This account of the logic of rules is stratified. The various layers consist of the logic of states, i.e. essentially classical logic (of the usual sort), the logic of agents and action types (or as I call them routines), and the logic of rules proper, as the top layer. It is assumed in this essay that the reader has an adequate grasp of classical logic.

The Logic of Analogy is a study of the valid logical forms of qualitative and quantitative analogical argument, and the rules pertaining to them. It investigates equally valid conflicting arguments, statistics-based arguments and their utility in science, arguments from precedent used in law-making or law-application, and examines subsumption in analogical terms. Included for purposes of illustration is a large section on Talmudic use of analogical reasoning.

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)

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.

In this paper we start an investigation of a logic called the logic of algebraic rules. The relation of derivability of this logic is defined on universal closures of special disjunctions of equations extending the relation of derivability of the usual equational logic. The paper contains some simple theorems and examples given in justification for the introduction of our logic. A number of open questions is posed.

Almost every state has its own legal system and there is a legal system in accordance with the social norms of the state. However, although states have autonomy with their own legal systems, the legal system of each state unites under larger legal systems. From this point of view, three major legal systems can be accepted; Anglo-Saxon Legal System (English Legal System, Common Law), Continental European Legal System (Legal system of European states based on Roman law), Social Legal System on (...) which Russian Federation is based. Apart from these systems, there is also an Islamic legal system with its unique structure. The reason why the Islamic legal system is unique is that it is based on both religious and rational sources. In general, the case-based method is used in the Anglo-Saxon Legal System, and the rule-based method is used in the Continental European Legal System. Which one of these two method used by Islamic law which takes its source from Qur’an and hadith well? Or is it use both methods? Our aim in this study, in addition to analysing the operation of these two methods in Islamic law from a logical point of view, is to analyse the integration of the rule-based method into the case-based method and the case-based method with the rule-based method. With this research it is to provide a logical proposal to extend the intended use of analogy in Islamic law. As a matter of fact, we think that different branches of science should come together in order to produce new suggestions in this field. (shrink)

The process of the liquidation of the Hetmanate lasted for decades because of its scale and the constant need of Russia in the Cossack Army. Empress Elizabeth’s regime continued the centralizing policies introduced by Tsar Peter I. In anticipation of the possible consequences of this centralization, in the early 1860s Hetman Kyrylo Rozumovskyi tried to strengthen local governance by reforming his administrative system and judiciary, outlining the justification of his measures in an appeal to Empress Catherine II, entitled “Petition of (...) the Little Russian Nobility and Officers, Together with the Hetman Concerning the Restoration of Various Old Rights of Little Russia.” At the same time, Catherine II’s State Secretary, Grigorii Teplov expressed opposite views to the positions expounded in the “Petition,” in his so-called “Notes on Violations in Little Russia.” The main theses of Teplov’s memorandum were used by the Empress in secret guidelines issued to Petr Rumiantsev in his position as the President of the Second Little Russian Collegium. The reasoning of the above-mentioned documents formed the basis for Rumiantsev for the liquidation of the Ukrainian state. (shrink)

This essay offers a new interpretation of Michael Hardt and Antonio Negri’s landmark work of critical social theory Empire. It develops an account of the politics of exile by situating this political strategy in terms of Hardt and Negri’s claim that it is no longer feasible to confront capitalist power head-on. It attends closely to Hardt and Negri’s account of Empire’s pyramidal structure, and the problems that this structure creates for the multitude’s passage from virtuality to actuality. It criticizes the (...) authors’ assertion that, under Empire, the institutions associated with modernity have become indistinguishable from one another in ‘society of control.’ This essay concludes by theorizing the historicity and power of the university, in an attempt to specify what the democratic self-organization of the multitude, together with its exodus from Empire, might look like. Accordingly, it contends that the production of knowledge and higher education depend upon an idea of absolute value which outstrips the commodity-form, and in so doing, provides an intimation of the multitude passing to the exterior of Empire. (shrink)

Go back to An-fang, the Peace Square at An-Fang, the Beginning Place at An-Fang, where all things start (… ) An-Fang was near a city, the only living city with a pre-atomic name (… ) The headquarters of the People Programmer was at An-Fang, and there the mistake happened: A ruby trembled. Two tourmaline nets failed to rectify the laser beam. A diamond noted the error. Both the error and the correction went into the general computer. Cordwainer SmithThe Dead Lady (...) of Clown Town, 1964. (shrink)

In this paper, we discuss semantical properties of the logic \(\textbf{GL}\) of provability. The logic \(\textbf{GL}\) is a normal modal logic which is axiomatized by the the Löb formula \( \Box (\Box p\supset p)\supset \Box p \), but it is known that \(\textbf{GL}\) can also be axiomatized by an axiom \(\Box p\supset \Box \Box p\) and an \(\omega \) -rule \((\Diamond ^{*})\) which takes countably many premises \(\phi \supset \Diamond ^{n}\top \) \((n\in \omega )\) and returns a (...) conclusion \(\phi \supset \bot \). We show that the class of transitive Kripke frames which validates \((\Diamond ^{*})\) and the class of transitive Kripke frames which strongly validates \((\Diamond ^{*})\) are equal, and that the following three classes of transitive Kripke frames, the class which validates \((\Diamond ^{*})\), the class which weakly validates \((\Diamond ^{*})\), and the class which is defined by the Löb formula, are mutually different, while all of them characterize \(\textbf{GL}\). This gives an example of a proof system _P_ and a class _C_ of Kripke frames such that _P_ is sound and complete with respect to _C_ but the soundness cannot be proved by simple induction on the height of the derivations in _P_. We also show Kripke completeness of the proof system with \((\Diamond ^{*})\) in an algebraic manner. As a corollary, we show that the class of modal algebras which is defined by equations \(\Box x\le \Box \Box x\) and \(\bigwedge _{n\in \omega }\Diamond ^{n}1=0\) is not a variety. (shrink)

I will describe the logics of a range of conditionals that behave like conditional probabilities at various levels of probabilistic support. Families of these conditionals will be characterized in terms of the rules that their members obey. I will show that for each conditional, →, in a given family, there is a probabilistic support level r and a conditional probability function P such that, for all sentences C and B, 'C → B' holds just in case P[B | C] (...) ≥ r. Thus, each conditional in a given family behaves like conditional probability above some specific support level. (shrink)

Natural languages are riddled with context-sensitivity. One and the same string of words can express many different meanings on occasion of use, and yet we understand one another effortlessly, on the fly. How do we do so? What fixes the meaning of context-sensitive expressions, and how are we able to recover the meaning so effortlessly? -/- This book offers a novel response: we can do so because we draw on a broad array of subtle linguistic conventions that determine the interpretation (...) of context-sensitive items. Contrary to the dominant tradition, which maintains that the meaning of context-sensitive language is underspecified by grammar and that interpretation relies on non-linguistic cues and speakers' intentions, this book argues that meaning is determined entirely by discourse conventions, rules of language that have largely been missed and the effects of which have been mistaken for extra-linguistic effects of an utterance situation on meaning. The linguistic account of context developed here sheds new light on the nature of linguistic content and the interaction between content and context. At the same time, it provides a novel model of context that should constrain and help evaluate debates across many subfields of philosophy where appeal to context has been common, often leading to surprising conclusions such as epistemology, ethics, value theory, metaphysics, metaethics, and logic. (shrink)

The Logic of Metaphor uses techniques from possible worlds semantics to provide formal truth-conditions for many grammatical classes of metaphors. It gives logically precise and practically useful syntactic and semantic rules for generating and interpreting metaphors. These rules are implemented in a working computer program. The book treats the lexicon as a conceptual network with semantics provided by an intensional predicate calculus. It gives rules for finding analogies in such networks. It shows how to syntactically and (...) semantically analyze texts containing metaphors and how to use structural similarities between parts of possible worlds to provide truth-conditions for metaphors. Meanings for metaphors are linked to the modal logics of identity and indiscernibility. The book shows how to extend deductive and abductive inference systems to handle metaphors. It shows how to handle novel metaphorical word-senses. The Logic of Metaphor will be useful to philosophers, logicians, linguists, and computer scientists. (shrink)

The knowledge norm of assertion is the subject of a lively debate on when someone is in a position to assert something. However, not much has been said about the logic that underlies such debate. In this paper, I propose a formalisation of the knowledge norm in a deontic logic that aims to be explanatory and conceptually sound. Afterwards, I investigate some problems that this formalisation makes visible. This reveals some significant limitations of the underlying logic: it (...) can neither contain Axiom 4 nor Axiom C4. Moreover, sentences of the form p and I have not asserted that p appear to licence a violation of deontic rules. (shrink)

We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics (...) of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers ∀ new and ∃ new which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels. (shrink)

Questions whether the logic of language underlying Habermas's theory of communicative action is in fact the defining feature of conversational practice.

I propose to bring fresh evidence here for my theory of knowledge and expand it in new directions. We shall arrive most swiftly at the centre of the theory, by going back to the point from which I started about twenty years ago. Upon examining the grounds on which science is pursued, I saw that its progress is determined at every stage by indefinable powers of thought. No rules can account for the way a good idea is found for (...) starting an inquiry; and there are no firm rules either for the verification or the refutation of the proposed solution of a problem. Rules widely current may be plausible enough, but scientific enquiry often proceeds and triumphs by contradicting them. Moreover, the explicit content of a theory fails to account for the guidance it affords to future discoveries. To hold a natural law to be true, is to believe that its presence may reveal itself in yet unknown and perhaps yet unthinkable consequences; it is to believe that such laws are features of a reality which as such will continue to bear consequences inexhaustibly. (shrink)

James Bohman has succeeded in reinvigorating the old debate over explanation and understanding by situating it within contemporary discussions about sociological indeterminacy and complexity. I argue that Bohman's preference for a paradigm based on Habermas's theory of communicative action is justifiable given the explanatory deficiencies of ethnomethodological, rational choice, rule-based, and functionalist methodologies. Yet I do not share his belief that the paradigm is preferable to less formalized models of interpretation.

Empiricism is a claim about the contents of the mind: its classic slogan is nihil est in intellectu quod non fuerit in sensu, ‘there is nothing in the mind (intellect, understanding) which is not first in the senses’. As such, it is not a claim about the fundamental nature of the world as material. I focus here on in an instance of what one might term the materialist appropriation of empiricism. One major component in the transition from a purely epistemological (...) claim about the mind and its contents, to an ontological claim about the nature of the world, is the new focus on brain-mind relations in the eighteenth century. Here I examine a Lockean trajectory as exemplified in Joseph Priestley’s 1777 Disquisitions Relating to Matter and Spirit. But Locke explicitly ruled out that his inquiry into the logic of ideas amounted to a ‘physical consideration of the mind’. What does it mean, then, for Priestley to present himself as continuing a Lockean tradition, while presenting mental processes as tightly identified with ‘an organical structure such as that of the brain’ (although he was not making a strict identity claim as we might understand it, post-Smart and Armstrong)? One issue here is that of Priestley's source of ‘empirical data’ regarding the correlation and indeed identification of mental and cerebral processes. David Hartley’s theory in his 1749 Observations on Man was, as is well known, republished in abridged form by Priestley, but he discards Hartley's 'vibratory neurophysiology' while retaining the associationist framework, although not because he disagreed with the former. Yet Hartley was at the very least, strongly agnostic about metaphysical issues (and it is difficult to study these authors while bracketing off religious considerations). One could see Locke and Hartley as articulating programs for the study of the mind which were more or less naturalistic (more strongly so in Hartley’s case) while avoiding ‘materialism’ per se; in contrast, Priestley bit the (materialist) bullet. In this paper I examine Priestley’s appropriation and reconstruction of this ‘micro-tradition’, while emphasizing its problems. (shrink)

Peirce algebras combine sets, relations and various operations linking the two in a unifying setting. This paper offers a modal perspective on Peirce algebras. Using modal logic a characterization of the full Peirce algebras is given, as well as a finite axiomatization of their equational theory that uses so-called unorthodox derivation rules. In addition, the expressive power of Peirce algebras is analyzed through their connection with first-order logic, and the fragment of first-order logic corresponding to Peirce (...) algebras is described in terms of bisimulations. (shrink)

Peirce algebras combine sets, relations and various operations linking the two in a unifying setting. This paper offers a modal perspective on Peirce algebras. Using modal logic as a characterization of the full Peirce algebras is given, as well as a finite axiomatization of their equational theory that uses so-called unorthodox derivation rules. In addition, the expressive power of Peirce algebras is analyzed through their connection with first-order logic and the fragment of first-order logic corresponding to (...) Peirce algebras is described in terms of bisimulations. (shrink)

The principle of belief persistence, or conservativity principle, states that ’\Nhen changing beliefs in response to new evidence, you should continue to believe as many of the old beliefs as possible' (Harman, 1986, p. 46). In particular, this means that if an individual gets new information, she has to accommodate it in her new belief set (the set of propositions she believes), and, if the new information is not inconsistent with the old belief set, then (1) the individual has to (...) maintain all the beliefs she previously had and (2) the change should be minimal in the sense that every proposition in the new belief set must be deducible from the union of the old belief set and the new information (see, e.g., Gardenfors, 1988; Stalnaker, 1984). We focus on this minimal notion of belief persistence and characterize it both semantically and syntactically. A ’possible world' semantic formalization of the principle easily comes to mind. The set of all the propositions that the individual believes corresponds to the set of states of the world that she considers possible and is a subset of the set of states that are not ruled out by the individual's information (or knowledge). It is required that, if the individual considers a state possible and her new information does not exclude this state, then she continue to consider it possible. Furthermore, if the individual regards a particular state as impossible, then she should continue to regard it as impossible unless her new information excludes all the states that she previously regarded as possible. This is closely related to the.. (shrink)

The study examines Russian philosopher Andrei Smirnov’s conception of the logic of sense as a way of providing exposition of the reasons for the Russian-Ukrainian war. The logic of sense is simultaneously a theory of rules of sense-setting and the very rules of sense-setting created by a culture and the ruling culture. Smirnov thinks that the reasons lie in the clash between common-human European culture and its logic of sense and all-human Russian culture and its (...) logic of sense. Smirnov sees no difference between Russians and Ukrainians and considers the Ukrainian culture an essential and indispensable part of Russian culture. Therefore, the expansion of the common-human European culture to Ukraine challenges the all-human Russian culture. (shrink)

James Bohman has succeeded in reinvigorating the old debate over explanation and understanding by situating it within contemporary discussions about sociological indeterminacy and complexity. I argue that Bohman's preference for a paradigm based on Habermas's theory of communicative action is justifiable given the explanatory deficiencies of ethnomethodological, rational choice, rule-based, and functionalist methodologies. Yet I do not share his belief that the paradigm is preferable to less formalized models of interpretation.

Without directly addressing the Demarcation Problem for logic—the problem of distinguishing logical vocabulary from others—we focus on distinctive aspects of logical vocabulary in pursuit of a second goal in the philosophy of logic, namely, proposing criteria for the justification of logical rules. Our preferred approach has three components. Two of these are effectively Belnap’s, but with a twist. We agree with Belnap’s response to Prior’s challenge to inferentialist characterisations of the meanings of logical constants. Belnap argued that (...) for a logical constant to exist, its rules must be conservative over a previously given consequence relation and guarantee the uniqueness of the constant. The twist is that we require logical vocabulary to be provably conservative, not over a previously given formal consequence relation, but over a previously given meaningful vocabulary of a language. Uniqueness is also a feature defined in terms of provability: if two syntactically distinguished expressions are governed by the same rules, then formulas with them as main operators are interderivable. Belnap’s criteria are not only those for the existence of a logical constant, but more: they are what distinguishes logical vocabulary from all other expressions. It is the defining mark of a logical constant that it is provably conservative over the fragment of the language which excludes it and that its rules guarantee its uniqueness. The third component is the topic neutrality of logic. The provable conservativeness of logic over a previously given vocabulary of a language is motivated, in part, by appeal to molecularism in the theory of meaning. Molecularity is a feature endorsed by the theory of meaning as a whole, so it does not distinguish logical constants from other expressions. The same is true for conservativeness. We argue that molecularity, and hence conservativeness, are implicit presuppositions of speakers’ use of language: the addition of new vocabulary to a language is presupposed to be conservative. But this presupposition is one that we should be able reflectively to endorse (such as when attempting a rational reconstruction of the use of the vocabulary). Thus, where conservativeness is found to fail, this defect should be remedied and the use of the vocabulary corrected (as in classical negation) or excised altogether (as in pejoratives). We go on to characterise a notion of topic neutrality, which we argue applies to logical vocabulary. We then note that reflective endorsement of the conservativeness of a topic neutral vocabulary requires a proof that the vocabulary is conservative relative to any base vocabulary. Thus we require that logical vocabulary be demonstrably conservative. Allying this requirement, distinctive of logic, to a general consideration about the commitments of assertion yields a mode of justification for logical constants akin to some conceptions of Harmony, i.e., to the idea that the consequences of assertion of a logical complex need to be warranted by the grounds and ought to be the strongest consequences warranted by the grounds. Intuitionistic logic acquires a somewhat familiar justification but emanating from a new motivation. (shrink)

Wolff's book defends the Kantian idea of a "general logic" whose principles underlie special systems of deductive logic. It thus undermines "logical pluralism," which tolerates the co-existence of divergent systems of modern logic without asking for consistent common principles. Part I of Wolff’s book identifies the formal language in which the most general principles of logic must be expressed. This language turns out to be a version of syllogistic language already used by Aristotle. The universal validity (...) of logical principles, as well as the translatability of other logical languages into this language, are shown to depend only on the meanings of its logical vocabulary. Part II of the book answers the metalogical question concerning the deductive relation between general logic and special logical systems, which also have their own (less general) principles. This part identifies the rules according to which logical rules can be derived from principles. The main result of the book is that the highest principles of logic and metalogics are provided by the syllogistic, when properly understood. (shrink)

This paper is taken from a long section of the _Tattva-cintā-maṇi_ by Gaṅgeśa that is devoted to proving the existence of—to use an inadequate word—“God” in a somewhat minimalist sense. The _īśvara_, the “Lord,” is for Gaṅgeśa, following Nyāya predecessors, a divine agent, a self, responsible for much, not all, of the order in the world. Unseen Force, _adṛṣṭa_, which is in effect _karman_ made by human action, is also a powerful agent as well as things’ intrinsic natures. Moreover, ordinary (...) selves, atoms, ether, and universals are uncreated. But the _īśvara_ brings about just desert in reincarnation in actualizing Unseen Force, and is responsible for a broad swathe of what some see as accidental arrangements as well as the forming of the macro elements from eternal, naturally _dis_joined atoms. Thus the cosmos in its general existence and structure is viewed in Nyāya as the work of the Lord. Gaṅgeśa’s argument runs: Earth and the like [_a_ (_pakṣa_) = earth and the like (_kṣity-ādi_)] have a conscious agent as a cause [S (_sādhya_) = having an agential cause (_sakartṛkatva_) (S_a_)], _since_ they are effects [H (_sādhana_) = being an effect [(_kāryatva_) (H_a_)], like a pot [_b_ (_dṛṣṭānta_) = a pot (H_b_,S_b_)]. And so the _vyāpti_ rule is: [H → S (_vyāpti_)] Whatever is an effect has an agential cause. For earth and the like, it is reasoned that only an omniscient _īśvara_ could be that cause. The argument was a target of Buddhists who pointed to counterexamples such as growing grass. Growing grass exhibits the prover property, being-an-effect, but not the property to be proved, having-an-agential-cause. The long section is dominated by Gaṅgeśa’s rebutting this and other potential defeaters, in particular, the _upādhi_, having-a-living-body (God does not have a living body but all the agential causes with which we are familiar do), along with a counterinference (_sat-pratipakṣa_), I_a_ & (x)(Ix → ¬Sx), where I = not-produced-by-an-agent-with-a-body. (shrink)

This paper deals with two issues in the field of reasoning by analogy in the law. The one issue is whether there exists such a thing as analogous rule application, or whether there is only the ‘normal’ application of a broadened rule. It is argued that if rules, as the entities made by a legislator, are distinguished from generalised solutions for cases, the idea of analogous application of rules makes sense. It is also shown how the so-called ‘reason-based (...) model of rule application’, in contrast to the traditional modus ponens or subsumption model, makes it easy to give a logical account of analogous rule application. The other issue is how to argue logically about whether two cases are sufficiently similar to adopt the outcome of the one case for the other case. Section 3 provides a general logical model to establish this. The model is based on the comparison of the reasons for and against a particular solution in the two cases. (shrink)

The aim of this paper is to offer a rigorous explication of statements ascribing ability to agents and to develop the logic of such statements. A world is said to be feasible iff it is compatible with the actual past-and-present. W is a P-world iff W is feasible and P is true in W (where P is a proposition). P is a sufficient condition for Q iff every P world is a Q world. P is a necessary condition for (...) Q iff Q is a sufficient condition forP. Each individual property S is shown to generate a rule for an agent X. X heeds S iff X makes all his future choices in accordance with S. (Note that X may heed S and yet fail to have it). S is a P-strategy for X iff X's heeding S together with P is a necessary and sufficient condition for X to have S. (P-strategies are thus rules which X is able to implement on the proviso P).Provisional opportunity: X has the opportunity to A provided P iff there is an S such that S is a P-strategy for X and X's implementing S is a sufficient condition for X's doing A. P is etiologically complete iff for every event E which P reports P also reports an etiological ancestry of E, and P is true. Categorical opportunity: X has the opportunity to A iff there is a P such that P is etiologically complete and X has the opportunity to A provided P. For X to have the ability to A there must not only be an appropriate strategy, but X must have a command of that strategy. X steadfastly intends A iff X intends A at every future moment at which his doing A is not yet inevitable. X has a command of S w.r.t. A and P iff X's steadfastly intending A together with P is a sufficient condition for X to implement S. Provisional ability: X can A provided P iff there is an S such that S is a P-strategy for X, X's implementing S is a sufficient condition for X's doing A, and X has a command of S w.r.t. A and P. Categorical ability: X can A iff there is a P such that P is etiologically complete and X can A provided P. X is free w.r.t. to A iff X can A and X can non- A. X is free iff there is an A such that X is free w.r.t. A. (shrink)

Two of Aristotle’s major legacies, namely, the theory of scientific syllogism and teleology seem to conflict on several planes. Indeed, an array of formal limitations prevents him from formalizing teleological explanations into scientific syllogisms, which are entirely absent from his works. To achieve this, Aristotle resorts to a different tool, the logic of ‘consequence’. This governs both the teleological relation between an end and a means that underlies necessity ‘from a hypothesis’—which is the necessity proper to living things—and a (...) different form of syllogisms, namely, syllogisms ‘from a hypothesis’. His guidelines in the first chapter of his Parts of Animals on the proper form of demonstration to be adopted in biology should be read as laying out the rules of inference for translating teleological explanations into syllogisms ‘from a hypothesis’. (shrink)

The status of the equality predicate as a logical constant is problematic. In the paper we look at the problem from the proof-theoretic standpoint and survey several ways of treating equality in formal systems of different sorts. In particular, we focus on the framework of sequent calculus and examine equality in the light of criteria of logicality proposed by Hacking and Došen. Both attempts were formulated in terms of sequent calculus rules, although in the case of Došen it has (...) a nonstandard character. It will be shown that equality can be characterised in a way which satisfies Došen’s criteria of logicality. In the case of Hacking’s approach the fully satisfying result can be obtained only for languages with a nonempty, finite set of predicate constants other than equality. Otherwise, cut elimination theorem fails to hold. (shrink)

Barbey & Sloman (B&S) relegate the logical rule of the excluded middle to a footnote. But this logical rule is necessary for natural sampling. Making the rule explicit in a logical tree can make a problem easier to solve. Examples are given of uses of the rule that are non-constructive and not reducible to a domain-specific module.

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)

Much has been said about the logical difference between rules and principles, yet few authors have focused on the distinct logical connectives linking the normative conditions of both norms. I intend to demonstrate that principles, unlike rules, are norms whose antecedents are linguistically formulated in a generic fashion, and thus logically described as inclusive disjunctions. This core feature incorporates the relevance criteria of normative antecedents into the world of principles and also explains their aptitude to conflict with opposing (...) norms, namely that their consequents are fulfilled to varying extents more frequently than those of rules. I conclude that the property of genericity should be predicated to the norm antecedent of principles, more precisely to the hypothetical action. This is of paramount importance to explain, in terms of logical implication and exclusion, the expansibility of competing principles, in contrast with the exclusive character of conflicting rules. (shrink)

Popper is well known for rejecting a logic of discovery, but he is only justified in rejecting the same type of logic of discovery that is denied by consequentialism. His own account of hypothesis generation, based on a natural selection analogy, involves an error-eliminative logic of discovery and the differences he admits between biological and conceptual evolution suggest an error-corrective logic of discovery. These types of logics of discovery are based on principles of plausibility that are (...) used in the generation as opposed to the preliminary evaluation of hypotheses. The normative relevance of these principles is grounded in the distinction between strategic and definitory rules. (shrink)

We investigate an axiomatization of the notion of common belief that makes use of no rules of inference and highlight the property of the set of accessibility relations that characterizes each axiom.

The medieval distinction between categorematic and syncategorematic words is usually given as the distinction between words which have signification or meaning in isolation from other words and those which have signification only when combined with other words . Some words, however, are classified as both categorematic and syncategorematic. One such word is Latin infinita ‘infinite’. Because infinita can be either categorematic or syncategorematic, it is possible to form sophisms using infinita whose solutions turn on the distinction between categorematic and syncategorematic (...) uses of infinita. As a result, medieval logicians were interested in identifying correct logical rules governing the categorematic and syncategorematic uses of the term. In this paper, we look at 13th–15th-century logical discussions of infinita used syncategorematically and categorematically. We also relate the distinction to other medieval distinctions with which it has often been conflated in modern times, and show how and where these conflations go wrong. (shrink)

I consider the idea of a propositional logic of location based on the following semantic framework, derived from ideas of Prior. We have a collection L of locations and a collection S of statements such that a statement may be evaluated for truth at each location. Typically one and the same statement may be true at one location and false at another. Given this semantic framework we may proceed in two ways: introducing names for locations, predicates for the relations (...) among them and an “at” preposition to express the value of statements at locations; or introduce statement operators which do not name locations but whose truth-conditional effect depends on the truth or falsity of embedded statements at various locations. The latter is akin to Prior’s approach to tense logic. In any logic of location there will be some basic operators which we can define. By ringing the changes on the topology of locations, different logical systems may be generated, and the challenge for the logician is then in each case to find operators, axioms and rules yielding a proof theory adequate to the semantics. The generality of the approach is illustrated with familiar and not so familiar examples from modal, tense and place logic, mathematics, and even the logic of games. To the memory of Ted Dawson, who introduced me to philosophy and the writings of Arthur Prior. (shrink)

“Probably Peirce’s best-known works are the first two articles in a series of six that originally were collectively entitled Illustrations of the Logic of Science and published in Popular Science Monthly from November 1877 through August 1878. The first is entitled ‘The Fixation of Belief’ and the second is entitled ‘How to Make Our Ideas Clear.’ In the first of these papers Peirce defended, in a manner consistent with not accepting naive realism, the superiority of the scientific method over (...) other methods of overcoming doubt and ‘fixing belief.’” — Robert Burch, “Charles Sanders Peirce,” entry in the Stanford Encyclopedia of Philosophy (2021 revision) -/- “Pragmatist epistemologies often explore how we can carry out inquiries in a self-controlled and fruitful way. (Where much analytic epistemology centres around the concept of knowledge, considered as an idealised end-point of human thought, pragmatist epistemology centres around the concept of inquiry, considered as the process of knowledge-seeking and how we can improve it.) So pragmatists often provide rich accounts of the capacities or virtues that we must possess in order to inquire well, and the rules or guiding principles that we should adopt. A canonical account is Peirce’s classic early paper ‘The Fixation of Belief‘. Here Peirce states that inquiry is a struggle to replace doubt with “settled belief“, and that the only method of inquiry that can make sense of the fact that at least some of us are disturbed by inconsistent beliefs, and will subsequently reflect upon which methods of fixing belief are correct is the Method of Science, which draws on the Pragmatic Maxim described above. This contrasts with three other methods of fixing belief: i) refusing to consider evidence contrary to one’s favored beliefs (the Method of Tenacity), ii) accepting an institution’s dictates (the Method of Authority), iii) developing the most rationally coherent or elegant-seeming belief-set (the A Priori Method).“ — Catherine Legg, “Pragmatism,” entry in the Stanford Encyclopedia of Philosophy (2021 revision). (shrink)