This paper identifies, explores, and resolves a tension between two common ideas about creativity. The first is that a project is less creative insofar as it adheres to a formula of some kind. The second is that a project’s being carried out within a set of constraints does not necessarily make it less creative. The tension arises from the fact that formulae and constraints operate in the same manner. They reduce the set of options for an agent undertaking some project. (...) How could it be that reducing an agent’s options both does and does not make their project less creative? I resolve this tension by arguing that the first idea is false and should be replaced with the claim that a project is less creative insofar as the decisions that comprise it are dictated by a formula rather than being attributable to the agent. Finally, I employ a case study to demonstrate that there are ways in which formulae and constraints can diminish creativity and ways in which they can have no effect, or perhaps even enhance creativity. (shrink)

Aylsworth and Castro have argued that, following Kant’s Formula of Humanity, using ChatGPT to write humanities essays constitutes a violation of a duty to cultivate one’s humanity. I first turn to a critical evaluation of their argument and then point to a further dimension in which the FH has a bearing on the ethics of AI use. My positive contribution is to propose that Kant’s Formula of Humanity can contribute to the ethics of LLM use when we focus on the (...) prohibition against treating others as mere means. Kant’s practical philosophy is in an excellent position to capture both the danger of treating others as mere means and to account for the value of being a means. AI may endanger both: the status of some workers as ends rather than mere means and the ability of workers to function as ends on their own terms. (shrink)

Isomorphism between formulae is defined with respect to categories formalizing equality of deductions in classical propositional logic and in the multiplicative fragment of classical linear propositional logic caught by proof nets. This equality is motivated by generality of deductions. Characterizations are given for pairs of isomorphic formulae, which lead to decision procedures for this isomorphism.

Organic chemists have been able to develop a robust, theoretical understanding of the phenomena they study; however, the primary theoretical devices employed in this field are not mathematical equations or laws, as is the case in most other physical sciences. Instead it is diagrams, and in particular structural formulas and potential energy diagrams, that carry the explanatory weight in the discipline. To understand how this is so, it is necessary to investigate both the nature of the diagrams employed in (...) organic chemistry and how these diagrams are used in the explanations of the discipline. I will begin this paper by characterizing some of the major ways that structural formulas used in organic chemistry. Next I will present a model of the explanations in organic chemistry and describe how both structural formulas and potential energy diagrams contribute to these explanations. This will be followed by several examples that support my abstract account of the role of diagrams in the explanations of organic chemistry. In particular, I will consider both the appeal to ‘hyperconjugation’ in the explanation of alkene stability and how the idea of ‘ring strain’ was developed to explain the relative stability of cyclic compounds. (shrink)

Stratified formulas were introduced by Quine as an alternative way to attack Russell’s paradox. Instead of limiting comprehension by size (as in ZF set theory, using its axiom scheme of separation), unlimited comprehension is given to formulas that are in some sense descended from formulas of typed set theory. By keeping variables in a stratified structure, the most common candidates for inconsistency such as {x∣x∉x} are eliminated. Under the usual syntax of set theory, the set of stratified (...) formulas form a formal language. We show that, unlike the full class of well-formed formulas of set theory, this language is not context-free, and extend the result to its complement. Therefore, much like the axioms of PA and ZF (under their usual axiomatizations), the theory NF (New Foundations) as a formal language is not context-free. We then introduce a nonstandard syntax of set theory and show that with this syntax there is a restricted class of formulas, the exo-stratified formulas, that is context-free and full (up to relabeling of variables). (shrink)

We propose a generalization of Sahlqvist formulas to polyadic modal languages by representing such languages in a combinatorial PDL style and thus, in particular, developing what we believe to be the right syntactic approach to Sahlqvist formulas at all. The class of polyadic Sahlqvist formulas PSF defined here expands essentially the so far known one. We prove first-order definability and canonicity for the class PSF.

Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use (...) the operational view of them as provided by uniform proof-search—the proof-theoretic foundation of logic programming (LP)—to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS. (shrink)

This Element defends a reading of Kant's formulas of the moral law in Groundwork of the Metaphysics of Morals. It disputes a long tradition concerning what the first formula attempts to do. The Element also expounds the Formulas of Humanity, Autonomy and the Realm of Ends, arguing that it is only the Formula of Humanity from which Kant derives general duties, and that it is only the third formula that represents a complete and definitive statement of the moral (...) principle as Kant derives it in the Groundwork. The Element also disputes the claim that the various formulas are 'equivalent', arguing that this claim is either false or else nonsensical because it is grounded on a false premise about what Kant thinks a moral principle is for. (shrink)

Kant’s most prominent formulation of the Categorical Imperative, known as the Formula of Universal Law (FUL), is generally thought to demand that one act only on maxims that one can will as universal laws without this generating a contradiction. Kant's view is standardly summarized as requiring the 'universalizability' of one's maxims and described in terms of the distinction between 'contradictions in conception' and 'contradictions in the will'. Focusing on the underappreciated significance of the simultaneity condition included in the FUL, I (...) argue, by contrast, that the principle is better read as requiring that one be able to will two things simultaneously without self-contradiction, namely, that a maxim be one's own and that it be a universal law. This amounts to a new interpretation of the FUL with significant interpretive and philosophical advantages. (shrink)

We give alternative characterizations of exact, extendible and projective formulas in intuitionistic propositional calculus IPC in terms of n -universal models. From these characterizations we derive a new syntactic description of all extendible formulas of IPC in two variables. For the formulas in two variables we also give an alternative proof of Ghilardi’s theorem that every extendible formula is projective.

In The Formula, Luke Dormehl takes readers inside the world of numbers, asking how we came to believe in the all-conquering power of algorithms; introducing the mathematicians, artificial intelligence experts and Silicon Valley entrepreneurs who are shaping this brave new world, and ultimately asking how we survive in an era where numbers can sometimes seem to create as many problems as they solve.

Related Works: Part I: Michael Zakharyaschev. Canonical Formulas for $K4$. Part I: Basic Results. J. Symbolic Logic, Volume 57, Issue 4 , 1377--1402. Project Euclid: euclid.jsl/1183744119 Part III: Michael Zakharyaschev. Canonical Formulas for K4. Part III: The Finite Model Property. J. Symbolic Logic, Volume 62, Issue 3 , 950--975. Project Euclid: euclid.jsl/1183745306.

This chapter deals with Kant’s second formulation of the categorical imperative: the formula of humanity as an end in itself (FH). It examines Kant’s claims that a categorical imperative presupposes something of absolute value; that this must have the status of an end in itself, and that humanity or rational nature is the only thing that could meet this condition. It argues that this end must be understood in the negative sense as something not to be acted against, rather than (...) in the positive sense as something to be attained. In light of this, it examines FH, which maintains that humanity (whether in one’s own person or that of others) is always to be respected as an end in itself and never treated merely as a means, and considers Kant’s application of this principle to the four examples that considered under FLN. (shrink)

The article defends three claims regarding the relation between the different formulas of the categorical imperative. On its prevailing reading, FUL gives different moral guidance than FH; left answered, this problem is an argument for adopting a competing perspective on FUL. The prohibitions and commands of the formulas should be taken to be extensionally the same; but FKE adds a dimension missing from the others, gained by uniting their perspectives, namely, bringing the variety of moral laws into systematic (...) unity. The grammatically ambiguous phrase inGMS, 4: 436.9–10 claims that FA alone unites the other formulas in itself. (shrink)

This paper provides a methodologically original construction of Kant’s “Formula of Universal Law”. A formal structure consisting of possible worlds and games—a “game frame”—is used to implement Kant’s concept of a maxim and to define the two tests FUL comprises: the “contradiction in conception” and “contradiction in the will” tests. The paper makes two contributions. Firstly, the model provides a formal account of the variables that are built into FUL: agents, maxims, intentions, actions, and outcomes. This establishes a clear benchmark (...) for understanding how the mechanics of FUL actually work. Secondly, the analysis of the resulting framework sheds new light on discussions about the implications of FUL. On the basis of this, we suggest a move to “comprehensive Kantianism’, which is the application of FUL to systems of maxims rather than to isolated maxims. (shrink)

SummaryThe daily motion of the Moon, and the synodic arcs and periods of Saturn and Jupiter are derived from the algebraic formulae given in the Pancasiddhantika.

We provide a new proof of the following Pałasińska's theorem: Every finitely generated protoalgebraic relation distributive equality free quasivariety is finitely axiomatizable. The main tool we use are ${\mathcal{Q}}$ Q -relation formulas for a protoalgebraic equality free quasivariety ${\mathcal{Q}}$ Q . They are the counterparts of the congruence formulas used for describing the generation of congruences in algebras. Having this tool in hand, we prove a finite axiomatization theorem for ${\mathcal{Q}}$ Q when it has definable principal ${\mathcal{Q}}$ Q (...) -subrelations. This is a property obtained by carrying over the definability of principal subcongruences, invented by Baker and Wang for varieties, and which holds for finitely generated protoalgebraic relation distributive equality free quasivarieties. (shrink)

This article examines dramatisation in Deleuze and Kierkegaard, encapsulated in the formulas ‘the stuttering of language’ and ‘the question that silences’. Deleuze’s differential approach actualises virtual ideas within a system of heterogenesis through the interplay between actual concepts and intensive dramas. Conversely, Kierkegaard’s existential approach, examined in The Concept of Anxiety, establishes an infinite loop between the psychological language of anxiety and the dogmatic language of hereditary sin, leading to a third existential language centred on the moment of appropriation. (...) The article concludes by contrasting Deleuze’s focus on expression through internal differences with Kierkegaard’s emphasis on making an existential choice. (shrink)

An oft-defended claim of a close relationship between Gentzen inference rules and the meaning of the connectives they introduce and eliminate has given rise to a whole domain called proof-theoretic semantics, see Schroeder- Heister (1991); Prawitz (2006). A branch of proof-theoretic semantics, mainly developed by Dosen (2019); Dosen and Petric (2011), isolates in a precise mathematical manner formulas (of a logic L) that have the same meaning. These isomorphic formulas are defined to be those that behave identically in (...) inferences. The aim of this paper is to investigate another type of recently discussed rules in the literature, namely grounding rules, and their link to the meaning of the connectives they provide the grounds for. In particular, by using grounding rules, we will refine the notion of isomorphic formulas through the notion of hyper-isomorphic formulas. We will argue that it is actually the notion of hyper-isomorphic formulas that identify those formulas that have the same meaning. (shrink)

Breastfeeding is analogous to pregnancy as an experience, in its exclusiveness to women, and in its cost and the effects it has on equitable share of labor. Therefore, the history of formula feeding provides useful insights into the future of full ectogenesis, which could evolve into a more severe version of what formula feeding is today: simplify life for some women and provide couples with a more equitable share of work at the cost of stigma, guilt and a daily diet (...) of studies purporting to show the benefits of natural pregnancy. Making pregnancy an optional route to motherhood would make women's life trajectory more similar to men's and thus put pressure on women to compete with men on the ground shaped by men's preferences. Despite being a treasured experience of many women today, bearing children could become the luxury of the few, the province of the very poor and a choice working women will pay a high price for as women who choose pregnancy become stigmatized as self-indulgent or unprofessional and penalized for it in the workplace. At the same time, scarce societal resources that could be used to support pregnant women and working mothers would instead be directed toward proving to women or even forcing them to gestate children "the right way." While not necessarily threatening on its own, when added to formula feeding, IVF, stem-cell produced ova and sex robots, full ectogenesis could diminish men's stake in women's wellbeing and even existence. (shrink)

This essay deals with the medical recipe as an epistemic genre that played an important role in the cross-cultural transmission of knowledge. The article first compares the development of the recipe as a textual form in Chinese and European premodern medical cultures. It then focuses on the use of recipes in the transmission of Chinese pharmacology to Europe in the second half of the seventeenth century. The main sources examined are the Chinese medicinal formulas translated—presumably—by the Jesuit Michael Boym (...) and published in Specimen Medicinae Sinicae (1682), a text that introduced Chinese pulse medicine to Europe. The article examines how the translator rendered the Chinese formulas into Latin for a European audience. Arguably, the translation was facilitated by the fact that the recipe as a distinct epistemic genre had developed, with strong parallels, in both Europe and China. Building on these parallels, the translator used the recipe as a shared textual format that would allow the transfer of knowledge between the two medical cultures. (shrink)

The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the →-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics. The ∨-free reducts of Heyting algebras give rise (...) to the -canonical formulas that we studied in an earlier work. Here we introduce the -canonical formulas, which are obtained from the study of the →-free reducts of Heyting algebras. We prove that every si-logic is axiomatizable by -canonical formulas. We also discuss the similarities and differences between these two kinds of canonical formulas. One of the main ingredients of these formulas is a designated subset D of pairs of elements of a finite subdirectly irreducible Heyting algebra A. When D=A2, we show that the -canonical formula of A is equivalent to the Jankov formula of A. On the other hand, when D=∅, the -canonical formulas produce a new class of si-logics we term stable si-logics. We prove that there are continuum many stable si-logics and that all stable si-logics have the finite model property. We also compare stable si-logics to splitting and subframe si-logics. (shrink)

The Barcan formula (BF) is commonly paraphrased as the schematic conditional that if it is possible that there be a phi then something or other is possibly a phi. It is validated by the most straightforward systems of quantified modal logic. It is also widely considered to pose a threat to the commonsensical metaphysical view that there are no non-actual (or ‘merely possible’) things. I show how BF can be cleared of such a charge by construing it as a bridge (...) principle connecting modality de dicto and modality de re while retaining a Russellian robust sense of reality in modal matters. (shrink)

In the predicate calculus, variables provide a flexible indexing service which selects the actual arguments to a predicate letter from among possible arguments that precede the predicate letter (in the parse of the formula). In the process of selection, the possible arguments can be permuted, repeated (used more than once), and skipped. If this service is withheld, so that arguments must be the immediately preceding ones, taken in the order in which they occur, the formula is said to be fluted. (...) Quine showed that if a fluted formula contains only homogeneous conjunction (conjoins only subformulas of equal arity), then the satisfiability of the formula is decidable. It remained an open question whether the satisfiability of a fluted formula without this restriction is decidable. This paper answers that question. (shrink)

A formula A is said to have the contraction property in a logic L if whenever A, A, Γ ⊨ L B also A, Γ & ; L B. In MLL and in MALL without the additive constants a formula has the contraction property if it is a theorem. Adding the mix rule does not change this fact. In MALL and in affine logic A has the contraction property if either A is provable of A is equivalent to the additive (...) constant 0. We present some general proof-theoretical principles from which all these results easily follow. (shrink)

Ritualistic formulae and acts pervade the political, legal, societal and religious life of the ancient world. In many instances there are striking similarities between the formulae of the Greco-Roman world and those of the Near East. Often illumination exists from one to the other. Here I wish to notice a few passages in Greek drama where I think such illumination is possible.

We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive formulae (...) are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames. (shrink)

The Modal Predicate Calculus gives rise to issues surrounding the Barcan formulas, their converses, and necessary existence. I examine these issues by means of the Quantified Argument Calculus, a recently developed, powerful formal logic system. Quarc is closer in syntax and logical properties to Natural Language than is the Predicate Calculus, a fact that lends additional interest to this examination, as Quarc might offer a better representation of our modal concepts. The validity of the Barcan formulas and their (...) converses is shown by Quarc to be a result of the specific incorporation of quantification in the Predicate Calculus, and not as reflecting a feature of the interaction of quantification and modality more generally. Necessary existence is shown to follow from the identification, in the Predicate Calculus on its canonical interpretation, of particular quantification, ascription of existence and the ‘there is’ construction, three constructions which are distinguished in both Quarc and Natural Language. The issues surrounding the Barcan formulas, their converses and necessary existence are thus shown to be an artefact of a specific logic system, not an essential feature of our relevant modal concepts or of formal logic. (shrink)

In 1995 Visser, van Benthem, de Jongh, and Renardel de Lavalette introduced NNIL-formulas, showing that these are exactly the formulas preserved under taking submodels of Kripke models. In this article we show that NNIL-formulas are up to frame equivalence the formulas preserved under taking subframes of frames, that NNIL-formulas are subframe formulas, and that subframe logics can be axiomatized by NNIL-formulas. We also define a new syntactic class of ONNILLI-formulas. We show that (...) these are the formulas preserved in monotonic images of frames and that ONNILLI-formulas are stable formulas as introduced by Bezhanishvili and Bezhanishvili in 2013. Thus, ONNILLI is a syntactically defined set of formulas axiomatizing all stable logics. This resolves a problem left open in 2013. (shrink)

This paper studies the semiotic,epistemological and historical aspects of Berzelianformulas in early nineteenth-century organicchemistry. I argue that Berzelian formulas wereenormously productive `paper tools' for representingchemical reactions of organic substances, and forcreating different pathways of reactions. Moreover, myanalysis of Jean Dumas's application of Berzelianformulas to model the creation of chloral from alcoholand chlorine exemplifies the role played by chemicalformulas in conceptual development (the concept ofsubstitution). Studying the dialectic of chemists'collectively shared goals and tools, I argue thatpaper tools, like laboratory instruments, (...) areresources whose possibilities are not exhausted byscientists' attempts to achieve existing goals, butrather whose applications generate new goals. The term`paper tools' is introduced to emphasize that thepragmatic and syntactic aspects of symbol systems arefully comparable to physical laboratory tools. (shrink)

We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (∧, →) homomorphisms, (∧, →, 0) homomorphisms, and (∧, →, ∨) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of Zakharyaschev’s (...) theorem that each intermediate logic can be axiomatized by canonical formulas. (shrink)

The notion of formulaicity has received increasing attention in disciplines and areas as diverse as linguistics, literary studies, art theory and art history. In recent years, linguistic studies of formulaicity have been flourishing and the very notion of formulaicity has been approached from various methodological and theoretical perspectives and with various purposes in mind. The linguistic approach to formulaicity is still in a state of rapid development of the field and they are arranged into three complementary parts. The first part, (...) with three chapters, presents new theoretical and methodological insights as well as their practical application in the development of custom-designed software tools for identification and exploration of formulaic language in texts. Two papers in the second part explore formulaic language in the context of language learning. Finally, the third part, with three chapters, showcases research on formulaic language conducted primarily from corpus linguistic, discourse studies and translation studies perspectives. The volume will be of interest to anyone involved in the study of formulaic language either from a theoretical or a practical perspective. (shrink)

The formula of universal law (FUL) is a natural starting point for philosophers interested in a Kantian perspective on the morality of abortion. I argue, however, that FUL does not yield much in the way of promising or substantive conclusions regarding the morality of abortion. I first reveal how two philosophers' (Hare's and Gensler's) attempts to use Kantian considerations of universality and prescriptivity fail to provide analyses of abortion that are either compelling or true to Kant=s understanding of FUL. I (...) then turn to some recent interpretations of Kant=s FUL contradiction in conception (CC) and contradiction in will (CW) tests. I argue that none of the interpretations of the CC testBincluding the practical interpretation favored by KorsgaardBdoes much to reveal moral problems with maxims of abortion. The CW test (as developed by Herman) is more helpful. Nevertheless, I argue that neither by considering abortion maxims as a subset of maxims of convenience killing, nor by considering such maxims as maxims of refusing to aid, can the CW test generate a general prohibition of abortion. At best, the CW test illuminates the abortion issue because by forcing us to think about how killing a fetus differs from killing other human beings, what attitudes we may reasonably have toward a fetus, and whether Kant's moral theory must be amended to do justice to the problem of abortion. But to pursue these questions, we must look beyond FUL; Kant’s formula of humanity and doctrine of virtue may well have more to offer. (shrink)

Kantians are increasingly deserting the universal law formula in favor of the humanity formula. The former, they argue, is open to various decisive objections; the two are not equivalent; and it is only by appealing to the humanity formula that Kant can reliably generate substantive implications from his theory of an acceptable sort. These assessments of the universal law formula, which clash starkly with Kant's own assessment of it, are based on various widely accepted interpretative assumptions. These assumptions, it is (...) argued in this article, depend on misleading translations of key terms; selective attention to Kant's concrete examples; not taking seriously Kant's theoretical claims about the relations among his various ideas; and a failure to take into account Kant's idiosyncratic definitions of key concepts. The article seeks to right these interpretative wrongs, and finds that the universal law formula is not open to many of the standard objections. (shrink)

We introduce relativized modal algebra homomorphisms and show that the category of modal algebras and relativized modal algebra homomorphisms is dually equivalent to the category of modal spaces and partial continuous p-morphisms, thus extending the standard duality between the category of modal algebras and modal algebra homomorphisms and the category of modal spaces and continuous p-morphisms. In the transitive case, this yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we (...) give an algebraic description of canonical, subframe, and cofinal subframe formulas, and provide a new algebraic proof of Zakharyaschev’s theorem that each logic over K4 is axiomatizable by canonical formulas. (shrink)

Ancient Greek mathematics developed the original feature of being deductive mathematics. This article attempts to give a explanation f or this achievement. The focus is on the use of a fixed system of linguistic formulae in Greek mathematical texts. It is shown that the structure of this system was especially adapted for the easy computation of operations of substitution on such formulae, that is, of replacing one element in a fixed formula by another, and it is further argued that such (...) operations of substitution were the main logical tool required by Greek mathematical deduction. The conclusion explains why, assuming the validity of the description above, this historical level is the best explanatory level for the phenomenon of Greek mathematical deduction. (shrink)

For each intermediate propositional logicJ, J * denotes the least predicate extension ofJ. By the method of canonical models, the strongly Kripke completeness ofJ *+D(=x(p(x)q)xp(x)q) is shown in some cases including:1. J is tabular, 2. J is a subframe logic. A variant of Zakharyashchev's canonical formulas for intermediate logics is introduced to prove the second case.

We study the computational complexity of the model checking problem for quantifier-free dependence logic ${(\mathcal{D})}$ formulas. We characterize three thresholds in the complexity: logarithmic space (LOGSPACE), non-deterministic logarithmic space (NL) and non-deterministic polynomial time (NP).

We characterize the modal logics of elementary classes of Kripke frames as precisely those modal logics that are axiomatized by modal axioms synthesized in a certain effective way from "quasi-positive" sentences of hybrid logic. These are pure positive hybrid sentences with arbitrary existential and relativized universal quantification over nominals. The proof has three steps. The first step is to use the known result that the modal logic of any elementary class of Kripke frames is also the modal logic of the (...) closure of this class under disjoint unions, generated subframes, bounded morphic images, and ultraroots. This latter class can be defined by the first-order sentences of a special syntactic form (called pseudo-equations by Goldblatt) that are valid in the former class. The second step is to translate these pseudo-equations into equivalent quasi-positive hybrid sentences. In the third and main step, we show that any quasi-positive sentence S generates an infinite set of modal formulas called "approximants," which together axiomatize a canonical modal logic that is sound and complete for the class of frames validating S. The proof is analogous to standard proofs of Sahlqvist's theorem. It generalizes to sets of quasi-positive sentences. The main result now follows. (shrink)

This note corrects an error in an earlier paper (Urquhart 2023)claiming that the algebra of three-variable formulas in the logic KR is finite.Here we show that the algebra of one-variable formulas in this logic is infinite.

Safe first-order formulas generalize the concept of a safe rule, which plays an important role in the design of answer set solvers. We show that any safe sentence is equivalent, in a certain sense, to the result of its grounding—to the variable-free sentence obtained from it by replacing all quantifiers with multiple conjunctions and disjunctions. It follows that a safe sentence and the result of its grounding have the same stable models, and that stable models of a safe sentence (...) can be characterized by a formula of a simple syntactic form. (shrink)

Related Works: Part I: Michael Zakharyaschev. Canonical Formulas for $K4$. Part I: Basic Results. J. Symbolic Logic, Volume 57, Issue 4 , 1377--1402. Project Euclid: euclid.jsl/1183744119 Part II: Michael Zakharyaschev. Canonical Formulas for K4. Part II: Cofinal Subframe Logics. J. Symbolic Logic, Volume 61, Issue 2 , 421--449. Project Euclid: euclid.jsl/1183745008.

We generalize the theory of canonical formulas for K4, the logic of transitive frames, to wK4, the logic of weakly transitive frames. Our main result establishes that each logic over wK4 is axiomatizable by canonical formulas, thus generalizing Zakharyaschev’s theorem for logics over K4. The key new ingredients include the concepts of transitive and strongly cofinal subframes of weakly transitive spaces. This yields, along with the standard notions of subframe and cofinal subframe logics, the new notions of transitive (...) subframe and strongly cofinal subframe logics over wK4. We obtain axiomatizations of all four kinds of subframe logics over wK4. We conclude by giving a number of examples of different kinds of subframe logics over wK4. (shrink)

In two recent articles I have argued that Kant’s legal and political philosophy can shed new light on his much-contested account of moral autonomy and that important changes in his political theory help to explain why in his later work the Formula of Autonomy disappears. In the present essay, I respond to comments by Sorin Baiasu and Marie Newhouse, who argue that the changes in Kant’s political theory fail to explain the disappearance of the Formula of Autonomy, since in both (...) phases Kant held that laws are given by the people’s representatives. I offer additional support for my original argument by developing a more detailed account of Kant’s conception of the relation between the legislating representatives and the people they are taken to represent. I argue that Kant’s conception of the proper representation relation changed, and that it changed so fundamentally that the analogy at the basis of the Formula of Autonomy was no longer apt, thus providing a plausible explanation for its disappearance. I also address several related comments and propose an explanation for why Kant dropped the very idea of autonomy as a property of the will. (shrink)

This chapter contains sections titled: Introduction Some Common Misunderstandings How Different Are PGW, FUL, and FLN? The Role of the Universal Law Formulas Issues Regarding the Maxim and its Universal Counterpart The Two Hegelian Objections Contradictions in Conception Contradictions in the Will Three Persistent Problems and One Very Modest Proposal Bibliography.

We give several new equivalences of NIP for formulas and new proofs of known results using Talagrand (Ann Probab 15:837–870, 1987) and Haydon et al. (in: Functional Analysis Proceedings, The University of Texas at Austin 1987–1989, Lecture Notes in Mathematics, Springer, New York, 1991). We emphasize that Keisler measures are more complicated than types (even in the NIP context), in an analytic sense. Among other things, we show that for a first order theory T and a formula $$\phi (x,y)$$, (...) the following are equivalent. (shrink)