I read Stefan Collini’s What are Universities For? last week with very mixed feelings. In the past, I’ve much admired his polemical essays on the REF, “impact”, the Browne Report, etc. in the London Review of Books and elsewhere: they speak to my heart. If you don’t know those essays, you can get some of their flavour from his latest article in the Guardian yesterday. But I found the book a disappointment. Perhaps the trouble is that Collini is too (...) decent, too high-minded, has too unrealistically exalted a view about what actually happens in universities which is too coloured by attitudes appropriate to the traditional humanities. And he is optimistic to the point of fantasy if he thinks that people are so susceptible to “the romance of ideas and the power of beauty” that they will want, or can be brought to want, lots and lots of universities in order to promote the ideas (as if they would suppose that the task of “conserving, understanding, extending and handing on to subsequent generations the intellectual, scientific, and artistic heritage of mankind” was clearly ill-served when there were only forty universities in England, as opposed to a hundred and whatever). (shrink)

We prove that the set of properties describable by a uniform sequence of first-order sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACE[n k ] (where n is the size of the universe). This set is also equal to the set of properties describable using an iterative definition for a finite set of relations of arity k. This is a refinement of the theorem PSPACE = (...) VAR[O[1]] [8]. We suggest some directions for exploiting this result to derive trade-offs between the number of variables and the quantifier depth in descriptive complexity. (shrink)

In 1990 J-L. Krivine introduced the notion of storage operators. They are $\lambda$ -terms which simulate call-by-value in the call-by-name strategy and they can be used in order to modelize assignment instructions. J-L. Krivine has shown that there is a very simple second order type in AF2 type system for storage operators using Gödel translation of classical to intuitionistic logic. In order to modelize the control operators, J-L. Krivine has extended the system AF2 to the classical logic. In his system (...) the property of the unicity of integers representation is lost, but he has shown that storage operators typable in the system AF2 can be used to find the values of classical integers. In this paper, we present a new classical type system based on a logical system called mixed logic. We prove that in this system we can characterize, by types, the storage operators and the control operators. (shrink)

In this paper, we present a propositional logic (called mixed logic) containing disjoint copies of minimal, intuitionistic and classical logics. We prove a completeness theorem for this logic with respect to a Kripke semantics. We establish some relations between mixed logic and minimal, intuitionistic and classical logics. We present at the end a sequent calculus version for this logic.

In order to define some interesting consequence relations, certain generalizations have been proposed in a many-valued semantic setting that have been useful for defining what have been called pure, mixed and ordertheoretic consequence relations. But these generalizations are insufficient to capture some other interesting relations, like other intersective mixed relations or relations with a conjunctive interpretation for multiple conclusions. We propose a broader framework to define these cases, and many others, and to set a common background that allows (...) for a direct compared analysis. At the end of the work, we illustrate some of these comparisons. (shrink)

This article explores areas of law loosely within English equity and trusts law that have not conventionally been subject to feminist debate, and within the context of a discussion about feminist method. The particular areas examined are whistleblowing and trustees’ powers of investment, each of which calls for consideration of decision-making processes which have an ethical content. These sites are chosen because they take debate outside the all too familiar locations of woman or ‘the body of woman’, including the family (...) home, where feminist analysis in relation to equity and trusts tends to stray. In exploring these chosen fields, Hobby’s (Feminist Perspectives on Equity and Trusts, Cavendish Publishing, 2001, pp. 219–255) critique of gender dimensions in relation to whistleblowing and Dunn’s (Feminist Perspectives on Equity and Trusts, Cavendish Publishing, 2001, pp. 179–196) account of women trustees’ investment decisions are evaluated and developed. The debate within feminist legal criticism of the possibility of integrating an ‘ethics of care’ and an ‘ethics of rights’ is acknowledged. Whilst acknowledging of the difficulty of adopting an approach that recognises ‘mixed logics’, it is concluded that women (and men) do integrate the ethics of care and the ethics of rights in their decision-making. It is argued that the effort should be made to trace the specific weave in particular circumstances, thereby paying suspicious attention to the values that underlie courts’ analyses of ethical choices. (shrink)

Logic is the study of the validity of arguments, which is to say the study of when a conclusion follows or does not follow from a set of premises. Logic is an ancient discipline pioneered by Aristotle and developed by some of the greatest thinkers in the Middle Ages. However, in the nineteenth century logic underwent a remarkable transformation into a precise branch of mathematics that changed the nature of logic, and the study of religion, forever. Both religious adherents and (...) critics of religion have sometimes claimed that logic and religion do not mix, but this is a puzzling claim since logic itself is not a set of substantive premises about the world which may come into conflict with religious claims, but a means of testing mathematically which premises come into conflict with which other premises. In fact, sense can be made of whether logic and religion mix, but the question, and the answer, turn out to be more complex than many have thought. (shrink)

We investigate complex algebras of the form arising from a frame where, and exhibit their abstract algebraic and logical counterparts.

Modal logics have proven to be a very successful tool for reasoning about games. However, until now, although logics have been put forward for games in both normal form and games in extensive form, and for games with complete and incomplete information, the focus in the logic community has hitherto been on games with pure strategies. This paper is a first to widen the scope to logics for games that allow mixed strategies. We present a modal (...) logic for games in normal form with mixed strategies, and demonstrate its soundness and strong completeness. Characteristic for our logic is a number of infinite rules. (shrink)

Variants of classical linear logics are presented based on the modal version of new structural rule !?mingle instead of the known rules !weakening and ?weakening. The cut-elimination theorems, the completeness theorems and a characteristic property named the mix separation principle are proved for these logics.

The problem of completeness for predicate modal logics is still under investigation, although some results have been obtained in the last few years . As far as we know, the case of multimodal logics has not been addressed at all. In this paper, we study the combination of modal logics in terms of combining their semantics. We demonstrate by a simple example that in this sense predicate modal logics are not so easily manipulated as propositional ones: (...) mixing two Kripke-complete predicate modal logics results in a Kripke-incomplete system. (shrink)

In this paper we solve the problem how to axiomatize a system of quantum computational gates known as the Poincaré irreversible quantum computational system. A Hilbert-style calculus is introduced obtaining a strong completeness theorem.

We consider a class of graphs embedded in $R^2$ as noncommutative proof-nets with an explicit exchange rule. We give two characterization of such proof-nets, one representing proof-nets as CW-complexes in a two-dimensional disc, the other extending a characterization by Asperti. As a corollary, we obtain that the test of correctness in the case of planar graphs is linear in the size of the data. Braided proof-nets are proof-nets for multiplicative linear logic with Mix embedded in $R^3$ . In order to (...) prove the cut-elimination theorem, we consider proof-nets in $R^2$ as projections of braided proof-nets under regular isotopy. (shrink)

This essay is basically divided into two parts. The first deals with the similarities between reductio ad absurdum arguments and slippery slope arguments. The chief example comes from Thomas Hobbes’s Leviathan, which advances an argument for the necessity of government for humane living. The second addresses some pedagogical concerns centered around another pair of arguments: the argument by complete enumeration and the argument by inductive generalization. The illustration for this pair comes from the arts. I finish with a suggestion that (...) pairs like the above can be as effectively used in shorter, non-regular critical reasoning or introductory logic courses as those in mid-term or summer courses. Such pairing can demonstrate a good use of mixing and matching deductive and non-deductive arguments in teaching logic. (shrink)

The aim of this paper is to define a λ-calculus typed in aMixed (commutative and non-commutative) Intuitionistic Linear Logic. The terms of such a calculus are the labelling of proofs of a linear intuitionistic mixed natural deduction NILL, which is based on the non-commutative linear multiplicative sequent calculus MNL [RuetAbrusci 99]. This linear λ-calculus involves three linear arrows: two directional arrows and a nondirectional one (the usual linear arrow). Moreover, the -terms are provided with seriesparallel orders on free variables. (...) We prove a normalization theorem which explicitly gives the behaviour of the order during the normalization procedure. (shrink)

There have been few attempts to answer the twin challenges for alethic pluralists to maintain standard accounts of the logical operators and of logical consequence in a sufficiently systematic and precise way. In this paper, I propose an account of logic and semantics on behalf of pluralists that answers both challenges in a sufficiently systematic and precise way. Crucially, the account accommodates mixed atomics, and its first-order extension also accommodates quantified sentences. Accordingly, pluralists can answer all the distinctively logical (...) challenges for their view. (shrink)

The article develops a critique of John Deely’s ontological realism, specifically in its relevance for the project of global semiotics. Deely, whose theorizations rely heavily on the pre-modern philosophical systems of Thomas Aquinas and the Latin scholastics, has made the most sustained attempt to give philosophical grounding to Charles Peirce’s famous intuition that “all this universe is perfused with signs, if it is not composed exclusively of signs”. The critique developsalong two main lines. Firstly, I contend that Deely’s account of (...) onto logical relations is unsatisfactory, for it assumes the very fact it promises to demonstrate: the identity between patterns of physical events, on one hand, and patt erns of thought, on the other. As it is never adequately explained what it means for the former to be identical with the latter, the mediating term – “ontological relation” – remains highly problematic. Secondly, I argue that even if this ontological paradigm is allowed to stand as proposed, it still fails to bridge the gap between the so-called “natural” signs and human-specific, language-based signification; for, as it posits a unity between the stipulable and the non-stipulable sign, the argument falls into a performative contradiction. (shrink)

Quotation exhibits characteristics of both use and mention. I argue against the recently popular pragmatic reductions of quotation to mere language use (Recanati 2001), and in favor of a truly hybrid account synthesizing and extending Potts (2007) and Geurts and Maier (2005), using a mention logic and a dynamic semantics with presupposition to establish a context-driven meaning shift. The main advantages are an account of error neutralization and shifted indexicality under quotation. The current paper addresses the problematic data involving quoted (...) non-constituents. (shrink)

A number of medieval Arabic logicians discussed inferences that combine the principles of propositional and term logic, for example: Whenever H is Z then Every J is DNo D is AWhenever H is Z then S...

Suszko’s problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski’s structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an (...) argument. In this article we give a more systematic perspective on Suszko’s problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those theorems to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos, and Wansing, for logics with various structural properties. We strengthen these results into exact rank results for nonpermeable logics. We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties. (shrink)

In a previous paper (see ‘Tolerant, Classical, Strict’, henceforth TCS) we investigated a semantic framework to deal with the idea that vague predicates are tolerant, namely that small changes do not affect the applicability of a vague predicate even if large changes do. Our approach there rests on two main ideas. First, given a classical extension of a predicate, we can define a strict and a tolerant extension depending on an indifference relation associated to that predicate. Second, we can use (...) these notions of satisfaction to define mixed consequence relations that capture non-transitive tolerant reasoning. Although we gave some empirical motivation for the use of strict and tolerant extensions, making use of them commits us to the view that sentences of the form ‘ p∨¬p ’ and ‘ p∧¬p ’ are not automatically valid or unsatisfiable, respectively. Some philosophers might take this commitment as a negative outcome of our previous proposal. We think, however, that the general ideas underlying our previous approach to vagueness can be implemented in a variety of ways. This paper explores the possibility of defining mixed notions of consequence in the more classical super/sub-valuationist setting and examines to what extent any of these notions captures non-transitive tolerant reasoning. (shrink)

Although a very recent topic in contemporary logic, the subject of combinations of logics has already shown its deep possibilities. Besides the pure philosophical interest offered by the possibility of defining mixed logic systems in which distinct operators obey logics of different nature, there are also several pragmatical and methodological reasons for considering combined logics. We survey methods for combining logics (integration of several logic systems into a homogeneous environment) as well as methods for decomposing (...) logics, showing their interesting properties and applications. (shrink)

Available for the first time in 20 years, here is the Rudolf Carnap's famous principle of tolerance by which everyone is free to mix and match the rules of ...

Deontic logic handles not only deontic modalities, but also alethic and temporal ones. In addition, individuals like authorities and addressees play an important role. R5-D5 is a system handling alethic, deontic and temporal modalities, whose adequacy has been proved in an earlier paper. Similarly for KD*UXY with sets of individuals . The present article is an attempt to construct a general system mixing R5-D5 and KD*UXY.

Monists say that the nature of truth is invariant, whichever sentence you consider; pluralists say that the nature of truth varies between different sets of sentences. The orthodoxy is that logic and logical form favour monism: there must be a single property that is preserved in any valid inference; and any truth-functional complex must be true in the same way as its components. The orthodoxy, I argue, is mistaken. Logic and logical form impose only structural constraints on a metaphysics of (...) truth. Monistic theories are not guaranteed to satisfy these constraints, and there is a pluralistic theory that does so. (shrink)

This article makes the following three programmatic points. First, an understanding of divine transcendence, prominent in Christian theology's apophatic strain, developed in tandem, both historically and logically, with ideas about creation that eventuated in a creation ex nihilo viewpoint. Such an account of divine transcendence, second, fosters an account of creation that typically mixes both natural and personalistic images and categories. The loss of such an account of transcendence since the early modern period, I suggest thirdly and in conclusion, is (...) therefore responsible in great part for the dualistic, mutually exclusive alternation between a deistic, interventionist God and pantheism so common in modern Christian thought. (shrink)

We will present all the mixed and impure disjoint three-valued logics based on the Strong Kleene schema. Some, but not all of them, are (inferentially) empty logics. We will also provide a recipe to build philosophical interpretations for each of these logics, and show why the kind of permeability that characterized them is not such a bad feature.

In this paper, questions regarding the cultural understanding of mixed race are explored, which have the ability to complicate the accepted portrayal of race in society as a black/white binary system. Thus, the acknowledgement of something other than this binary system offers new ways of theorizing about race, particularly concerning the sociopolitical implications of mixed-race designation. This paper argues that the visually mixed-race person has a certain direct ability to challenge the binary and its racist logic. Furthermore, (...) this paper goes on to offer a unique interpretation of where power for working against a racially oppressive system lies within critical mixed-race theory. (shrink)

It's natural to think that the principles expressed by the statements "Promises ought to be kept" and "We ought to help those in need" are defeasible. But how are we to make sense of this defeasibility? On one proposal, moral principles have hedges or built-in unless clauses specifying the conditions under which the principle doesn't apply. On another, such principles are contributory and, thus, do not specify which actions ought to be carried out, but only what counts in favor or (...) against them. Drawing on a defeasible logic framework, this paper sets up three models: one model for each proposal, as well as a third model capturing a mixed view on principles that combines them. It then explores the structural connections between the three models and establishes some equivalence results, suggesting that the seemingly different views captured by the models are closer than standardly thought. (shrink)

We give some applications of mixed support forcing iterations to the topics of disjoint stationary sequences and internally approachable sets. In the first half of the paper we study the combinatorial content of the idea of a disjoint stationary sequence, including its relation to adding clubs by forcing, the approachability ideal, canonical structure, the proper forcing axiom, and properties related to internal approachability. In the second half of the paper we present some consistency results related to these ideas. We (...) construct a model in which a disjoint stationary sequence exists at the successor of an arbitrary regular uncountable cardinal. We also construct models in which the properties of being internally stationary, internally club, and internally approachable are distinct. (shrink)

In the ordered Abelian group of reals with the integers as a distinguished subgroup, the projection of a finite intersection of generalized halfspaces is a finite intersection of generalized halfspaces. The result is uniform in the integer coefficients and moduli of the initial generalized halfspaces.

Alethic pluralism holds that there are many ways of being true. Such a view has been challenged to make sense of the standard account of logical validity as necessary truth preservation. In this paper, the recent solution elaborated by Aaron Cotnoir, based on an algebraic approach, is shown to be untenable. Some reflections about the relation of many-valued logics with truth pluralism are also discussed.

In this essay we develop a theory of discourse reports. The theory provides a common set of structural and interpretive principles that together account for the truth conditions of direct, indirect and mixed reports. A distinguishing feature of our view is the assumption that the complement sentence of a report divides exclusively and exhaustively into regions that characterize the content of the reported utterance and regions that characterize the form of the utterance. This assumption implies that mixed reports (...) do not imply full direct or indirect reports. We discuss a classical David‐sonian proposal by Herman Cappelen and Ernest Lepore which conflicts with our theory on this latter point and show that it is unable to account for many of the phenomena we discuss. (shrink)

Contingency and accident are two important notions in philosophy and philosophical logic. Their meanings are so close that they are mixed up sometimes, in both daily life and academic research. This indicates that it is necessary to study them in a unified framework. However, there has been no logical research on them together. In this paper, we propose a language of a bimodal logic with these two concepts, investigate its model-theoretical properties such as expressivity and frame definability. We axiomatize (...) this logic over various classes of frames, whose completeness proofs are shown with the help of a crucial schema. The interactions between contingency and accident can sharpen our understanding of both notions. Then we extend the logic to a dynamic case: public announcements. By finding the required reduction axioms, we obtain a complete axiomatization, which gives us a good application to Moore sentences. (shrink)

Alethic pluralists argue truth is a metaphysically robust higher-order property that is multiply realized by a set of diverse and domain-specific subvening alethic properties. The higher-order truth property legitimizes mixed inferences and accounts for a univocal truth predicate. Absent of this higher-order property, pluralists lack an account of the validity of mixed inferences and an adequate semantics for the truth predicate and thereby appear forced to abandon the central tenets of alethic pluralism. I argue the use of many-valued (...) logics to support pluralism fails to address the pluralist’s metaphysical problem regarding mixed inferences and mixed truth functional connectives. The high degree of heterogeneity of the alethic realizers challenges the plausibility of a single higher-order functional property. A functional property with such a heterogeneous base cannot be projectable at a theoretically significant level. The problem with mixed inferences and truth functions is but one symptom of the deeper projectability problem. (shrink)

To develop critical thinking skills, higher order ethical reasoning, a better grasp of the implications of ethical decisions, and a basis for ethical knowledge, it is necessary to explore the philosophical premises foundational to one’s ethical persuasion. No philosophical premises are more important than those pertaining to the nature of human personhood and business’ responsibility to respect the inherent value of human beings. Philosophical naturalism assigns the essence of human personhood strictly to causal interactions of physical matter. Substance dualism, on (...) the other hand, posits both a physical aspect and an immaterial substance to personhood, interacting within the totality of each being. This paper argues for the logical superiority of substance dualism in achieving the overriding objective of discerning ethical knowledge. Substance dualism offers a better explanation – and one that more closely follows the way most people commonly experience themselves and others–than naturalism for free agency and accountability, meaningful moral standards, confidence in knowing what ethical decisions to make, and the moral drive residing in conscience. (shrink)

David Earl Over is a leading cognitive scientist and, with his firm grounding in philosophical logic, he also exerts a powerful influence on the psychology of reasoning. He is responsible for not only a large body of empirical work and accompanying theory, but for advancing a major shift in thinking about reasoning, commonly known as the 'new paradigm' in the psychology of human reasoning. Over's signature mix of philosophical logic and experimental psychology has inspired generations of researchers, psychologists, and philosophers (...) alike over more than a quarter of a century. The chapters in this volume, written by a leading group of contributors including a number who helped shape the psychology of reasoning as we know it today, each take their starting point from the key themes of Over's ground-breaking work. The essays in this collection explore a wide range of central topics--such as rationality, bias, dual processes, and dual systems--as well as contemporary psychological and philosophical theories of conditionals. It concludes with an engaging new chapter, authored by David Over himself, which details and analyses the new paradigm psychology of reasoning. This book is therefore important reading for scholars, researchers, and advanced students in psychology, philosophy, and the cognitive sciences, including those who are not familiar with Over's thought already. (shrink)

Following the speech act theory, we take hypotheses and assertions as linguistic acts with different illocutionary forces. We assume that a hypothesis is justified if there is at least a scintilla of evidence for the truth of its propositional content, while an assertion is justified when there is conclusive evidence that its propositional content is true. Here we extend the logical treatment for assertions given by Dalla Pozza and Garola (1995, Erkenntnis, 43, 81–109) by outlining a pragmatic logic for assertions (...) and hypotheses. On the basis of this extension we analyse the standard logical opposition relations for assertions and hypotheses. We formulate a pragmatic square of oppositions for assertions and a hexagon of oppositions for hypotheses. Finally, we give a mixed hexagon of oppositions to point out the opposition relations for assertions and hypotheses. (shrink)

This mixed-methods case study describes the experiences of a rural health organization in Canada that was a pioneer in undergoing institutionally driven radical change. This change was advocated by senior managers and physicians with the strong backing of the government. The senior managers and physicians made a strong case for the radical change and argued that a focus on efficiency and wellness would lead to improved service and quality of patient-care. However, this radical change initiative was resisted by nurses (...) and support staff who perceived that these changes were being driven by market-based institutional logics and questioned their ethical appropriateness in a public system. They also expressed a lack of trust given the large-scale layoffs in a prior restructuring. These findings run counter to extant theory by highlighting the role of agency despite institutional pressures. Specifically, change implementers not only face the burden of justifying ethical appropriateness of institutional logics, but also are required to engage in persuasive discourse that these institutional logics protect the interests of the members. (shrink)

We show that every MALL proof-structure [9] satisfies the property of softness, originally a categorical notion introduced by Joyal. Furthermore, we show that the notion of hereditary softness precisely captures Girard’s algebraic restriction of the technical condition on proof-structures. Relying on this characterization, we prove a MALL+Mix sequentialization theorem by a proof-theoretical method, using Girard’s notion of jump. Our MALL+Mix correctness criterion subsumes the Danos/Fleury-Retoré criterion [6] for MLL+Mix.

This paper attempts to outline the logical structure of imperatives. It criticizes the prevailing view that this structure is isomorphic with that for indicatives. For "mixed" imperatives with constituents in both indicative and imperative moods (e.G., Conditional imperatives with indicative antecedents) there are features unique to imperatives. These features are specified, And consequences of them are traced. Finally, Formation rules for imperatives are stated.

In this paper, we study the introduction of modal past temporal operators in Temporal Equilibrium Logic, an hybrid formalism that mixes linear-time modalities and logic programs interpreted under stable models and their characterisation in terms of Equilibrium Logic. We show that Kamp’s translation can also be used to translate the new extension of TEL with past operators into Quantified Equilibrium Logic. Additionally, we provide a method for removing past operators that consists in replacing past-time subformulas by fresh auxiliary atoms, obtaining (...) an equivalent formula, modulo the original alphabet. (shrink)

In this paper we investigate first order common knowledge logics; i.e., modal epistemic logics based on first order logic with common knowledge operators. It is shown that even rather weak fragments of first order common knowledge logics are not recursively axiomatizable. This applies, for example, to fragments which allow to reason about names only; that is to say, fragments the first order part of which is based on constant symbols and the equality symbol only. Then formal properties (...) of "quantifying into" epistemic contexts are investigated. The results are illustrated by means of epistemic representations of Nash Equilibria for finite games with mixed strategies. (shrink)