We refine the interpolation property of the {&, v, ~, A, E}-fragment of classical first-order log... more We refine the interpolation property of the {&, v, ~, A, E}-fragment of classical first-order logic, showing that if [G is satisfiable] and [D is ot logically true] and G|- D then there is an interpolant c, constructed using only non-logical vocabulary common to both members of G and members of D, such that (i) G entails c in the first-order version of Kleene’s strong three-valued logic (K3), and (ii) c entails D in the first-order version of Priest’s Logic of Paradox (LP). The proof proceeds via a careful analysis of derivations in a cut-free sequent calculus for first-order classical logic. Lyndon’s strengthening falls out of an observation regarding such derivations and the steps involved in the construction of interpolants. The proof is then extended to cover the {&, v, ~, A, E}-fragment of classical first-order logic with identity. Keywords: Craig–Lyndon Interpolation Theorem (for classical first-order logic); Kleene’s strong 3-valued logic;Priest’s Logic of Paradox; Belnap’s f...
This paper presents a thoroughgoing interpretation of a weak relevant logic built over the Dunn-B... more This paper presents a thoroughgoing interpretation of a weak relevant logic built over the Dunn-Belnap four-valued semantics in terms of the communication of information in a network of sites of knowledge production (laboratories). The knowledge communicated concerns experimental data and the regularities tested using it. There have been many nods to interpretations similar to ours - for example, in Dunn (1976), Belnap (1977). The laboratory interpretation was outlined in Bilkova et al. (2010). Our system is built on the Routley--Meyer semantics for relevant logic equipped with a four-valued valuation of formulas, where labs stand in for situations, and the four values reflect the complexity of assessing results of experiments. This semantics avoids using the Routley star, on the cost of introducing a further relation, required in evaluating falsity assignments of implication. We can however provide a natural interpretation of two accessibility relations - confirmation and refutatio...
When we look closely at textbooks on model theory, we find that there are three different account... more When we look closely at textbooks on model theory, we find that there are three different accounts of what a model or structure is. One of these is highly language dependent, so that the same structure cannot be the interpretation of two different languages or signatures. The other two definitions do not fall foul of that dependence but all textbooks tie the notion of homomorphism so closely to language (signature) that only structures interpreting the same language (signature) are isomorphic. Although this follows the practice in universal algebra, it is highly unnatural. The aim here is to present a notion of homomorphism better consonant with intuition and with what the less cautious authors of textbooks say when they speak informally.
285 which, in a way, is just fine and dandy, but unless he is committed to the primacy of idiolec... more 285 which, in a way, is just fine and dandy, but unless he is committed to the primacy of idiolects and the authoritativeness of agents regarding their own idiolects, the agent's beliefs regarding precisifications of the vague predicates of her language have no part to play in providing a semantics of vague terms, not that Bennett suggests otherwise, despite calling his paper 'Standpoint semantics'. Speakers of a language can have all sorts of beliefs, including mistaken beliefs, about the meanings of expressions in their (native and other) languages. In one sense, these beliefs are distinct from how they take the world to be—represented by some subset of W. But for most, if not all, of what we have beliefs about, there are standards of correctness, albeit perhaps transcending all possibility of verification. When Bennett offers us a predicate grounding theory for the colour terms red, orange, pink, peach and purple, he presents it for all the world as though he were mak...
There are distinctive methodological and conceptual challenges in rare and severe event (RSE) for... more There are distinctive methodological and conceptual challenges in rare and severe event (RSE) forecast verification, that is, in the assessment of the quality of forecasts of rare but severe natural hazards such as avalanches, landslides or tornadoes. While some of these challenges have been discussed since the inception of the discipline in the 1880s, there is no consensus about how to assess RSE forecasts. This article offers a comprehensive and critical overview of the many different measures used to capture the quality of categorical, binary RSE forecasts-forecasts of occurrence and nonoccurrence-and argues that of skill scores in the literature there is only one adequate for RSE forecasting. We do so by first focusing on the relationship between accuracy and skill and showing why skill is more important than accuracy in the case of RSE forecast verification. We then motivate three adequacy constraints for a measure of skill in RSE forecasting. We argue that of skill scores in the literature only the Peirce skill score meets all three constraints. We then outline how our theoretical investigation has important practical implications for avalanche forecasting, basing our discussion on a study in avalanche forecast verification using the nearestneighbour method (Heierli et al., 2004). Lastly, we raise what we call the "scope challenge"; this affects all forms of RSE forecasting and highlights how and why working with the right measure of skill is important not only for local binary RSE forecasts but also for the assessment of different diagnostic tests widely used in avalanche risk management and related operations, including the design of methods to assess the quality of regional multi-categorical avalanche forecasts.
From Introduction: In a 1968 article, ‘Probability Measures of Fuzzy Events’, Lotfi Zadeh pro-pos... more From Introduction: In a 1968 article, ‘Probability Measures of Fuzzy Events’, Lotfi Zadeh pro-posed accounts of absolute and conditional probability for fuzzy sets (Zadeh, 1968).
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
The paper tries to spell out a connection between deductive logic and rationality, against Harman... more The paper tries to spell out a connection between deductive logic and rationality, against Harman's arguments that there is no such connection, and also against the thought that any such connection would preclude rational change in logic. One might not need to connect logic to rationality if one could view logic as the science of what preserves truth by a certain kind of necessity (or by necessity plus logical form); but the paper points out a serious obstacle to any such view.
6 Russell objected to coherence theories of truth that there could be more than one maximal consi... more 6 Russell objected to coherence theories of truth that there could be more than one maximal consistent set of beliefs. Likewise, there can be more than one maximal consistent set of Fregean thoughts and it would seem that the world must somehow play a role in determining the set which is the set of true thoughts (facts). Perhaps because he was concerned as much with mathematical truths as empirical ones, Frege says little regarding contingency and modality generally. Crispin Wright has objected persuasively that coherence theories have a hard job in accommodating contingency say in the sense of genuine chance events ('Truth: A traditional debate reviewed', in S. Blackburn and K. Simmons (eds.), Truth (Oxford: Oxford University Press, 1999), pp. 203-38. at pp. 221-2). That is not the objection here. The objection here to a Fregean identity theory is more fundamental: it is that we are given no clue as to what fi xes the set of truths. The coherence-theorist has a story to tell about that; Wright's complaint is that that story is defective.
The British Journal for the Philosophy of Science, 1997
This article begins by outlining some of the history-beginning with brief remarks of Quine's-of w... more This article begins by outlining some of the history-beginning with brief remarks of Quine's-of work on conditional assertions and conditional events. The upshot of the historical narrative is that diverse works from various starting points have circled around a nexus of ideas without convincingly tying them together. Section 3 shows how ideas contained in a neglected article of de Finetti's lead to a unified treatment of the topics based on the identification of conditional events as the objects of conditional bets. The penultimate section explores some of the consequences of the resulting logic of conditional events while the last defends it.
Uncertainty and vagueness/imprecision are not the same: one can be certain about events described... more Uncertainty and vagueness/imprecision are not the same: one can be certain about events described using vague predicates and about imprecisely specified events, just as one can be uncertain about precisely specified events. Exactly because of this, a question arises about how one ought to assign probabilities to imprecisely specified events in the case when no possible available evidence will eradicate the imprecision (because, say, of the limits of accuracy of a measuring device). Modelling imprecision by rough sets over an approximation space presents an especially tractable case to help get one's bearings. Two solutions present themselves: the first takes as upper and lower probabilities of the event X the (exact) probabilities assigned X's upper and lower rough-set approximations; the second, motivated both by formal considerations and by a simple betting argument, is to treat X's rough-set approximation as a conditional event and assign to it a point-valued (conditional) probability. With rough sets over an approximation space we get a lot of good behaviour. For example, in the first construction mentioned the lower probabilities are n-monotone, for every n ∈ N +. When we examine other models of approximation/imprecision/vagueness, and in particular, proximity spaces, we lose a lot of that good behaviour. In the literature there is not (even) agreement on the definition of upper and lower approximations for events (subsets) in the underlying domain. Betting considerations suggest one choice and, again, ways to assign upper and lower and point-valued probabilities, but nothing works well.
Various natural deduction formulations of classical, minimal, intuitionist, and intermediate prop... more Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid (i) to which properties of theories result in the presence of which rules of inference, and (ii) to restrictions on the sets of formulas to which the rules may be employed, restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used througho...
As yet we haven't established that there is an upper limit to information added; we're just playi... more As yet we haven't established that there is an upper limit to information added; we're just playing safe, introducing what is, with a little bit of thought, a fairly obvious qualification.
International Journal of Approximate Reasoning, 1996
This paper investigates four questions. What are the logical presuppositions underlying classical... more This paper investigates four questions. What are the logical presuppositions underlying classical probability that have a role to play in David Lewis's proof of triviality concerning probabilities of conditionals and conditional probabilities? To what extent and how are they avoided in fuzzy logics when we treat semantic evaluations as the analogues of probability distributions? The introduction into the classical setting of conditional events (or assertions)-as opposed to implications-as a class of objects whose probabilities are equated with conditional probabilities has been the object of much recent investigation. To what extent, if any, can fuzzy logics accommodate the analogues of conditional events? How is triviality avoided in conditional event algebras?
The propositional doxastic logics investigated here are minimal in the sense that they make very ... more The propositional doxastic logics investigated here are minimal in the sense that they make very limited claims about what holds good of rational or justified belief on the basis of the meaning of those terms. Indeed some of the logics allow for the truth of total scepticism, the view that there are no rational or justified beliefs. The logics are subject to constraints such as that any doxastic logic must be believable in its own terms and that any proposition which must, according to a doxastic logic, be believed (not believed) must itself be a theorem of (be refuted by) the logic. Two techniques are used to establish completeness, one employing possible-worlds models in which there may be several or no accessibility relations, the other using probability distributions and a maximal-probability conception of belief.
The paper develops a formal model for interpreting vague languages in a setting similar to that o... more The paper develops a formal model for interpreting vague languages in a setting similar to that of supervaluation semantics. Two modes of semantic variability are modelled, corresponding to different aspects of vagueness: one mode arises where there can be multiple definitions of a term involving different attributes or different logical combinations of attributes. The other relates to the threshold of applicability of a vague term with respect to the magnitude of relevant observable values. The truth of a proposition depends on both the possible world and the precisification with respect to which it is evaluated. Structures representing both possible worlds and precisifications are specified in terms of primitive functions representing observable measurements, so that the semantics is grounded upon an underlying theory of physical reality. On the basis of this semantics, the acceptability of a proposition to an agent is characterised in terms of a combination of the agent’s beliefs...
Intervals in boolean algebras enter into the study of conditional assertions (or events) in two w... more Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, Łukasiewicz's threevalued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates.
Cambridge Studies in Probability, Induction, and Decision Theory THE DESIGN INFERENCE ELIMINATING... more Cambridge Studies in Probability, Induction, and Decision Theory THE DESIGN INFERENCE ELIMINATING CHANCE THROUGH SMALL PROBABILITIES LIAM EMBSKI .3' cw are ... In Personal Knowledge Michael Polanyi (1962, p. 33) considers stones placed in a garden. ...
We refine the interpolation property of the {&, v, ~, A, E}-fragment of classical first-order log... more We refine the interpolation property of the {&, v, ~, A, E}-fragment of classical first-order logic, showing that if [G is satisfiable] and [D is ot logically true] and G|- D then there is an interpolant c, constructed using only non-logical vocabulary common to both members of G and members of D, such that (i) G entails c in the first-order version of Kleene’s strong three-valued logic (K3), and (ii) c entails D in the first-order version of Priest’s Logic of Paradox (LP). The proof proceeds via a careful analysis of derivations in a cut-free sequent calculus for first-order classical logic. Lyndon’s strengthening falls out of an observation regarding such derivations and the steps involved in the construction of interpolants. The proof is then extended to cover the {&, v, ~, A, E}-fragment of classical first-order logic with identity. Keywords: Craig–Lyndon Interpolation Theorem (for classical first-order logic); Kleene’s strong 3-valued logic;Priest’s Logic of Paradox; Belnap’s f...
This paper presents a thoroughgoing interpretation of a weak relevant logic built over the Dunn-B... more This paper presents a thoroughgoing interpretation of a weak relevant logic built over the Dunn-Belnap four-valued semantics in terms of the communication of information in a network of sites of knowledge production (laboratories). The knowledge communicated concerns experimental data and the regularities tested using it. There have been many nods to interpretations similar to ours - for example, in Dunn (1976), Belnap (1977). The laboratory interpretation was outlined in Bilkova et al. (2010). Our system is built on the Routley--Meyer semantics for relevant logic equipped with a four-valued valuation of formulas, where labs stand in for situations, and the four values reflect the complexity of assessing results of experiments. This semantics avoids using the Routley star, on the cost of introducing a further relation, required in evaluating falsity assignments of implication. We can however provide a natural interpretation of two accessibility relations - confirmation and refutatio...
When we look closely at textbooks on model theory, we find that there are three different account... more When we look closely at textbooks on model theory, we find that there are three different accounts of what a model or structure is. One of these is highly language dependent, so that the same structure cannot be the interpretation of two different languages or signatures. The other two definitions do not fall foul of that dependence but all textbooks tie the notion of homomorphism so closely to language (signature) that only structures interpreting the same language (signature) are isomorphic. Although this follows the practice in universal algebra, it is highly unnatural. The aim here is to present a notion of homomorphism better consonant with intuition and with what the less cautious authors of textbooks say when they speak informally.
285 which, in a way, is just fine and dandy, but unless he is committed to the primacy of idiolec... more 285 which, in a way, is just fine and dandy, but unless he is committed to the primacy of idiolects and the authoritativeness of agents regarding their own idiolects, the agent's beliefs regarding precisifications of the vague predicates of her language have no part to play in providing a semantics of vague terms, not that Bennett suggests otherwise, despite calling his paper 'Standpoint semantics'. Speakers of a language can have all sorts of beliefs, including mistaken beliefs, about the meanings of expressions in their (native and other) languages. In one sense, these beliefs are distinct from how they take the world to be—represented by some subset of W. But for most, if not all, of what we have beliefs about, there are standards of correctness, albeit perhaps transcending all possibility of verification. When Bennett offers us a predicate grounding theory for the colour terms red, orange, pink, peach and purple, he presents it for all the world as though he were mak...
There are distinctive methodological and conceptual challenges in rare and severe event (RSE) for... more There are distinctive methodological and conceptual challenges in rare and severe event (RSE) forecast verification, that is, in the assessment of the quality of forecasts of rare but severe natural hazards such as avalanches, landslides or tornadoes. While some of these challenges have been discussed since the inception of the discipline in the 1880s, there is no consensus about how to assess RSE forecasts. This article offers a comprehensive and critical overview of the many different measures used to capture the quality of categorical, binary RSE forecasts-forecasts of occurrence and nonoccurrence-and argues that of skill scores in the literature there is only one adequate for RSE forecasting. We do so by first focusing on the relationship between accuracy and skill and showing why skill is more important than accuracy in the case of RSE forecast verification. We then motivate three adequacy constraints for a measure of skill in RSE forecasting. We argue that of skill scores in the literature only the Peirce skill score meets all three constraints. We then outline how our theoretical investigation has important practical implications for avalanche forecasting, basing our discussion on a study in avalanche forecast verification using the nearestneighbour method (Heierli et al., 2004). Lastly, we raise what we call the "scope challenge"; this affects all forms of RSE forecasting and highlights how and why working with the right measure of skill is important not only for local binary RSE forecasts but also for the assessment of different diagnostic tests widely used in avalanche risk management and related operations, including the design of methods to assess the quality of regional multi-categorical avalanche forecasts.
From Introduction: In a 1968 article, ‘Probability Measures of Fuzzy Events’, Lotfi Zadeh pro-pos... more From Introduction: In a 1968 article, ‘Probability Measures of Fuzzy Events’, Lotfi Zadeh pro-posed accounts of absolute and conditional probability for fuzzy sets (Zadeh, 1968).
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
The paper tries to spell out a connection between deductive logic and rationality, against Harman... more The paper tries to spell out a connection between deductive logic and rationality, against Harman's arguments that there is no such connection, and also against the thought that any such connection would preclude rational change in logic. One might not need to connect logic to rationality if one could view logic as the science of what preserves truth by a certain kind of necessity (or by necessity plus logical form); but the paper points out a serious obstacle to any such view.
6 Russell objected to coherence theories of truth that there could be more than one maximal consi... more 6 Russell objected to coherence theories of truth that there could be more than one maximal consistent set of beliefs. Likewise, there can be more than one maximal consistent set of Fregean thoughts and it would seem that the world must somehow play a role in determining the set which is the set of true thoughts (facts). Perhaps because he was concerned as much with mathematical truths as empirical ones, Frege says little regarding contingency and modality generally. Crispin Wright has objected persuasively that coherence theories have a hard job in accommodating contingency say in the sense of genuine chance events ('Truth: A traditional debate reviewed', in S. Blackburn and K. Simmons (eds.), Truth (Oxford: Oxford University Press, 1999), pp. 203-38. at pp. 221-2). That is not the objection here. The objection here to a Fregean identity theory is more fundamental: it is that we are given no clue as to what fi xes the set of truths. The coherence-theorist has a story to tell about that; Wright's complaint is that that story is defective.
The British Journal for the Philosophy of Science, 1997
This article begins by outlining some of the history-beginning with brief remarks of Quine's-of w... more This article begins by outlining some of the history-beginning with brief remarks of Quine's-of work on conditional assertions and conditional events. The upshot of the historical narrative is that diverse works from various starting points have circled around a nexus of ideas without convincingly tying them together. Section 3 shows how ideas contained in a neglected article of de Finetti's lead to a unified treatment of the topics based on the identification of conditional events as the objects of conditional bets. The penultimate section explores some of the consequences of the resulting logic of conditional events while the last defends it.
Uncertainty and vagueness/imprecision are not the same: one can be certain about events described... more Uncertainty and vagueness/imprecision are not the same: one can be certain about events described using vague predicates and about imprecisely specified events, just as one can be uncertain about precisely specified events. Exactly because of this, a question arises about how one ought to assign probabilities to imprecisely specified events in the case when no possible available evidence will eradicate the imprecision (because, say, of the limits of accuracy of a measuring device). Modelling imprecision by rough sets over an approximation space presents an especially tractable case to help get one's bearings. Two solutions present themselves: the first takes as upper and lower probabilities of the event X the (exact) probabilities assigned X's upper and lower rough-set approximations; the second, motivated both by formal considerations and by a simple betting argument, is to treat X's rough-set approximation as a conditional event and assign to it a point-valued (conditional) probability. With rough sets over an approximation space we get a lot of good behaviour. For example, in the first construction mentioned the lower probabilities are n-monotone, for every n ∈ N +. When we examine other models of approximation/imprecision/vagueness, and in particular, proximity spaces, we lose a lot of that good behaviour. In the literature there is not (even) agreement on the definition of upper and lower approximations for events (subsets) in the underlying domain. Betting considerations suggest one choice and, again, ways to assign upper and lower and point-valued probabilities, but nothing works well.
Various natural deduction formulations of classical, minimal, intuitionist, and intermediate prop... more Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid (i) to which properties of theories result in the presence of which rules of inference, and (ii) to restrictions on the sets of formulas to which the rules may be employed, restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used througho...
As yet we haven't established that there is an upper limit to information added; we're just playi... more As yet we haven't established that there is an upper limit to information added; we're just playing safe, introducing what is, with a little bit of thought, a fairly obvious qualification.
International Journal of Approximate Reasoning, 1996
This paper investigates four questions. What are the logical presuppositions underlying classical... more This paper investigates four questions. What are the logical presuppositions underlying classical probability that have a role to play in David Lewis's proof of triviality concerning probabilities of conditionals and conditional probabilities? To what extent and how are they avoided in fuzzy logics when we treat semantic evaluations as the analogues of probability distributions? The introduction into the classical setting of conditional events (or assertions)-as opposed to implications-as a class of objects whose probabilities are equated with conditional probabilities has been the object of much recent investigation. To what extent, if any, can fuzzy logics accommodate the analogues of conditional events? How is triviality avoided in conditional event algebras?
The propositional doxastic logics investigated here are minimal in the sense that they make very ... more The propositional doxastic logics investigated here are minimal in the sense that they make very limited claims about what holds good of rational or justified belief on the basis of the meaning of those terms. Indeed some of the logics allow for the truth of total scepticism, the view that there are no rational or justified beliefs. The logics are subject to constraints such as that any doxastic logic must be believable in its own terms and that any proposition which must, according to a doxastic logic, be believed (not believed) must itself be a theorem of (be refuted by) the logic. Two techniques are used to establish completeness, one employing possible-worlds models in which there may be several or no accessibility relations, the other using probability distributions and a maximal-probability conception of belief.
The paper develops a formal model for interpreting vague languages in a setting similar to that o... more The paper develops a formal model for interpreting vague languages in a setting similar to that of supervaluation semantics. Two modes of semantic variability are modelled, corresponding to different aspects of vagueness: one mode arises where there can be multiple definitions of a term involving different attributes or different logical combinations of attributes. The other relates to the threshold of applicability of a vague term with respect to the magnitude of relevant observable values. The truth of a proposition depends on both the possible world and the precisification with respect to which it is evaluated. Structures representing both possible worlds and precisifications are specified in terms of primitive functions representing observable measurements, so that the semantics is grounded upon an underlying theory of physical reality. On the basis of this semantics, the acceptability of a proposition to an agent is characterised in terms of a combination of the agent’s beliefs...
Intervals in boolean algebras enter into the study of conditional assertions (or events) in two w... more Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, Łukasiewicz's threevalued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates.
Cambridge Studies in Probability, Induction, and Decision Theory THE DESIGN INFERENCE ELIMINATING... more Cambridge Studies in Probability, Induction, and Decision Theory THE DESIGN INFERENCE ELIMINATING CHANCE THROUGH SMALL PROBABILITIES LIAM EMBSKI .3' cw are ... In Personal Knowledge Michael Polanyi (1962, p. 33) considers stones placed in a garden. ...
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