Two-dimensional semantics, which can represent the distinction between a priority and necessity, has wielded considerable influence in the philosophy of language. In this paper, I axiomatize the dagger operator of Stalnaker’s “Assertion” in the formal context of two-dimensional modal logic. The language contains modalities of actuality, necessity, and a priority, but is also able to represent diagonalization, a conceptually important operation in a variety of contexts, including models of the relative a priori and a posteriori often appealed (...) to Bayesian and Gricean contexts. Finally, I sketch the prospects for extending this two-dimensional upgrade to other kinds of modal logics for natural language. (shrink)

In this paper, we axiomatize the deontic logic in Fusco (2015), which uses a Stalnaker-inspired account of diagonal acceptance and a two-dimensional account of disjunction to treat Ross’s Paradox and the Puzzle of Free Choice Permission. On this account, disjunction-involving validities are a priori rather than necessary. We show how to axiomatize two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general two-dimensional rules. These completeness results (...) help make explicit the restrictions Fusco’s account must place on free-choice inferences. They are also of independent interest, as they raise difficult questions about how to “lift” a Kripke frame for a one-dimensional modal logic into two dimensions. (shrink)

Since Kripke, philosophers have distinguished a priori true statements from necessarily true ones. A statement is a priori true if its truth can be established before experience, and necessarily true if it could not have been false according to logical or metaphysical laws. This distinction can be captured formally using two-dimensional semantics. There is a natural way to extend the notions of apriority and necessity so they can also apply to questions. Questions either can or cannot be resolved before (...) experience, and either are or are not about necessary facts. Classical two-dimensionalism has no account of question meanings, so it has to be combined with a framework for question semantics in order to capture these observations. It is shown in [14] how two-dimensional semantics can be combined with inquisitive semantics, in which questions are analyzed in terms of information. The present paper investigates the logic of two-dimensional inquisitive semantics, and provides a complete proof system. (shrink)

In this paper, we axiomatize the deontic logic in Fusco 2015, which uses a Stalnaker-inspired account of diagonal acceptance and a two-dimensional account of disjunction to treat Ross’s Paradox and the Puzzle of Free Choice Permission. On this account, disjunction-involving validities are a priori rather than necessary. We show how to axiomatize two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general two-dimensional rules. These completeness results (...) help make explicit the restrictions Fusco’s account must place on free-choice inferences. They are also of independent interest, as they raise difficult questions about how to ‘lift’ a Kripke frame for a one- dimensional modal logic into two dimensions. (shrink)

We provide a Hilbert-style axiomatization of the logic of , as well as a two-dimensional semantics with respect to which our logics are sound and complete. Our completeness results are quite general, pertaining to all such actuality logics that extend a normal and canonical modal basis. We also show that our logics have the strong finite model property and permit straightforward first-order extensions.

This paper offers a two dimensional variation of Standard Deontic Logic SDL, which we call 2SDL. Using 2SDL we can show that we can overcome many of the difficulties that SDL has in representing linguistic sets of Contrary-to-Duties (known as paradoxes) including the Chisholm, Ross, Good Samaritan and Forrester paradoxes. We note that many dimensional logics have been around since 1947, and so 2SDL could have been presented already in the 1970s. Better late than never! As a (...) detailed case study illustrating the power of 2SDL, we examine the system DL of Deontic Logic of Andrew Jones and Ingmar Pörn offered in 1985 to solve the Chisholm paradox of Contrary to Duties. The critical examination is done using logics and methods available in 1985 and solutions are proposed using what was available in 1985. (shrink)

This paper offers a two dimensional variation of Standard Deontic Logic SDL, which we call 2SDL. Using 2SDL we can show that we can overcome many of the difficulties that SDL has in representing linguistic sets of Contrary-to-Duties (known as paradoxes) including the Chisholm, Ross, Good Samaritan and Forrester paradoxes. We note that many dimensional logics have been around since 1947, and so 2SDL could have been presented already in the 1970s. Better late than never! As a (...) detailed case study illustrating the power of 2SDL, we examine the system DL of Deontic Logic of Andrew Jones and Ingmar Pörn offered in 1985 to solve the Chisholm paradox of Contrary to Duties. The critical examination is done using logics and methods available in 1985 and solutions are proposed using what was available in 1985. (shrink)

Epistemic two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. While this theory is usually presented in an informal manner, I take some steps in formalizing it in this paper. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that captures the relevant ideas of epistemic two-dimensional semantics. (...) I also describe some properties of the logic that are interesting from a philosophical perspective, and apply it to the so-called nesting problem. (shrink)

We introduce a two‐dimensional metric (interval) temporal logic whose internal and external time flows are dense linear orderings. We provide a suitable semantics and a sequent calculus with axioms for equality and extralogical axioms. Then we prove completeness and a semantic partial cut elimination theorem down to formulas of a certain type.

This paper offers a two dimensional variation of Standard Deontic Logic SDL, which we call 2SDL. Using 2SDL we can show that we can overcome many of the difficulties that SDL has in representing linguistic sets of Contrary-to-Duties (known as paradoxes) including the Chisholm, Ross, Good Samaritan and Forrester paradoxes. We note that many dimensional logics have been around since 1947, and so 2SDL could have been presented already in the 1970s. Better late than never! As a (...) detailed case study illustrating the power of 2SDL, we examine the system DL of Deontic Logic of Andrew Jones and Ingmar Pörn offered in 1985 to solve the Chisholm paradox of Contrary to Duties. The critical examination is done using logics and methods available in 1985 and solutions are proposed using what was available in 1985. (shrink)

In this paper we present tableau methods for two-dimensional modal logics. Although models for such logics are well known, proof systems remain rather unexplored as most of their developments have been purely axiomatic. The logics herein considered contain first-order quantifiers with identity, and all the formulas in the language are doubly-indexed in the proof systems, with the upper indices intuitively representing the actual or reference worlds, and the lower indices representing worlds of evaluation—first and second dimensions, respectively. The tableaux (...) modulate over different notions of validity such as local, general, and diagonal, besides being general enough for several two-dimensional logics proposed in the literature. We also motivate the introduction of a new operator into two-dimensional languages and explore some of the philosophical questions raised by it concerning the relations there are between actuality, necessity, and the a priori, that seem to undermine traditional intuitive interpretations of two-dimensional operators. (shrink)

Ontological commitments and other problems concerning existence arise in connection with various aspects of logical theories. The semantics of quantification theory is usually formulated in such a manner that theorems are all and only those formulae which come out true under all interpretations in all non-empty domains. There are several approaches to include the empty domain. Paradoxically this apparent semantic extension means surrendering several formulae which are valid and intuitively plausible.

Awareness logic is a type of belief logic in which an agent's beliefs are restricted to those sentences that the agent is aware of. Awareness logic is a successful way to circumvent the problem of omniscience so that actual belief is modelled in a reasonable way. In this paper, we suggest a new method modelling awareness and actual belief by using two-dimensional logics. We show that the two-dimensional logics are flexible tools. Different types of concepts (...) of awareness can be easily modelled by this method. (shrink)

We consider a formal language whose logical syntax involves both modal and tense propositional operators, as well as sortal quantifiers, sortal identities and (second order) quantifiers over sortals. We construct an intensional semantics for the language and characterize a formal logical system which we prove to be sound and complete with respect to the semantics. Conceptualism is the philosophical background of the semantic system.

Two-dimensional semantics is a framework that helps us better understand some of the most fundamental issues in philosophy: those having to do with the relationship between the meaning of words, the way the world is, and our knowledge of the meaning of words. This selection of new essays by some of the world's leading authorities in this field sheds fresh light both on foundational issues regarding two-dimensional semantics and on its specific applications. Contributors: Richard Breheny, Alex Byrne, David (...) Chalmers, Martin Davies, Gareth Evans, Manuel Garcia-Carpintero, Josep Maci`, Martine Nida-Rumelin, Christopher Peacocke, James Pryor, Francois Recanati, Scott Soames, Cara Spencer, Robert Stalnaker, Kai-Yee Wong, Stephen Yablo. (shrink)

We show the first examples of recursively enumerable (even decidable) two-dimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nesting-depth.

ABSTRACT We introduce a two-dimensional temporal logic as a formalism which enables the description of both the history of a world and the evolution of an observer's views about the history. We apply such formalism to the description of certain problems that occur in historical database systems due to updates. The historical dimension describes the history of a world according to an observer's view at a certain moment in time. The transaction dimension describes the evolution of an observer's (...) view; changes can occur in the observer's view of the present and future as well as in his views of the past. The database is seens as an observer of the history of the world. Besides describing the history of the world, the database system can also execute temporal specifications in the form of temporal actions. The two-dimensional formalism is used to describe the effects of updates in the execution of temporal actions. The detection of some problematic effects and its implementation are discussed. (shrink)

The two-dimensional modal logic of Davies and Humberstone [3] is an important aid to our understanding the relationship between actuality, necessity and a priori knowability. I show how a cut-free hypersequent calculus for 2D modal logic not only captures the logic precisely, but may be used to address issues in the epistemology and metaphysics of our modal concepts. I will explain how the use of our concepts motivates the inference rules of the sequent calculus, and then (...) show that the completeness of the calculus for Davies–Humberstone models explains why those concepts have the structure described by those models. The result is yet another application of the completeness theorem. (shrink)

This paper recalls some applications of two-dimensional modal logic from the 1980s, including work on the logic of Actually and on a somewhat idealized version of the indicative/subjunctive distinction, as well as on absolute and relative necessity. There is some discussion of reactions this material has aroused in commentators since. We also survey related work by Leslie Tharp from roughly the same period.

Many authors have noted that there are types of English modal sentences cannot be formalized in the language of basic first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who is actually rich could have been poor.” In response to this lack of expressive power, many authors have discussed extensions of first-order modal logic with two-dimensional operators. But claims about the relative expressive power of these extensions (...) are often justified only by example rather than by rigorous proof. In this paper, we provide proofs of many of these claims and present a more complete picture of the expressive landscape for such languages. (shrink)

Why is two-dimensional semantics important? One can think of it as the most recent act in a drama involving three of the central concepts of philosophy: meaning, reason, and modality. First, Kant linked reason and modality, by suggesting that what is necessary is knowable a priori, and vice versa. Second, Frege linked reason and meaning, by proposing an aspect of meaning (sense) that is constitutively tied to cognitive signi?cance. Third, Carnap linked meaning and modality, by proposing an aspect of (...) meaning (intension) that is constitutively tied to possibility and necessity. (shrink)

We present two-dimensional tableau systems for the actuality, fixedly, and up-arrow operators. All systems are proved sound and complete with respect to a two-dimensional semantics. In addition, a decision procedure for the actuality logics is discussed.

We consider a version of so called T x W logic for historical necessity in the sense of R.H. Thomason (1984), which is somewhat special in three respects: (i) it is explicitly based on two-dimensional modal logic in the sense of Segerberg (1973); (ii) for reasons of applicability to interesting fields of philosophical logic, it conceives of time as being discrete and finite in the sense of having a beginning and an end; and (iii) it utilizes (...) the technique of systematic frame constants in order to handle the problem of irreflexivity in tense logics, well known since Gabbay (1981). Axiomatizations are given for two infinite hierarchies of two-dimensional modal tense logics, one without and one with the characteristic operators for historical necessity and possibility. Strong and weak completeness results are obtained for both hierarchies as well as a result to the effect that two approaches to their semantics are equivalent, much in the spirit of Di Maio and Zanardo (1996) and von Kutschera (1997). (shrink)

. David Chalmerss version of two-dimensional semantics is an attempt at setting up a unified semantic framework that would vindicate both the Fregean and the Kripkean semantic intuitions. I claim that there are three acceptable ways of carrying out such a project, and that Chalmerss theory does not coherently fit any of the three patterns. I suggest that the theory may be seen as pointing to the possibility of a double reading for many linguistic expressions (a double reading which, (...) however, is not easily identified with straightforward semantic ambiguity). (shrink)

In this paper I put forward a representationalist theory of conscious experience based on Robert Stalnaker's version of two-dimensional modal semantics. According to this theory the phenomenal character of an experience correlates with a content equivalent to what Stalnaker calls the diagonal proposition. I show that the theory is closely related both to functionalist theories of consciousness and to higher-order representational theories. It is also more compatible with an anti-Cartesian view of the mind than standard representationalist theories.

In this article, a qualitative notion of subjective plausibility and its revision based on a preorder relation are implemented in higher-order logic. This notion of plausibility is used for modeling pragmatic aspects of communication on top of traditional two-dimensional semantic representations.

Philosophical views about the logical structure of time are typically divided between proponents of A and B theories, based on McTaggart's A and B series. Drawing on Paul Ricoeur's hermeneutic phenomenology, I develop and defend McTaggart's thesis that the C series and the A series working together give a consistent description of temporal experience, provided that the two series are treated as distinct dimensions internal to time. In the proposed two-dimensional model, the C series expresses a nesting order of (...) the constitutive states of a world, whereas ontological continuity and change are properties of the A series. This, I argue, allows for limited backward causation. (shrink)

Two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. Usually, this theory is presented in an informal manner. In this thesis, I take first steps in formalizing it, and use the formalization to present some considerations in favor of two-dimensional semantics. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, (...) and apriority that captures the relevant ideas of two-dimensional semantics. I use this to show that some criticisms of two-dimensional semantics that claim that the theory is incoherent are not justified. I also axiomatize the logic, and compare it to the most important proposals in the literature that define similar logics. To indicate that two-dimensional semantics is a plausible semantic theory, I give an argument that shows that all theorems of the logic can be philosophically justified independently of two-dimensional semantics. (shrink)

Murray MacBeath, in his essay ``Time's Square'', describes a fictitious scenariowhere various physical observations made by the participants would, he claims, invitethe interpretation that time for them is two-dimensional. In the present paper, however, Iargue that such observations come close to underdetermining the hypothesis of time's twodimensionality;for a rival hypothesis - that, under certain circumstances, the observationscan be explained in terms of the familiar time dilation effects predicted by special relativity- almost fits the evidence as well. That is, under (...) certain (albeit artificial) circumstances theworld can already behave almost as though it were temporally two-dimensional. (shrink)

I review and reconsider some of the themes of ‘Two notions of necessity’ (Davies and Humberstone, 1980) and attempt to reach a deeper understanding and appreciation of Gareth Evans’s reflections (in ‘Reference and contingency’, 1979) on both modality and reference. My aim is to plot the relationships between the notions of necessity that Humberstone and I characterised in terms of operators in two-dimensional modal logic, the notions of superficial and deep necessity that Evans himself described, and the epistemic (...) notion of a priority. (shrink)

This paper presents the model of ‘bounded revision’ that is based on two-dimensional revision functions taking as arguments pairs consisting of an input sentence and a reference sentence. The key idea is that the input sentence is accepted as far as (and just a little further than) the reference sentence is ‘cotenable’ with it. Bounded revision satisfies the AGM axioms as well as the Same Beliefs Condition (SBC) saying that the set of beliefs accepted after the revision does not (...) depend on the reference sentence (although the posterior belief state does depend on it). Bounded revision satisfies the Darwiche–Pearl (DP) axioms for iterated belief change. If the reference sentence is fixed to be a tautology or a contradiction, two well-known one-dimensional revision operations result. Bounded revision thus naturally fills the space between conservative revision (also known as natural revision) and moderate revision (also known as lexicographic revision). I compare this approach to the two-dimensional model of ‘revision by comparison’ investigated by Fermé and Rott (Artif Intell 157:5–47, 2004 ) that satisfies neither the SBC nor the DP axioms. I conclude that two-dimensional revision operations add substantially to the expressive power of qualitative approaches that do not make use of numbers as measures of degrees of belief. (shrink)

Starting from the notions of q-entailment and p-entailment, a two-dimensional notion of entailment is developed with respect to certain generalized q-matrices referred to as B-matrices. After showing that every purely monotonic singleconclusion consequence relation is characterized by a class of B-matrices with respect to q-entailment as well as with respect to p-entailment, it is observed that, as a result, every such consequence relation has an inferentially four-valued characterization. Next, the canonical form of B-entailment, a two-dimensional multiple-conclusion notion of (...) entailment based on B-matrices, is introduced, providing a uniform framework for studying several different notions of entailment based on designation, antidesignation, and their complements. Moreover, the two-dimensional concept of a B-consequence relation is defined, and an abstract characterization of such relations by classes of B-matrices is obtained. Finally, a contribution to the study of inferential many-valuedness is made by generalizing Suszko’s Thesis and the corresponding reduction to show that any B-consequence relation is, in general, inferentially four-valued. (shrink)

In this paper we revisit Pavel Tichý’s novel distinction between one-dimensional and two-dimensional conception of inference, which he presented in his book Foundations of Frege’s Logic (1988), and later in On Inference (1999), which was prepared from his manuscript by his co-author Jindra Tichý. We shall focus our inquiry not only on the motivation behind the introduction of this non-classical concept of inference, but also on further inspection of selected Tichý’s arguments, which we see as the most (...) compelling or simply most effective in providing support for his two-dimensional account of inference. Main attention will be given to exposing the failure of one-dimensional theory of inference in its explanation of indirect (reductio ad absurdum) proofs. Lastly, we discuss shortly the link between two-dimensional inference and deduction apparatus of Tichý’s Transparent Intensional Logic. (shrink)

In this paper I argue that the two-dimensional character of Frege’s Begriffsschrift plays an epistemological role in his argument for the analyticity of arithmetic. First, I motivate the claim that its two-dimensional character needs a historical explanation. Then, to set the stage, I discuss Frege’s notion of a Begriffsschrift and Kant’s epistemology of mathematics as synthetic a priori and partly grounded in intuition, canvassing Frege’s sharp disagreement on these points. Finally, I argue that the two-dimensional character of (...) Frege’s notations play the epistemological role of facilitating our grasp of logical truths independently of intuition. The rest of this paper critically evaluates Frege’s view and discusses Macbeth’s account. (shrink)

Let ${{\mathcal J}\,(\mathbb M^2)}$ denote the σ-ideal associated with two-dimensional Miller forcing. We show that it is relatively consistent with ZFC that the additivity of ${{\mathcal J}\,(\mathbb M^2)}$ is bigger than the covering number of the ideal of the meager subsets of ω ω. We also show that Martin’s Axiom implies that the additivity of ${{\mathcal J}\,(\mathbb M^2)}$ is 2 ω .Finally we prove that there are no analytic infinite maximal antichains in any finite product of ${\mathfrak{P}{(\omega)}/{\rm fin}}$.

This paper concerns the applications of two-dimensional modal semantics to the explanation of the contents of speech and thought. Different interpretations and applications of the apparatus are contrasted. First, it is argued that David Kaplan's two-dimensional semantics for indexical expressions is different from the use that I made of a formally similar framework to represent the role of contingent information in the determination of what is said. But the two applications are complementary rather than conflicting. Second, my interpretation (...) of the apparatus is contrasted with that of David Chalmers, Frank Jackson, and David Lewis. It is argued that this difference reflects a contrast between internalist and externalist approaches to the problem of intentionality. (shrink)