Harmony and modality

It is argued that the meaning of the modal connectives must be given inferentially, by the rules for the assertion of formulae containing them, and not semantically by reference to possible worlds. Further, harmony confers transparency on the inferentialist account of meaning, when the introduction-rule specifies both necessary and sufficient conditions for assertion, and the elimination-rule does no more than exhibit the consequences of the meaning so conferred. Hence, harmony is not to be identified with normalization, since the standard modal natural deduction rules, though normalizable, are not in this sense harmonious. Harmonious rules for modality have lately been formulated, using labelled deductive systems

The philosophy of logic

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Obligations, Sophisms and Insolubles

The focus of the paper is a sophism based on the proposition ‘This is Socrates’ found in a short treatise on obligational casus attributed to William Heytesbury. First, the background to the puzzle in Walter Burley’s traditional account of obligations (the responsio antiqua), and the objections and revisions made by Richard Kilvington and Roger Swyneshed, are presented. All six types of obligations described by Burley are outlined, including sit verum, the type used in the sophism. Kilvington and Swyneshed disliked the dynamic nature of the responsio antiqua, and Kilvington proposed a revision to the rules for irrelevant propositions. This allowed him to use a form of reasoning, the “disputational meta-argument”, which is incompatible with Burley’s rules. Heytesbury explicitly rejected Kilvington’s revision and the associated meta-argument. Swyneshed also revised Burley’s account of obligations, formulating the so-called responsio nova, characterised by the apparently surprising thesis that a conjunction can be denied both of whose conjuncts are granted. On closer inspection, however, his account is found to be less radical than first appears

‘Everything true will be false’: Paul of Venice’s two solutions to the insolubles

In his Quadratura, Paul of Venice considers a sophism involving time and tense which appears to show that there is a valid inference which is also invalid. His argument runs as follows: consider this inference concerning some proposition A: A will signify only that everything true will be false, so A will be false. Call this inference B. Then B is valid because the opposite of its conclusion is incompatible with its premise. In accordance with the standard doctrine of ampliation, Paul takes A to be equivalent to 'Everything that is or will be true will be false'. But he proceeds to argue that it is possible that B's premise ('A will signify only that everything true will be false') could be true and its conclusion false, so B is not only valid but also invalid. Thus A and B are the basis of an insoluble. In his Logica Parva, a self-confessedly elementary text aimed at students and not necessarily representing his own view, and in the Quadratura, Paul follows the solution found in the Logica Oxoniensis, which posits an implicit assertion of its own truth in insolubles like B. However, in the treatise on insolubles in his Logica Magna, Paul develops and endorses Swyneshed's solution, which stood out against this ''multiple-meanings'' approach in offering a solution that took insolubles at face value, meaning no more than is explicit in what they say. On this account, insolubles imply their own falsity, and that is why, in so falsifying themselves, they are false. We consider how both types of solution apply to B and how they complement each other. On both, B is valid. But on one (following Swyneshed), B has true premises and false conclusion, and contradictories can be false together; on the other (following the Logica Oxoniensis), the counterexample is rejected

Swyneshed, Paradox and the Rule of Contradictory Pairs

Roger Swyneshed, in his treatise on insolubles (logical paradoxes), dating from the early 1330s, drew three notorious corollaries of his solution. The third states that there is a contradictory pair of propositions both of which are false. This appears to contradict the Rule of Contradictory Pairs, which requires that in every such pair, one must be true and the other false. Looking back at Aristotle's treatise De Interpretatione, we find that Aristotle himself, immediately after defining the notion of a contradictory pair, gave counterexamples to the rule. Thus Swyneshed's solution to the logical paradoxes is not contrary to Aristotle's teaching, as many of Swyneshed's contemporaries claimed. Dialetheism, the contemporary claim that some propositions are both true and false, is wedded to the Rule, and in consequence divorces denial from the assertion of the contradictory negation

Harmonic inferentialism and the logic of identity

Inferentialism claims that the rules for the use of an expression express its meaning without any need to invoke meanings or denotations for them. Logical inferentialism endorses inferentialism specically for the logical constants. Harmonic inferentialism, as the term is introduced here, usually but not necessarily a subbranch of logical inferentialism, follows Gentzen in proposing that it is the introduction-rules whch give expressions their meaning and the elimination-rules should accord harmoniously with the meaning so given. It is proposed here that the logical expressions are those which can be given schematic rules that lie in a specific sort of harmony, general-elimination (ge) harmony, resulting from applying a certain operation, the ge-procedure, to produce ge-rules in accord with the meaning defined by the I-rules. Griffiths (2014) claims that identity cannot be given such rules, concluding that logical inferentialists are committed to ruling identity a non-logical expression. It is shown that the schematic rules for identity given in Read (2004), slightly amended, are indeed ge-harmonious, so confirming that identity is a logical notion.PostprintPeer reviewe

Proof-theoretic validity

This work is supported by Research Grant AH/F018398/1 (Foundations of Logical Consequence) from the Arts and Humanities Research Council, UK.The idea of proof-theoretic validity originated in the work of Gerhard Gentzen, when he suggested that the meaning of each logical expression was encapsulated in its introduction-rules, and that the elimination-rules were justified by the meaning so given. It was developed by Dag Prawitz in a series of articles in the early 1970s, and by Michael Dummett in his William James lectures of 1976, later published as The Logical Basis of Metaphysics. The idea had been attacked in 1960 by Arthur 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 paper 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.PostprintNon peer reviewe

Anti-Exceptionalism about Logic

Anti-exceptionalism about logic is the doctrine that logic does not require its own epistemology, for its methods are continuous with those of science. Although most recently urged by Williamson, the idea goes back at least to Lakatos, who wanted to adapt Popper's falsicationism and extend it not only to mathematics but to logic as well. But one needs to be careful here to distinguish the empirical from the a posteriori. Lakatos coined the term 'quasi-empirical' `for the counterinstances to putative mathematical and logical theses. Mathematics and logic may both be a posteriori, but it does not follow that they are empirical. Indeed, as Williamson has demonstrated, what counts as empirical knowledge, and the role of experience in acquiring knowledge, are both unclear. Moreover, knowledge, even of necessary truths, is fallible. Nonetheless, logical consequence holds in virtue of the meaning of the logical terms, just as consequence in general holds in virtue of the meanings of the concepts involved; and so logic is both analytic and necessary. In this respect, it is exceptional. But its methodologyand its epistemology are the same as those of mathematics and science in being fallibilist, and counterexamples to seemingly analytic truths are as likely as those in any scientic endeavour. What is needed is a new account of the evidential basis of knowledge, one which is, perhaps surprisingly, found in Aristotle

The Calculators on the insolubles : Bradwardine, Kilvington, Heytesbury, Swyneshed and Dumbleton

The present work was funded by Leverhulme Trust Research Project Grant RPG-2016-333: “Theories of Paradox in Fourteenth-Century Logic: Edition and Translation of Key Texts”.The most exciting and innovative period in the discussion of the insolubles (i.e., logical paradoxes) before the twentieth century occurred in the second quarter of the fourteenth in Oxford, and at its heart were many of the Calculators. It was prompted by Thomas Bradwardine's iconoclastic ideas about the insolubles in the early 1320s. Framed largely within the context of the theory of (logical) obligations, it was continued by Richard Kilvington, Roger Swyneshed, William Heytesbury and John Dumbleton, each responding in different ways to Bradwardine's analysis, particularly his idea that propositions had additional hidden and implicit meanings. Kilvington identified an equivocation in what was said; Swyneshed preferred to modify the account of truth rather than signification; Heytesbury exploited the respondent's role in obligational dialogues to avoid Bradwardine's tendentious closure postulate on signification; and Dumbleton relied on other constraints on signification to give new life to two long-standing accounts of insolubles that Bradwardine had summarily dismissed. The present paper focusses on the central thesis of each thinker's response to the insolubles and their interaction.Postprin

The liar and the new t-schema

Desde que Tarski publicó su estudio sobre el concepto de verdad en los años 30, ha sido una práctica  ortodoxa  el considerar que toda instancia del esquema T es verdadera. Sin embargo, algunas instancias del  esquema  son falsas. Éstas incluyen las instancias paradójicas ejemplificadas por la oración del mentiroso.  Aquí se  demuestra que un esquema mejor permite un tratamiento uniforme de la verdad en el que las  paradojas  semánticas resultan ser simplemente falsas.Since Tarski published his study of the concept of truth in the 1930s, it has been orthodox practice to suppose that every instance of the T-schema is true. However, some instances of the schema are false. These include the paradoxical instances exemplified by the Liar sentence. It is shown that a better schema allows a uniform treatment of truth in which the semantic paradoxes turn out to be simply false