Modal Logic

A brief overview of the system S5 in modal logic as defined by Brian F. Chellas, author of "Modal Logic: An Introduction." The history and usage of modal logic are given mention, along with some applications. Very much a draft. Written for PhileInSophia on July 5, 2021.

Prior to Kripke's seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano-Tarski notion of logical truth. According to this interpretation, `it is necessary that A' is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some views in (...) modal metaphysics. In particular, we present an hitherto unpublished solution to problems 41 and 42 from Friedman's 102 problems, which uses a different method of proof from the solution presented in the paper of Tadeusz Prucnal. (shrink)

This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the property of knowledge in epistemic logic, I argue that rational intuition can be codified by a modal operator governed by the modal $\mu$-calculus. Via correspondence results between fixed point modal propositional logic and the bisimulation-invariant fragment of monadic second-order logic, a precise translation can then be provided between the notion of 'intuition-of', i.e., the cognitive phenomenal properties of thoughts, and the modal operators regimenting the (...) notion of 'intuition-that'. I argue that intuition-that can further be shown to entrain conceptual elucidation, by way of figuring as a dynamic-interpretational modality which induces the reinterpretation of both domains of quantification and the intensions and hyperintensions of mathematical concepts that are formalizable in monadic first- and second-order formal languages. Hyperintensionality is countenanced via a topic-sensitive epistemic two-dimensional truthmaker semantics. (shrink)

Possibility semantics offers an elegant framework for a semantic analysis of modal logic that does not recruit fully determinate entities such as possible worlds. The present papers considers the application of possibility semantics to the modeling of the indeterminacy of the future. Interesting theoretical problems arise in connection to the addition of object-language determinacy operator. We argue that adding a two-dimensional layer to possibility semantics can help solve these problems. The resulting system assigns to the two-dimensional determinacy operator a well-known (...) logic (coinciding with the logic of universal modalities under global consequence). The paper concludes with some preliminary inroads into the question of how to distinguish two-dimensional possibility semantics from the more established branching framework. (shrink)

Cathoristic logic is a multi-modal logic where negation is replaced by a novel operator allowing the expression of incompatible sentences. We present the syntax and semantics of the logic including complete proof rules, and establish a number of results such as compactness, a semantic characterisa- tion of elementary equivalence, the existence of a quadratic-time decision pro- cedure, and Brandom’s incompatibility semantics property. We demonstrate the usefulness of the logic as a language for knowledge representation.

It is often assumed that pluralities are rigid, in the sense of having all and only their actual members necessarily. This assumption is operative in standard approaches to modal plural logic. I argue that a sceptical approach towards the assumption is warranted.

Ontological arguments like those of Gödel (1995) and Pruss (2009; 2012) rely on premises that initially seem plausible, but on closer scrutiny are not. The premises have modal import that is required for the arguments but is not immediately grasped on inspection, and which ultimately undermines the simpler logical intuitions that make the premises seem plausible. Furthermore, the notion of necessity that they involve goes unspecified, and yet must go beyond standard varieties of logical necessity. This leaves us little reason (...) to believe the premises, while their implausible existential import gives us good reason not to. (shrink)

The dogma that the propositional logic of metaphysical modality is S5 is rebutted in related installments (previously published and unpublished essays).

Possible worlds are commonly seen as an interpretation of modal operators such as "possible" and "necessary". Here, we develop possible world semantics (PWS) which can be expressed in basic set theory and first-order logic, thus offering a reductionist account of modality. Specifically, worlds are understood as complete sets of statements and possible worlds are sets whose statements are consistent with a set of conceptual laws. We introduce the construction calculus (CC), a set of axioms and rules for truth, possibility, worldness (...) and consistency. We show that CC allows to prove fundamental theorems about necessity, possibility, impossibility and contigency, thus demonstrating prima-facie plausibility of PWS. Finally, we discuss the explanatory power of our approach and draw connections between our account and established philosophical conceptions of possible worlds. (shrink)

Some central epistemological notions are expressed by sentential operators O that entail the possibility of knowledge in the sense that 'Op' entails 'It is possible to know that p'. We call these modal-epistemological notions. Using apriority and being in a position to know as case studies, we argue that the logics of modal epistemological notions are extremely weak. In particular, their logics are not normal and do not include any closure principles.

These lecture notes were composed while teaching a class at Stanford and studying the work of Brian Chellas (Modal Logic: An Introduction, Cambridge: Cambridge University Press, 1980), Robert Goldblatt (Logics of Time and Computation, Stanford: CSLI, 1987), George Hughes and Max Cresswell (An Introduction to Modal Logic, London: Methuen, 1968; A Companion to Modal Logic, London: Methuen, 1984), and E. J. Lemmon (An Introduction to Modal Logic, Oxford: Blackwell, 1977). The Chellas text influenced me the most, though the order of (...) presentation is inspired more by Goldblatt.2 My goal was to write a text for dedicated undergraduates with no previous experience in modal logic. The text had to meet the following desiderata: (1) the level of difficulty should depend on how much the student tries to prove on his or her own—it should be an easy text for those who look up all the proofs in the appendix, yet more difficult for those who try to prove everything themselves; (2) philosophers (i.e., colleagues) with a basic training in logic should be able to work through the text on their own; (3) graduate students should find it useful in preparing for a graduate course in modal logic; (4) the text should prepare people for reading advanced texts in modal logic, such as Goldblatt, Chellas, Hughes and Cresswell, and van Benthem, and in particular, it should help the student to see what motivated the choices in these texts; (5) it should link the two conceptions of logic, namely, the conception of a logic as an axiom system (in which the set of theorems is constructed from the bottom up through proof sequences) and the conception of a logic as a set containing initial ‘axioms’ and closed under ‘rules of inference’ (in which the set of theorems is constructed from the top down, by carving out the logic from the set of all formulas as the smallest set closed under the rules); finally, (6) the pace for the presentation of the completeness theorems should be moderate—the text should be intermediate between Goldblatt and Chellas in this regard (in Goldblatt, the completeness proofs come too quickly for the undergraduate, whereas in Chellas, too many unrelated.... (shrink)

This paper is a study of a distinctively chemical notion of possibility. This is the notion of possibility that occurs in chemical discourses when chemists speak of the possibility or impossibility of achieving a given result through chemical means. This notion pertains to the possibility of processes, not of compounds, so it differs from the kind of chemical possibility mentioned in Wittgenstein's Philosophical Investigations or the kinds discussed in the literature on Putnam's Twin Earth argument. I argue that this process-oriented (...) notion of possibility cannot be reduced in a simple way to physical possibility, and that standard possible worlds semantics does not allow a natural analysis of this notion. I suggest an extension of possible worlds semantics that may overcome this limitation. I finish by pointing out some open questions about chemical possibility. (shrink)

We consider a modal language over crisp frames and formulas evaluated on a finite MTL-chain (a linearly ordered commutative integral residuated lattice). We first show that the basic modal abstract logic with constants for the values of the MTL-chain is the maximal abstract logic satisfying Compactness, the Tarski Union Property and strong invariance for bisimulations. Finally, we improve this result by replacing the Tarski Union Property by a relativization property.

Propositional modal logic over relational frames is naturally extended with propositional quantifiers by letting them range over arbitrary sets of worlds of the relevant frame. This is also known as second-order propositional modal logic. The propositionally quantified modal logic of a class of relational frames is often not axiomatizable, although there are known exceptions, most notably the case of frames validating the strong modal logic$\mathrm {S5}$. Here, we develop new general methods with which many of the open questions in this (...) area can be answered. We illustrate the usefulness of these methods by applying them to a range of examples, which provide a detailed picture of which normal modal logics define classes of relational frames whose propositionally quantified modal logic is axiomatizable. We also apply these methods to establish new results in the multimodal case. (shrink)

We give a theory of epistemic modals in the framework of possibility semantics and axiomatize the corresponding logic, arguing that it aptly characterizes the ways in which reasoning with epistemic modals does, and does not, diverge from classical modal logic.

Fragmentalism allows incompatible facts to constitute reality in an absolute manner, provided that they fail to obtain together. In recent years, the view has been extensively discussed, with a focus on its formalisation in model-theoretic terms. This paper focuses on three formalisations: Lipman’s approach, the subvaluationist interpretation, and a novel view that has been so far overlooked. The aim of the paper is to explore the application of these formalisations to the alethic modal case. This logical exploration will allow us (...) to study (i) cases of metaphysical incompatibility between modal facts and (ii) cases of modal dialetheias. In turn, this will enrich our understanding of the role of impossibility in the fragmentalist framework. (shrink)

This paper draws attention to a proof of the necessity of identity given by Arthur Prior. In its simplicity, it is comparable to a proof of Quine's, popularised by Kripke, but it is slightly different. Prior's Polish notation is transcribed into a more familiar idiom. Prior's proof is followed by a proof of the necessity of difference, possibly the first such proof in the literature, which is also repeated here and transcribed. The paper concludes with a brief discussion of Prior's (...) views on identity and difference over time. (shrink)

Rumfitt has given two arguments that in unilateralist verificationist theories of meaning, truth collapses into correct assertibility. In the present paper I give similar arguments that show that in unilateral falsificationist theories of meaning, falsehood collapses into correct deniability. According to bilateralism, meanings are determined by assertion and denial conditions, so the question arises whether it succumbs to similar arguments. I show that this is not the case. The final section considers the question whether a principle central to Rumfitt's first (...) argument, ‘It is assertible that if and only if it is assertible that it is assertible that ’, is one that bilateralists can reject, and concludes that they cannot. It follows that the logic of assertibility and deniability, according to a result by Williamson, is the little known modal logic K4 studied by Sobociński. The paper ends with a plaidoyer for bilateralists to adopt this logic. (shrink)

This paper argues that modal realism has a problem with mathematical impossibilities. Due to the peculiar way it treats both propositions and mathematical objects, modal realism cannot distinguish the content of different mathematically impossible beliefs. While one might be happy to identify all logically impossible beliefs, there are many different mathematically impossible beliefs, none of which is a belief in a logical contradiction. The fact that it cannot distinguish these beliefs speaks against adopting modal realism.

In this work, we illustrate applications of a semantic framework for non-congruential modal logic based on hyperintensional models. We start by discussing some philosophical ideas behind the approach; in particular, the difference between the set of possible worlds in which a formula is true (its intension) and the semantic content of a formula (its hyperintension), which is captured in a rigorous way in hyperintensional models. Next, we rigorously specify the approach and provide a fundamental completeness theorem. Moreover, we analyse examples (...) of non-congruential systems that can be semantically characterized within this framework in an elegant and modular way. Finally, we compare the proposed framework with some alternatives available in the literature. In the light of the results obtained, we argue that hyperintensional models constitute a basic, general and unifying semantic framework for (non-congruential) modal logic. (shrink)

We investigate a recently devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely combinatorial (...) characterisation of polyhedrally complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov–Fine formulas of ‘starlike trees’ all of which are polyhedrally complete. The polyhedral completeness theorem for these ‘starlike logics’ is the second main result of this paper. (shrink)

Some philosophers advance the claim that the phenomena of logical omniscience and of the indiscernibility of metaphysical statements, which arise in (certain) interpretations of normal modal logic, provide strong reasons in favour of impossible world approaches. These two specific lines of argument will be presented and discussed in this paper. Contrary to the recent much-held view that the characteristics of these two phenomena provide us with strong reasons to adopt impossible world approaches, the view defended here is that no such (...) ‘knock-down arguments’ do emanate on those grounds. This is not to rule out that there cannot be any other good reasons for assuming impossible world semantics. However, the discussion of a further argument for impossible worlds will suggest that different attempts to argue for them likely present intertwined problems. (shrink)

A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational approach to modelling vague language. This approach is formally equivalent to an account of logical consequence according (...) to which α1, ..., αn entails β just in case □α1, ..., □αn entails □β in the modal logic S5. This raises a natural question: If we start with a different underlying modal logic, can we generate a strongly classical logic? This paper explores this question. In particular, it discusses four related technical issues: (1) Which base modal logics generate strongly classical logics and which generate weakly classical logics? (2) Which base logics generate themselves? (3) How can we directly characterize the logic generated from a given base logic? (4) Given a logic that can be generated, which base logics generate it? The answers to these questions have philosophical interest. They can help us to determine whether there is a plausible supervaluational approach to modelling vague language that yields the usual meta-rules. They can also help us to determine the feasibility of other philosophical projects that rely on an analogous formalism, such as the project of defining logical consequence in terms of the preservation of an epistemic status. (shrink)

A fundamental question asked in modal logic is whether a given theory is consistent. But consistent with what? A typical way to address this question identifies a choice of background knowledge axioms (say, S4, D, etc.) and then shows the assumptions codified by the theory in question to be consistent with those background axioms. But determining the specific choice and division of background axioms is, at least sometimes, little more than tradition. This paper introduces generic theories for propositional modal logic (...) to address consistency results in a more robust way. As building blocks for background knowledge, generic theories provide a standard for categorical determinations of consistency. We argue that the results and methods of this paper help to elucidate problems in epistemology and enjoy sufficient scope and power to have purchase on problems bearing on modalities in judgement, inference, and decision making. (shrink)

Classical theists hold that God is omnipotent. But now suppose a critical atheologian were to ask: Can God create a stone so heavy that even he cannot lift it? This is the dilemma of the stone paradox. God either can or cannot create such a stone. Suppose that God can create it. Then there's something he cannot do – namely, lift the stone. Suppose that God cannot create the stone. Then, again, there's something he cannot do – namely, create it. (...) Either way, God cannot be omnipotent. Among the variety of known theological paradoxes, the paradox of the stone is especially troubling because of its logical purity. It purports to show that one cannot believe in both God and the laws of logic. In the face of the stone paradox, how should the contemporary analytic theist respond? Ought they to revise their belief in theology or their belief in logic? Ought they to lose their religion or lose their mind? (shrink)

Havstad (2022) argues that the argument from inductive risk for the claim that non-epistemic values have a legitimate role to play in the internal stages of science is deductively valid. She also defends its premises and thus soundness. This is, as far as we are aware, the best reconstruction of the argument from inductive risk in the existing literature. However, there is a small flaw in this reconstruction of the argument from inductive risk which appears to render the argument invalid. (...) This flaw is superficial, and a small amendment to it rescues the claim of validity. (shrink)

El presente trabajo pretende explicitar que los operadores modales, como construcciones lógicas, insertos en las interpretaciones modales (MI) de la mecánica cuántica son usados de manera informal sin una semántica modal adecuada. Primero se estudiarán en detalle los motivos por los que ninguna lógica cuántica puede ofrecer una base apropiada para formalizar estos operadores en contextos mecánico-cuánticos. A continuación, se presentará el enfoque de las historias cuánticas como una nueva lógica cuántica (NQL) intrínsecamente booleana como posible herramienta para formalizar operadores (...) modales en el contexto de las MI. Tras comentar brevemente las principales MI, se propondrá abordar un estudio de las modalidades presentes en ellas desde esta NQL prestando siempre especial atención al problema de la medida como hilo conductor. Finalmente, se pretende señalar cuáles han de ser los primeros pasos necesarios para ofrecer una exposición lógicamente consistente de las modalidades en contextos cuánticos relevantes con NQL como herramienta, así como señalar las ventajas que esta perspectiva podría llegar a ofrecer. (shrink)

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; (...) if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations. (shrink)

We present a new puzzle about logical truth, necessity, and moral responsibility. We defend one solution to the puzzle. A corollary of our preferred solution is that prominent arguments for the incompatibility of determinism and moral responsibility are invalid.

A counterpossible is a counterfactual whose antecedent is impossible. The vacuity thesis says all counterpossibles are true solely because their antecedents are impossible. Recently, some have rejected the vacuity thesis by citing purported non-vacuous counterpossibles in science. One limitation of this work, however, is that it is not grounded in experimental data. Do scientists actually reason non-vacuously about counterpossibles? If so, what is their basis for doing so? We presented biologists (N = 86) with two counterfactual formulations of a well-known (...) model in biology, the antecedents of which contain what many philosophers would characterize as a metaphysical impossibility. Participants consistently judged one counterfactual to be true, the other to be false, and they explained that they formed these judgments based on what they perceived to be the mathematical relationship between the antecedent and consequent. Moreover, we found no relationship between participants’ judgments about the (im)possibility of the antecedent and whether they judged a counterfactual to be true or false. These are the first experimental results on counterpossibles in science with which we are familiar. We present a modal semantics that can capture these judgments, and we deal with a host of potential objections that a defender of the vacuity thesis might make. (shrink)

Sorensen (1992) has provided two modal-logical schemas to reconstruct the logical structure of two types of destructive thought experiments: the Necessity Refuter and the Possibility Refuter. The schemas consist of five propositions which Sorensen claims but does not prove to be inconsistent.We show that the five propositions, as presented by Sorensen, are not inconsistent, but by adding a premise (and a logical truth), we prove that the resulting sextet of premises is inconsistent. Häggqvist (2009) has provided a different modal-logical schema (...) (Counterfactual Refuter), which is equivalent to four premises, again claimed to be inconsistent. We show that this schema also is not inconsistent, for similar reasons. Again, we add another premise to achieve inconsistency. The conclusion is that all three modal-logical reconstructions of the arguments that accompany thought experiments, two by Sorensen and one by Häggqvist, have now been made rigorously correct. This may inaugurate new avenues to respond to destructive thought experiments. (shrink)

This paper investigates inquisitive extensions of normal modal logic with an existential modal operator taken as primitive. The semantics of the existential modality is generalized to apply to questions, as well as statements. When the generalized existential modality is applied to a question, the result is a statement that roughly expresses that each way of resolving the question is consistent with the available information. I study the resulting logic both from a semantic and from a proof-theoretic point of view. I (...) argue that it can be used for reasoning about a general notion of ignorance, and for reasoning about choice-offering permissions and obligations. The main technical results are sound and complete axiomatizations, both for the class of all Kripke frames, and for any class of frames corresponding to a canonical normal modal logic. (shrink)

The received view says that possibility is the dual of necessity: a proposition is (metaphysically, logically, epistemically etc.) possible iff it is not the case that its negation is (metaphysically, logically, epistemically etc., respectively) necessary. This reading is usually taken for granted by modal logicians and indeed seems plausible when dealing with logical or metaphysical possibility. But what about epistemic possibility? We argue that the dual definition of epistemic possibility in terms of epistemic necessity generates tension when reasoning about non-idealized (...) agents and is a problem of concern for most hyperintensional epistemic logics that alleviate the problem of logical omniscience. The tension is particularly evident when knowledge is taken as a primitive to define other epistemic concepts, such as justification and belief, as done in the knowledge-first tradition. We propose a non-dual interpretation of epistemic possibility, employing a hyperintensionality filter similar to the one that makes the corresponding epistemic necessity operator hyperintensional. We employ the proposed semantics to model Stalnaker's belief as epistemic possibility of knowledge and provide a sound and complete axiomatization for a hyperintensional version of his bimodal logic of knowledge and belief. (shrink)

Stephen Read presented harmonious inference rules for identity in classical predicate logic. I demonstrate here how this approach can be generalised to a setting where predicate logic has been extended with epistemic modals. In such a setting, identity has two uses. A rigid one, where the identity of two referents is preserved under epistemic possibility, and a non-rigid one where two identical referents may differ under epistemic modality. I give rules for both uses. Formally, I extend Quantified Epistemic Multilateral Logic (...) with two identity signs. I argue that a uniform meaning for identity tout court can be given by adopting Maria Aloni’s account of reference using conceptual covers. We obtain a harmonious set of rules for identity that is sound and complete for Aloni’s model theory. (shrink)

This essay considers Kant’s theory of modality in light of a debate in contemporary modal metaphysics and modal logic concerning the Barcan formulas. The comparison provides a new and fruitful perspective on Kant’s complex and sometimes confusing claims about possibility and necessity. Two central Kantian principles provide the starting point for the comparison: that the possible must be grounded in the actual and that existence is not a real predicate. Both are shown to be intimately connected to the Barcan formulas, (...) and Kant’s views on what he distinguishes as three different kinds of modality are then considered in light of this connection. (shrink)

This article proposes the use of temporal logic for an analysis of instrumentality inspired by the work of G.H. von Wright. The first part of the article contains the philosophical foundations. We discuss von Wright’s general theory of agency and his account of instrumentality. Moreover, we propose several refinements to this framework via rigorous definitions of the core notions involved. In the second part, we develop a logical system called Temporal Logic of Action and Expectations (TLAE). The logic is inspired (...) by a fragment of propositional dynamic logic based on indeterministic time. The system is proven to be weakly complete relative to its given semantics. We then employ TLAE to formalise and analyse the instrumentality relations defined in the first part of the paper. Last, we point out philosophical implications and possible extensions of our work. (shrink)

A novel fatalistic argument that combines elements of modal, temporal, and epistemic logic to prove that the fixed past is not compatible with the open future has recently been presented by Lampert (Analysis 82(3):426–434, 2022). By the construction of a countermodel, it is shown that his line of reasoning is defective. However, it is also explained how Lampert’s argument could be corrected if it were supported with an extra premise regarding the temporal status of a priori knowledge. This additional assumption—which (...) was tacit in the original presentation—is shown to be the weakest link, and it is argued that, despite Lampert’s assurance, it can be plausibly rejected in a roughly Ockhamist framework. Thus, it is concluded that the fixed past poses no threat to the open future; however, a few different lessons about knowledge, necessity, and time are drawn from careful reflection on this novel argument. (shrink)

We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of *being a necessity*, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about (...) their logical behaviour. We also wield the framework to probe the conditions under which a logicist account of necessities is possible, in which the theory is completely reducible to logic. (shrink)

Upward monotonic semantics for necessity modals give rise to Ross’s Puzzle: they predict that □φ entails □(φ ∨ ψ), but common intuitions about arguments of this form suggest they are invalid. It is widely assumed that the intuitive judgments involved in Ross’s Puzzle can be explained in terms of the licensing of ‘Diversity’ inferences: from □(φ ∨ ψ), interpreters infer that the truth of each disjunct (φ, ψ) is compatible with the relevant set of worlds. I introduce two pieces of (...) data that this analysis fails to explain. Analyzing this data, I argue, suggests that necessity modals with embedded disjunctions license ‘Independence’ inferences: from the truth of □(φ ∨ ψ), interpreters infer that φ-without-ψ and ψ-without-φ are each compatible with the relevant set of worlds. I outline a bilateral inquisitive semantics for necessity modals that predicts the validity of the Independence inferences. I then argue that the resulting theory should be understood as one on which disjunctions denote pluralities of propositions, and necessity modals behave like collective predicates applied to these pluralities. (shrink)

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...) for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)

The concept of hypodox is dual to the concept of paradox. Whereas a paradox is incompatibly overdetermined, a hypodox is underdetermined. Indeed, many particular paradoxes have dual hypodoxes. So, naively the dual of Russell’s Paradox is whether the set of all sets that are members of themselves is self-membered. The dual of the Liar Paradox is the Truth-teller, and a hypodoxical dual of the Heterological paradox is whether ‘autological’ is autological. I provide some analysis of the duality and I search (...) for modal hypodoxes as duals to known paradoxes. I also try to continue this search further by mapping the relations between some paradoxes and hypodoxes in duality squares. This investigation increases the known extension of the concept of hypodox, and the duality relation between paradoxes and hypodoxes assists in finding some of these. (shrink)

In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploščica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then (...) extend each of the three representations to lattices with a multiplicative unary modality; the representing structures, like so-called graph-based frames, add a second relation of accessibility interacting with compatibility. The three representations generalize possibility semantics for classical modal logics to non-classical modal logics, motivated by a recent application of modal orthologic to natural language semantics. (shrink)

Jaakko Hintikka proposed treating objectual perception sentences, such as “Alice sees Bob,” as de re propositional perception sentences. Esa Saarinen extended Hintikka’s idea to eventive perception sentences, such as “Alice sees Bob smile.” These approaches, elegant as they may be, are not philosophically neutral, for they presuppose, controversially, that the content of all perceptual experiences is propositional in nature. The aim of this paper is to propose a formal treatment of objectual and eventive perception sentences that builds on Hintikka’s modal (...) approach to propositional attitude ascriptions while avoiding controversial assumptions on the nature of perceptual experiences. Despite being simple and theoretically frugal, our approach is powerful enough to express a variety of interesting philosophical views about propositional, objectual, and eventive perception sentences, thus enabling the study of their inferential relationships. (shrink)

We present epistemic multilateral logic, a general logical framework for reasoning involving epistemic modality. Standard bilateral systems use propositional formulae marked with signs for assertion and rejection. Epistemic multilateral logic extends standard bilateral systems with a sign for the speech act of weak assertion (Incurvati and Schlöder 2019) and an operator for epistemic modality. We prove that epistemic multilateral logic is sound and complete with respect to the modal logic S5 modulo an appropriate translation. The logical framework developed provides the (...) basis for a novel, proof-theoretic approach to the study of epistemic modality. To demonstrate the fruitfulness of the approach, we show how the framework allows us to reconcile classical logic with the contradictoriness of so-called Yalcin sentences and to distinguish between various inference patterns on the basis of the epistemic properties they preserve. (shrink)

It is a widely held principle that no one is able to do something that would require the past to have been different from how it actually is. This principle of the fixity of the past has been presented in numerous ways, playing a crucial role in arguments for logical and theological fatalism, and for the incompatibility of causal determinism and the ability to do otherwise. I will argue that, assuming bivalence, this principle is in conflict with standard views about (...) knowledge and the semantics for ‘actually’. I also consider many possible responses to the argument. (shrink)

This paper investigates and develops generalizations of two-dimensional modal logics to any finite dimension. These logics are natural extensions of multidimensional systems known from the literature on logics for a priori knowledge. We prove a completeness theorem for propositional n-dimensional modal logics and show them to be decidable by means of a systematic tableau construction.

Discovering the differences between the various systems of modal logics was one of the advantages of inventing Kripke semantics. One of the most obvious examples is interpreting the necessity of provability in provability logic. According to Boolos in The Logic of Provability, by discovering this logic, we can say that the understanding of new issues in the field of argument was opened. In this paper, with a formal approach and with a descriptive-analytical and comparative method, the axiomatic propositional systems of (...) the GL, Grz, and H, and their possible world semantics based on Kripke semantics are studied, as well as the sequent calculus of GL (in Peano arithmetic) and GLS (in the standard model) were introduced. Finally, the meta-theorems of soundness, consistency, and completeness of the GL were interpreted and proved. (shrink)