In Zielonka (1981a, 1989), I found an axiomatics for the product-free calculus L of Lambek whose only rule is the cut rule. Following Buszkowski (1987), we shall call such an axiomatics linear. It was proved that there is no finite axiomatics of that kind. In Lambek's original version of the calculus (cf. Lambek, 1958), sequent antecedents are non empty. By dropping this restriction, we obtain the variant L 0 of L. This modification, introduced in the early 1980s (see, (...) e.g., Buszkowski, 1985; Zielonka, 1981b), did not gain much popularity initially; a more common use of L 0 has only occurred within the last few years (cf. Roorda, 1991: 29). In Zielonka (1988), I established analogous results for the restriction of L 0 to sequents without left (or, equivalently, right) division. Here, I present a similar (cut-rule) axiomatics for the whole of L 0. This paper is an extended, corrected, and completed version of Zielonka (1997). Unlike in Zielonka (1997), the notion of rank of an axiom is introduced which, although inessential for the results given below, may be useful for the expected non-finite-axiomatizability proof. (shrink)
In this paper we show that, in Gentzen systems, there is a close relation between two of the main characters in algebraic logic and proof theory respectively: protoalgebraicity and the cut rule. We give certain conditions under which a Gentzen system is protoalgebraic if and only if it possesses the cut rule. To obtain this equivalence, we limit our discussion to what we call regular sequent calculi, which are those comprising some of the structural rules and some logical (...) rules, in a sense we make precise. We note that this restricted set of rules includes all the usual rules in the literature. We also stress the difference between the case of two-sided sequents and the case of many-sided sequents, in which more conditions are needed. (shrink)
An axiomatics of the product-free syntactic calculus L ofLambek has been presented whose only rule is the cut rule. It was alsoproved that there is no finite axiomatics of that kind. The proofs weresubsequently simplified. Analogous results for the nonassociativevariant NL of L were obtained by Kandulski. InLambek's original version of the calculus, sequent antecedents arerequired to be nonempty. By removing this restriction, we obtain theextensions L 0 and NL 0 ofL and NL, respectively. Later, the finiteaxiomatization problem (...) for L 0 andNL 0 was partially solved, viz., for formulas withoutleft (or, equivalently, right) division and an (infinite) cut-ruleaxiomatics for the whole of L 0 has been given. Thepresent paper yields an analogous axiomatics forNL 0. Like in the author's previous work, the notionof rank of an axiom is introduced which, although inessentialfor the results given below, may be useful for the expectednonfinite-axiomatizability proof. (shrink)
ABSTRACT: A detailed presentation of Stoic theory of arguments, including truth-value changes of arguments, Stoic syllogistic, Stoic indemonstrable arguments, Stoic inference rules (themata), including cut rules and antilogism, argumental deduction, elements of relevance logic in Stoic syllogistic, the question of completeness of Stoic logic, Stoic arguments valid in the specific sense, e.g. "Dio says it is day. But Dio speaks truly. Therefore it is day." A more formal and more detailed account of the Stoic theory of deduction can be found (...) in S. Bobzien, Stoic Syllogistic, OSAP 1996. (shrink)
ABSTRACT: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental rules which (...) establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments (cut type rules and antilogism). The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed (similar to tableau methods). The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented, and similarities to relevance logic are pointed out. (shrink)
A simple method is provided for translating proofs in Grentzen's LK into proofs in Gentzen's LJ with the Peirce rule adjoined. A consequence is a simpler cut elimination operator for LJ + Peirce that is primitive recursive.
It has been maintained by Smullyan that the importance of cut-free proofs does not stem from cut elimination per se but rather from the fact that they satisfy the subformula property. In accordance with such a viewpoint in this paper we introduce <span class='Hi'>analytic</span> cut trees, a system from which cuts cannot be eliminated but satisfying the subformula property. Like tableaux <span class='Hi'>analytic</span> cut trees are a refutation system but unlike tableaux they have a single inference rule (a form (...) of the <span class='Hi'>analytic</span> cut rule) and several branch closure rules. The main advantage of <span class='Hi'>analytic</span> cut trees over tableaux is efficiency: while <span class='Hi'>analytic</span> cut trees can simulate tableaux with an increase in complexity by at most a constant factor, tableaux cannot polynomially simulate <span class='Hi'>analytic</span> cut trees. Indeed <span class='Hi'>analytic</span> cut trees are intrinsically more efficient than any cut-free system. (shrink)
The motivation for Core Logic is explained. Its system of proof is set out. It is then shown that, although the system has no Cut rule, its relation of deducibility obeys Cut with epistemic gain.
We provide a constructive, direct, and simple proof of the completeness of the cut-free part of the hypersequential calculus for G¨odel logic (thereby proving both completeness of the calculus for its standard semantics, and the admissibility of the cut rule in the full calculus). We then extend the results and proofs to derivations from assumptions, showing that such derivations can be confined to those in which cuts are made only on formulas which occur in the assumptions.
We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G 0(these logics are the smallest normal modal logics containing K and the schemata A A, A A and A ( A); A A and AA; A A and ((A A) A) A resp.) with the following properties: the calculi for S4.04 and G 0are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, (...) the so-called analytical cut-rule.In addition we show that G 0is not compact (and therefore not canonical), and we proof with the tableau-method that G 0is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G 0is decidable and also characterised by the class of all frames for G 0. (shrink)
In this paper we study families of resource aware logics that explore resource restriction on rules; in particular, we study the use of controlled cut-rule and introduce three families of parameterised logics that arise from different ways of controlling the use of cut. We start with a formulation of classical logic in which cut is non-eliminable and then impose restrictions on the use of cut. Three Cut-and-Pay families of logics are presented, and it is shown that each family provides (...) an approximation process for full propositional classical logic when the control over the use of cut is progressively weakened. A sound and complete semantics is given for each component of each of the three families of approximated logics. One of these families is shown to possess the uniform substitution property, a new result for approximated reasoning. A tableau based decision procedure is presented for each element of the approximation families and the complexity of each decision procedure is studied. We show that there are families in which every component logic can be decided polynomially. (shrink)
In this paper we present a sequent calculus for propositional dynamic logic built using an enriched version of the tree-hypersequent method and including an infinitary rule for the iteration operator. We prove that this sequent calculus is theoremwise equivalent to the corresponding Hilbert-style system, and that it is contraction-free and cut-free. All results are proved in a purely syntactic way.
Normal 0 false false false EN-US ZH-TW X-NONE /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} The Mohist conceives the dao -following issue as “ how we can put dao in words and speeches into practice.” The dao -following issue is the Mohist counterpart of Wittgenstein’s rule-following problem. This paper aims to shed light on the rule-following issue in terms of the Mohist answer (...) to the dao-following problem. The early Mohist takes fa （法 , standard ） and the later Mohist takes lei （類 , analogy ） as the key to the dao -following issue. I argue that the way of fa is not viable. Fa comes in various forms, but all of them are regarded as being cut off from everyday life and therefore subject to various interpretations and, hence, incapable of action-guiding. On the other hand, the Mohist lei represents a kind of life world action drama. A lei situates various elements of action in the context of an everyday life scene. I argue that lei is more qualified than fa in answering to the dao-following issue. I also show that lei substantializes what McDowell calls the “course between a Scylla and a Charybdis” hinted in terms of Wittgenstein’s idea of “custom,” “practice,” and “institution” in regard to the rule-following problem. . (shrink)
We present a cut-free tableau calculus with histories and variables for the EXPTIME-complete multi-modal logic of common knowledge (LCK). Our calculus constructs the tableau using only one pass, so proof-search for testing theoremhood of ϕ does not exhibit the worst-case EXPTIME-behaviour for all ϕ as in two-pass methods. Our calculus also does not contain a “finitized ω-rule” so that it detects cyclic branches as soon as they arise rather than by worst-case exponential branching with respect to the size of (...) ϕ. Moreover, by retaining the rooted-tree form from traditional tableaux, our calculus becomes amenable to the vast array of optimisation techniques which have proved essential for “practical” automated reasoning in very expressive description logics. Our calculus forms the basis for developing a uniform framework for the family of all fix-point logics of common knowledge. However, there is still no “free lunch” as, in the worst case, our method exhibits 2EXPTIME-behaviour. A prototype implementation can be found at twb.rsise.anu.edu.au which allows users to test formulae via a simple graphical interface. (shrink)
In 1983, Valentini presented a syntactic proof of cut elimination for a sequent calculus GLSV for the provability logic GL where we have added the subscript V for “Valentini”. The sequents in GLSV were built from sets, as opposed to multisets, thus avoiding an explicit contraction rule. From a syntactic point of view, it is more satisfying and formal to explicitly identify the applications of the contraction rule that are ‘hidden’ in these set based proofs of cut elimination. (...) There is often an underly ing assumption that the move to a proof of cut elimination for sequents built from multisets is easy. Recently, however, it has been claimed that Valentini’s arguments to eliminate cut do not terminate when applied to a multiset formulation of GLSV with an explicit rule of contraction. The claim has led to much confusion and various authors have sought new proofs of cut elimination for GL in a multiset setting. Here we refute this claim by placing Valentini’s arguments in a formal setting and proving cut elimination for sequents built from multisets. The formal setting is particularly important for sequents built from multisets, in order to accurately account for the interplay between the weakening and contraction rules. Furthermore, Valentini’s original proof relies on a novel induction parameter called “width” which is computed ‘globally’. It is diffi cult to verify the correctness of his induction argument based on “width”. In our formulation however, verification of the induction argument is straight forward. Finally, the multiset setting also introduces a new complication in the the case of contractions above cut when the cut formula is boxed. We deal with this using a new transformation based on Valentini’s original arguments. Finally, we show that the algorithm purporting to show the non termi nation of Valentini’s arguments is not a faithful representation of the original arguments, but is instead a transformation already known to be insufficient. (shrink)
In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of cut admissibility. We (...) then show that for most of the systems under consideration the labelling mechanism can be avoided by choosing an appropriate way of structuring theories. One peculiar feature of our proof systems is the use of restart rules which allow to re-ask the original goal of a deduction. In case of K, K4, S4 and G, we can eliminate such a rule, without loosing completeness. In all the other cases, by dropping such a rule, we get an intuitionistic variant of each system. The present results are part of a larger project of a goal directed proof theory for non-classical logics; the purpose of this project is to show that most implicational logics stem from slight variations of a unique deduction method, and from different ways of structuring theories. Moreover, the proof systems we present follow the logic programming style of deduction and seem promising for proof search [Gabbay and Reyle 84, Miller et al. 91]. (shrink)
The dead donor rule justifies current practice in organ procurement for transplantation and states that organ donors must be dead prior to donation. The majority of organ donors are diagnosed as having suffered brain death and hence are declared dead by neurological criteria. However, a significant amount of unrest in both the philosophical and the medical literature has surfaced since this practice began forty years ago. I argue that, first, declaring death by neurological criteria is both unreliable and unjustified (...) but further, the ethical principles which themselves justify the dead donor rule are better served by abandoning that rule and instead allowing individuals who have suffered severe and irreversible brain damage to become organ donors, even though they are not yet dead and even though the removal of their organs would be the proximal cause of death. (shrink)
In this paper I argue that the most prominent and familiar features of Wittgenstein’s rule following considerations generate a powerful argument for the thesis that most of our concepts are innate, an argument that echoes a Chomskyan poverty of the stimulus argument. This argument has a significance over and above what it tells us about Wittgenstein’s implicit commitments. For, it puts considerable pressure on widely held contemporary views of concept learning, such as the view that we learn concepts by (...) constructing prototypes. This should lead us to abandon our general default hostility to concept nativism and be much more sceptical of claims made on behalf of learning theories. (shrink)
The aim of this paper is to discover whether or not a solitary individual, a human being isolated from birth, could become a rule-follower. The argumentation against this possibility rests on the claim that such an isolate could not become aware of a normative standard, with which her actions could agree or disagree. As a consequence, theorists impressed by this argumentation adopt a view on which the normativity of rules arises from corrective practices in which agents engage in a (...) community. However, it has been suggested that an isolated individual could engage in such a practice by herself. Three prospective examples of such cases are considered, and the possibility of solitary rule-following is vindicated. Furthermore, the nature of the goals at which rule-following practices generally aim is clarified. (shrink)
A modest solution to the problem(s) of rule-following is defended against Kripkensteinian scepticism about meaning. Even though parts of it generalise to other concepts, the theory as a whole applies to response-dependent concepts only. It is argued that the finiteness problem is not nearly as pressing for such concepts as it may be for some other kinds of concepts. Furthermore, the modest theory uses a notion of justification as sensitivity to countervailing conditions in order to solve the justification problem. (...) Finally, in order to solve the normativity problem, it relies on substantial specifications of normal conditions such as those that have been proposed by Crispin Wright and Mark Johnston, rather than on Philip Pettit's functionalist specification. This theory is modest in that it does not meet the demands of Kripke's sceptic in full. Arguments are provided as to why this is not needed. (shrink)
The paper explicates a version of dispositionalism and defends it against Kripke's objections (in his "Wittgenstein on Rules and Private Language") that 1) it leaves out the normative aspect of a rule, 2) it cannot account for the directness of the knowledge one has of what one meant, and 3) regarding rules for computable functions of numbers, a) there are numbers beyond one's capacity to consider and b) there are people who are disposed to make systematic mistakes in computing (...) values of functions they understand perfectly well. (shrink)
What is objectivity? What is the rule of law? Are the operations of legal systems objective? If so, in what ways and to what degrees are they objective? Does anything of importance depend on the objectivity of law? These are some of the principal questions addressed by Matthew H. Kramer in this lucid and wide-ranging study that introduces readers to vital areas of philosophical enquiry.
Many philosophers believe that agents are self-ruled only when ruled by their (authentic) selves. Though this view is rarely argued for explicitly, one tempting line of thought suggests that self-rule is just obviously equivalent to rule by the self . However, the plausibility of this thought evaporates upon close examination of the logic of ‘self-rule’ and similar reflexives. Moreover, attempts to rescue the account by recasting it in negative terms are unpromising. In light of these problems, this (...) paper instead proposes that agents are self-ruled only when not ruled by others. One reason for favouring this negative social view is its ability to yield plausible conclusions concerning various manipulation cases that are notoriously problematic for nonsocial accounts of self-rule. A second reason is that the account conforms with ordinary usage. It is concluded that self-rule may be best thought of as an essentially social concept. (shrink)
The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle (...) in Posterior analytics is used to distinguish between conceptions that share the same name but are substantively different: for example the search for a broader genus including all mathematical objects; the search for a common character of different species of mathematical objects; and the effort to treat magnitudes as numbers. (shrink)
The view that psychological episodes have a physical nature (physicalism) and the view that they have a mental nature (Cartesian dualism) can be distinguished from the view that they have a purely normative nature. I explore some strands of a distinct, fourth view: psychological episodes are what they are because of the actual and possible relations of defeasible justification in which they stand; defeasible justification is an internal relation; it is not at bottom a normative matter; rule-following presupposes such (...) internal relations; to follow a rule is not to break it. (shrink)
Given (1) Wittgensteins externalist analysis of the distinction between following a rule and behaving in accordance with a rule, (2) prima facie connections between rule-following and psychological capacities, and (3) pragmatic issues about training, it follows that most, even all, future artificially intelligent computers and robots will not use language, possess concepts, or reason. This argument suggests that AIs traditional aim of building machines with minds, exemplified in current work on cognitive robotics, is in need of substantial (...) revision. (shrink)
As it is known, there is no rule satisfying Additivity in the complete domain of bankruptcy problems. This paper proposes a notion of partial Additivity in this context, to be called µ-additivity. We find that µ-additivity, together with two quite compelling axioms, anonymity and continuity, identify the Minimal Overlap rule, introduced by Neill (1982).
In recent works, Chomsky has once more endorsed a computational view of rulefollowing, whereby to follow a rule is to operate certain computations on a subject’s mental representations. As is well known, this picture does not conform to what we may call the grammatical conception of rule-following outlined by Wittgenstein, whereby an elucidation of the concept of rule-following is aimed at by isolating grammatical statements regarding the phrase ‘to follow a rule’. As a result, Chomskyan and (...) Wittgensteinian treatments of topics immediately connected with rule-following, namely linguistic competence and understanding, are utterly different from one another. There are two possible stances that computationalists like Chomsky may adopt with regard to the discrepancy between the two aforementioned modes of dealing with rule-following, namely a conciliatory and a non-conciliatory attitude. According to the former attitude, grammatical remarks on and computationallyoriented theories of rule-following investigate one and the same topic although admittedly at different levels, namely a conceptual and an empirical one. According to the latter attitude, grammatical remarks are just a preliminary step in the investigation of rule-following which scientific advancement, presently represented by computationally-oriented theories on this matter, is well entitled to put aside. In what follows, however, I will try to show that both stances are problematic. The conciliatory attitude simply does not work, for it hardly copes with the fact that the concept of rule-following does not supervene, even weakly, on the property of rule-following, namely the property instantiated in the mental/cerebral phenomena that computationally-oriented theories of rule-following study. To take the contrary attitude, on the other hand, is to end up with another disappointing result, namely that the computational treatment of rule-following ultimately deals with something different from that which we wished to gain knowledge of when we began our inquiry into rule-following.. (shrink)
Agents which perform inferences on the basis of unreliable information need an ability to revise their beliefs if they discover an inconsistency. Such a belief revision algorithm ideally should be rational, should respect any preference ordering over the agent’s beliefs (removing less preferred beliefs where possible) and should be fast. However, while standard approaches to rational belief revision for classical reasoners allow preferences to be taken into account, they typically have quite high complexity. In this paper, we consider belief revision (...) for agents which reason in a simpler logic than full first-order logic, namely rule-based reasoners. We show that it is possible to define a contraction operation for rule-based reasoners, which we call McAllester contraction, which satisfies all the basic Alchourrón, Gärdenfors and Makinson (AGM) postulates for contraction (apart from the recovery postulate) and at the same time can be computed in polynomial time. We prove a representation theorem for McAllester contraction with respect to the basic AGM postulates (minus recovery), and two additional postulates. We then show that our contraction operation removes a set of beliefs which is least preferred, with respect to a natural interpretation of preference. Finally, we show how McAllester contraction can be used to define a revision operation which is also polynomial time, and prove a representation theorem for the revision operation. (shrink)
I argue that rule consequentialism sometimes requires us to act in ways that we lack sufficient reason to act. And this presents a dilemma for Parfit. Either Parfit should concede that we should reject rule consequentialism (and, hence, Triple Theory, which implies it) despite the putatively strong reasons that he believes we have for accepting the view or he should deny that morality has the importance he attributes to it. For if morality is such that we sometimes have (...) decisive reason to act wrongly, then what we should be concerned with, practically speaking, is not with the morality of our actions, but with whether our actions are supported by sufficient reasons. We could, then, for all intents and purposes just ignore morality and focus on what we have sufficient reason to do, all things considered. So if my arguments are cogent, they show that Parfit’s Triple Theory is either false or relatively unimportant in that we can, for all intents and purposes, simply ignore its requirements and just do whatever it is that we have sufficient reason to do, all things considered. (shrink)
Most plausible moral theories must address problems of partial acceptance or partial compliance. The aim of this paper is to examine some proposed ways of dealing with partial acceptance problems as well as to introduce a new Rule Utilitarian suggestion. Here I survey three forms of Rule Utilitarianism, each of which represents a distinct approach to solving partial acceptance issues. I examine Fixed Rate, Variable Rate, and Optimum Rate Rule Utilitarianism, and argue that a new approach, Maximizing (...) Expectation Rate Rule Utilitarianism, better solves partial acceptance problems. (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.
In the attempt of defending an interpretation of David Hume's moral and political philosophy connected to classical utilitarianism, intervenes in a key way the so called problem of the " Sensitive Knave " raised by this author at the end of his more utilitarian work, the Enquiry Concerning the Principles of Morals. According to the classic interpretation of this fragment, the utilitarian rationality in politics would clash with morality turning useless the latter. Therefore, in the political area the defense of (...) a moral utilitarianism would be an auto contradictory task. In order to show that, first, Hume does not say anything similar to this and second, that even indicates the way of overcoming this apparent contradiction between morality and rationality, we analyze briefly the arguments from which there comes basically this "anti-utilitarian" standard interpretation, and we defend an interpretation of the humean discussion on the problem of the supposed conflict between morality and rationality, or of rational incentives for immoral behavior, which allows to explain better Hume's position on this problem. Finally, we propose an instance of overcoming the contradiction morality/ rationality by a rule adjusted utilitarianism centered on the idea of the "progressive development of artificial institutions of reinforcement of morality", that Hume himself would suggest in other places in which he approaches the topic of the apparent contradiction between "morality" and "knavery". We propose also possible lines of future development of this idea, between them its use to clarify the relation of David Hume's thought with certain forms of contemporary liberalism. (shrink)
In an incendiary 2010 Nature article, M. A. Nowak, C. E. Tarnita and E. O. Wilson present a savage critique of the best known and most widely used framework for the study of social evolution, W. D. Hamilton’s theory of kin selection. Over a hundred biologists have since rallied to the theory’s defence, but Nowak et al. maintain that their arguments ‘stand unrefuted’. Here I consider the most contentious claim Nowak et al. defend: that Hamilton’s rule, the core explanatory (...) principle of kin selection theory, ‘almost never holds’. I first distinguish two versions of Hamilton’s rule in contemporary theory: a special version (HRS) that requires restrictive assumptions, and a general version (HRG) that does not. I then show that Nowak et al. are most charitably construed as arguing that HRS almost never holds, while HRG buys its generality at the expense of explanatory power. While their arguments against HRS are fairly uncontroversial, their arguments against HRG are more contentious, yet these have been largely overlooked in the ensuing furore. I consider the arguments for and against the explanatory value of HRG, with a view to assessing what exactly is at stake in the debate. I suggest that the debate hinges on issues concerning the causal interpretability of regression coefficients, and concerning the explanatory function Hamilton’s rule is intended to serve. (shrink)
People often have a strong intuitive sense that we ought to rescue those in serious need, even in cases where we could produce better outcomes by acting in other ways. It has become common in such cases to refer to this as the Rule of Rescue. Within the medical field this rule has predominantly been discussed in relation to decisions about whether to fund particular treatments. Whilst in this setting the arguments in favour of the Rule of (...) Rescue have generally been found to be unconvincing, there are some reasons for thinking that it may have more of a role to play at the clinical level. In this article we examine three lines that such reasoning might take. In each case we argue that the reasons given do not support the adoption of a Rule of Rescue in clinical practice. (shrink)
Usual derivations of Lilders's projection rule show that Liuders's rule is the rule required by quantum statistics to calculate the final state after an ideal (minimally disturbing) measurement. These derivations are at best inconclusive, however, when it comes to interpreting Liuders's rule as a description of individual state transformations. In this paper, I show a natural way of deriving Liiders's rule from well-motivated and explicit physical assumptions referring to individual systems. This requires, however, the introduction (...) of a concept of individual state which is not standard. (shrink)
David Gauthier and Edward McClennen have claimed that it could be rational to form an intention to A because it maximizes utility to intend to A, and that acting on such an intention could be rational even if it maximizes utility not to A. Michael Bratman has objected to this way of thinking, claiming that it is equivalent to the familiar rule-utilitarian mistake of rule-worship. The purpose of this paper is to argue that, so long as one is (...) aware at the time of forming an intention to A that it maximizes utility not to A, then acting on that intention need not be rule worship, but the result of a rational refusal to reconsider an issue which has already been adequately considered. (shrink)
We give a simple and direct proof that super-consistency implies the cut-elimination property in deduction modulo. This proof can be seen as a simplification of the proof that super-consistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that proceed by proving the completeness of the cut-free calculus. As an application, we compare our work with the cut-elimination theorems in higher-order logic that involve V-complexes.
The general chemistry curriculum includes a prelude that consumes nearly all of the first semester and occupies the first third of the typical textbook. This necessary prelude to the main event is comparable in scope to precalculus though not broken out as a formal ‘prechemistry’ course. Atomic orbitals account for much of this prelude-to-chemistry. By tradition, orbital theory is conveyed to the student in three disjunct pieces, presented in the following illogical order: the Pauli principle, the Aufbau principle, and Hund’s (...)rule. (Often the n + l rule is tossed into the mix as well, though with no fixed place in the scheme). In the early twentieth century, as various researchers announced new insights into the atom at unpredictable intervals, no one could have been faulted for teaching orbitals in such a manner, catch-as-catch-can. A hundred years on, the vestiges of that (presumed) practice look wrong, and are indefensible. In the approach advocated here, orbitals would be taught as a single hierarchical rule-set, with the parts coherently sequenced as Aufbau–Hund–Pauli (and with Madelung’s n + l rule rehabilitated as part of Aufbau, no longer a free-floating mnemonic aid only). Logic aside, pragmatism offers its own argument for adopting this scheme: A tighter approach to Aufbau can lighten the ‘prechemistry’ burden significantly and bring the student that much sooner to chemistry itself. (shrink)
This essay investigates Xunzi’s political philosophy of ba dao (Hegemonic Rule). It argues that Xunzi’s practical philosophy of ba dao was developed in the course of resolving the tension between theory and practice latent in Mencius’s account of ba dao . Its central claim is that contra Mencius who remained torn between his ideal political theory of ba dao and the practical utility and moral value of ba dao , Xunzi creatively re-appropriated ba dao as a “morally decent” (if (...) not morally ideal) statecraft, within the parameter of practical Confucian philosophy. After examining the moral and political value of ba dao in both domestic and international governance, the essay concludes by arguing that Xunzi’s defense of ba dao should be understood in the context of what I call “negative Confucianism,” without which the realization of the Confucian moral-political ideal (or positive Confucianism) is impossible. (shrink)
This paper offers an appraisal of Phillip Pettit’s approach to the problem how a finite set of examples can serve to represent a determinate rule, given that indefinitely many rules can be extrapolated from any such set. Negatively, I argue that Pettit’s so-called ethocentric theory of rule-following fails to deliver the solution to this problem that he sets out to provide. More constructively, I consider what further provisions are needed in order to advance Pettit’s distinctive general approach to (...) the problem. I conclude that what is needed is a ‘no-priority’ account of rule-exemplification: that is, an account that (a) affirms the constitutive role of agents’ responses in the exemplification of rules but (b) denies the explanatory priority given to such responses in Pettit’s theory. (shrink)
Kant’s Critique of Pure Reason contains an original and powerful semantics of singular cognitive reference which has important implications for epistemology and for philosophy of science. Here I argue that Kant’s semantics directly and strongly supports Newton’s Rule 4 of Philosophy in ways which support Newton’s realism about gravitational force. I begin with Newton’s Rule 4 of Philosophy and its role in Newton’s justification of realism about gravitational force (§2). Next I briefly summarize Kant’s semantics of singular cognitive (...) reference (§3), and then show that it is embedded in and strongly supports Newton’s Rule 4, and that it rules out not only Cartesian physics (per Harper) but also Cartesian, infallibilist presumptions about empirical justification generally (§4). This result exposes a key fallacy in Bas van Fraassen’s original argument for his anti-realist Constructive Empiricism (§5). (shrink)
I. Recent years have witnessed a great resurgence of interest in the writings of the later Wittgenstein, especially with those passages roughly, Philosophical Investigations p)I 38 ââ¬â 242 and Remarks on the Foundations of mathematics, section VI that are concerned with the topic of rules. Much of the credit for all this excitement, unparalleled since the heyday of Wittgenstein scholarship in the early IIJ6os, must go to Saul Kripke's I4rittgenstein on Rules and Private Language. It is easy to explain why. (...) To begin with, the dialectic Kripke uncovered from Wittgenstein's.. (shrink)
Semantic holists view what one's terms mean as function of all of one's usage. Holists will thus be coherentists about semantic justification: showing that one's usage of a term is semantically justified involves showing how it coheres with the rest of one's usage. Semantic atomists, by contrast, understand semantic justification in a foundationalist fashion. Saul Kripke has, on Wittgenstein's behalf, famously argued for a type of skepticism about meaning and semantic justification. However, Kripke's argument has bite only if one understands (...) semantic justification in foundationalist terms. Consequently, Kripke's arguments lead not to a type of skepticism about meaning, but rather to the conclusion that one should be a coherentist about semantic justification, and thus a holist about semantic facts. (shrink)
Abstract: This paper argues that most of the alleged straight solutions to the sceptical paradox which Kripke (1982) ascribed to Wittgenstein can be regarded as the first horn of a dilemma whose second horn is the paradox itself. The dilemma is proved to be a by-product of a foundationalist assumption on the notion of justification, as applied to linguistic behaviour. It is maintained that the assumption is unnecessary and that the dilemma is therefore spurious. To this end, an alternative conception (...) of the justification of linguistic behaviour is outlined, a conception that vindicates some of the insights behind Kripke's Wittgenstein's sceptical solution of the paradox. This alternative conception is defended against two objections (both familiar from McDowell's works): (1) that it would imply that for the linguistic community there is no authority, no standard to meet and, therefore, no possibility of error and (2) that it would lead to a kind of idealism. (shrink)
This paper employs some outcomes (for the most part due to David Lewis) of the contemporary debate on the metaphysics of dispositions to evaluate those dispositional analyses of meaning that make use of the concept of a disposition in ideal conditions. The first section of the paper explains why one may find appealing the notion of an ideal-condition dispositional analysis of meaning and argues that Saul Kripke’s well-known argument against such analyses is wanting. The second section focuses on Lewis’ work (...) in the metaphysics of dispositions in order to call attention to some intuitions about the nature of dispositions that we all seem to share. In particular, I stress the role of what I call ‘Actuality Constraint’. The third section of the paper maintains that the Actuality Constraint can be used to show that the dispositions with which ideal-condition dispositional analyses identify my meaning addition by ‘+’ do not exist (in so doing, I develop a suggestion put forward by Paul Boghossian). This immediately implies that ideal-condition dispositional analyses of meaning cannot work. The last section discusses a possible objection to my argument. The point of the objection is that the argument depends on an illicit assumption. I show (1) that, in fact, the assumption in question is far from illicit and (2) that even without this assumption it is possible to argue that the dispositions with which ideal-condition dispositional analyses identify my meaning addition by ‘+’ do not exist. (shrink)
INTRODUCTORY ESSAY: COMMUNAL AGREEMENT AND OBJECTIVITY Christopher M. Leich and Steven H. Holtzman In this essay we shall take up certain questions raised ...
According to a standard criticism, Robert Brandom's “normative pragmatics”, i.e. his attempt to explain normative statuses in terms of practical attitudes, faces a dilemma. If practical attitudes and their interactions are specified in purely non-normative terms, then they underdetermine normative statuses; but if normative terms are allowed into the account, then the account becomes viciously circular. This paper argues that there is no dilemma, because the feared circularity is not vicious. While normative claims do exhibit their respective authors' practical attitudes (...) and thereby contribute towards establishing the normative statuses they are about, this circularity is not a mark of Brandom's explanatory strategy but a feature of social practice of which we theorists partake. (shrink)
Through detailed and trenchant criticism of standard interpretations of some of the key arguments in analytical philosophy over the last sixty years, this book ...
This article presents a sequent calculus for a negative free logic with identity, called N . The main theorem (in part 1) is the admissibility of the Cut-rule. The second part of this essay is devoted to proofs of soundness, compactness and completeness of N relative to a standard semantics for negative free logic.
We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using these axiomatizations, we prove interpolation theorems for basic propositional logic.
From the point of view of proof-theoretic semantics, it is argued that the sequent calculus with introduction rules on the assertion and on the assumption side represents deductive reasoning more appropriately than natural deduction. In taking consequence to be conceptually prior to truth, it can cope with non-well-founded phenomena such as contradictory reasoning. The fact that, in its typed variant, the sequent calculus has an explicit and separable substitution schema in form of the cut rule, is seen as a (...) crucial advantage over natural deduction, where substitution is built into the general framework. (shrink)
Bi-intuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent ‘cut-free’ sequent calculus has recently been shown to fail cut-elimination. We present a new cut-free sequent calculus for bi-intuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in (...) tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose. (shrink)
Given the harmony principle for logical operators, compositionality ought to ensure that harmony should obtain at the level of whole contents. That is, the role of a content qua premise ought to be balanced exactly by its role as a conclusion. Frege's contextual definition of propositional content happens to exploit this balance, and one appeals to the Cut rule to show that the definition is adequate.We show here that Frege's definition remains adequate even when one relevantizes logic by abandoning (...) an unrestricted Cut rule. The proof exploits the fact that in the relevantized logic, which abandons the unrestricted rule of Cut, any failure of the transitivity of deduction is offset by the epistemic gain involved in learning that a stronger-than-expected result holds. (shrink)
We continue a series of papers on a family of many-valued modal logics, a family whose Kripke semantics involves many-valued accessibility relations. Earlier papers in the series presented a motivation in terms of a multiple-expert semantics. They also proved completeness of sequent calculus formulations for the logics, formulations using a cut rule in an essential way. In this paper a novel cut-free tableau formulation is presented, and its completeness is proved.
Sometimes the fact that something is the law can be justified by the law. For example, the Sarbanes-Oxley Act is the law because it was enacted by Congress pursuant to the Commerce Clause. But eventually legal justification of law ends. The ultimate criteria of validity in a legal system cannot themselves be justified by law. According to H.L.A. Hart, justification of these ultimate criteria is still available, by reference to social facts concerning official acceptance - facts about what Hart calls (...) the "rule of recognition" for the system. -/- Drawing upon criticisms of sociological accounts of the law that can be found in the writings of Hans Kelsen, I argue in this essay that Hart's approach cannot account for statements about the law that assert the independence of legal validity from rule of recognition facts. I offer as an alternative a legal quietist approach, which can account for such statements. For the quietist, legal justification exhausts the possible justification for law. If our judgments about the law are fundamental, in the sense that they cannot be justified by other judgments about the law, then they have no justification (which is not to say that they should be abandoned). I argue that legal quietism is exemplified - if somewhat imperfectly - in Kelsen's writings, and I end the essay by exploring some difficulties that the quietist approach must face. (shrink)
Deep inference is a natural generalisation of the one-sided sequent calculus where rules are allowed to apply deeply inside formulas, much like rewrite rules in term rewriting. This freedom in applying inference rules allows to express logical systems that are difficult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. However, the same freedom also makes it harder to carry out this analysis, in particular it (...) is harder to design cut elimination procedures. In this paper we see a cut elimination procedure for a deep inference system for classical predicate logic. As a consequence we derive Herbrand's Theorem, which we express as a factorisation of derivations. (shrink)
A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic (...) with equality in which also cuts on the equality axioms are eliminated. (shrink)
An (n, k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)-ary quantifiers form a natural class of Gentzen-type systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantifier is introduced. The semantics for these systems is provided using two-valued non-deterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion of coherence (...) to characterize strong cutelimination in such systems. We show that the following properties of a canonical system G with arbitrary (n, k)-ary quantifiers are equivalent: (i) G is coherent, (ii) G admits strong cut-elimination, and (iii) G has a strongly characteristic two-valued generalized non-deterministic matrix. In addition, we define simple calculi, an important subclass of canonical calculi, for which coherence is equivalent to the weaker, standard cut-elimination property. (shrink)
Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive (...) relevant implication. The proof technique-attributable to Gentzen-that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen's sequent calculus LK without any use of mix. (shrink)
We use a deep embedding of the display calculus for relation algebras RA in the logical framework Isabelle/HOL to formalise a machine-checked proof of cut-admissibility for RA. Unlike other “implementations”, we explicitly formalise the structural induction in Isabelle/HOL and believe this to be the first full formalisation of cutadmissibility in the presence of explicit structural rules.
We continue our work [5] on the logic of multisets (or on the multiset semantics of linear logic), by interpreting further the additive disjunction . To this purpose we employ a more general class of processes, called free, the axiomatization of which requires a new rule (not compatible with the full LL), the cancellation rule. Disjunctive multisets are modeled as finite sets of multisets. The -Horn fragment of linear logic, with the cut rule slightly restricted, is sound (...) with respect to this semantics. Another rule, which is a slight modification of cancellation, added to HF makes the system sound and complete. (shrink)
In [4], I proved that the product-free fragment L of Lambek's syntactic calculus (cf. Lambek [2]) is not finitely axiomatizable if the only rule of inference admitted is Lambek's cut-rule. The proof (which is rather complicated and roundabout) was subsequently adapted by Kandulski [1] to the non-associative variant NL of L (cf. Lambek [3]). It turns out, however, that there exists an extremely simple method of non-finite-axiomatizability proofs which works uniformly for different subsystems of L (in particular, for (...) NL). We present it below to the use of those who refer to the results of [1] and [4]. (shrink)
An (n, k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)-ary quantifiers form a natural class of Gentzen-type systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantifier is introduced. The semantics for these systems is provided using two-valued non-deterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion of coherence (...) to characterize strong cutelimination in such systems. We show that the following properties of a canonical system G with arbitrary (n, k)-ary quantifiers are equivalent: (i) G is coherent, (ii) G admits strong cut-elimination, and (iii) G has a strongly characteristic two-valued generalized non-deterministic matrix. (shrink)
The fixed point combinator (Y) is an important non-proper combinator, which is defhable from a combinatorially complete base. This combinator guarantees that recursive equations have a solution. Structurally free logics (LC) turn combinators into formulas and replace structural rules by combinatory ones. This paper introduces the fixed point and the dual fixed point combinator into structurally free logics. The admissibility of (multiple) cut in the resulting calculus is not provable by a simple adaptation of the similar proof for LC with (...) proper combinators. The novelty of our proof—beyond proving the cut for a newly extended calculus–is that we add a fourth induction to the by-and-large Gentzen-style proof. (shrink)
In this paper we provide cut-free tableau calculi for the intuitionistic modal logics IK, ID, IT, i.e. the intuitionistic analogues of the classical modal systems K, D and T. Further, we analyse the necessity of duplicating formulas to which rules are applied. In order to develop these calculi we extend to the modal case some ideas presented by Miglioli, Moscato and Ornaghi for intuitionistic logic. Specifically, we enlarge the language with the new signs Fc and CR near to the usual (...) signs T and F. In this work we establish the soundness and completeness theorems for these calculi with respect to the Kripke semantics proposed by Fischer Servi. (shrink)
We use a deep embedding of the display calculus for relation algebras ÆRA in the logical framework Isabelle/HOL to formalise a machine-checked proof of cut-admissibility for ÆRA. Unlike other “implementations”, we explicitly formalise the structural induction in Isabelle/HOL and believe this to be the first full formalisation of cutadmissibility in the presence of explicit structural rules.
This book discusses theories of legal reasoning and provides an overall view of the rhetoric of legal justification. It shows how and why lawyers arguments can be rationally persuasive even though rarely, if ever, logically conclusive or compelling. It examines the role of "legal syllogism" and universality of legal reasoning, looking at arguments of consequentialism and principle, and concludes by questioning the infallibility of judges as lawmakers.
Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a “canonical system” to first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with (...) unary quantifiers and show that it remains constructive. Then we provide semantics for such canonical systems using 2-valued non-deterministic matrices extended to languages with quantifiers, and prove that the following properties are equivalent for a canonical system G: (1) G admits Cut-Elimination, (2) G is coherent, and (3) G has a characteristic 2-valued non-deterministic matrix. (shrink)
Axiomatics which do not employ rules of inference other than the cut rule are given for commutative product-free Lambek calculus in two variants: with and without the empty string. Unlike the former variant, the latter one turns out not to be finitely axiomatizable in that way.
