The article presents proofs of the context freeness of a family of typelogical grammars, namely all grammars that are based on a uni- ormultimodal logic of pure residuation, possibly enriched with thestructural rules of Permutation and Expansion for binary modes.

Stephen Mumford develops his view of sport spectatorship partly through a rejection of an argument he attributes to Best, which distinguishes between two categories of sports, the ‘purposive’ and the ‘aesthetic’, on the basis of the claim that they have different principal aims. This paper considers the principal aim argument and one feature of Mumford’s rejection of that argument, namely, Best’s observation that the distinctions to which he draws attention are based on logical differences. The paper argues that Mumford (...) misconstrues Best’s argument by taking it to be about the intentions of players and athletes, while it is actually about a specific feature of the rules of each sport. (shrink)

This article is concerned with the role of moral principles, specifically the Fundamental Principles of Olympism, in the judgements of the International Olympic Committee (IOC) on matters of performance enhancement. The article begins with two pairs of distinctions, that between moral judgements and morally-laden judgements, and that between the moral judgement of cases and the ethical environment of a society. The article is concerned with working through the implications of those distinctions in the context of the IOC's judgements on performance (...) enhancement. The article favours a particularist account of the moral judgement of cases, while preserving a place for meaningful general moral statements as contributions to the ethical environment of a society, but not as general action-guiding statements or principles that can be applied to judgements in specific cases. The article illustrates the implications of this conclusion in the context of the decision-making of the IOC on performance enhancement (the case of Alain Baxter is considered). It is argued that there is a danger of the decision-making of the IOC suffering deep confusion over the difference between general and particular statements, the nature of reasons and the logic of what it is to apply a general action-guiding statement. (shrink)

In this short paper, using category theory, we argue that logical rules can be seen as fractions and logics as limit sketches.

This paper tackles foundational issues regarding the justification of logical rules. It is argued that standard accounts from both proof-theoretical and semantical points of view do not su ffi ce to account for the justification of basic logical rules. By way of response, an analysis of logical inference as acts taking place in dialogical situations is provided. In turn, this makes way for an internal justification of logical rules at the termination of dialogue, that can be (...) formalized in terms of co-constructive logic for proof and refutation [author reference omitted]. (shrink)

In his remarks from later period, Ludwig Wittgenstein is frequently concerned with so-called external roots of our logical operations. He asks questions like: ‘How is anything like logical necessity possible?‘; ‚How is possible anything like following a logical rule under normal circumstances?‘; ‚Where is the compelling force of a logical proof coming from?‘; etc. In the philosophical community, it is generally accepted that the later Wittgenstein’s remarks deal with these questions, but the philosophical motivation behind (...) these remarks is still not clear. Instead, there is a growing disagreement among various interpretations over these remarks. The present consideration is built upon the belief that Wittgenstein’s remarks try to communicate a new sense to us. I argue that, contrary to the generally accepted view, there is a new and positive sense in these remarks. My aim in this paper is to clarify the philosophical motivation behind them. (shrink)

In this paper I construct a system of semantics for classical and intuitionistic propositional logic based on epistemic norms governing belief expansion. Working in the AGM-framework of belief change, I give a generalisation of Gärdenfors’ notion of belief systems which can be defined without reference to a logical consequence operator by using a version of the Ramsey Test. These belief expansion systems can then be used to define epistemic models which are sound and complete for either classical or intuitionistic (...) propositional logic depending on which of the two notions of epistemic validity, identified by Levi and Arló-Costa, is used. Finally, I offer a discussion on how these results can be understood as providing a model theory within the framework of logical expressivism. (shrink)

The use of conventional logical connectives either in logic, in mathematics, or in both cannot determine the meanings of those connectives. This is because every model of full conventional set theory can be extended conservatively to a model of intuitionistic set plus class theory, a model in which the meanings of the connectives are decidedly intuitionistic and nonconventional. The reasoning for this conclusion is acceptable to both intuitionistic and classical mathematicians. En route, I take a detour to prove that, (...) given strictly intuitionistic principles, classical negation cannot exist. (shrink)

As far as logic is concerned, the conclusion of Michael Dummett's manifestability argument is that intuitionistic logic, as first developed by Heyting, satisfies the semantic requirements of antirealism. The argument may be roughly sketched as follows: since we cannot manifest a grasp of possibly justification-transcendent truth conditions, we must countenance conditions which are such that, at least in principle and by the very nature of the case, we are able to recognize that they are satisfied whenever they are. Intuitionistic logic (...) satisfies the semantic requirement that we should either eschew the notion of truth altogether and replace it by provability in principle, or constrain it by provability in principle (Dummett [1973] 1978). (Handout: 1-5). Some philosophers have argued that the traditional antirealist desideratum of decidability in principle is too weak, so that semantic antirealism properly construed must be committed to effective decidability. As such, it either leads to strict finitism (Wright [1982] 1993) [Handout: 6] or to a much stronger kind of logical revisionism than the one considered by intuitionists (whether or not they accept the manifestability argument): substructural logics, and in particular linear logics, rather than intuitionistic logic, satisfy the semantic requirements of strict antirealism (Dubucs and Marion 2004) [Handout: 8-9]. I shall develop two different kinds of replies. The first is concerned with the notion of meaning per se and looks to strict finitism directly, although not on the ground that it would provide a correct way of dealing with Soritic-type paradoxes (the original and primary focus of discussion in Wright [1982] 1993). The second is concerned with the justification of structural and logical rules in a natural deduction system à la Gentzen. It will deal in particular with the criticism of the structural rules of Weakening and Contraction [Handout: 10 and 11]. The first kind of reply, which Dummett has partially taken into consideration, is that if we jettison the effectively vs. in principle distinction, as applied to manifestability-type arguments, we end up with an unsatisfactory explanation of how the meaning of statements covering the practically unsurveyable or pro tempora undecided cases is fixed. The idea is that if we have a method which may be used over some small range, we have determined a way of applying the method everywhere in principle and that this is enough as far as fixing meaning is concerned. Decidability in principle is just what we need with respect to manifestation of grasp of meaning. In this perspective, antirealism shouldn't be strict and manifestability-type arguments need not be applied as far as the strict finitist would want to [Handout: 7]. It follows that there is no reason to think that practical feasibility should be built in assertibility conditions and proofs be construed dynamically as acts in the strict antirealist's acception of that term [Handout: 8]. I shall then look at one radical antirealist principle disqualifying structural rules, namely Token Preservation, arguing against Bonnay and Cozic's criticisms of Dubucs and Marion (Bonnay and Cozic 3 2007) that some conceptual support may be provided for Token Preservation, which doesn't rely on a causal misreading of the turnstile [Handout: 13, 14]. I shall then assess the merits and limits of radical antirealism and the logic of feasible proofs with respect to the original Dummettian argument in favour of semantic antirealism (provided it has indeed revisionist implications for logic), whether the radical antirealist merely stipulates what human feasibility amounts to, or dispenses with structural rules in order to argue in favour of a curb on the epistemic idealizations they unwarrantedly embed. It will be noted here that there is a great difference, conceptually speaking, between the rejection of classical logic via the curbing of the epistemic idealizations embedded in structural rules, and the rejection of classical logic via the criticism of the introduction and elimination rules which fix the meaning of the classical constants. The kind of logical revisionism envisaged by intuitionists from Heyting on is in many respects stronger than the one envisaged by advocates of linear logic, should they ground their arguments on an endorsement of strict antirealism. The crucial case of excluded middle is telling. Because of the splitting of the constants in linear logic (Handout: 12], two distinct laws of excluded middle may be formulated (Handout: 15), but the traditional arguments of Brouwer and Heyting against the classical law yield a stronger revision than the substructural revision with its admission of these two (very) weak versions of the law. (Project: a clearer conception is needed of how the introduction and elimination rules for the logical connectives in the intuitionistic calculus depend on the structural rules which the radical antirealist wishes to reject.). (shrink)

The inversion principle for logical rules expresses a relationship between introduction and elimination rules for logical constants. Hallnäs & Schroeder-Heister proposed the principle of definitional reflection, which embodies basic ideas of inversion in the more general context of clausal definitions. For the context of admissibility statements, this has been further elaborated by Schroeder-Heister. Using the framework of definitional reflection and its admissibility interpretation, we show that, in the sequent calculus of minimal propositional logic, the left introduction rules are (...) admissible when the right introduction rules are taken as the definitions of the logical constants and vice versa. This generalizes the well-known relationship between introduction and elimination rules in natural deduction to the framework of the sequent calculus. (shrink)

The inversion principle for logical rules expresses a relationship between introduction and elimination rules for logical constants. Hallnäs & Schroeder-Heister proposed the principle of definitional reflection, which embodies basic ideas of inversion in the more general context of clausal definitions. For the context of admissibility statements, this has been further elaborated by Schroeder-Heister . Using the framework of definitional reflection and its admissibility interpretation, we show that, in the sequent calculus of minimal propositional logic, the left introduction rules (...) are admissible when the right introduction rules are taken as the definitions of the logical constants and vice versa. This generalizes the well-known relationship between introduction and elimination rules in natural deduction to the framework of the sequent calculus. (shrink)

In this chapter, I discuss some problems of Kant’s logic corpus while recognizing its richness and potential value. I propose and explain a methodic way to approach it. I then test the proposal by showing how we may use various mate- rials from the corpus to construct a Kantian demonstration of the formal rules of thinking (or judging) that lie at the base of Kant’s Metaphysical Deduction. The same proposal can be iterated with respect to other topics. The said demonstration (...) will have cleared the path for such iterations. (shrink)

Antoine Arnauld and Pierre Nicole were philosophers and theologians associated with Port-Royal Abbey, a centre of the Catholic Jansenist movement in seventeenth-century France. Their enormously influential Logic or the Art of Thinking, which went through five editions in their lifetimes, treats topics in logic, language, theory of knowledge and metaphysics, and also articulates the response of 'heretical' Jansenist Catholicism to orthodox Catholic and Protestant views on grace, free will and the sacraments. In attempting to combine the categorical theory of the (...) proposition with a Cartesian account of knowledge, their Logic represents the classical view of judgment which inspired the modern transformation in logic and semantic theory by Frege, Russell, Wittgenstein and recent philosophers. This edition presents a new translation of the text, together with a historical introduction and suggestions for further reading. (shrink)

We explore the consequences, for logical system-building, of taking seriously the aim of having irredundant rules of inference, and a preference for proofs of stronger results over proofs of weaker ones. This leads one to reconsider the structural rules of REFLEXIVITY, THINNING, and CUT. REFLEXIVITY survives in the minimally necessary form $\varphi:\varphi$. Proofs have to get started. CUT is subject to a CUT-elimination theorem, to the effect that one can always make do without applications of CUT. So CUT is (...) redundant, and should not be a rule of the system. CUT-elimination, however, in the context of the usual forms of logical rules, requires the presence, in the system, of THINNING. But THINNING, it turns out, is not really necessary. Provided only that one liberalizes the statement of certain logical rules in an appropriate way, one can make do without CUT or THINNING. From the methodological point of view of this study, the logical rules ought to be framed in this newly liberalized form. These liberalized logical rules determine the system of core logic. Given any intuitionistic Gentzen proof of $\Delta:\varphi$, one can determine from it a Core proof of some subsequent of $\Delta:\varphi$. Given any classical Gentzen proof of $\Delta:\varphi$, one can determine from it a classical Core proof of some subsequent of $\Delta:\varphi$. In both cases the Core proof is of a result at least as strong as that of the Gentzen proof; and the only structural rule used is $\varphi:\varphi$. (shrink)

The logical omniscience problem, whereby standard models of epistemic logic treat an agent as believing all consequences of its beliefs and knowing whatever follows from what else it knows, has received plenty of attention in the literature. But many attempted solutions focus on a fairly narrow specification of the problem: avoiding the closure of belief or knowledge, rather than showing how the proposed logic is of philosophical interest or of use in computer science or artificial intelligence. Sentential epistemic logics, (...) as opposed to traditional possible worlds approaches, do not suffer from the problems of logical omniscience but are often thought to lack interesting epistemic properties. In this paper, I focus on the case of rule-based agents, which play a key role in contemporary AI research but have been neglected in the logical literature. I develop a framework for modelling monotonic, nonmonotonic and introspective rule-based reasoners which have limited cognitive resources and prove that the resulting models have a number of interesting properties. An axiomatization of the resulting logic is given, together with completeness, decidability and complexity results. (shrink)

Visser's rules form a basis for the admissible rules of . Here we show that this result can be generalized to arbitrary intermediate logics: Visser's rules form a basis for the admissible rules of any intermediate logic for which they are admissible. This implies that if Visser's rules are derivable for then has no nonderivable admissible rules. We also provide a necessary and sufficient condition for the admissibility of Visser's rules. We apply these results to some specific intermediate logics and (...) obtain that Visser's rules form a basis for the admissible rules of, for example, De Morgan logic, and that Dummett's logic and the propositional Gödel logics do not have nonderivable admissible rules. (shrink)

We discuss a `negative' way of defining frame classes in (multi)modal logic, and address the question of whether these classes can be axiomatized by derivation rules, the `non-ξ rules', styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If Λ is a derivation system having a set of axioms that are special Sahlqvist formulas and Λ+ is the extension of Λ with a set of non-ξ rules, then Λ+ (...) is strongly sound and complete with respect to the class of frames determined by the axioms and the rules. (shrink)

In the theory of meaning, it is common to contrast truth-conditional theories of meaning with theories which identify the meaning of an expression with its use. One rather exact version of the somewhat vague use-theoretic picture is the view that the standard rules of inference determine the meanings of logical constants. Often this idea also functions as a paradigm for more general use-theoretic approaches to meaning. In particular, the idea plays a key role in the anti-realist program of Dummett (...) and his followers. In the theory of truth, a key distinction now is made between substantial theories and minimalist or deflationist views. According to the former, truth is a genuine substantial property of the truth-bearers, whereas according to the latter, truth does not have any deeper essence, but all that can be said about truth is contained in T-sentences (sentences having the form: ‘P’ is true if and only if P). There is no necessary analytic connection between the above theories of meaning and truth, but they have nevertheless some connections. Realists often favour some kind of truth-conditional theory of meaning and a substantial theory of truth (in particular, the correspondence theory). Minimalists and deflationists on truth characteristically advocate the use theory of meaning (e.g. Horwich). Semantical anti-realism (e.g. Dummett, Prawitz) forms an interesting middle case: its starting point is the use theory of meaning, but it usually accepts a substantial view on truth, namely that truth is to be equated with verifiability or warranted assertability. When truth is so understood, it is also possible to accept the idea that meaning is closely related to truth-conditions, and hence the conflict between use theories and truth-conditional theories in a sense disappears in this view. (shrink)

Uniform infinite bases are defined for the single-conclusion and multiple-conclusion admissible rules of the implication–negation fragments of intuitionistic logic and its consistent axiomatic extensions . A Kripke semantics characterization is given for the structurally complete implication–negation fragments of intermediate logics, and it is shown that the admissible rules of this fragment of form a PSPACE-complete set and have no finite basis.

Large portions of mathematics such as algebra and geometry can be formalized using first-order axiomatizations. In many cases it is even possible to use a very well-behaved class of first-order axioms, namely, what are called coherent or geometric implications. Such class of axioms can be translated to inference rules that can be added to a sequent calculus while preserving its structural properties. In this work, this fundamental result is extended to their infinitary generalizations as extensions of sequent calculi for both (...) classical and intuitionistic infinitary logic. As an application, a simple proof of the infinitary Barr’s theorem without the axioms of choice is shown. (shrink)

A unified and modular falsification-aware single-succedent Gentzen-style framework is introduced for classical, paradefinite, paraconsistent, and paracomplete logics. This framework is composed of two special inference rules, referred to as the rules of explosion and excluded middle, which correspond to the principle of explosion and the law of excluded middle, respectively. Similar to the cut rule in Gentzen’s LK for classical logic, these rules are admissible in cut-free LK. A falsification-aware single-succedent Gentzen-style sequent calculus fsCL for classical logic is formalized (...) based on the proposed framework. The calculus fsCL is obtained from the existing falsification-aware single-succedent Gentzen-style sequent calculus GN4 for Nelson’s paradefinite (or paraconsistent) four-valued logic N4 by adding the rules of explosion and excluded middle. A falsification-aware single-succedent Gentzen-style sequent calculus GN3 for Nelson’s paracomplete three-valued logic N3 is also obtained from GN4 by adding the rule of explosion. The cut-elimination theorems for fsCL, GN3, and some of their neighbors as well as the Glivenko theorem for fsCL are proved. (shrink)

Syntactic logics do not suffer from the problems of logical omniscience but are often thought to lack interesting properties relating to epistemic notions. By focusing on the case of rule-based agents, I develop a framework for modelling resource-bounded agents and show that the resulting models have a number of interesting properties.

We provide five semantic preservation properties which apply to the various rules -- primitive, derived and admissible -- of Hilbert-style axiomatizations of relevant logics. These preservation properties are with respect to the Routley-Meyer semantics, and consist of various truth- preservations and validity-preservations from the premises to the conclusions of these rules. We establish some deduction theorems, some persistence theorems and some soundness and completeness theorems, for these preservation properties. We then apply the above ideas, as best we can, to the (...) classical sentential and predicate calculi, to normal and non- normal modal logics, and to many- valued logics. (shrink)

This paper surveys the various forms of Deduction Theorem for a broad range of relevant logics. The logics range from the basic system B of Routley-Meyer through to the system R of relevant implication, and the forms of Deduction Theorem are characterized by the various formula representations of rules that are either unrestricted or restricted in certain ways. The formula representations cover the iterated form,A 1 .A 2 . ... .A n B, the conjunctive form,A 1&A 2 & ...A n (...) B, the combined conjunctive and iterated form, enthymematic version of these three forms, and the classical implicational form,A 1&A 2& ...A n B. The concept of general enthymeme is introduced and the Deduction Theorem is shown to apply for rules essentially derived using Modus Ponens and Adjunction only, with logics containing either (A B)&(B C) .A C orA B .B C .A C. (shrink)

A full separation theorem for the derivable rules of intuitionistic linear logic without bounds, 0 and exponentials is proved. Several structural consequences of this theorem for subreducts of (commutative) residuated lattices are obtained. The theorem is then extended to the logic LR+ and its proof is extended to obtain the finite embeddability property for the class of square increasing residuated lattices.

In this paper, the relationship between the e-value of a complex hypothesis, H, and those of its constituent elementary hypotheses, Hj, j = 1… k, is analyzed, in the independent setup. The e-value of a hypothesis H, ev, is a Bayesian epistemic, credibility or truth value defined under the Full Bayesian Significance Testing mathematical apparatus. The questions addressed concern the important issue of how the truth value of H, and the truth function of the corresponding FBST structure M, relate to (...) the truth values of its elementary constituents, Hj, and to the truth functions of their corresponding FBST structures Mj, respectively. (shrink)

An algorithm recognizing admissibility of inference rules in generalized form (rules of inference with parameters or metavariables) in the intuitionistic calculus H and, in particular, also in the usual form without parameters, is presented. This algorithm is obtained by means of special intuitionistic Kripke models, which are constructed for a given inference rule. Thus, in particular, the direct solution by intuitionistic techniques of Friedman's problem is found. As a corollary an algorithm for the recognition of the solvability of (...) class='Hi'>logical equations in H and for constructing some solutions for solvable equations is obtained. A semantic criterion for admissibility in H is constructed. (shrink)

Logic was revered in the thirteenth century, perhaps more highly than it has been revered before or since. In the Muslim East, logic was an integral part of the syllabus of schools and found to be especially helpful for legal studies. It was at this time that The Canons of Logic was composed by Najm al-Din al-Katibi, a scholar of the Shafi'i school of law. The Rules of Logic is the most widely read introduction to logic in the Arabic-speaking world. (...) It has probably enjoyed a longer shelf-life than any other logic textbook ever written, having been in use by madrasa students from the early fourteenth century up until the present day. Building on the theories of Avicenna, al-Razi, and other pioneers of logic, al-Katibi discusses the many pitfalls of building arguments and setting out unambiguous claims in natural language. The enduring nature of the text is a testament to al-Katibi and his impact on concepts of formal discourse and argument. This new translation of The Canons of Logic provides new life to an influential text that has shaped the work of scholars of logic for centuries. (shrink)

We reflect on lessons that the lottery and preface paradoxes provide for the logic of uncertain inference. One of these lessons is the unreliability of the rule of conjunction of conclusions in such contexts, whether the inferences are probabilistic or qualitative; this leads us to an examination of consequence relations without that rule, the study of other rules that may nevertheless be satisfied in its absence, and a partial rehabilitation of conjunction as a ‘lossy’ rule. A second (...) lesson is the possibility of rational inconsistent belief; this leads us to formulate criteria for deciding when an inconsistent set of beliefs may reasonably be retained. (shrink)

The logic B+ is Routley and Meyer’s basic positive logic. We define the logics BK+ and BK'+ by adding to B+ the K rule and to BK+ the characteristic S4 axiom, respectively. These logics are endowed with a relatively strong non-constructive negation. We prove that all the logics defined lack the K axiom and the standard paradoxes of consistency.

The interplay of introduction and elimination rules for propositional connectives is often seen as suggesting a distinguished role for intuitionistic logic. We prove three formal results about intuitionistic propositional logic that bear on that perspective, and discuss their significance.

In this paper, we will develop an algebraic study of substructural propositional logics over FLew, i.e. the logic which is obtained from intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Lukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework.

Originally published in 1968. This is a critical study of the concept of 'rule' featuring in law, ethics and much philosophical analysis which the author uses to investigate the concept of 'rationality'. The author indicates in what manner the modes of reasoning involved in reliance upon rules are unique and in what fashion they provide an alternative both to the modes of logico-mathematical reasoning and to the modes of scientific reasoning. This prepares the groundwork for a methodology meeting the (...) requirements of the fields using rules such as law and ethics which could be significant for communications theory and the use of computers in normative fields. Other substantive issues related to the mainstream of legal philosophy are discussed - theories of interpretation, the notion of purpose and the requirements of principled decision-making. The book utilizes examples drawn from English and American legal decisions to suggest how the positions of legal positivism and of natural law are equally artificial and misleading. (shrink)

In a recent paper, Michael Friedman and Hilary Putnam argued that the Luders rule is ad hoc from the point of view of the Copenhagen interpretation but that it receives a natural explanation within realist quantum logic as a probability conditionalization rule. Geoffrey Hellman maintains that quantum logic cannot give a non-circular explanation of the rule, while Jeffrey Bub argues that the rule is not ad hoc within the Copenhagen interpretation. As I see it, all four (...) are wrong. Given that there is to be a projection postulate, there are at least two natural arguments which the Copenhagen advocate can offer on behalf of the Luders rule, contrary to Friedman and Putnam. However, the argument which Bub offers is not a good one. At the same time, contrary to Hellman, quantum logic really does provide an explanation of the Luders rule, and one which is superior to that of the Copenhagen account, since it provides an understanding of why there should be a projection postulate at all. (shrink)

This paper deals with the question of the logicality of modal logics from a proof-theoretic perspective. It is argued that if Dos̆en’s analysis of logical constants as punctuation marks is embraced, it is possible to show that all the modalities in the cube of normal modal logics are indeed logical constants. It will be proved that the display calculus for each displayable modality admits a purely structural presentation based on double-line rules which, following Dos̆en’s analysis, allows us to (...) claim that the corresponding modal operators are logical constants. (shrink)

This dissertation develops, justifies, and examines the jurisprudential implications of a non-monotonic theory of common law legal reasoning. Legal rules seem to have exceptions but identifying all of them is difficult. This hinders attempts to formalize legal rules using classical logics. Non-monotonic logics allow defeasible inference, permitting rules that hold generally but can be defeated in the presence of exceptions. This ameliorates the problem of characterizing all exceptions to a rule, because exceptions can be added piecemeal while the (...) class='Hi'>rule remains. The first portion of the dissertation rebuts a prominent criticism leveled at a large class of theories of legal reasoning that includes my theory. The charge is that no coherent theory can recognize both the difference between distinguishing and overruling, and the constraint of precedent. The critics argue that is more important than and that can only be explained by monotonic legal rules. Drawing on cognitive psychology as well as legal theory, I show that coherent theories, such as my own, can accommodate both and. The second chapter provides motivation for understanding precedential constraint in terms of non-monotonic default rules and introduces my positive theory, which elaborates on John Horty's work in treating legal rules as prioritized defaults involving reasons. I motivate and implement a relaxation of Horty's restrictions on the form of rules to allow a more fine-grained characterization of precedent. Finally, I explore the relationship between these relaxations and the concept of precedent. The final section explains how my theory fits into the traditional jurisprudential ecosystem. I demonstrate that, contrary to assertions in the legal theory literature, this non-monotonic approach is entirely compatible with positivism's commitment to extracting rules from authoritative legal sources, namely court opinions. I also suggest how it might be attractive to law and economics theorists, pragmatists, and followers of Dworkin. (shrink)

We outline a pragmatic-normative understanding of logic as a discipline that is completely anchored in the sphere of action, rules, means and ends: We characterize inferring as a speech act which is in need of regulation and we connect inferential rules with consequence relations. Furthermore, we present a scenario which illustrates how one actually assesses or can in principle assess the quality of logical rules with respect to justificatory questions. Finally, we speculate on the origin of logical rules (...) as a means of supporting our practice of inferring. (shrink)

In our previous paper [5], we have studied Kripke-type semantics for propositional logics without the contraction rule. In this paper, we will extend our argument to predicate logics without the structure rules. Similarly to the propositional case, we can not carry out Henkin's construction in the predicate case. Besides, there exists a difficulty that the rules of inference () and () are not always valid in our semantics. So, we have to introduce a notion of normal models.