Roger Swyneshed, in his treatise on insolubles (logical paradoxes), dating from the early 1330s, ... more 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
In the late 1970s, Paul Spade edited three treatises, on Consequences, Insolubles and Obligations... more In the late 1970s, Paul Spade edited three treatises, on Consequences, Insolubles and Obligations, which he attributed to an otherwise unknown fourteenth-century logician whom he named as Robert Fland. We question this reading of the name and argue that his real name was Robert Eland. Moreover, we suggest that he should be identified with Eland the dialectician, whose Sophismata is mentioned in an account book at Merton College in 1367, and whose renown as a logician was disdainfully recorded some two hundred years later by the bibliographer John Bale. In 1976, Paul Vincent Spade published a treatise on Consequences which he attributed to a previously unknown logician of the fourteenth century, Robert Fland. 1 This attribution was made on the basis of the explicits of three treatises preserved in a single manuscript, Bruges ms 497, ff. 41 r-46 r. 2 The other treatises, on Insolubles and on Obligations, were attributed in the ms to the same author, and were edited by Spade in 1978 and 1980. 3 Spade commented in his introduction to his edition of the Insolubles that "there are no external references to Fland by name" 4 and in his introduction to the Consequences that "not even his name is certain". 5
The standard rules for identity do not seem to lie in harmony, where the elimination-rule is just... more The standard rules for identity do not seem to lie in harmony, where the elimination-rule is justified by the meaning given to a logical operator by its introduction-rule. However, harmonious rules can be given for identity, showing that identity is a logical notion even on the inferentialist conception. These new rules are shown to be sound and complete. The present paper omits §3 of the original published version [7], which responded to a mistaken objection by the referee, as was shown by Michael Kremer in [3].
Conclusion References Sophism 47: 'You know that the king is seated' "Suppose this hypothesis, th... more Conclusion References Sophism 47: 'You know that the king is seated' "Suppose this hypothesis, that if the king is seated, you know that the king is seated, and if the king is not seated, you know that the king is not seated." Kilvington proceeds to prove, first, that you know that the king is
Roger Swyneshed, in his treatise on insolubles (logical paradoxes), dating from the early 1330s, ... more Roger Swyneshed, in his treatise on insolubles (logical paradoxes), dating from the early 1330s, drew three notorious corollaries from his solution. The third states that there is a contradictory pair of propositions both of which are false. This appears to contradict what Whitaker, in his iconoclastic reading of Aristotle’s De Interpretatione, dubbed “The Rule of Contradictory Pairs” (RCP), which requires that in every such pair, one must be true and the other false. Whitaker argued that, immediately after defining the notion of a contradictory pair, in which one statement affirms what the other denies of the same thing, Aristotle himself gave counterexamples to the rule. This gives some credence to Swyneshed’s claim that his solution to the logical paradoxes is not contrary to Aristotle’s teaching, as many of Swyneshed’s contemporaries claimed. Insolubles are false, he said, because they falsify themselves; and their contradictories are false because they falsely deny that the ins...
Anti-exceptionalism about logic is the doctrine that logic does not require its own epistemology,... more 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 meanin...
General-elimination harmony articulates Gentzen's idea that the eliminationrules are justified if... more General-elimination harmony articulates Gentzen's idea that the eliminationrules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the E-rules allow one to infer no more and no less than the I-rules justify. Pfenning and Davies call the rules locally complete if the E-rules are strong enough to allow one to infer the original judgement. A method is given of generating harmonious general-elimination rules from a collection of I-rules. We show that the general-elimination rules satisfy Pfenning and Davies' test for local completeness, but question whether that is enough to show that they are stable. Alternative conditions for stability are considered, including equivalence between the introduction-and elimination-meanings of a connective, and recovery of the grounds for assertion, finally generalizing the notion of local completeness to capture Dummett's notion of stability satisfactorily. We show that the general-elimination rules meet the last of these conditions, and so are indeed not only harmonious but also stable.
Intentionality, Cognition, and Mental Representation in Medieval Philosophy, 2015
In his recent study, Concepts, Fodor identifies five nonnegotiable constraints on any theory of c... more In his recent study, Concepts, Fodor identifies five nonnegotiable constraints on any theory of concepts. These theses were all shared by the standard medieval theories of concepts. However, those theories were cognitivist, in contrast with Fodor's: concepts are definitions, a form of natural knowledge. The medieval theories were formed under two influences, from Aristotle by way of Boethius, and from Augustine. The tension between them resulted in the Ockhamist notion of a natural language, concepts as signs. Thus conventional signs, spoken and written, signify things in the world by the mediation of concepts which themselves form a language of thought, signifying those things naturally by their similarity. Indeed, later thinkers realised that everything signifies itself and what is like it naturally in a broad sense by means of the concept of its natural likeness.
The Western Ontario Series in Philosophy of Science
Logical pluralism is the claim that different accounts of validity can be equally correct. Beall ... more Logical pluralism is the claim that different accounts of validity can be equally correct. Beall and Restall have recently defended this position. Validity is a matter of truth-preservation over cases, they say: the conclusion should be true in every case in which the premises are true. Each logic specifies a class of cases, but differs over which cases should be considered. I show that this account of logic is incoherent. Validity indeed is truth-preservation, provided this is properly understood. Once understood, there is one true logic, relevance logic. The source of Beall and Restall's error is a recent habit of using a classical metalanguage to analyse non-classical logics generally, including relevance logic.
Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In ... more Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they may be said to exhibit "general-elimination" harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I-and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions.
Bradwardine's solution to the the logical paradoxes depends on the idea that every sentence signi... more Bradwardine's solution to the the logical paradoxes depends on the idea that every sentence signifies many things, and its truth depends on things' being wholly as it signifies. This idea is underpinned by his claim that a sentence signifies everything that follows from what it signifies. But the idea that signification is closed under entailment appears too strong, just as logical omniscience is unacceptable in the logic of knowledge. What is needed is a more restricted closure principle. A clue can be found in speech act pluralism, whereby indirect speech reports are closed under intersubstitutivity of co-referential terms. The conclusion is that solving the semantic paradoxes does not require revision of logic, thus saving logic from paradox.
... Barbara Celarent Darii Ferio Baralipton Celantes Dabitis Fapesmo Frisesomorum; Cesare Camestr... more ... Barbara Celarent Darii Ferio Baralipton Celantes Dabitis Fapesmo Frisesomorum; Cesare Camestres Festino Baroco; Darapti Felapton Disamis Datisi Bocardo Ferison. ... Here is an example of reduction, reducing Camestres (aee in Figure II) to Celarent: ...
The correspondence theory of truth has experienced something of a revival recently in the form of... more The correspondence theory of truth has experienced something of a revival recently in the form of the Truthmaker Axiom: whatever is true, something makes it true. We consider various postulates which have been proposed to characterize truthmaking, in particular, the Disjunction Thesis (DT), that whatever makes a disjunction true must make one or other disjunct true. In conjunction with certain other assumptions, DT leads to triviality. We show that there are elaborations of truthmaking on which DT holds (which must therefore take steps to avoid the triviality); but that there are more plausible accounts of truthmaking on which DT fails.
What makes necessary truths true? I argue that all truth supervenes on how things are, and that n... more What makes necessary truths true? I argue that all truth supervenes on how things are, and that necessary truths are no exception. What makes them true are proofs. But if so, the notion of proof needs to be generalized to include verification-transcendent proofs, proofs whose correctness exceeds our ability to verify it. It is incumbent on me, therefore, to show that arguments, such as Dummett's, that verification-truth is not compatible with the theory of meaning, are mistaken. The answer is that what we can conceive and construct far outstrips our actual abilities. I conclude by proposing a proof-theoretic account of modality, rejecting a claim of Armstrong's that modality can reside in non-modal truthmakers.
Roger Swyneshed, in his treatise on insolubles (logical paradoxes), dating from the early 1330s, ... more 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
In the late 1970s, Paul Spade edited three treatises, on Consequences, Insolubles and Obligations... more In the late 1970s, Paul Spade edited three treatises, on Consequences, Insolubles and Obligations, which he attributed to an otherwise unknown fourteenth-century logician whom he named as Robert Fland. We question this reading of the name and argue that his real name was Robert Eland. Moreover, we suggest that he should be identified with Eland the dialectician, whose Sophismata is mentioned in an account book at Merton College in 1367, and whose renown as a logician was disdainfully recorded some two hundred years later by the bibliographer John Bale. In 1976, Paul Vincent Spade published a treatise on Consequences which he attributed to a previously unknown logician of the fourteenth century, Robert Fland. 1 This attribution was made on the basis of the explicits of three treatises preserved in a single manuscript, Bruges ms 497, ff. 41 r-46 r. 2 The other treatises, on Insolubles and on Obligations, were attributed in the ms to the same author, and were edited by Spade in 1978 and 1980. 3 Spade commented in his introduction to his edition of the Insolubles that "there are no external references to Fland by name" 4 and in his introduction to the Consequences that "not even his name is certain". 5
The standard rules for identity do not seem to lie in harmony, where the elimination-rule is just... more The standard rules for identity do not seem to lie in harmony, where the elimination-rule is justified by the meaning given to a logical operator by its introduction-rule. However, harmonious rules can be given for identity, showing that identity is a logical notion even on the inferentialist conception. These new rules are shown to be sound and complete. The present paper omits §3 of the original published version [7], which responded to a mistaken objection by the referee, as was shown by Michael Kremer in [3].
Conclusion References Sophism 47: 'You know that the king is seated' "Suppose this hypothesis, th... more Conclusion References Sophism 47: 'You know that the king is seated' "Suppose this hypothesis, that if the king is seated, you know that the king is seated, and if the king is not seated, you know that the king is not seated." Kilvington proceeds to prove, first, that you know that the king is
Roger Swyneshed, in his treatise on insolubles (logical paradoxes), dating from the early 1330s, ... more Roger Swyneshed, in his treatise on insolubles (logical paradoxes), dating from the early 1330s, drew three notorious corollaries from his solution. The third states that there is a contradictory pair of propositions both of which are false. This appears to contradict what Whitaker, in his iconoclastic reading of Aristotle’s De Interpretatione, dubbed “The Rule of Contradictory Pairs” (RCP), which requires that in every such pair, one must be true and the other false. Whitaker argued that, immediately after defining the notion of a contradictory pair, in which one statement affirms what the other denies of the same thing, Aristotle himself gave counterexamples to the rule. This gives some credence to Swyneshed’s claim that his solution to the logical paradoxes is not contrary to Aristotle’s teaching, as many of Swyneshed’s contemporaries claimed. Insolubles are false, he said, because they falsify themselves; and their contradictories are false because they falsely deny that the ins...
Anti-exceptionalism about logic is the doctrine that logic does not require its own epistemology,... more 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 meanin...
General-elimination harmony articulates Gentzen's idea that the eliminationrules are justified if... more General-elimination harmony articulates Gentzen's idea that the eliminationrules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the E-rules allow one to infer no more and no less than the I-rules justify. Pfenning and Davies call the rules locally complete if the E-rules are strong enough to allow one to infer the original judgement. A method is given of generating harmonious general-elimination rules from a collection of I-rules. We show that the general-elimination rules satisfy Pfenning and Davies' test for local completeness, but question whether that is enough to show that they are stable. Alternative conditions for stability are considered, including equivalence between the introduction-and elimination-meanings of a connective, and recovery of the grounds for assertion, finally generalizing the notion of local completeness to capture Dummett's notion of stability satisfactorily. We show that the general-elimination rules meet the last of these conditions, and so are indeed not only harmonious but also stable.
Intentionality, Cognition, and Mental Representation in Medieval Philosophy, 2015
In his recent study, Concepts, Fodor identifies five nonnegotiable constraints on any theory of c... more In his recent study, Concepts, Fodor identifies five nonnegotiable constraints on any theory of concepts. These theses were all shared by the standard medieval theories of concepts. However, those theories were cognitivist, in contrast with Fodor's: concepts are definitions, a form of natural knowledge. The medieval theories were formed under two influences, from Aristotle by way of Boethius, and from Augustine. The tension between them resulted in the Ockhamist notion of a natural language, concepts as signs. Thus conventional signs, spoken and written, signify things in the world by the mediation of concepts which themselves form a language of thought, signifying those things naturally by their similarity. Indeed, later thinkers realised that everything signifies itself and what is like it naturally in a broad sense by means of the concept of its natural likeness.
The Western Ontario Series in Philosophy of Science
Logical pluralism is the claim that different accounts of validity can be equally correct. Beall ... more Logical pluralism is the claim that different accounts of validity can be equally correct. Beall and Restall have recently defended this position. Validity is a matter of truth-preservation over cases, they say: the conclusion should be true in every case in which the premises are true. Each logic specifies a class of cases, but differs over which cases should be considered. I show that this account of logic is incoherent. Validity indeed is truth-preservation, provided this is properly understood. Once understood, there is one true logic, relevance logic. The source of Beall and Restall's error is a recent habit of using a classical metalanguage to analyse non-classical logics generally, including relevance logic.
Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In ... more Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they may be said to exhibit "general-elimination" harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I-and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions.
Bradwardine's solution to the the logical paradoxes depends on the idea that every sentence signi... more Bradwardine's solution to the the logical paradoxes depends on the idea that every sentence signifies many things, and its truth depends on things' being wholly as it signifies. This idea is underpinned by his claim that a sentence signifies everything that follows from what it signifies. But the idea that signification is closed under entailment appears too strong, just as logical omniscience is unacceptable in the logic of knowledge. What is needed is a more restricted closure principle. A clue can be found in speech act pluralism, whereby indirect speech reports are closed under intersubstitutivity of co-referential terms. The conclusion is that solving the semantic paradoxes does not require revision of logic, thus saving logic from paradox.
... Barbara Celarent Darii Ferio Baralipton Celantes Dabitis Fapesmo Frisesomorum; Cesare Camestr... more ... Barbara Celarent Darii Ferio Baralipton Celantes Dabitis Fapesmo Frisesomorum; Cesare Camestres Festino Baroco; Darapti Felapton Disamis Datisi Bocardo Ferison. ... Here is an example of reduction, reducing Camestres (aee in Figure II) to Celarent: ...
The correspondence theory of truth has experienced something of a revival recently in the form of... more The correspondence theory of truth has experienced something of a revival recently in the form of the Truthmaker Axiom: whatever is true, something makes it true. We consider various postulates which have been proposed to characterize truthmaking, in particular, the Disjunction Thesis (DT), that whatever makes a disjunction true must make one or other disjunct true. In conjunction with certain other assumptions, DT leads to triviality. We show that there are elaborations of truthmaking on which DT holds (which must therefore take steps to avoid the triviality); but that there are more plausible accounts of truthmaking on which DT fails.
What makes necessary truths true? I argue that all truth supervenes on how things are, and that n... more What makes necessary truths true? I argue that all truth supervenes on how things are, and that necessary truths are no exception. What makes them true are proofs. But if so, the notion of proof needs to be generalized to include verification-transcendent proofs, proofs whose correctness exceeds our ability to verify it. It is incumbent on me, therefore, to show that arguments, such as Dummett's, that verification-truth is not compatible with the theory of meaning, are mistaken. The answer is that what we can conceive and construct far outstrips our actual abilities. I conclude by proposing a proof-theoretic account of modality, rejecting a claim of Armstrong's that modality can reside in non-modal truthmakers.
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