions of logical consequence are extensionally equivalent to versions of first-order provability and model-theoretic consequence. Every model-theoretic interpretation has a substitutional counterpart, but not vice versa. In particular, in contrast to the model-theoretic account, there is a trivial intended interpretation on the substitutional account, namely, the homophonic interpretation that does not substitute anything. Applications to free logic, and theories and languages other than set theory are sketched. (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)

In this paper I argue against the commonly received view that Kripke's formal Possible World Semantics (PWS) reflects the adoption of a metaphysical interpretation of the modal operators. I consider in detail Kripke's three main innovations vis-à-vis Carnap's PWS: a new view of the worlds, variable domains of quantification, and the adoption of a notion of universal validity. I argue that all these changes are driven by the natural technical development of the model theory and its related notion (...) of validity: they are dictated by merely formal considerations, not interpretive concerns. I conclude that Kripke's model theoretic semantics does not induce a metaphysical reading of necessity, and is formally adequate independently of the specific interpretation of the modal operators. (shrink)

The aim of this paper is to see whether Robinson and Kajiyama’s critiques of Nāgārjuna’s discourse of catuṣkoṭi, as contradicting formal logic, while following a dialectical formula is plausible. According to them, the 3rd koṭi is a violation of the law of non-contradiction, while the 4th koṭi, a violation of the law of excluded middle. Yet, since catuṣkoṭi can be interpreted as containing different perspectives in its expression of each koṭi, the critique of violating the law of non-contradiction fails. (...) Further, should a proper presupposition be added, the problem of excluded middle is resolved. Owing to the presence of this premise postulated as a real entity possessing intrinsic nature, it becomes perfectly valid, irrespective of all-out refutation of every option, In this context, catuṣkoṭi can be seen as an ‘expedient’, representing different views of separate schools or perspectives without violating the principle of formal logic, nor postulating any hierarchical dialectics elevating into the higher grades among these koṭis. Likewise, catuṣkoṭi in Nāgārjuna can be viewed as revealing the absurdity of postulating any identity or precondition for causal relations in perceiving intrinsic substance. (shrink)

The Oxford Calculator Roger Swyneshed put forward three provocative claims in his treatise on insolubles, written in the early 1330s, of which the second states that there is a formally valid inference with true premises and false conclusion. His example deployed the Liar paradox as the conclusion of the inference: ‘The conclusion of this inference is false, so this conclusion is false’. His account of insolubles supported his claim that the conclusion is false, and so the premise, referring to the (...) conclusion, would seem to be true. But what is his account of validity that can allow true premises to lead to a false conclusion? This article considers Roger’s own account, as well as that of Paul of Venice, writing some sixty years later, whose account of the truth and falsehood of insolubles followed Roger’s closely. Paul endorsed Roger’s three claims. But their accounts of validity were different. The question is whether these accounts are coherent and support Paul’s claim in his Logica Magna that he endorsed all the normal rules of inference. (shrink)

Central topics in medieval logic are here treated in a way that is congenial to the modern reader, without compromising historical reliability. The achievements of medieval logic are made available to a wider philosophical public then the medievalists themselves. The three genres of logica moderna arising in a later Middle Ages are covered: obligations, insolubles and consequences - the first time these have been treated in such a unified way. The articles on obligations look at the role of logical consistence (...) in medieval disputation techniques. Those on insolubles concentrate on medieval solutions to the Liar Paradox. There is also a systematic account of how medieval authors described the logical content of an inference, and how they thought that the validity of an inference could be guaranteed. (shrink)

Several recent results bring into focus the superintuitionistic nature of most notions of proof-theoretic validity, but little work has been done evaluating the consequences of these results. Proof-theoretic validity claims to offer a formal explication of how inferences follow from the definitions of logic connectives (which are defined by their introduction rules). This paper explores whether the new results undermine this claim. It is argued that, while the formal results are worrying, superintuitionistic inferences are valid because (...) the treatments of atomic formulas are insufficiently general, and a resolution to this issue is proposed. (shrink)

Biomarkers can be powerful tools to guide diagnosis, treatment, and research. However, prudent use of biomarkers requires formal validation efforts. Although the data needed for biomarker validation has traditionally been hard to access, new research initiatives can ease this process.

This paper claims that there is a plausible sense in which validity is a matter of truth preservation relative to interpretations of the sentences that occur in an argument, although it is not the sense one might have in mind. §1 outlines three independent problems: the first is the paradox of the sorites, the second concerns the fallacy of equivocation, and the third arises in connection with the standard treatment of indexicals. §2 elucidates the claim about validity, while (...) §§3-5 show how the three problems outlined can be handled in accordance with it. §6 explains how the claim squares with the traditional idea that validity is related to formality, and in particular with a broadly accepted definition based on that idea, the model-theoretic definition of logical consequence. Unlike other works on the subject, this paper does not focus on necessity. It is not its intention to provide a characterization of necessity that conforms to some ideal of rigour or to some pre-theoretical understanding of validity. What follows can be taken as conditional on the assumption that such a characterization can be provided. (shrink)

In this paper we make some observations about Natural Deduction derivations [Prawitz, 1965, van Dalen, 1986, Bell and Machover, 1977]. We assume the reader is familiar with it and with proof-theory in general. Our development will be simple, even simple-minded, and concrete. However, it will also be evident that general ideas motivate our examples, and we think both our specific examples and the ideas behind them are interesting and may be useful to some readers. In a sentence, the bare technical (...) content of this paper is: Extending natural deduction with global well-formedness conditions can neatly and cheaply capture classical and intermediate logics. The interest here is in the ‘neatly’ and ‘cheaply’. By ‘neatly’ we mean ‘preserving proof-normalisation’,1 and ‘maintaining the subformula property’, and by ‘cheaply’ we mean ‘preserving the formal structure of deductions’ (so that a deduction in the original system is still, formally, a deduction in the extended system, and in particular it requires no extra effort to write just because it is in the extended system). To illustrate what we have in mind consider intuitionistic first-order logic (FOL) [van Dalen, 1986] as a paradigmatic example of a formal notion of deduction. A natural deduction derivation (or deduction) is an inductively defined tree structure where each node contains an instance of a formula. A deduction is valid when each successive node follows from its predecessors in accordance with some predetermined inference rules. A particular attraction of Natural Deduction is its clean and economical presentation. Here for example are deduction (fragments) proving A ∧ B from A and B, and ∀x. (P (x) ∧ Q(x)) from ∀x. P (x) and ∀x. Q(x): ∀x. P (x) (∀E). (shrink)

The interplay of philosophical ambitions and technical reality have given birth to rich and interesting approaches to explain the oft-claimed special character of mathematical and logical knowledge. Two projects stand out both for their audacity and their innovativeness. These are logicism and proof-theoretic semantics. This dissertation contains three chapters exploring the limits of these two projects. In both cases I find the formal results offer a mixed blessing to the philosophical projects. Chapter 1. Is a logicist bound to the (...) claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. I re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. I conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. Chapter 2. There have been several recent results bringing into focus the super-intuitionistic nature of most notions of proof-theoretic validity. But there has been very little work evaluating the consequences of these results. In this chapter, I explore the question of whether these results undermine the claim that proof-theoretic validity shows us which inferences follow from the meaning of the connectives when defined by their introduction rules. It is argued that the super-intuitionistic inferences are valid due to the correspondence between the treatment of the atomic formulas and more complex formulas. And so the goals of proof-theoretic validity are not undermined. Chapter 3. Prawitz (1971) conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister (2019). This chapter resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The chapter further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic.Abstract prepared by Will Stafford extracted partially from the dissertation. (shrink)

Since the time, about a century ago, when DeMorgan, Boole and Jevons, inaugurated the study of the logic of numerically definite reasoning, no one has been concerned to establish the validity rules for a very general type of numerically definite inference which is a strong analogue of the classical syllogism. The reader will readily agree that the traditional rules of syllogistic inference cannot even begin to decide whether the following proportionally quantified syllogism is a valid argument: at most 4/7 (...) p are at most 2/3 not-m and at least 3/5 s are precisely 7/8 m, hence, at most 1/3 s are p. Likely reasons, of broad methodological interest, why this readily definable type of formal inference has been neglected will presently be proposed. Among the masters of twentieth century logic, Hilbert and Bernays have, to be sure, meticulously shown how such numerically definite propositions as “at least 7 individuals have the property p” can be expressed in the predicate calculus with the use of the identity relation. They do not, since their concern with such propositions is indirect, proceed to establish a systematic calculus of classes whose “size” or extent is numerically definite. As we shall always notice, traditional logic and even ordinary Boolean algebra prefer to generalize about the laws governing the interrelationship of classes in inference systems without making any hypotheses concerning class “size” or extent except by the classical quantifier “some”, which quantifier admits the measure or extent of a class only in terms of the proposition that a given class contains or does not contain at least one member. Such a restricted admission of the relevance for inference of class size is among the fundamental reasons for the neglect of proportional quantifiers, i.e., quantifiers where numerical value is any rational number r such that 0/100 ≤ r ≤ 100/100. It can, of course, also be pointed out that the traditional syllogism and ordinary Boolean algebra do not i.e. measure, numerically define, the degree of inclusion of one class in another. Inclusion is taken either as total or partial, with no discrimination of the degree of partial inclusion. (shrink)

The mistake Mill made by confusing what we can desire and what deserves to be desired or must be desired has been discussed extensively in the literature. However, the meta-normative intuition that remains in the background of the argument has been less discussed. This intuition, though, aims in the right direction, as it points at the idea that the desirable emerges from the desirable. My purpose here is to defend this claim by arguing that it provides a way to avoid (...) some of the classical objections to utilitarianism. (shrink)

According to the ideological surround model of research, a more “objective” psychology of religion requires efforts to bring etic social scientific and emic religious perspectives into formal dialog. This study of 245 Iranian university students illustrated how the dialogical validity of widely used etic measures of religion can be assessed by examining an emic religious perspective on psychology. Integrative Self-Knowledge and Self-Control Scales recorded two aspects of the “Perfect Man” as described by the Iranian Muslim philosopher Mortazā Motahharī. (...) Use of these instruments in correlation and multiple regression procedures identified Intrinsic, Extrinsic Personal, Religious Interpretation, Extrovertive Mysticism, Prayer Fulfillment, Universality, Connectedness, and Religiosity Scales as adaptive in their implications for a Muslim psychology of religion. Religious Crisis had maladaptive and Extrinsic Social, Introvertive Mysticism, and Quest Scales had ambiguous implications. These data illustrated how etic forms of understanding can clarify and can be clarified by emic insights. (shrink)

In jurisprudential literature, the adjective ‘defeasible’ appears as a predicate of many terms: concepts, laws, rules, reasoning, justification, proof, and so on. In this paper, we analyze the effects of some versions of the thesis of the defeasibility of legal norms on the reconstruction of the notion of legal validity. We analyze some possible justifications of this thesis considered as a claim concerning validity, and enquire into two possible sets of problems related to the defeasibility of the criteria (...) of identification of a legal system. We also provide a formalization of some options regarding defeasible criteria of identification, which can be used as a tool for meta-jurisprudential analysis. Finally, the thesis according to which defeasibility is better conceived of as a feature of legal application is examined and questioned. (shrink)

This is a thesis in formal semantics. It consists of two parts corresponding to the distinction, due to Richard Montague, between universal grammar and specific semantic theories. The first part concerns universal grammar and is intended to provide a precise and unified conceptual framework within which different theories of formal semantics can be represented and compared. ;The second part of the thesis is concerned with intensional logic, i.e., with the logical analysis of discourse involving so called oblique contexts. (...) These contexts are characterized by the apparent failure of certain classically valid principles of substitutivity. Two kinds of reactions among logicians to the appearance of oblique contexts in natural languages are distinguished: (1) the intensional point of view, according to which oblique contexts are seen as genuine counterexamples to classical logic and to the related semantical principle of extensionality: the principle that the denotation of an expression is a function of the denotations of its parts; and (2) the extensional point of view, according to which the conflict between oblique contexts and classical logic is only apparent. The extensional point of view therefore calls for analyses of oblique contexts that preserve the laws of classical logic and the principle of extensionality. In this thesis, we are concerned with the precise formulation of and comparison between different programs for the formal semantic representation of oblique contexts. In Chapter 5, we are concerned with Church's program of representing oblique contexts within a formal language whose semantics is fully extensional and based on Frege's distinction between sense and denotation. The ambiguity between the "direct" and "indirect" denotation of an expression A is avoided by letting A be replaced in oblique contexts by an expression which denotes A's ordinary sense. We provide a possible-worlds semantics for a version of Church's formal language with the following features: (i) Objectual quantification into oblique constructions is accounted for. The treatment of quantification requires for each entity x an essence of x, by which we mean a sense which determines x in those worlds in which x exists and is undefined in all other worlds. (ii) The variables range over actually existing entities only. It is allowed that entities of certain types are contingent, i.e., the same entity may exist in one world and fail to exist in other worlds. (iii) In order to combine a proper treatment of the cross-world identification of functions with varying domains, partial functions are allowed. (iv) Senses are allowed to be partially defined. For example, the sense of a proper name is undefined in a world in which the denotation of the name does not exist. (v) Senses of functions from objects of type a to objects of type b are identified with functions from senses of objects of type a to senses of objects of type b. In Chapters 6 and 7, we pursue the Russellian Alternative in intensional semantics, where a sense/denotation distinction is avoided. Apparent failures of substitutivity principles for singular terms are explained along the lines of Russell's theory of definite descriptions. Failures of substitutivity of coextensional sentences and predicates are explained by Russell's doctrine that sentences and predicate denote propositions and propositional functions , respectively. Russellian semantics is fully extensional in the sense that the Russellian denotation of a designator is a function of the Russellian denotations of its parts. In Chapter 6, a theory of Russellian propositions is developed within set theory. This theory is then applied in Chapter 7 to the task of providing a set-theoretic semantics for Russellian type theory. (shrink)

Adopting a different method from the previous scholars, this article deduces the remaining 23 valid syllogisms just taking the syllogism AEE-4 as the basic axiom. The basic idea of this study is as follows: firstly, make full use of the trichotomy structure of categorical propositions to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we deduce the remaining 23 valid categorical syllogisms by taking just one (...) syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. We hope that the results and methods will provide a good mathematical paradigm for studying other kinds of syllogistic logics, and that the project will appeal to specialists in logic, linguistic semantics, computational semantics, cognitive science and artificial intelligence. (shrink)