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)

본 논문의 목적은 19세기와 20세기의 아리스토텔레스적 논리학에서 제시된 정언삼단논법의 타당성 규칙을 설명하고 타당성 검사 테크닉을 소개하는 것이다. 필자는 우선 ‘주연 혹은 부주연의 지위에 있는 명사’라는 말이 무엇을 의미하는지를 설명하였다. 다음으로 필자는 ‘주연의 지위에 있는 명사’라는 개념이 중세 후기의 ‘분포 상정’ 이론에서 등장하는 "분포를 상정하고 있는 명사‘라는 개념의 활용에서 나온 결과물이라고 주장하는 기치의 견해와 ‘주연의 지위에 있는 명사’라는 개념은 아리스토텔레스의 의 연구에서 직접적으로 성장했으며, 분포 상정 연구의 발전과는 독립적으로 (혹은 분포 상정에 관한 연구가 시작되기 전에) 생겨났다고 주장하는 파슨즈의 견해를 살펴보았다. 이어서 (...) 필자는 정언삼단논법의 타당성 규칙들을 긍정-부정 관련 규칙과 주연-부주연 관련 규칙으로 요약하면서 각 규칙의 존립 근거를 설명하였다. 마지막으로 필자는 정언삼단논법의 타당성규칙을 구체적인 사례에 적용하는 테크닉을 소개하였다. (shrink)

In a recent contribution to legal ontology, Kenneth Ehrenberg identifies a puzzle concerning _the basic validity rule_ of legal systems: If formal institutions require a codified foundational constitutive rule, then legal systems cannot be formal institutions, since their foundational constitutive rule is necessarily an uncodified basic validity rule. To solve this puzzle, Ehrenberg suggests taking this rule as ‘a foundational and self-identifying institutional fact’. Here, I challenge his solution and the very existence of this puzzle. By (...) arguing, contra Ehrenberg, that the basic validity rule is not the foundational constitutive rule of a legal system, but only the foundational constitutive rule of legal validity; that its codification in fact contributes towards the institutionalisation of a legal system; and that the true foundational constitutive rule of a legal system is the formal group structure of the higher-constitutional organisations, I submit an alternative account of the institutional nature of legal systems. (shrink)

By “the” modal fallacy one commonly means the following argument pattern.

Can judging that an agent blamelessly broke a rule lead us to claim, paradoxically, that no rule was broken at all? Surprisingly, it can. Across seven experiments, we document and explain the phenomenon of excuse validation. We found when an agent blamelessly breaks a rule, it significantly distorts people’s description of the agent’s conduct. Roughly half of people deny that a rule was broken. The results suggest that people engage in excuse validation in order to avoid (...) indirectly blaming others for blameless transgressions. Excuse validation has implications for recent debates in normative ethics, epistemology and the philosophy of language. These debates have featured thought experiments perfectly designed to trigger excuse validation, inhibiting progress in these areas. (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)

In this paper, we propose a computational interpretation of the generalized Kreisel–Putnam rule, also known as the generalized Harrop rule or simply the Split rule, in the style of BHK semantics. We will achieve this by exploiting the Curry–Howard correspondence between formulas and types. First, we inspect the inferential behavior of the Split rule in the setting of a natural deduction system for intuitionistic propositional logic. This will guide our process of formulating an appropriate program that (...) would capture the corresponding computational content of the typed Split rule. Our investigation can also be reframed as an effort to answer the following question: is the Split rule constructively valid in the sense of BHK semantics? Our answer is positive for the Split rule as well as for its newly proposed general version called the S rule. (shrink)

Prawitz conjectured that the proof-theoretically valid logic is intuitionistic logic. Recent work on proof-theoretic validity has disproven this. In fact, it has been shown that proof-theoretic validity is not even closed under substitution. In this paper, we make a minor modification to the definition of proof-theoretic validity found in Prawitz’s 1973paper ‘Towards a foundation of a general proof theory’ and refined by Schroeder-Heister in ‘Validity concepts in proof-theoretic semantics’ (2006). We will call the new notion generalized proof-theoretic validity and (...) show that the logic of generalized proof-theoretic validity is intuitionistic logic. (shrink)

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)

In case when the criteria for the validity of interpretative directives are not clearly defined and a political dispute arises, the dispute participant may refer to such interpretative directives, the use of which will determine the content of the law and determine the outcome of the political dispute. In this way a crisis of the legal order in Poland has occurred. Therefore, the aim of this article is to draw attention to the consequences of the undefined status of the directives (...) of legal interpretation, the resulting need for legitimacy of the interpretation of the law, the role of legal views (legal doctrine) possible in this respect and the validation role of the interpretation custom in legal doctrine. The article consists of four parts. The first one describes the phenomena that have disrupted the traditional approach to the sources of law. The second part describes the traditional approach to the sources of the law. The third one proves that the traditional approach to the sources of law, in which the main role is fulfilled by normative acts, is incorrect, this part also indicates that the key role in this respect is played by spontaneous rules of law interpretation. The fourth part indicates the need to legitimize the interpretation of the law and the role of the interpretation custom in judicial doctrine. The research material includes parliamentary practice in Poland, practice of law interpretation and scientific studies on law interpretation. The research material was examined mainly using the method of linguistic analysis. (shrink)

The semantical structures called T x W frames were introduced in (Thomason, 1984) for the Ockhamist temporal-modal language, $[Unrepresented Character]_{o}$ , which consists of the usual propositional language augmented with the Priorean operators P and F and with a possibility operator ◇. However, these structures are also suitable for interpreting an extended language, $[Unrepresented Character]_{so}$ , containing a further possibility operator $\lozenge^{s}$ which expresses synchronism among possibly incompatible histories and which can thus be thought of as a cross-history 'simultaneity' operator. (...) In the present paper we provide an infinite set of axioms in $[Unrepresented Character]_{so}$ , which is shown to be strongly complete for T x W-validity. Von Kutschera (1997) contains a finite axiomatization of T x W-validity which however makes use of the Gabbay Irreflexivity Rule (Gabbay, 1981). In order to avoid using this rule, the proof presented here develops a new technique to deal with reflexive maximal consistent sets in Henkin-style constructions. (shrink)

The past decade has seen the gradual emergence of what might be called a postmodern perspective on nursing research. However, the development of a coherent postmodern critique of the modernist position has been hampered by some misunderstandings and misrepresentations of postmodern epistemology by a number of writers, leading to a fractured and distorted view of postmodern nursing research. This paper seeks to distinguish between judgemental relativist and epistemic relativist or ironist positions, and regards the latter as offering the most coherent (...) critique of modernist/(post)positivist nursing research. The writings of poststructuralist philosophers, including Barthes, Lyotard, Derrida, Foucault and Rorty are examined, and a number of criteria for a postmodern ironist concept of research validity or trustworthiness are suggested. Whilst these writers reject the idea of Method as a guarantee of valid research, they nevertheless believe that value judgements can and must be made, and turn to notions of ironism, différance, and the differend. Ultimately, the postmodern ironist reader of the research report must make a judgement without criteria, based on her own practical wisdom or ‘prudence’. (shrink)

If someone unintentionally breaks the rules, do they break the rules? In the abstract, the answer is obviously “yes.” But, surprisingly, when considering specific examples of unintentional, blameless rule-breaking, approximately half of people judge that no rule was broken. This effect, known as excuse validation, has previously been observed in American adults. Outstanding questions concern what causes excuse validation, and whether it is peculiar to American moral psychology or cross-culturally robust. The present paper studies the phenomenon cross-culturally, focusing (...) on Korean and American adults, and proposes a new explanation of why people engage in excuse validation, in terms of competing forces in human norm-psychology. The principal findings are that Americans and Koreans engaged in excuse validation at similar levels, and older adults were more likely to engage in excuse validation. (shrink)

The most influential theory of law in current analytic legal philosophy is legal positivism, which generally understands law to be a kind of institution. The most influential theory of institutions in current analytic social philosophy is that of John Searle. One would hope that the two theories are compatible, and in many ways they certainly are. But one incompatibility that still needs ironing out involves the relation of the social rule that undergirds the validity of any legal system (H.L.A. (...) Hart's rule of recognition) to Searle's notion of codification: the idea that institutions need official declarations of their constitutive rules in order to enjoy the full benefits of institutions. The incompatibility arises from the fact that, in order to do its institutional work, the basic validity rule must be codified in Searle's sense—yet, given the particular role it has in legal positivism, it may be impossible to codify in the Searlean sense. In this paper I develop the incompatibility in detail, consider and reject consigning the basic validity rule to Searle's “Background” capacities that support institutional facts, and conclude that the best route to eliminating it while doing a minimum of damage to the two theories is to make a slight emendation to Searle's theory of institutions. (shrink)

What accounts for how we know that certain rules of reasoning, such as reasoning by Modus Ponens, are valid? If our knowledge of validity must be based on some reasoning, then we seem to be committed to the legitimacy of rule-circular arguments for validity. This paper raises a new difficulty for the rule-circular account of our knowledge of validity. The source of the problem is that, contrary to traditional wisdom, a universal generalization cannot be inferred just on (...) the basis of reasoning about an arbitrary object. I argue in favor of a more sophisticated constraint on reasoning by universal generalization, one which undermines a rule-circular account of our knowledge of validity. (shrink)

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)

The revisionary approach to semantic paradox is commonly thought to have a somewhat uncomfortable corollary, viz. that, on pain of triviality, we cannot affirm that all valid arguments preserve truth (Beall2007, Beall2009, Field2008, Field2009). We show that the standard arguments for this conclusion all break down once (i) the structural rule of contraction is restricted and (ii) how the premises can be aggregated---so that they can be said to jointly entail a given conclusion---is appropriately understood. In addition, we (...) briefly rehearse some reasons for restricting structural contraction. (shrink)

One lists the distinct pairs of categorical premises formulable via only the positive terms, S,P,M, by constructing a six by six matrix obtained by pairing the six categorical P-premises, A, O, A, O, where P* ∈ {P,P′}, with the six, similar, categorical S-premises. One shows how five rules of valid syllogism, select only 15 distinct PCPs that entail logical consequences belonging to the set L+: = {A, O, A, E, O, I}. The choice of admissible LCs can be regarded (...) as a condition separated from the conditions contained in the RofVS: the usual eight PCPs that generate valid syllogisms are obtained when the only admissible LCs belong to the set L: = {A, E, O, I} and no existential imports are addressed. A 64 PCP-matrix obtains when both PCPs and LCs may contain indefinite terms—the positive, S,P,M, terms, and their complementary sets, S′,P′, M′, in the universe of discourse, U, called the negative terms. Now one can accept eight LCs: A, I, where P* ∈ {P,P′}, S* ∈ {S,S′}, and there are 32 conclusive PCPs, entailing precise, “one partitioning subset of U” LCs. The four rules of conclusive syllogisms predict the less precise LCs, left after eliminating the middle term from the exact LCs. The RofCS also predict that the other 32 PCPs of the 64 PCP-matrix are non-conclusive. The RofVS and the RofCS are generalized, and arguments are given, for also accepting as valid syllogisms the conclusive syllogisms formulable via positive terms which entail the LCs A and O. (shrink)

The article deals with the issue of initial impossibility of performance of an obligation and the influence of such impossibility of performance on the validity of the legal transaction that establishes such an obligation. The legal doctrine convincingly demonstrates that for Roman lawyers the rule Impossitionbilium nulla obligatio est merely meant that nobody can be obliged to perform something that cannot be performed; however, it did not necessarily follow that a contract establishing such an obligation was void. Modern civil (...) codifications overlook another significant feature of the Roman law – the uncontested application of the principle Impossibilium nulla obligatio est only in case of stipulatio, whereas modern contract law has its origins in the Roman law of consensual contracts. (shrink)

Several authors have argued that a version of Curry's paradox involving validity motivates rejecting the structural rule of contraction. This paper criticizes two recently suggested alternative responses to “validity Curry.” There are three salient stages in a validity Curry derivation. Rejecting contraction blocks the first, while the alternative responses focus on the second and third. I show that a distinguishing feature of validity Curry, as contrasted with more familiar forms of Curry's paradox, is that paradox arises already at the (...) first stage. Accordingly, blocking the second or third stages won't suffice for resolving the paradox. (shrink)

We argue in this note that the principles of validity need fresh understanding to explain the elements of private ordering exhibited in the vast swathes of informal markets around the world. Informal markets, by definition, lie outside formal legal systems and yet display a tenacious stability in their norms. We show that these norms are valid because they are driven by reputation rather than any higher order of law. In doing so, reputation drives efficacy into validity. By remolding Kelsen’s (...) ideas on validity into Hart’s internal point of view, we show how reputation drives efficacy into validity. We illustrate this idea through a case study on a centuries-old footwear cluster in Agra (India). (shrink)

Beall and Murzi :143–165, 2013) introduce an object-linguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules. As a consequence, they suggest that revisionary approaches to semantic paradox must be substructural. In response to Beall and Murzi, Field :1–19, 2017) has argued that naïve validity principles do not admit of a coherent reading and that, for this reason, a non-classical solution to the semantic paradoxes need not be substructural. The aim of this paper (...) is to respond to Field’s objections and to point to a coherent notion of validity which underwrites a coherent reading of Beall and Murzi’s principles: grounded validity. The notion, first introduced by Nicolai and Rossi, is a generalisation of Kripke’s notion of grounded truth, and yields an irreflexive logic. While we do not advocate the adoption of a substructural logic, we take the notion of naïve validity to be a legitimate semantic notion that points to genuine expressive limitations of fully structural revisionary approaches. (shrink)

Rule A: if it's metaphysically necessary that p, we may validly infer that no one is even partly morally responsible for the fact that p. Our principal aim in this article is to highlight the importance of this rule and to respond to two recent challenges to it. We argue that rule A is more important to contemporary theories of moral responsibility than has previously been recognized. We then consider two recent challenges to the rule and (...) argue that neither challenge successfully undermines the rule's initial appeal. (shrink)

Tarski's Undefinability of Truth Theorem comes in two versions: that no consistent theory which interprets Robinson's Arithmetic (Q) can prove all instances of the T-Scheme and hence define truth; and that no such theory, if sound, can even express truth. In this note, I prove corresponding limitative results for validity. While Peano Arithmetic already has the resources to define a predicate expressing logical validity, as Jeff Ketland has recently pointed out (2012, Validity as a primitive. Analysis 72: 421-30), no theory (...) which interprets Q closed under the standard structural rules can define nor express validity, on pain of triviality. The results put pressure on the widespread view that there is an asymmetry between truth and validity, viz. that while the former cannot be defined within the language, the latter can. I argue that Vann McGee's and Hartry Field's arguments for the asymmetry view are problematic. (shrink)

Legal reasoning settles morally pressing matters through a technique that largely bypasses open-ended moral argument. That technique makes central what certain persons validly decided in the past, for example in creating statutes, judicial resolutions, contracts, or wills. Identifying valid decisions is a lawyerly skill and, echoing legal practice, legal philosophy has paid considerable attention to validity criteria. But it has neglected to explore validity's point: whether, and if so exactly how, the special technique of validity contributes to a legal (...) system's ability to realise justice and human rights. Even the most sensible defences of the moral need for positive law have insufficiently probed this key step in the argument. This book lays bare the workings of legal validity, and shows why, and within what general limits, this technique equips legal systems with privileged tools to foster human well-being. In so doing, the book explains how power-conferring norms setting out validity criteria endow agents' intentions with 'performative' effect ('I hereby.') and can secure the requirement of justice that there be widespread convergence on specific patterns of conduct shaped through a cooperative effort. Legal meaning is accordingly systemic, resulting from the valid acts of multiple agents whose decisions can, by virtue of exercising legal power, also exercise moral power (authority). In short, the book articulates why it is morally necessary that validity does not primarily turn on moral considerations. (shrink)

This preprint was accepted for publication in Pramana – Journal of Physics. The final published version is available via the journal. Abstract: In this paper, the ontological aspects of the Many-Worlds Interpretation (MWI) of quantum mechanics are studied by employing measure theoretical methods of the ergodic theory. Before everything it is shown that the frequency interpretation of the Born rule is independent of the measurement postulate and hence, must be extended to the statistical ontology of the MWI. Then, based (...) on Birkhoff’s ergodic theorem we will prove that except for a Lebesgue null set of extremely rare parallel worlds, the Born rule will be refuted in all of the parallel worlds created in a long-term process of consecutive experiments in the MWI of quantum mechanics. This reduces the unlimited ontology of the MWI to an extremely confined but still uncountable set of possible parallel worlds in long-term processes and lets us acknowledge a pseudo-deterministic mechanism of quantum mechanics in the statistical picture of the theory. Hence, we conclude that the common understanding of the unconstrained MWI ontology, even if considered valid, cannot survive in long-term consecutive observations and must soon collapse into a more constrained interpretation of quantum mechanics. Keywords: Many-Worlds Interpretation, Born Rule, Measure Theory, Lebesgue Measure, Birkhoff’s Ergodic Theorem, Born-Vaidman Rule, Statistical Interpretation. (shrink)

Any theory of truth must find a way around Curry’s paradox, and there are well-known ways to do so. This paper concerns an apparently analogous paradox, about validity rather than truth, which JC Beall and Julien Murzi call the v-Curry. They argue that there are reasons to want a common solution to it and the standard Curry paradox, and that this rules out the solutions to the latter offered by most “naive truth theorists.” To this end they recommend a radical (...) solution to both paradoxes, involving a substructural logic, in particular, one without structural contraction. In this paper I argue that substructuralism is unnecessary. Diagnosing the “v-Curry” is complicated because of a multiplicity of readings of the principles it relies on. But these principles are not analogous to the principles of naive truth, and taken together, there is no reading of them that should have much appeal to anyone who has absorbed the morals of both the ordinary Curry paradox and the second incompleteness theorem. (shrink)

En el presente trabajo se aborda el problema fundamental que concierne a la pragmática trascendental del lenguaje referido al alcance normativo de las reglas del discurso. Luego de presentar las características fundamentales que estas reglas manifiestan en la pragmática trascendental (I), se desarrolla la discusión en torno a su alcance normativo (II). Reconstruyendo y criticando la solución de Karl-Otto Apel a esta problemática (III) se presenta esquemáticamente una propuesta de solución (IV) y finalmente se extraen algunas conclusiones que pueden ser (...) tenidas en cuenta para futuros desarrollos de esta propuesta (V). (shrink)

The idea of proof-theoretic validity originated in the work of Gentzen, when he suggested that the meaning of each logical expression was encapsulated in its introduction-rules. The idea was developed by Prawitz and Dummett, but came under attack by Prior under the soubriquet 'analytic validity'. Logical truths and logical consequences are deemed analytically valid by virtue of following, in a way which the present chapter clarifies, from the meaning of the logical constants. But different logics are based on different (...) rules, confer different meanings and so validate different theorems and consequences, some of which are arguably not true or valid at all. It seems to follow that some analytic statements are in fact false. The moral is that we must be careful what rules we adopt and what meanings we use our rules to determine. (shrink)

Machine generation of arithmetic word problems (AWPs) is challenging as these problems require the correct use of quantities and mathematical relationships among them. While state-of-the-art deep-learning (DL) models excel at generating text with language variations, the mathematical validity of generated problems often remains unchecked. Metrics such as BLEU-4, METEOR, and ROUGE-L exist to assess the language quality of generated problems, but checking the end-to-end mathematical validity of AWPs is less explored. This work focuses on transfer-case (TC)-AWPs (problems involving object transfer (...) among agents). Though we train them with a dataset of valid problems, DL systems generate valid, near-valid, and invalid problems. Near-valid cases are invalid problems that are grammatically correct but mathematically incorrect. The proposed work focuses on validity-checking of TC-AWPs and repairing the near-valid cases. Detecting valid/near-valid problems requires manual effort and is error-prone. Encoding the relevant domain knowledge as an ontology is very helpful in these tasks. We propose leveraging an extended TC-ontology, previously developed to solve TC-AWPs, for automated validity-checking and repairing near-valid problems. We construct a problem-specific representation (ontology Assertional-Box) of an auto-generated problem by leveraging a sentence-classifier and BERT language models (LMs). The training set for these LMs is problem-texts where sentence-parts are annotated with ontology class-names. The proposed approach ensures that TC-AWPs produced in the output are always valid. We also briefly discuss how our ontology-based approach can be adapted to generate TC-AWPs that contain multiple object-transfers and are guaranteed to be valid. Adopting this approach to generate other types of AWPs is interesting future work. (shrink)

The Old Nyāya believes that a cause has a causal power of some kind, and it is possible to have valid cognition of a causal event. But Nāgārjuna (2nd century) challenged the very idea of causality. Also, he attacked the concept of epistemic instruments (_pramāṇa_). Śrīharṣa (12th century) too found counterexamples to the Nyāya definition of valid cognition. These attacks raised fundamental questions about the Naiyāyika’s take on the validity of causal statements. In 14 th century, Gaṅgeśa defended (...) the Nyāya theories of causality by shifting the starting point, and redefined causality. He completely bypassed the ontological issues raised by Nāgārjuna, and sought to answer an epistemic question: Is the knowledge about a causal event possible? In this essay, I shall critically examine his answer. He adopted the Old Nyāya view that a valid cognition leads one to pragmatic success, and stated that all pragmatically successful cognitions about causal events were valid. But he was aware of the fact that success could also be accidental. His ‘inferential defeater’ has the potential to rule out cases of accidental success or epistemic luck in the inferential context, although his definition of valid cognition seems to apply to those. In this essay, I shall formulate a definition of ‘epistemic defeater’ by modifying Gaṅgeśa’s definition of ‘inferential defeater’, and shall use the new concept for modifying Gaṅgeśa’s definition of valid cognition. This modified definition of valid cognition—I shall argue—will be able to rule out most of the problematic cases cited by Śrīharṣa. (shrink)

Rejecting the Cut rule has been proposed as a strategy to avoid both the usual semantic paradoxes and the so-called v-Curry paradox. In this paper we consider if a Cut-free theory is capable of accurately representing its own notion of validity. We claim that the standard rules governing the validity predicate are too weak for this purpose and we show that although it is possible to strengthen these rules, the most obvious way of doing so brings with it a (...) serious problem: an internalized version of Cut can be proved for a Curry-like sentence. We also evaluate a number of possible ways of escaping this difficulty. (shrink)

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. These (...) practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun¬termodels are studied in formal semantics (model theory). The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole’s first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole’s symbolic derivations. It is well known, of course, that Boole’s logical analysis of compound terms (involving ‘not’, ‘and’, ‘or’, ‘except’, etc.) contributed to the enlargement of the class of propositions and arguments formally treatable in logic. The present study shows, in addition, that Boole made significant contributions to the study of deduc¬tive reasoning. He identified the role of logical axioms (as opposed to inference rules) in formal deductions, he conceived of the idea of an axiomatic deductive sys¬tem (which yields logical truths by itself and which yields consequences when ap¬plied to arbitrary premises). Nevertheless, surprisingly, Boole’s attempt to imple¬ment his idea of an axiomatic deductive system involved striking omissions: Boole does not use his own formal deductions to establish validity. Boole does give symbolic derivations, several of which are vitiated by “Boole’s Solutions Fallacy”: the fallacy of supposing that a solution to an equation is necessarily a logical consequence of the equation. This fallacy seems to have led Boole to confuse equational calculi (i.e., methods for gen-erating solutions) with deduction procedures (i.e., methods for generating consequences). The methodological confusion is closely related to the fact, shown in detail below, that Boole had adopted an unsound criterion of validity. It is also shown that Boole totally ignored the countermodel criterion of invalid¬ity. Careful examination of the text does not reveal with certainty a test for invalidity which was adopted by Boole. However, we have isolated a test that he seems to use in this way and we show that this test is ineffectual in the sense that it does not serve to identify invalid arguments. We go beyond the simple goal stated above. Besides comparing Boole’s earliest criteria of validity and invalidity with those traditionally (and still generally) employed, this paper also investigates the framework and details of THE MATHEMATICAL ANALYSIS OF LOGIC. (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)

This paper considers the classification in terms of "relative validity" of arguments that are taken to be good arguments in an informal setting, despite not being strictly or absolutely valid. It employs Timothy Smiley’s framework for relative validity in his ‘A tale of two tortoises’ (1995). Everyday arguments can rely on rules of inference that are not formally valid: such rules can be presumed to be good ones in the context, just as context can justify suppressed premises relied (...) upon in other arguments. The paper goes on to draw on Smiley’s discussion of suppressed rules and relative consequence in considering arguments involving vague predicates. In particular, it examines Delia Graff Fara’s account of relative validity within the framework of a supervaluationist theory of vagueness, arguing that Smiley’s discussion suggests objections to Fara’s account and allows us to respond to the criticisms that she levels at the supervaluationist theory of vagueness. Further features and merits of Smiley’s framework are thereby revealed. (shrink)

Purpose of this article is to assess the validity of the Razian conception of the rule of law by subjecting it to the acid test of Michel Troper's 'realist theory of interpretation'. The author argues that, in light of the Wittgensteinian view of rule-following, a serious indeterminacy can be seen as inherent in both this conception of the rule of law and Troper's theory of interpretation.

Rules are meant to apply equally to all within their jurisdiction. However, some rules are frequently broken without consequence for most. These rules are only occasionally enforced, often at the discretion of a third-party observer. We propose that these rules—whose violations are frequent, and enforcement is rare—constitute a unique subclass of explicitly codified rules, which we call ‘phantom rules’ (e.g., proscribing jaywalking). Their apparent punishability is ambiguous and particularly susceptible to third-party motives. Across six experiments, (N = 1440) we validated (...) the existence of phantom rules and found evidence for their motivated enforcement. First, people played a modified Dictator Game with a novel frequently broken and rarely enforced rule (i.e., a phantom rule). People enforced this rule more often when the “dictator” was selfish (vs. fair) even though the rule only proscribed fractional offers (not selfishness). Then we turned to third person judgments of the U.S. legal system. We found these violations are recognizable to participants as both illegal and commonplace (Experiment 2), differentiable from violations of prototypical laws (Experiments 3) and enforced in a motivated way (Experiments 4a and 4b). Phantom rule violations (but not prototypical legal violations) are seen as more justifiably punished when the rule violator has also violated a social norm (vs. rule violation alone)—unless the motivation to punish has been satiated (Experiment 5). Phantom rules are frequently broken, codified rules. Consequently, their apparent punishability is ambiguous, and their enforcement is particularly susceptible to third party motives. (shrink)

Logical inferentialists hold that the meaning of logical operators is given by their rules of inference. Arthur Prior cast doubt on this by introducing rules for his tonk operator that would allow for the derivation of any sentence whatsoever from any sentence whatsoever. The obvious inferentialist reply was to require some constraints on the defining rules, such as conservativeness (Belnap) or harmony (Dummett). In this paper, I will defend and investigate a different constraint: rules define a classical logical operator just (...) in case they translate into an explicit definition in pure classical second-order logic. The right-hand side of this criterion will be found (i) to be philosophically principled in taking the idea of rules as definitions perfectly seriously, (ii) to explain how the semantic meaning of the operators can be determined from their rules, (iii) to be local in a similar sense as harmony, (iv) to validate the intuitionistic natural deduction rules and the intuitionistic/classical sequent calculus rules as defining the classical propositional operators while ruling out Prior’s rules for tonk, (v) to make clear why already the intuitionistic natural deduction rules define the classical meaning of logical operators so long as metavariables are interpreted as expressing classical propositions, (vi) to validate the classical natural deduction rules as analytic, (vii) to entail consistency, and, in the case of propositional operators (not quantifiers), (viii) to be decidable and (ix) to determine precisely those operators to be definable by rules that correspond to truth-functions. (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)

The first learning game to be developed to help students to develop and hone skills in constructing proofs in both the propositional and first-order predicate calculi. It comprises an autotelic (self-motivating) learning approach to assist students in developing skills and strategies of proof in the propositional and predicate calculus. The text of VALIDITY consists of a general introduction that describes earlier studies made of autotelic learning games, paying particular attention to work done at the Law School of Yale University, called (...) the ALL Project (Accelerated Learning of Logic). Following the introduction, the game of VALIDITY is described, first with reference to the propositional calculus, and then in connection with the first-order predicate calculus with identity. Sections in the text are devoted to discussions of the various rules of derivation employed in both calculi. Three appendices follow the main text; these provide a catalogue of sequents and theorems that have been proved for the propositional calculus and for the predicate calculus, and include suggestions for the classroom use of VALIDITY in university-level courses in mathematical logic. (shrink)

This thesis challenges a general prejudice against visualization in the history of logic and mathematics, by providing a semantic analysis of two graphical representation systems--a traditional Venn diagram representation system and an extension of it. While Venn diagrams have been used to solve problems in set theory and to test the validity of syllogisms in logic, they have not been considered valid proofs but heuristic tools for finding valid formal proofs. ;I present Venn diagrams which have been used (...) in logic as a formal system of representation equipped with its own syntax and semantics. I also show that Venn-I, with the rules of transformation that I specify, is sound and complete. The soundness of Venn-I refutes a worry that mathematicians and logicians have about graphical representation: Fallacies often have arisen from the misuse of diagrams in mathematical proofs, especially in geometry. However, the validity of the transformation rules assures us that the correct application of these rules will not lead to fallacies. ;I extend Venn-I so that this new system has a general way to convey disjunctive information. A new formation rule is introduced and new transformation rules are added. I call this extended system Venn-II. The semantics of Venn-II is formalized as an extension of the semantics of Venn-I. The soundness and completeness of Venn-II are also proven. I then show that Venn-II and a monadic first-order language are equivalent in their expressiveness. ;The analysis of Venn-I and Venn-II leads to interesting issues which have their analogues in other deductive systems. I conclude the thesis with a discussion of some fundamental differences between graphical systems and linguistic systems. (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)