Online research in philosophy

We investigate the philosophical significance of the existence of different semantic systems with respect to which a given deductive system is sound and complete. Our case study will be Corcoran’s deductive system D for Aristotelian syllogistic and some of the different semantic systems for syllogistic that have been proposed in the literature. We shall prove that they are not equivalent, in spite of D being sound and complete with respect to each of them. Beyond the specific case of syllogistic, the (...) goal is to offer a general discussion of the relations between informal notions—in this case, an informal notion of deductive validity—and logical apparatuses such as deductive systems and (model-theoretic or other) semantic systems that aim at offering technical, formal accounts of informal notions. Specifically, we will be interested in Kreisel’s famous ‘squeezing argument’; we shall ask ourselves what a plurality of semantic systems (understood as classes of mathematical structures) may entail for the cogency of specific applications of the squeezing argument. More generally, the analysis brings to the fore the need for criteria of adequacy for semantic systems based on mathematical structures. Without such criteria, the idea that the gap between informal and technical accounts of validity can be bridged is put under pressure. (shrink)

A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is (...) to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete. (shrink)

Here is a familiar history: modal logics (see [13]) were around for some time before a semantic framework was found for them (by Kripke and others).1 This framework did at least two Very Good Things for modal logics: 1) it connected the powerful mathematical tools of model theory to these logics, allowing a variety of technical results to be proven, and 2) it connected modal logics (more) firmly to philosophy, allowing their application to the understanding of metaphysics, tense, scientific laws, (...) modal verbs, and so on. This application crucially depends not only on simply having Kripke-style semantic frameworks, but on interpreting the frameworks in various ways. The points of evaluation of the framework, in various interpretations, might be metaphysically possible worlds, say, or times, or morally allowable outcomes, or fictions of some sort, or . . . whathaveyou; these interpretations give modal logics an ‘applied semantics’ as opposed to merely ‘pure semantics’. Relevant logics have a similar history. The logics themselves were articulated and explored (see [1]) before semantic frameworks were found for them [11, 24, 25, 26, 28, 31]. Of these frameworks, the one that’s had the most mileage put on it is the Routley-Meyer one, and that’s the one we’re most concerned to.. (shrink)

We present the inconsistency-adaptive deontic logic DP r , a nonmonotonic logic for dealing with conflicts between normative statements. On the one hand, this logic does not lead to explosion in view of normative conflicts such as O A ∧ O ∼ A , O A ∧ P ∼ A or even O A ∧ ∼ O A . On the other hand, DP r still verifies all intuitively reliable inferences valid in Standard Deontic Logic ( SDL ). DP r (...) interprets a given premise set ‘as normally as possible’ with respect to SDL . Whereas some SDL -rules are verified unconditionally by DP r , others are verified conditionally. The latter are applicable unless they rely on formulas that turn out to behave inconsistently in view of the premises. This dynamic process is mirrored by the proof theory of DP r. (shrink)

The conceptual spaces approach has recently emerged as a novel account of concepts. Its guiding idea is that concepts can be represented geometrically, by means of metrical spaces. While it is generally recognized that many of our concepts are vague, the question of how to model vagueness in the conceptual spaces approach has not been addressed so far, even though the answer is far from straightforward. The present paper aims to fill this lacuna.

We propose a revision operator on a stratified belief base, i.e., a belief base that stores beliefs in different strata corresponding to the value an agent assigns to these beliefs. Furthermore, the operator will be defined as to perform the revision in such a way that information is never lost upon revision but stored in a stratum or layer containing information perceived as having a lower value. In this manner, if the revision of one layer leads to the rejection of (...) some information to maintain consistency, instead of being withdrawn it will be kept and introduced in a different layer with lower value. Throughout this development we will follow the principle of minimal change, being one of the important principles proposed in belief change theory, particularly emphasized in the AGM model. Regarding the reasoning part from the stratified belief base, the agent will obtain the inferences using an argumentative formalism. Thus, the argumentation framework will decide which information prevails when sentences of different layers are used for entailing conflicting beliefs. We will also illustrate how inferences are changed and how the status of arguments can be modified after a revision process. (shrink)

Classical mereology (CM) is usually taken to be formulated in a tenseless language, and is therefore associated with a four-dimensionalist metaphysics. This paper presents three ways one might integrate the core idea of flat plenitude, i.e., that every suitable condition or property has exactly one mereological fusion, with a tensed logical setting. All require a revised notion of mereological fusion. The candidates differ over how they conceive parthood to interact with existence in time, which connects to the distinction between endurance (...) and perdurance. Similar issues arise for the integration of mereology with modality, and much of our discussion applies to this project as well. (shrink)

We study a range of issues connected with the idea of replacing one formula by another in a fixed (linguistic) context. The replacement core of a consequence relation ⊢ is the relation holding between a set of formulas { A 1 , ..., A m , ...} and a formula B when for every context C (·), we have C ( A 1 ), ..., C ( A m ), ... ⊢ C ( B ). Section 1 looks at some (...) differences between which inferences are lost on passing to the replacement cores of the classical and intuitionistic consequence relations. For example, we find that while the inference from A and B to , sanctioned by both these initial consequence relations, is retained on passage to the replacement core in the classical case, it is lost in the intuitionistic case. Further discussion of these two (and some other) logics occupies Sections 3 and 4. Section 2 looks at the m = 1 case, describing A as replaceable by B according to ⊢ when B is a consequence of A by the replacement core of ⊢, and inquiring as to which choices of ⊢ render this induced replaceability relation symmetric. Section 5 investigates further conceptual refinements— such as a contrast between horizontal and vertical replaceability—suggested by some work of R. B. Angell and R. Harrop (and a comment on the latter by T. J. Smiley) in the 1950s and 1960s. Appendix 1 examines a related aspect of term-for-term replacement in connection with identity in predicate logic. Appendix 2 is a repository for proofs which would otherwise clutter up Section 3. (shrink)

For deductive reasoning to be justified, it must be guaranteed to preserve truth from premises to conclusion; and for it to be useful to us, it must be capable of informing us of something. How can we capture this notion of information content, whilst respecting the fact that the content of the premises, if true, already secures the truth of the conclusion? This is the problem I address here. I begin by considering and rejecting several accounts of informational content. I (...) then develop an account on which informational contents are indeterminate in their membership. This allows there to be cases in which it is indeterminate whether a given deduction is informative. Nevertheless, on the picture I present, there are determinate cases of informative (and determinate cases of uninformative) inferences. I argue that the model I offer is the best way for an account of content to respect the meaning of the logical constants and the inference rules associated with them without collapsing into a classical picture of content, unable to account for informative deductive inferences. (shrink)

Call an argument a ‘happy sorites’ if it is a sorites argument with true premises and a false conclusion. It is a striking fact that although most philosophers working on the sorites paradox find it at prima facie highly compelling that the premises of the sorites paradox are true and its conclusion false, few (if any) of the standard theories on the issue ultimately allow for happy sorites arguments. There is one philosophical view, however, that appears to allow for at (...) least some happy sorites arguments: strict finitism in the philosophy of mathematics. My aim in this paper is to explore to what extent this appearance is accurate. As we shall see, this question is far from trivial. In particular, I will discuss two arguments that threaten to show that strict finitism cannot consistently accept happy sorites arguments, but I will argue that (given reasonable assumptions on strict finitistic logic) these arguments can ultimately be avoided, and the view can indeed allow for happy sorites arguments. (shrink)

We describe a variety of sets internal to models of intuitionistic set theory that (1) manifest some of the crucial behaviors of potentially infinite sets as described in the foundational literature going back to Aristotle, and (2) provide models for systems of predicative arithmetic. We close with a brief discussion of Church’s Thesis for predicative arithmetic.

This paper begins an analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.

In J Philos Logic 34:155–192, 2005 , Leitgeb provides a theory of truth which is based on a theory of semantic dependence. We argue here that the conceptual thrust of this approach provides us with the best way of dealing with semantic paradoxes in a manner that is acceptable to a classical logician. However, in investigating a problem that was raised at the end of J Philos Logic 34:155–192, 2005 , we discover that something is missing from Leitgeb’s original definition. (...) Moreover, we show that once the appropriate repairs have been made, the resultant definition is equivalent to a version of the supervaluation definition suggested in J Philos 72:690–716, 1975 and discussed in detail in J Symb Log 51(3):663–681, 1986 . The upshot of this is a philosophical justification for the simple supervaluation approach and fresh insight into its workings. (shrink)

A curious feature of Belnap’s “useful four-valued logic”, also known as first-degree entailment (FDE), is that the overdetermined value B (both true and false) is treated as a designated value. Although there are good theoretical reasons for this, it seems prima facie more plausible to have only one of the four values designated, namely T (exactly true). This paper follows this route and investigates the resulting logic, which we call Exactly True Logic.

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P , then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of (...) it, which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases. (shrink)

Some left-nested indicative conditionals are hard to interpret while others seem fine. Some proponents of the view that indicative conditionals have No Truth Values (NTV) use their view to explain why some left-nestings are hard to interpret: the embedded conditional does not provide the truth conditions needed by the embedding conditional. Left-nestings that seem fine are then explained away as cases of ad hoc, pragmatic interpretation.

Rabern and Rabern (2008) and Uzquiano (2010) have each presented increasingly harder versions of ‘the hardest logic puzzle ever’ (Boolos 1996), and each has provided a two-question solution to his predecessor’s puzzle. But Uzquiano’s puzzle is different from the original and different from Rabern and Rabern’s in at least one important respect: it cannot be solved in less than three questions. In this paper we solve Uzquiano’s puzzle in three questions and show why there is no solution in two. Finally, (...) to cement a tradition, we introduce a puzzle of our own. (shrink)

Jeff Paris (2001) proves a generalized Dutch Book theorem. If a belief state is not a generalized probability (a kind of probability appropriate for generalized distributions of truth-values) then one faces ‘sure loss’ books of bets. In <span class='Hi'>Williams</span> (manuscript) I showed that Joyce’s (1998) accuracy-domination theorem applies to the same set of generalized probabilities. What is the relationship between these two results? This note shows that (when ‘accuracy’ is treated via the Brier Score) both results are easy corollaries of (...) the core result that Paris appeals to in proving his dutch book theorem (Minkowski’s separating hyperplane theorem). We see that every point of accuracy-domination defines a dutch book, but we only have a partial converse. (shrink)

There is no consensus as to whether a Liar sentence is meaningful or not. Still, a widespread conviction with respect to Liar sentences (and other ungrounded sentences) is that, whether or not they are meaningful, they are useless. The philosophical contribution of this paper is to put this conviction into question. Using the framework of assertoric semantics, which is a semantic valuation method for languages of self-referential truth that has been developed by the author, we show that certain computational problems, (...) called query structures, can be solved more efficiently by an agent who has self-referential resources (amongst which are Liar sentences) than by an agent who has only classical resources; we establish the computational power of self-referential truth. The paper concludes with some thoughts on the implications of the established result for deflationary accounts of truth. (shrink)

The naive theory of vagueness holds that the vagueness of an expression consists in its failure to draw a sharp boundary between positive and negative cases. The naive theory is contrasted with the nowadays dominant approach to vagueness, holding that the vagueness of an expression consists in its presenting borderline cases of application. The two approaches are briefly compared in their respective explanations of a paramount phenomenon of vagueness: our ignorance of any sharp boundary between positive and negative cases. These (...) explanations clearly do not provide any ground for choosing the dominant approach against the naive theory. The decisive advantage of the former over the latter is rather supposed to consist in its immunity to any form of sorites paradox. But another paramount phenomenon of vagueness is higher-order vagueness: the expressions (such as ‘borderline’ and ‘definitely’) introduced in order to express in the object language the vagueness of the object language are themselves vague. Two highly plausible claims about higher-order vagueness are articulated and defended: the existence of “definitely ω ” positive and negative cases and the “radical” character of higher-order vagueness itself. Using very weak logical principles concerning vague expressions and the ‘definitely’-operator, it is then shown that, in the presence of higher-order vagueness as just described, the dominant approach is subject to higher-order sorites paradoxes analogous to the original ones besetting the naive theory, and therefore that, against the communis opinio , it does not fare substantially better with respect to immunity to any form of sorites paradox. (shrink)

Free Semantics is based on normalized natural deduction for the weak relevant logic DW and its near neighbours. This is motivated by the fact that in the determination of validity in truth-functional semantics, natural deduction is normally used. Due to normalization, the logic is decidable and hence the semantics can also be used to construct counter-models for invalid formulae. The logic DW is motivated as an entailment logic just weaker than the logic MC of meaning containment. DW is the logic (...) focussed upon, but the results extend to MC. The semantics is called ‘free semantics’ since it is disjunctively and existentially free in that no disjunctive or existential witnesses are produced, unlike in truth-functional semantics. Such ‘witnesses’ are only assumed in generality and are not necessarily actual. The paper sets up the free semantics in a truth-functional style and gives a natural deduction interpetation of the meta-logical connectives. We then set out a familiar tableau-style system, but based on natural deduction proof rather than truth-functional semantics. A proof of soundness and completeness is given for a reductio system, which is a transform of the tableau system. The reductio system has positive and negative rules in place of the elimination and introduction rules of Brady’s normalized natural deduction system for DW. The elimination-introduction turning points become closures of threads of proof, which are at the points of contradiction for the reductio system. (shrink)

Both I and Belnap, motivated the “Belnap-Dunn 4-valued Logic” by talk of the reasoner being simply “told true” ( T ), and simply “told false” ( F ), which leaves the options of being neither “told true” nor “told false” ( N ), and being both “told true” and “told false” ( B ). Belnap motivated these notions by consideration of unstructured databases that allow for negative information as well as positive information (even when they conflict). We now experience this (...) on a daily basis with the Web. But the 4-valued logic is deductive in nature, and its matrix is discrete: there are just four values. In this paper I investigate embedding the 4-valued logic into a context of probability. Jøsang’s Subjective Logic introduced uncertainty to allow for degrees of belief, disbelief, and uncertainty. We extend this so as to allow for two kinds of uncertainty—that in which the reasoner has too little information (ignorance) and that in which the reasoner has too much information (conflicted). Jøsang’s “Opinion Triangle” becomes an “Opinion Tetrahedron” and the 4-values can be seen as its vertices. I make/prove various observations concerning the relation of non-classical “probability” to non-classical logic. (shrink)

The classical theory of semantic information ( ESI ), as formulated by Bar-Hillel and Carnap in 1952, does not give a satisfactory account of the problem of what information, if any, analytically and/or logically true sentences have to offer. According to ESI , analytically true sentences lack informational content, and any two analytically equivalent sentences convey the same piece of information. This problem is connected with Cohen and Nagel’s paradox of inference: Since the conclusion of a valid argument is contained (...) in the premises, it fails to provide any novel information. Again, ESI does not give a satisfactory account of the paradox. In this paper I propose a solution based on the distinction between empirical information and analytic information . Declarative sentences are informative due to their meanings. I construe meanings as structured hyperintensions , modelled in Transparent Intensional Logic as so-called constructions . These are abstract, algorithmically structured procedures whose constituents are sub-procedures. My main thesis is that constructions are the vehicles of information. Hence, although analytically true sentences provide no empirical information about the state of the world, they convey analytic information , in the shape of constructions prescribing how to arrive at the truths in question. Moreover, even though analytically equivalent sentences have equal empirical content, their analytic content may be different. Finally, though the empirical content of the conclusion of a valid argument is contained in the premises, its analytic content may be different from the analytic content of the premises and thus convey a new piece of information. (shrink)

The modal logic of Gödel sentences, termed as GS , is introduced to analyze the logical properties of ‘true but unprovable’ sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk’s Logic, where modality can be interpreted as ‘true and provable’. As we show, GS and Grzegorczyk’s Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS . GS is also an extended system of the logic of ‘Essence and Accident’ proposed (...) by Marcos (Bull Sect Log 34(1):43–56, 2005 ). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic. (shrink)

Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they may (...) be said to exhibit “general-elimination” harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I- and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions. (shrink)

This paper investigates a generalized version of inquisitive semantics . A complete axiomatization of the associated logic is established, the connection with intuitionistic logic and several intermediate logics is explored, and the generalized version of inquisitive semantics is argued to have certain advantages over the system that was originally proposed by Groenendijk ( 2009 ) and Mascarenhas ( 2009 ).

Quine has argued that modal logic began with the sin of confusing use and mention. Anderson and Belnap, on the other hand, have offered us a way out through a strategy of nominalization. This paper reviews the history of Lewis’s early work in modal logic, and then proves some results about the system in which “ A is necessary” is intepreted as “ A is a classical tautology.”.

The trilattice is a natural generalization of the well-known bilattice . Cut-free, sound and complete sequent calculi for truth entailment and falsity entailment in are presented.