This paper studies, by way of an example, the intuitionistic propositional connective * defined in the language of second order propositional logic by * ≡ ∃Q. In full topological models * is not generally definable but over Cantor-space and the reals it can be classically shown that *↔ ⅂⅂P; on the other hand, this is false constructively, i.e. a contradiction with Church's thesis is obtained. This is comparable with some well-known results on the completeness of intuitionistic first-order predicate (...) logic. Over [0, 1], the operator * is undefinable. We show how to recast this argument in terms of intuitive intuitionistic validity in some parameter. The undefinability argument essentially uses the connectedness of [0, 1]; most of the work of recasting consists in the choice of a suitable intuitionistically meaningful parameter, so as to imitate the effect of connectedness. Parameters of the required kind can be obtained as so-called projections of lawless sequences. (shrink)

This paper studies, by way of an example, the intuitionistic propositional connective * defined in the language of second order propositional logic by. In full topological models * is not generally definable, but over Cantor-space and the reals it can be classically shown that; on the other hand, this is false constructively, i.e. a contradiction with Church's thesis is obtained. This is comparable with some well-known results on the completeness of intuitionistic first-order predicate logic.Over [0, 1], the operator (...) * is (constructively and classically) undefinable. We show how to recast this argument in terms of intuitive intuitionistic validity in some parameter. The undefinability argument essentially uses the connectedness of [0, 1]; most of the work of recasting consists in the choice of a suitable intuitionistically meaningful parameter, so as to imitate the effect of connectedness. (shrink)

Relational, operator, and predicate systems are distinguished on the basis that they correspond to the three possible pair-wise bracketings into two constituents of the three parts of a proposition: relation, subject, and object. It is asserted that the verbal propositional modality (left hemisphere) uses a predicate grammar, while the spatial-image operational modality (right hemisphere) uses an operator grammar. Verbal propositional memory has the capacity for extensive propositional embedding while spatial operational memory does not.

It is a widely held view in philosophy that propositions perform a plethora of different theoretical roles. Amongst other things, they are believed to be the semantic values of sentences in contexts, the objects of attitudes, the contents of illocutionary acts, and the referents of that-clauses. This assumption is often combined with the claim that propositions have their truth-values eternally. In this paper I aim to show that these two assumptions are incompatible: propositions cannot both fulfill the mentioned roles and (...) be eternally true or false. Following Kaplan and Lewis’s Operator Argument, I argue that compositional semantic values of sentences in contexts do not correspond to eternal propositions. Thus, either we regard the non-eternal entities that in fact realize the semantic role of propositions as also fulfilling the remaining propositional roles, or we abandon the assumption that there is a unique realizer of all the roles. The Operator Argument has recently come under attack, mainly for its tense-logical assumptions. However, rejecting these assumptions is not a sufficient defense of the compatibility of the two claims, since the extensional alternative to the tense-logical framework does not allow us to universally retain eternal propositions as compositional semantic values of sentences either. (shrink)

Drawing on our Associative-Propositional Evaluation (APE) Model, we argue for the usefulness of distinguishing between basic operating principles of learning processes (associative linking vs. propositional reasoning) and secondary features pertaining to the conditions of their operation (automatic vs. controlled). We review empirical evidence that supports the joint operation of associative and propositional processes in the formation of new associations.

In orthodox quantum mechanics, it has virtually become the custom to identify properties of a physical system with operationally testable propositions about the system. The causes and consequences of this practice are explored mathematically in this paper. Among other things, it is found that such an identification imposes severe constraints on the admissible states of the physical system.

A simple model accounting for semantic properties of propositional attitude operators in negative contexts with no reference to possible worlds is proposed. Verbs occurring in such operators denote relations between individuals and specific sets of sentences (of a given natural language) and their negation is defined as the complement within a specific set of cognitively determined sentences. This approach avoids in particular the problem of intensionality of propositional attitude operators and allows to use many tools from the generalised (...) quantifier theory. In that way the negation giving rise to factive presuppositions and to the neg-raising is defined in a natural way. (shrink)

As consistency is such an important topic in logic, researchers have for a long time investigated how to attain and maintain it. But consistency can also be studied from the point of view of its opposite, inconsistency. The problem with inconsistency in classical logic is that by the principle of explosion a single inconsistency leads to triviality. Paraconsistent logics were introduced to get around this problem by defining logics in such a way that the explosion principle does not apply to (...) them. Another approach stays in the classical framework and evaluates the amount of inconsistency in a set of formulas. The great bulk of this work has been done for propositional logic and presents many interesting issues about inconsistency. A previous paper introduced the concept of generalized propositional logic (GPL) to provide a uniform method for measuring inconsistency in logics that allow the application of unary operator pairs, such as for modality, time, and space, to propositional logic formulas. The universality of GPL manifests itself in the fact that such an operator pair is evaluated in a uniform manner across all such logics. The difference lies solely in the choice of a frame for each logic. But some logics also contain nonunary operators. For example, temporal logics typically contain a binary Until operator. The purpose of this paper is to show how to extend generalized propositional logic to extended generalized propositional logic (EGPL) by adding nonunary operator pairs and measure inconsistency in such logics. The universality of EGPL manifests itself in the fact that once the evaluation of the nonunary operators is given, it carries over to all such logics. For example, the temporal Until operator becomes applicable to modal logic. Furthermore, while relative inconsistency measures were previously considered for GPL, they are now extended to EGPL and a new approach removes an undesirable feature from the previous version. Also, this paper provides results about various properties of the new inconsistency measures. Many examples and explanations are given to illustrate the issues involving this extension. (shrink)

The addition of "actually" operators to modal languages allows us to capture important inferential behaviours which cannot be adequately captured in logics formulated in simpler languages. Previous work on modal logics containing "actually" operators has concentrated entirely upon extensions of KT5 and has employed a particular modeltheoretic treatment of them. This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing "actually" operators, the weakest of which are conservative extensions of (...) K, using a novel generalisation of the standard semantics. (shrink)

Tautology is interpreted as a necessary condition for the workability of an operations system. This condition suggests the following possibilities: the stable solvability of balance equations between available resources and requests for them; the calculation of potential and kinetic of the system together with the estimation of the contribution of every operation to the kinetic of the system; the construction of a deadlockless infinite cyclic process for performance of some works.

It is argued that propositions cannot be the compositional semantic values of sentences (in context) simply due to issues stemming from the compositional semantics of modal operators (or modal quantifiers). In particular, the fact that the arguments for double indexing generalize to multiple indexing exposes a fundamental tension in the default philosophical conception of semantic theory. This provides further motivation for making a distinction between two sentential semantic contents—what (Dummett 1973) called “ingredient sense” and “assertoric content”.

Propositional quantifiers are added to a propositional modal language with two modal operators. The resulting language is interpreted over so-called products of Kripke frames whose accessibility relations are equivalence relations, letting propositional quantifiers range over the powerset of the set of worlds of the frame. It is first shown that full second-order logic can be recursively embedded in the resulting logic, which entails that the two logics are recursively isomorphic. The embedding is then extended to all sublogics (...) containing the logic of so-called fusions of frames with equivalence relations. This generalizes a result due to Antonelli and Thomason, who construct such an embedding for the logic of such fusions. (shrink)

The Operator Argument against eternalism holds that having non-vacuous tense operators in the language is incompatible with the claim that every proposition has its truth-value eternally. Assuming that (1) there are non-vacuous tense operators, (2) tense operators operate on propositions and (3) tense operators which operate on eternal entities are vacuous, it may be argued that eternalism is false. In this paper, I examine the Operator Argument. The goal is threefold. First, I want to present some aspects of the debate (...) in a more elaborate way, especially those concerning formal matters. Secondly, I will argue that eternalism can escape the Operator Argument. There are two main strategies for handling the Operator Argument. The first one is based on replacing temporal operators with object-language quantifiers. The second rejects the identification of compositional semantic value with assertoric content. My third goal is to show that none of them is as good as the strategy that adopts Timestamp Semantics (Fritz in Philosl Stud 176:2933–2959, 2019). I am going to argue that the quantificational treatment of tenses is compatible with temporalism and that the arguments for rejecting the identification of compositional semantic value with assertoric content provide, in fact, a motivation for the temporalist position. At the end, I will develop Timestamp Semantics by providing a novel formalization of it, and defend it against three potential counter-arguments. (shrink)

// tl;dr A Proposition is a Way of Thinking // -/- This chapter is about type-theoretic approaches to propositional content. Type-theoretic approaches to propositional content originate with Hintikka, Stalnaker, and Lewis, and involve treating attitude environments (e.g. "Nate thinks") as universal quantifiers over domains of "doxastic possibilities" -- ways things could be, given what the subject thinks. -/- This chapter introduces and motivates a line of a type-theoretic theorizing about content that is an outgrowth of the recent literature (...) on epistemic modality, according to which contentful thought is broadly "informational" in its nature and import. The general idea here is that an object of thought is not a way *the world* could be, but rather a way *one's perspective* could be (with respect to a relevant representational question). I will spend the middle part of this chapter motivating and developing a version of this strategy that is, I’ll argue, well-suited to explaining clear phenomena concerning the attribution of perspectival attitudes -- in particular, attitudes towards loosely information-sensitive propositions -- with which extant approaches struggle. My overarching goal here will be to motivate a distinctive version of the "informational" approach -- the "Flexible Types" approach, which is based on the theory proposed in Charlow (2020). According to the Flexible Types approach, propositional attitude verbs are quantifiers over sets of possibilities, but a possibility is a type-flexible notion -- sometimes a possible world, sometimes a perspective, sometimes a set of possible worlds, sometimes a set of perspectives. -/- After introducing the Flexible Types approach, this chapter circles back to more traditional concerns for the analysis of propositions as types of possibilities -- Frege's Puzzle and the problem of Logical Omniscience. Here too the Flexible Types approach bears fruit. Although there are certainly significant differences -- I note some in the concluding section -- the gist of this theory is Hinitkkan or Lewisian in spirit (if not quite in letter). We can make progress on addressing the challenges for the analysis of propositional content in terms of types of possibilities, through empirically driven refinement of our notion of what kind of thing a "doxastic possibility" is. (shrink)

Robert Stalnaker has recently advocated propositional contingentism, the claim that it is contingent what propositions there are. He has proposed a philosophical theory of contingency in what propositions there are and sketched a possible worlds model theory for it. In this paper, such models are used to interpret two propositional modal languages: one containing an existential propositional quantifier, and one containing an existential propositional operator. It is shown that the resulting logic containing an existential quantifier is (...) not recursively axiomatizable, as it is recursively isomorphic to second-order logic, and a natural candidate axiomatization for the resulting logic containing an existential operator is shown to be incomplete. (shrink)

The paper presents a family of propositional epistemic logics such that languages of these logics are extended by quantification over modal operators or over agents of knowledge and extended by predicate symbols that take modal operators as arguments. Denote this family by \}\). There exist epistemic logics whose languages have the above mentioned properties :311–350, 1995; Lomuscio and Colombetti in Proceedings of ATAL 1996. Lecture Notes in Computer Science, vol 1193, pp 71–85, 1996). But these logics are obtained from (...) first-order modal logics, while a logic of \}\) can be regarded as a propositional multi-modal logic whose language includes quantifiers over modal operators and predicate symbols that take modal operators as arguments. Among the logics of \}\) there are logics with a syntactical distinction between two readings of epistemic sentences: de dicto and de re. We show the decidability of logics of \}\) with the help of the loosely guarded fragment of first-order logic. Namely, we generalize LGF to a higher-order decidable loosely guarded fragment. The latter fragment allows us to construct various decidable propositional epistemic logics with quantification over modal operators. The family of this logics coincides with \}\). There are decidable propositional logics such that these logics implicitly contain quantification over agents of knowledge, but languages of these logics are usual propositional epistemic languages without quantifiers and predicate symbols :345–378, 1993). Some logics of \}\) can be regarded as counterparts of logics defined in Grove and Halpern :345–378, 1993). We prove that the satisfiability problem for these logics of \}\) is Pspace-complete using their counterparts in Grove and Halpern :345–378, 1993). (shrink)

Proposition and sentence are two separate entities indicating their specific purposes, definitions and problems. A proposition is a logical entity. A proposition asserts that something is or not the case, any proposition may be affirmed or denied, all proportions are either true (1’s) or false (0’s). All proportions are sentences but all sentences are not propositions. Propositions are factual contains three terms: subject, predicate and copula and are always in indicative or declarative mood. While sentence is a grammatical entity, a (...) unit of language that expresses a complete thought; a sentence may express a proposition, but is distinct from the proposition it may be used to express: categories, declarative sentences, exclamatory, imperative and interrogative sentences. Not all sentences are propositions, propositions express sentence. Sentence is a proposition only in condition when it bears truth values i.e. true or false. We use English sentences governed by imprecise rule to state the precise rules of proposition. In logic we use sentence as logical entity having propositional function but grammatical sentences are different from logical sentences while the former are having only two divisions namely subject and predicate and may express wishes, orders, surprise or facts and also have multiple subjects and predicates and the latter must be in a propositional form which states quantity of the subject and the quality of the proposition and multiple subjects and multiple predicate make the proposition multiple. (shrink)

LK is much more difficult than NK, and to make matters worse, Gentzen's intention is still unclear when it comes to that system (LK). -/- The second and third volumes of the series titled What is LK? conduct the detailed survey of each inference-figure in a toe-to-toe way, as it were, which most mathematicians looked through. -/- The present volume, Vol.3, looks deeper into those operational inference-figures which concerns propositional logic.

In the previous paper with a similar title :311–344, 2018), we presented a family of propositional epistemic logics whose languages are extended by two ingredients: by quantification over modal operators or over agents of knowledge and by predicate symbols that take modal operators as arguments. We denoted this family by \}\). The family \}\) is defined on the basis of a decidable higher-order generalization of the loosely guarded fragment of first-order logic. And since HO-LGF is decidable, we obtain the (...) decidability of logics of \}\). In this paper we construct an alternative family of decidable propositional epistemic logics whose languages include ingredients and. Denote this family by \}\). Now we will use another decidable fragment of first-order logic: the two variable fragment of first-order logic with two equivalence relations +2E) [the decidability of FO\+2E was proved in Kieroński and Otto :729–765, 2012)]. The families \}\) and \}\) differ in the expressive power. In particular, we exhibit classes of epistemic sentences considered in works on first-order modal logic demonstrating this difference. (shrink)

History witnesses alternative approaches to “the proposition”. The proposition has been referred to as the object of belief, disbelief, and doubt: generally as the object of propositional attitudes, that which can be said to be believed, disbelieved, understood, etc. It has also been taken to be the object of grasping, judging, assuming, affirming, denying, and inquiring: generally as the object of propositional actions, that which can be said to be grasped, judged true or false, assumed for reasoning purposes, (...) etc. The proposition has also been taken to be the subject of truth and falsity: generally as the subject of propositional properties, that which can be said to be true, false, tautological, informative, inconsistent,etc. It has also been taken as the subject and object of logical relations, e.g. that which can be said to imply, be implied, contradict, be contradicted, etc. Prima facie, such properties and relations are non-mental and objective. It has also been taken to be the resultants or products of propositional operations, usually mental or linguistic; e.g. judging, affirming, and denying have been held to produce propositions called judgments, affirmations, and negations, respectively. Propositions have also been taken to be certain declarative sentences. Finally,propositions have been taken to be meanings of certain declarative sentences. This essay is an informal, selective, and incomplete survey of alternative approaches to “the proposition” with special attention to the views of the late American philosopher Peter Hare (1935–2008)and of those who influenced him. DOI: 10.5007/1808-1711.2011v15n1p51. (shrink)

Intuitionistic propositional logic with Galois negations ( \(\mathsf {IGN}\) ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair \((\lnot,{\sim })\) and dual Galois pair \((\dot{\lnot },\dot{\sim })\) of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for \(\mathsf {IGN}\) are developed. A Hilbert-style axiomatic system \(\mathsf {HN}\) is given for \(\mathsf {IGN}\), and Galois negation logics are defined as extensions of \(\mathsf {IGN}\). We give the (...) bi-tense logic \(\mathsf {S4N}_t\) which is obtained from the minimal tense extension of the modal logic \(\mathsf {S4}\) by adding tense operators. We give a new extended Gödel translation \(\tau \) and prove that \(\mathsf {IGN}\) is embedded into \(\mathsf {S4N}_t\) by \(\tau \). Moreover, every Kripke-complete Galois negation logic _L_ is embedded into its tense companion \(\tau (L)\). (shrink)

Chapter 1 provides a world-semantics for intentional propositional operators. Impossible worlds of various kinds are deployed to ensure that the operators behave in various important ways; notably they fail to be closed under entailment.

We present our calculus of higher-level rules, extended with propositional quantification within rules. This makes it possible to present general schemas for introduction and elimination rules for arbitrary propositional operators and to define what it means that introductions and eliminations are in harmony with each other. This definition does not presuppose any logical system, but is formulated in terms of rules themselves. We therefore speak of a foundational account of proof-theoretic harmony. With every set of introduction rules a (...) canonical elimination rule, and with every set of elimination rules a canonical introduction rule is associated in such a way that the canonical rule is in harmony with the set of rules it is associated with. An example given by Hazen and Pelletier is used to demonstrate that there are significant connectives, which are characterized by their elimination rules, and whose introduction rule is the canonical introduction rule associated with these elimination rules. Due to the availabiliy of higher-level rules and propositional quantification, the means of expression of the framework developed are sufficient to ensure that the construction of canonical elimination or introduction rules is always possible and does not lead out of this framework. (shrink)

Logical and philosophical literature provides different classifications of reasoning. In the Polish literature on the subject, for instance, there are three popular ones accepted by representatives of the Lvov-Warsaw School: Jan Łukasiewicz, Tadeusz Czeżowski and Kazimierz Ajdukiewicz (Ajdukiewicz in Logika pragmatyczna [Pragmatic Logic]. PWN, Warsaw (1965, 2nd ed. 1974). Translated as: Pragmatic Logic. Reidel & PWN, Dordrecht, 1975). The author of this paper, having modified those classifications, distinguished the following types of reasoning: (1) deductive and (2) non-deductive, and additionally two (...) types of them in each of the two, depending on the manner of combining their premises with the conclusion through the relation of classical logical entailment. Consequently, the four types of reasoning: unilateral deductive (incl. its sub-types: deductive inference and proof), bilateral deductive (incl. complete induction), and reductive (incl. the sub-types: explanation and verification), logically nonvaluable (incl. inference by analogy, statistic inference), correspond to four operators of derivability. They are defined formally on the ground of Tarski’s axiomatic theory of deductive systems, by means of the consequence operation _Cn_ (Tarski in Monatshefte Math Phys 37:361–404, 1930a, C R Soc Sci Lett Vars 23:22–29, 1930b). Also, certain metalogical properties of these operators are given, as well as their relations with Tarski’s consequence operations \(Cn^+\) ( \(Cn^+ = Cn\) ) and dual consequences \(Cn^{-1}\) (Słupecki in Zeszyty Naukowe Uniwersytetu Wrocławskiego Seria B Nr 3:33–40, 1959, Słupecki et al. in Stud Log 29:76–123, 1971, Wybraniec-Skardowska, in: Wybraniec-Skardowska, Bryll (eds) Z badań nad teorią zdań odrzuconych [Studies in the Theory of Rejected Propositions], Series B, Studia i Monografie, Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Opole, 1969), and \(Cn^-\) (Wójcicki in Bull Sect Log 2(2):54–57, 1973)). (shrink)

This paper combines propositional dynamic logic ) with propositional inquisitive logic ). The result of this combination is a logical system \ that conservatively extends both \ and \, and, moreover, allows for an interaction of the question-forming operator from \ with the structured modalities from \. We study this system from a semantic as well as a syntactic point of view. These two perspectives are linked via a completeness proof, which also shows that \ is decidable.

Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements. In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study those logical properties and relations that (...) depend upon parts of statements that are not themselves statements on their own, such as the subject and predicate of a statement. The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, which studies logical operators and connectives that are used to produce complex statements whose truth-value depends entirely on the truth-values of the simpler statements making them up, and in which it is assumed that every statement is either true or false and not both. However, there are other forms of propositional logic in which other truth-values are considered, or in which there is consideration of connectives that are used to produce statements whose truth-values depend not simply on the truth-values of the parts, but additional things such as their necessity, possibility or relatedness to one another. (shrink)

Unintentional plagiarism abounds at universities. The literature offers several explanations for students’ difficulties with acquiring standards of good academic practice. In this paper, I propose an alternative account: unintentional plagiarism can only be understood in the context of implicit but irreconcilable forms of knowledge. While higher education institutions mainly operate within the framework of propositional epistemology, institutions of primary and secondary education tend to furnish students with encyclopedic epistemology. Accordingly, universities and institutions of pre-college education tend to propagate conflicting (...) assumptions regarding the nature of knowledge. Put simply, propositional epistemology is characterized by dialogue, agency and the exchange of ideas, which are ideally made explicit in academic writing. By contrast, encyclopedic epistemology tends to present knowledge monologically, leaving the scholarly conversation around it out. It is highly likely that the hardwired legacy of encyclopedic epistemology among students impedes the acquisition of the dialogism of academic inquiry at universities, resulting in cases of unintentional plagiarism. (shrink)

In this paper I shall present some of the results I have obtained on modal theories which contain quantifiers for propositions. The paper is in two parts: in the first part I consider theories whose non-quantificational part is S5; in the second part I consider theories whose non-quantificational part is weaker than or not contained in S5. Unless otherwise stated, each theory has the same language L. This consists of a countable set V of propositional variables pl, pa, ... (...) , the operators v (or), ~ (not) and □ (necessarily), the universal quantifier (p), p a propositional variable, and brackets ( and ). The formulas of L are then defined in the usual way. (shrink)

ABSTRACT Section 1 discusses properties and propositions, and some of the motivation for an account in which property instantiation and propositional truth behave ‘naively’. Section 2 generalizes a standard Kripke construction for naive properties and propositions, in a language with modal operators but no conditionals. Whereas Kripke uses a 3-valued value space, the generalized account allows for a broad array of value spaces, including the unit interval [0,1]. This is put to use in Section 3, where I add to (...) the language a conditional suitable for restricting quantification. The shift from a value space based on the ‘mini-space’ {0,, 1} to one based on the ‘mini-space’ [0,1] leads to more satisfactory results than I was able to achieve in previous work: a vast variety of paradoxical sentences can now be treated very simply. In Section 4 I make a further addition to the language, a conditional modeled on the ordinary English conditional, paying particular attention to how it interacts with the restricted quantifier conditional. This is all done in the [0,1] framework, and two alternatives are considered for how the ordinary conditional is to be handled; one of them results from adding a tweak to a construction by Ross Brady. Section 5 discusses a further alternative, a standard relevance conditional (for the ordinary conditional, perhaps for use with a different quantifier-restricting conditional), but argues that it is not promising. Section 6 discusses the identity conditions of properties and propositions (again in the setting of a value space based on [0,1]); the issue of achieving naivety for coarse-grained properties is seen to be more complicated than some brief remarks in Field [2010] suggested, but a way to get a fair degree of coarse-grainedness is shown. (shrink)

According to the standard definition, a first-order theory is categorical if all its models are isomorphic. The idea behind this definition obviously is that of capturing semantic notions in axiomatic terms: to be categorical is to be, in this respect, successful. Thus, for example, we may want to axiomatically delimit the concept of natural number, as it is given by the pre-theoretic semantic intuitions and reconstructed by the standard model. The well-known results state that this cannot be done within first-order (...) logic, but it can be done within second-order one. Now let us consider the following question: can we axiomatically capture the semantic concept of conjunction? Such question, to be sure, does not make sense within the standard framework: we cannot construe it as asking whether we can form a first-order (or, for that matter, whatever-order) theory with an extralogical binary propositional operator so that its only model (up to isomorphism) maps the operator on the intended binary truth-function. The obvious reason is that the framework of standard logic does not allow for extralogical constants of this type. But of course there is also a deeper reason: an existence of a constant with this semantics is presupposed by the very definition of the framework1. Hence the question about the axiomatic capturability of concunction, if we can make sense of it at all, cannot be asked within the framework of standard logic, we would have to go to a more abstract level. To be able to make sense of the question we would have to think about a propositional ‘proto-language’, with uninterpreted logical constants, and to try to search out axioms which would fix the denotations of the constants as the intended truth-functions. Can we do this? It might seem that the answer to this question is yielded by the completeness theorem for the standard propositional calculus: this theorem states that the axiomatic delimitation of the calculus and the semantic delimitation converge to the same result.. (shrink)

Jeffrey King has recently argued: (i) that the semantic value of a sentence at a context is (or determines) a function from possible worlds to truth values, and (ii) that this undermines Jason Stanley's argument against the rigidity thesis, the claim that no rigid term has the same content as a non-rigid term. I show that King's main argument for (i) fails, and that Stanley's argument is consistent with the claim that the semantic value of a sentence at a context (...) is (or determines) a function from worlds to truth values. (shrink)

This paper aims to clarify the connection between the logic of temporal distinctions and the temporal features of propositions. Contra Prior, it argues that the adoption of tense operators does not commit one to the view that propositions can change their truth value over time.

This presentation of a system of propositional logic is a foundational paper for systems of illocutionary logic. The language contains the illocutionary force operators '' for assertion and ' ' for supposition. Sentences occurring in proofs of the deductive system must be prefixed with one of these operators, and rules of take account of the forces of the sentences. Two kinds of semantic conditions are investigated; familiar truth conditions and commitment conditions. Accepting a statement A or rejecting A commits (...) a person to accepting some statements (the symbol '' marks this value), to rejecting some statements (), and will leave the person uncommitted with respect to others (). Commitment valuations assign the values to sentences of ; such a valuation is conceived as reflecting the beliefs/knowledge of a particular person. This paper explores the relations between truth conditions and commitment conditions, and between semantic concepts defined in terms of these conditions. (shrink)

In formal semantics based on modern type theories, some sentences may be interpreted as judgements and some as logical propositions. When interpreting composite sentences, one may want to turn a judgemental interpretation or an ill-typed semantic interpretation into a proposition in order to obtain an intended semantics. For instance, an incorrect judgement $$a:A$$ may be turned into its propositional form $$\textsc {is}(A,a)$$ and an ill-typed application p(a) into $$\textsc {do}(p,a)$$, so that the propositional forms can take part in (...) logical compositions that interpret composite sentences, especially those that involve negations and conditionals.In this paper, we propose an operator not that facilitates such a transformation. Introducing not axiomatically, with five axiomatic laws to govern its behaviour, we shall use it to define $$\textsc {is}$$ and $$\textsc {do}$$ and give examples to illustrate its use in semantic interpretation. The introduction of not into type theories is logically consistent – this is justified by showing that not can be defined by means of the heterogeneous equality $$\textrm{JMeq}$$ so that all of the axiomatic laws for not become provable. Therefore, since the extension with $$\textrm{JMeq}$$ preserves logical consistency, so does the extension with not. We shall also study conditions under which $$\textsc {is}$$ and $$\textsc {do}$$ operators can be used safely without the risk of over-generation. (shrink)

David Kaplan observed in Kaplan that the principle \\) cannot be verified at a world in a standard possible worlds model for a quantified bimodal propositional language. This raises a puzzle for certain interpretations of the operator Q: it seems that some proposition p is such that is not possible to query p, and p alone. On the other hand, Arthur Prior had observed in Prior that on pain of contradiction, ∀p is Q only if one true proposition is (...) Q and one false proposition is Q. The two observations are related: ∀p is elusive in that it is not possible for the proposition to be uniquely Q. Kaplan based his model-theoretic observation on Cantor’s theorem, but there is a less well-known link between this simple set-theoretic observation and Prior’s remark. We generalize the link to develop a heuristic designed to move from Cantor’s theorem to the observation that a variety of sentences of the bimodal language express propositions that cannot be Q uniquely. We highlight the analogy between some of these results and some set-theoretic antinomies and suggest that the phenomenon is richer than one may have anticipated. (shrink)

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express metalogical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the logics of formal inconsistency and by the logics of formal undeterminedness. LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of (...) the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core—in the case studied here, this common core is classical positive propositional logic. The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of LFI and undeterminedness, namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion one at a time. The last sections offer an algebraic account for such logics by adapting the swap structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite nondeterministic matrices. (shrink)

This paper shows how propositional dynamic logic can be interpreted as a logic for multi-agent belief revision. For that we revise and extend the logic of communication and change of [9]. Like LCC, our logic uses PDL as a base epistemic language. Unlike LCC, we start out from agent plausibilities, add their converses, and build knowledge and belief operators from these with the PDL constructs. We extend the update mechanism of LCC to an update mechanism that handles belief change (...) as relation substitution, and we show that the update part of this logic is more expressive than either that of LCC or that of doxastic/epistemic PDL with a belief change modality. It is shown that the properties of knowledge and belief are preserved under any update, and that the logic is complete. © 2008 Springer-Verlag Berlin Heidelberg. (shrink)

We propose a new perspective on logics of computation by combining instantial neighborhood logic \ with bisimulation safe operations adapted from \. \ is a recent modal logic, based on an extended neighborhood semantics which permits quantification over individual neighborhoods plus their contents. This system has a natural interpretation as a logic of computation in open systems. Motivated by this interpretation, we show that a number of familiar program constructors can be adapted to instantial neighborhood semantics to preserve invariance (...) for instantial neighborhood bisimulations, the appropriate bisimulation concept for \. We also prove that our extended logic \ is a conservative extension of dual-free game logic, and its semantics generalizes the monotone neighborhood semantics of game logic. Finally, we provide a sound and complete system of axioms for \, and establish its finite model property and decidability. (shrink)

Natural language provides motivation for studying modal backwards-looking operators such as “now”, “then” and “actually” that evaluate their argument formula at some previously considered point instead of the current one. This paper investigates the expressive power over models of both propositional and first-order basic modal language enriched with such operators. Having defined an appropriate notion of bisimulation for first-order modal logic, I show that backwards-looking operators increase its expressive power quite mildly, contrary to beliefs widespread among philosophers of language (...) and formal semanticists. That in turn presents a strong argument for the use of operator-based systems in the semantics of natural language, instead of systems with explicit quantification over worlds and times that have become a de-facto standard for such applications. The popularity of such explicit-quantification systems is shown to be based on the misinterpretation of a claim by Cresswell, which led many philosophers and linguists to assume that introducing “now” and “then” is expressively equivalent to explicitly quantifying over worlds and times. (shrink)

In a paper recently published in this quarterly I argued that modern quantum physics, as an integrated system of laws, supplements and completes in purely quantitative terms the fragmentary order of first-person experience, removing in a unique way ambiguities otherwise encountered at the level of common-sense things; and I contended that the choice of a different selective operator—purpose or value, let us say, rather than quantity—might entail an entirely different range and system of order. It is now my intention to (...) follow up this contention and to indicate at least in outline form some of the implications and possibilities of such a program; for it is possible that this undertaking may bring into new perspective some of the crucial problems which have vexed philosophers at all times when they have dealt with normative propositions and moral judgments. (shrink)

This paper considers the question of what knowing a logical rule consists in. I defend the view that knowing a logical rule is having propositional knowledge. Many philosophers reject this view and argue for the alternative view that knowing a logical rule is, at least at the fundamental level, having a disposition to infer according to it. To motivate this dispositionalist view, its defenders often appeal to Carroll’s regress argument in ‘What the Tortoise Said to Achilles’. I show that (...) this dispositionalist view, and the regress that supposedly motivates it, operate with the wrong picture of what is involved in knowing a logical rule. In particular I show that it gives us the wrong picture of the relation between knowing a logical rule and actions of inferring according to it, as well as of the way in which knowing a logical rule might be a priori. (shrink)

In this work we define contingency logic with arbitrary announcement. In contingency logic, the primitive modality contingency formalises that a proposition may be true but also may be false, so that if it is non-contingent then it is necessarily true or necessarily false. To this logic one can add dynamic operators to describe change of contingency. Our logic has operators for public announcement and operators for arbitrary public announcement, as in the dynamic epistemic logic called arbitrary public announcement logic. However, (...) our language primitive is the more suitable notion of public announcement whether, instead of the public announcement that. We compare the expressive power of our logic and its various fragments to related dynamic epistemic logics. We further present an axiomatisation and show its completeness by adapting a method to demonstrate completeness of arbitrary public announcement logic. Various extensions are also shown to be complete with respect to the corresponding frame classes. (shrink)

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the (...) validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core– in the case studied here, this common core is classical positive propositional logic (CPL + ). The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formal inconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices. (shrink)

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of (...) "and", "or", "not" and "conditioning" are then defined on these conditional events with variable conditions. The probabilities of the disjunction and of the conjunction of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed. (shrink)

Two approaches to the study of diagnostic thinking are compared, one mainly propositional, namely that of Patel and Groen (1986), the other mainly semantic, that of Lemieux and Bordage (1986). Patel and Groen analyzed the linear dimension of cardiologists' discourses while solving a case of acute bacterial endocarditis, that is, the before and after propositional rules. A secondary analysis of two of their pothophysiological protocols is done using structural semantic techniques from Lemieux and Bordage where the vertical dimension (...) of the discourses is analyzed, that is, the levels of meaning or semantic axes. Contrary to Patel and Groen's position, making an accurate diagnosis is not explained in terms of pure forward reasoning through networks of causal rules, but by means of networks of semantic qualities abstracted from the symptoms and signs. The semantic qualities operate in the clinician's mind in terms of binary oppositions (e.g., sudden-gradual, unilateral-bilateral) whereby each pair of properties constitutes a semantic axis (e.g., sudden-gradual). The successful diagnosticians are those who use the most diversified and pertinent set of semantic axes and, therefore, have a deeper representation of the problem. (shrink)