We study the linear Lindenbaum algebra of Basic Propositional Calculus, called linear basic algebra.

We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using these axiomatizations, we prove interpolation theorems for basic propositional logic.

In this paper we prove a bounded translation of intuitionistic propositional logic into basic propositional logic. Our new theorem, compared with the translation theorem in [1], has the advantage that it gives an effective bound on the translation, depending on the complexity of formulas.

We present an axiom system for basic hybrid logic extended with propositional quantifiers (a second-order extension of basic hybrid logic) and prove its (basic and pure) strong completeness with respect to general models.

We present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E1, a theory axiomatized by T → ⊥. The intersection CPC ∩ E1 is axiomatizable by the Principle of the Excluded Middle A V ∨ ⌝A. If B (...) is a formula such that → B is not derivable, then the lattice of formulas built from one propositional variable p using only the binary connectives, is isomorphically preserved if B is substituted for p. A formula → B is derivable exactly when B is provably equivalent to a formula of the form → A) →. (shrink)

The logic BKc6 is the basic constructive logic in the ternary relational semantics adequate to consistency understood as absolute consistency, i.e., non-triviality. Negation is introduced in BKc6 with a negation connective. The aim of this paper is to define the logic BKc6F. In this logic negation is introduced via a propositional falsity constant. We prove that BKc6 and BKc6F are definitionally equivalent. Then, we show how to extend BKc6F within the spectrum of logics delimited (...) by contractionless intuitionistic logic. All logics defined in the paper are paraconsistent logics. (shrink)

The logic BKc1 is the basic constructive logic in the ternary relational semantics adequate to consistency understood as the absence of the negation of any theorem. Negation is introduced in BKc1 with a negation connective. The aim of this paper is to define the logic BKc1F. In this logic negation is introduced via a propositional falsity constant. We prove that BKc1 and BKc1F are definitionally equivalent.

Let ℒ and ? be propositional languages over Basic Propositional Calculus, and ℳ = ℒ∩?. Weprove two different but interrelated interpolation theorems. First, suppose that Π is a sequent theory over ℒ, and Σ∪ {C⇒C′} is a set of sequents over ?, such that Π,Σ⊢C⇒C′. Then there is a sequent theory Φ over ℳ such that Π⊢Φ and Φ, Σ⊢C⇒C′. Second, let A be a formula over ℒ, and C 1, C 2 be formulas over ?, such (...) that A∧C 1⊢C 2. Then there exists a formula B over ℳ such that A⊢B and B∧C 1⊢C 2. (shrink)

This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then (...) treated, including polynomial simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find this comprehensive treatment an excellent guide to this expanding interdisciplinary area. (shrink)

In this paper by using a model-theoretic approach, we prove Craig interpolation property for Formal Propositional Logic, FPL, Basic propositional logic, BPL and the uniform left-interpolation property for FPL. We also show that there are countably infinite extensions of FPL with the uniform interpolation property.

Prawitz proposed certain notions of proof-theoretic validity and conjectured that intuitionistic logic is complete for them [11, 12]. Considering propositional logic, we present a general framework of five abstract conditions which any proof-theoretic semantics should obey. Then we formulate several more specific conditions under which the intuitionistic propositional calculus turns out to be semantically incomplete. Here a crucial role is played by the generalized disjunction principle. Turning to concrete semantics, we show that prominent proposals, including Prawitz’s, (...) satisfy at least one of these conditions, thus rendering IPC semantically incomplete for them. Only for Goldfarb’s [1] proof-theoretic semantics, which deviates from standard approaches, IPC turns out to be complete. Overall, these results show that basic ideas of proof-theoretic semantics for propositional logic are not captured by IPC. (shrink)

As anyone who is familiar with the literature knows, there is a great deal of controversy concerning which, if any, of the extant theories of propositional modal logic correctly formalizes the logic of certain logical concepts such as analyticity and logical necessity. Most of the controversy concerns certain principles that involve iterated modalities (where one modal operator occurs within the scope of another). For example, there is considerable disagreement about whether the principle(□p⊃□□p)should be considered valid. However, when (...) philosophers and logiciansapplymodal logic to concrete problems, they rarely need principles which involve iterated modalities. For most practical purposes, principles involving only one layer of modalities are all that are needed. This suggests that if we try to construct a theory of modal logic in which there are no iterated modalities, we can avoid most of the controversy and still have a theory that is strong enough for all of the normal uses to which modal logic is put. (shrink)

Using the theory of BL-algebras, it is shown that a propositional formula ϕ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ˜˜ϕ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ→˜φ)) → ψ, then ϕ is derivable in in classical logic if and only if ˜˜ ϕ is derivable in SBL. Axiomatic extensions of Basic (...) Logic are in correspondence with subvarieties of the variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras. (shrink)

The logic BKc1 is the basic constructive logic for weak consistency in the ternary relational semantics without a set of designated points. In this paper, a number of extensions of B Kc1 defined with a propositional falsity constant are defined. It is also proved that weak consistency is not equivalent to negation-consistency or absolute consistency in any logic included in positive contractionless intermediate logic LC plus the constructive negation of BKc1 and the contraposition axioms.

We introduce a Gentzen style formulation of Basic Propositional Calculus(BPC), the logic that is interpreted in Kripke models similarly tointuitionistic logic except that the accessibility relation of eachmodel is not necessarily reflexive. The formulation is presented as adual-context style system, in which the left hand side of a sequent isdivided into two parts. Giving an interpretation of the sequents inKripke models, we show the soundness and completeness of the system withrespect to the class of Kripke models. (...) The cut-elimination theorem isproved in a syntactic way by modifying Gentzen's method. Thisdual-context style system exemplifies the effectiveness of dual-contextformulation in formalizing various non-classical logics. (shrink)

Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by (...) means of growing computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed. (shrink)

The language of the basic logic of proofs extends the usual propositional language by forming sentences of the sort x is a proof of F for any sentence F. In this paper a complete axiomatization for the basic logic of proofs in Heyting Arithmetic HA was found.

Two cut-free sequent calculi which are conservative extensions of Visser's Formal Propositional Logic are introduced. These satisfy a kind of subformula property and by this property the interpolation theorem for FPL are proved. These are analogies to Aghaei-Ardeshir's calculi for Visser's Basic Propositional Logic.

In the paper, we apply Kooi and Tamminga’s correspondence analysis to some conventional and functionally incomplete fragments of classical propositional logic. In particular, the paper deals with the implication, disjunction, and negation fragments. Additionally, we consider an application of correspondence analysis to some connectiveless fragment with certain basic properties of the logical consequence relation only. As a result of the application, one obtains a sound and complete natural deduction system for any binary extension of each fragment in (...) question. With the focus on exclusive disjunction we comparatively study the proposed systems. Finally, we discuss Segerberg’s systems for connectiveless and negation fragments and compare them with our systems. (shrink)

This paper introduces sequent systems for Visser's two propositional logics: Basic Propositional Logic (BPL) and Formal Propositional Logic (FPL). It is shown through semantical completeness that the cut rule is admissible in each system. The relationships with Hilbert-style axiomatizations and with other sequent formulations are discussed. The cut-elimination theorems are also demonstrated by syntactical methods.

A common misconception among logicians is to think that intuitionism is necessarily tied-up with single conclusion calculi. Single conclusion calculi can be used to model intuitionism and they are convenient, but by no means are they necessary. This has been shown by such influential textbook authors as Kleene, Takeuti and Dummett, to cite only three. If single conclusions are not necessary, how do we guarantee that only intuitionistic derivations are allowed? Traditionally one insists on restrictions on particular rules: implication right, (...) negation right and universal quantification right are required to be single conclusion rules. In this note we show that instead of a cardinality restriction such as one-conclusion-only, we can use a notion of dependency between formulae to enforce the constructive character of derivations. The system we obtain, called FIL for full intuitionistic logic, satisfies basic properties such as soundness, completeness and cut elimination. We present two motivating applications of FIL and discuss some future work. (shrink)

_Logic: The Basics_ is an accessible introduction to several core areas of logic. The first part of the book features a self-contained introduction to the standard topics in classical logic, such as: · mathematical preliminaries · propositional logic · quantified logic · English and standard ‘symbolic translations’ · tableau procedures. Alongside comprehensive coverage of the standard topics, this thoroughly revised second edition also introduces several philosophically important nonclassical logics, free logics, and modal logics, and gives (...) the reader an idea of how they can take their knowledge further. With its wealth of exercises, _Logic: The Basics _is a useful textbook for courses ranging from the introductory level to the early graduate level, and also as a reference for students and researchers in philosophical logic. (shrink)

Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they (...) can take their knowledge further and the philosophical questions around logic. -/- Logic: the Basics is essential reading for first-year undergraduate philosophy students on standard introductory logic courses. (shrink)

Jeffrey King, Scott Soames, and others have recently challenged the familiar identification of a Russellian proposition, such as the proposition that Brutus stabbed Caesar, with an ordered sequence constructed out of objects, properties, and relations. There is, as they point out, a surplus of candidate sequences available that are each equally serviceable. If so, any choice among these candidates will be arbitrary. In this paper, I show that, unless a controversial assumption is made regarding the nature of nonsymmetrical relations, none (...) of the proffered candidate sequences are in fact adequate to the play the role. Moreover, as I argue, the most promising alternative theory of relations—one that avoids the problematic assumption and, in addition, fits most naturally into the sequentialist’s framework—fails to meet a basic requirement: it cannot distinguish between the proposition that Brutus stabbed Caesar and the proposition that Caesar stabbed Brutus. The upshot is that the conspicuously structured entities that are widely assumed to be up to the task of “playing the proposition role” shed no light on the very structure they are invoked to represent. (shrink)

The author puts forth an approach to propositional attitude contexts based upon the view that one does not have beliefs of ordinary extensional entitiessimpliciter. Rather, one has beliefs of such entities as presented in various manners. Roughly, these are treated as beliefs of ordered pairs — the first member of which is the ordinary extensional entity and the second member of which is a predicate that it satisfies. Such an approach has no difficulties with problems involving identity, such as (...) of The Morning Star and The Evening Star (section 1). Given the second members of the pairs, the modes of presentation, it is quite natural to allow exportation everywhere. There is no need for essentialism. (One also can have non-essentialistic modal logic if one grants analyticity or the like.) (section 2). Given that the second member of the pair need only be one that is satisfied by the entity that is the first member (and need not be specificative), the method has no difficulties when one is concerned only with discriminations (and not specifications) (section 3). When this method is combined with the Frege-Carnap method of descriptions, fictional entities can be accommodated; Goodman''s unicorn-picture and the like can be brought within a Tarskian semantics; and Geach''s difficulties with intentional identity appear to be handled (section 4). Given the author''s ordered pair construals, there appears to be no additional need for notional construals; i.e., the author''s one unified method appears satisfactory for dealing with both traditionalde re (relational) andde dicto (notional) construals. The Paradox of the Knower and the like do not appear formulatable against the author''s approach. (section 5). The author also argues against the basic principles behind the Church-Langford translation argument (section 6). (shrink)

"A valuable collection both for original source material as well as historical formulations of current problems."-- The Review of Metaphysics "Much more than a mere collection of papers . . . a valuable addition to the literature."-- Mathematics of Computation An anthology of fundamental papers on undecidability and unsolvability by major figures in the field, this classic reference opens with Godel's landmark 1931 paper demonstrating that systems of logic cannot admit proofs of all true assertions of arithmetic. Subsequent papers (...) by Godel, Church, Turing, and Post single out the class of recursive functions as computable by finite algorithms. Additional papers by Church, Turing, and Post cover unsolvable problems from the theory of abstract computing machines, mathematical logic, and algebra, and material by Kleene and Post includes initiation of the classification theory of unsolvable problems. Suitable for graduate and undergraduate courses. 1965 ed. (shrink)

Essential Logic offers: BL Readability. A dialogue-like yet challenging style makes this introductory logic textbook engaging and interesting. BL Essentials. Deductive and inductive reasoning, formal and informal logic are placed within a philosophical perspective. BL Rigor. A careful sequence of learning steps communicates the essential skills of reasoning and directs students to write, support, and argue by connecting criticism to key concepts. BL Relevance. Explanations and examples take students' lives into consideration and are designed for students with (...) diverse backgrounds and a wide range of experiences. BL A Theme. Traditional concepts are integrated with a discussion of modern technological issues and the world view of modern science. A unique chapter on Logic and Hope addresses questions students often ask and suggests a global perspective. BL Controversy. Students are encouraged to defend and critique positions--including those presented by the author. A unique final chapter on Fuzzy Logic is framed as a debate between Western and Eastern philosophy. BL Exercises. Students gain confidence in recognizing arguments, structuring them into premises and conclusions, identifying and critiquing informal fallacies, while learning to create, follow, and appreciate symbolic reasoning trails. BL Coverage. Chapters cover Argument Recognition and Language Analysis, Inductive Reasoning, Structuring Informal Fallacies, Symbolic Translation, Truth Tables, Formal Proofs of Validity, Quantification, and the basics of Fuzzy Set Theory and Propositional Logic. (shrink)

We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set (...) is in general undecidable, and can be -complete. These classifications are exact. For first order theories even finite sets of premises can generate such consequence sets in either calculus. (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

Topic of the paper is Q-logic - a logic of agency in its temporal and modal context. Q-logic may be considered as a basal logic of agency since the most important stitoperators discussed in the literature can be defined or axiomatized easily within its semantical and syntactical framework. Its basic agent dependent operator, the Q-operator (also known as Δ- or cstit-operator), which has been discussed independently by E v. Kutschera and B. E Chellas, is investigated (...) here in respect of its relation to other temporal and modal operators. The main result of the paper, then, is a completeness result for a calculus of Q-logic with respect to a semantics defined on the tree-approach to agency as introduced and developed by, among others, E v. Kutschera and N. D. Belnap. (shrink)

We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order (...) class='Hi'>logic and SOPML. (shrink)

Topic of the paper is Q-logic – a logic of agency in its temporal and modal context. Q-logic may be considered as a basal logic of agency since the most important stit-operators discussed in the literature can be defined or axiomatized easily within its semantical and syntactical framework. Its basic agent dependent operator, the Q-operator (also known as Δ- or cstit-operator), which has been discussed independently by F. v. Kutschera and B. F. Chellas, is investigated (...) here in respect of its relation to other temporal and modal operators. The main result of the paper, then, is a completeness result for a calculus of Q-logic with respect to a semantics defined on the tree-approach to agency as introduced and developed by, among others, F. v. Kutschera and N. D. Belnap. (shrink)

This talk shows how propositional dynamic logic (PDL) can be interpreted as a logic for multi-agent knowledge update and belief revision, or as a logic of preference change, if the basic relations are read as preferences instead of plausibilities. Our point of departure is the logic of communication and change (LCC) 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 epistemic/doxastic PDL with a belief change modality. Next, we show that the properties of knowledge and belief are preserved under any update, unlike in LCC. We prove completeness of the logic and give examples of its use. If there is time, we will also look at the preference interpretation of the system, and at preference change scenarios that can be modelled with it. (shrink)

At least some of us, at least some of the time—when not in the grip of radical sceptical doubt—are inclined to believe that we know, for example, that if we infer a conclusion from two true premises, one a conditional whose consequent is that conclusion and the other the antecedent of that conditional, then our conclusion must be true, or that we know similar things about other simple patterns of inference. If we do indeed have knowledge of this sort, it (...) is what I mean by logical knowledge. Logical knowledge is, roughly speaking, knowledge about logic—such as knowledge that a certain principle of inference necessarily preserves truth, or that every proposition of a certain form must be true—and so is not the same thing as knowledge that is gained by using logic, i.e. inferential knowledge. That is not to say, of course, that logical knowledge can't be inferential. On the contrary, it is barely open to question that—if there is any logical knowledge at all—there is a lot of inferential logical knowledge. For example, if we know that the introduction and elimination principles for the conditional are truth-preserving, we can surely get to know, by inference, that the principle of hypothetical syllogism is so too, not to mention other, less obvious and more recondite, examples of putative logical knowledge. (shrink)

This article concerns the treatment of propositional quantification in a framework of labelled natural deduction for modal logic developed by Basin, Matthews and Viganò. We provide a detailed analysis of a basic calculus that can be used for a proof-theoretic rendering of minimal normal multimodal systems with quantification over stable domains of propositions. Furthermore, we consider variations of the basic calculus obtained via relational theories and domain theories allowing for quantification over possibly unstable domains of propositions. (...) The main result of the article is that fragments of the labelled calculi not exploiting reductio ad absurdum enjoy the Church–Rosser property and the strong normalization property; such result is obtained by combining Girard’s method of reducibility candidates and labelled languages of lambda calculus codifying the structure of modal proofs. (shrink)

In the context of truth-functional propositional many-valued logics, Hájek’s Basic Fuzzy Logic BL [14] plays a major rôle. The completeness theorem proved in [7] shows that BL is the logic of all continuous t -norms and their residua. This result, however, does not directly yield any meaningful interpretation of the truth values in BL per se . In an attempt to address this issue, in this paper we introduce a complete temporal semantics for BL. Specifically, we (...) show that BL formulas can be interpreted as modal formulas over a flow of time, where the logic of each instant is Łukasiewicz, with a finite or infinite number of truth values. As a main result, we obtain validity with respect to all flows of times that are non-branching to the future, and completeness with respect to all finite linear flows of time, or to an appropriate single infinite linear flow of time. It may be argued that this reduces the problem of establishing a meaningful interpretation of the truth values in BL logic to the analogous problem for Łukasiewicz logic. (shrink)

In this note we provide a simple proof of the incompactness over Routley-Meyer B-frames of the A1 fragment of the second order propositional relevant language.

We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order (...) class='Hi'>logic and SOPML. (shrink)

This chapter provides a brief introduction to propositional epistemic logic and its applications to epistemology. No previous exposure to epistemic logic is assumed. Epistemic-logical topics discussed include the language and semantics of basic epistemic logic, multi-agent epistemic logic, combined epistemic-doxastic logic, and a glimpse of dynamic epistemic logic. Epistemological topics discussed include Moore-paradoxical phenomena, the surprise exam paradox, logical omniscience and epistemic closure, formalized theories of knowledge, debates about higher-order knowledge, and issues (...) of knowability raised by Fitch’s paradox. The references and recommended readings provide gateways for further exploration. (shrink)

Peter Hanks defends a new theory about the nature of propositional content, according to which the basic bearers of representational properties are particular mental or spoken actions. He explains the unity of propositions and provides new solutions to a long list of puzzles and problems in philosophy of language.

In sections 1 through 5, I develop in detail what I call the standard theory of worlds and propositions, and I discuss a number of purported objections. The theory consists of five theses. The first two theses, presented in section 1, assert that the propositions form a Boolean algebra with respect to implication, and that the algebra is complete, respectively. In section 2, I introduce the notion of logical space: it is a field of sets that represents the propositional (...) structure and whose space consists of all and only the worlds. The next three theses, presented in sections 3, 4, and 5, respectively, guarantee the existence of logical space, and further constrain its structure. The third thesis asserts that the set of propositions true at any world is maximal consistent; the fourth thesis that any two worlds are separated by a proposition; the fifth thesis that only one proposition is false at every world. In sections 6 through 10, I turn to the problem of reduction. In sections 6 and 7, I show how the standard theory can be used to support either a reduction of worlds to propositions or a reduction of propositions to worlds. A number of proposition-based theories are developed in section 6, and compared with Adams's world-story theory. A world-based theory is developed in section?, and Stalnaker's account of the matter is discussed. Before passing judgment on the proposition based and world-based theories, I ask in sections 8 and 9 whether both worlds and propositions might be reduced to something else. In section 8, I consider reductions to linguistic entities; in section 9, reductions to unfounded sets. After rejecting the possibility of eliminating both worlds and propositions, I return in section 10 to the possibility of eliminating one in favor of the other. I conclude, somewhat tentatively, that neither worlds nor propositions should be reduced one to the other, that both worlds and propositions should be taken as basic to our ontology. (shrink)