In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding ‘exactly true’ and ‘non-falsity’ versions of four-valued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135; Shramko (2019). Dual-Belnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). First-degree entailment and its relatives. Studia Logica, 105(6), (...) 1291–1317] to the case of infectious logics, namely to the case of Deutsch's logic Sfde introduced in Deutsch [Relevant analytic entailment. The Relevance Logic Newsletter, 2(1), 26–44; The completeness of S. Studia Logica, 38(2), 137–147]. The particular systems obtained in this way are Setl and Snfl. We present them in the form of sequent calculi and prove corresponding soundness and completeness theorems. We illuminate the connection between Setl, Snfl and two well-known systems, strong Kleene three-valued logic K3 and Priest's Logic of Paradox LP. This connection allows us to investigate the characterisation of the entailment relations associated with Setl and Snfl as well as to introduce the notion of ‘infectious analogue’ of a certain logic. We also study implicative extensions of Setl and Snfl and prove soundness and completeness theorems for them as well. (shrink)

In this paper we re-assess the philosophical foundation of Exactly True Logic ($$\mathcal {ET\!L}$$ ET L ), a competing variant of First Degree Entailment ($$\mathcal {FDE}$$ FDE ). In order to do this, we first rebut an argument against it. As the argument appears in an interview with Nuel Belnap himself, one of the fathers of $$\mathcal {FDE}$$ FDE, we believe its provenance to be such that it needs to be taken seriously. We submit, however, that the (...) argument ultimately fails, and that $$\mathcal {ET\!L}$$ ET L cannot easily be dismissed. We then proceed to give an overview of the research that was inspired by this logic over the last decade, thus providing further motivation for the study of $$\mathcal {ET\!L}$$ ET L and, more generally, of $$\mathcal {FDE}$$ FDE -related logics that result from semantical analyses alternative to Belnap’s canonical one. We focus, in particular, on philosophical questions that these developments raise. (shrink)

In this paper we present a solution of the axiomatization problem for the Fmla-Fmla versions of the Pietz and Rivieccio exactly true logic and the non-falsity logic dual to it. To prove the completeness of the corresponding binary consequence systems we introduce a specific proof-theoretic formalism, which allows us to deal simultaneously with two consequence relations within one logical system. These relations are hierarchically organized, so that one of them is treated as the basic for the (...) resulting logic, and the other is introduced as an extension of this basic relation. The proposed bi-consequences systems allow for a standard Henkin-style canonical model used in the completeness proof. The deductive equivalence of these bi-consequence systems to the corresponding binary consequence systems is proved. We also outline a family of the bi-consequence systems generated on the basis of the first-degree entailment logic up to the classic consequence. (shrink)

In this paper we present a solution of the axiomatization problem for the Fmla-Fmla versions of the Pietz and Rivieccio exactly true logic and the non-falsity logic dual to it. To prove the completeness of the corresponding binary consequence systems we introduce a specific proof-theoretic formalism, which allows us to deal simultaneously with two consequence relations within one logical system. These relations are hierarchically organized, so that one of them is treated as the basic for the (...) resulting logic, and the other is introduced as an extension of this basic relation. The proposed bi-consequences systems allow for a standard Henkin-style canonical model used in the completeness proof. The deductive equivalence of these bi-consequence systems to the corresponding binary consequence systems is proved. We also outline a family of the bi-consequence systems generated on the basis of the first-degree entailment logic up to the classic consequence. (shrink)

An exact truthmaker for A is a state which, as well as guaranteeing A’s truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as exact truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for exact truthmaking and characterise the resulting (...) notion of entailment, showing that it is compact and decidable. We then investigate the effect of various restrictions on the semantics. We also formulate a sequent-style proof system for exact entailment and give soundness and completeness results. (shrink)

The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth. Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued (...) logics, for example, the logic of truth and exact truth or the logic of truth and material equivalence. These combined systems are consequence relations which allow us to express implications involving more than one of these features of propositions. (shrink)

A common reaction to Hegel’s suggestion that we collapse Kant’s distinction between form and content is that, since such a move would also deprive us of any way of distinguishing the merely logical from the real possibility of our concepts, it is incoherent and ought to be rejected. It is true that these two distinctions are intimately related in Kant, such that if one goes, the other does as well. But it is less obvious that giving them up as (...) Kant conceives them is as incoherent a proposal as many of Hegel’s critics think. It has been the point of a recent account of Hegel’s idealism to demonstrate that his critique of Kant’s dichotomy between form and content or concept and intuition does not commit him to the view that human cognition is materially creative of its content in the manner of a God-like or intuitive intellect—does not, in other words, signify a return to a pre-Critical metaphysics. This is the interpretative stance I adopt here as well, in hope of giving defenders of Kant reasons for taking Hegel more seriously. (shrink)

By using the notions of exact truth and exact falsity, one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the Strong Kleene schema. Besides familiar logics such as Strong Kleene logic, the Logic of Paradox and First Degree Entailment, the resulting class of all Strong Kleene generalizations of classical logic also contains a (...) host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus that is sound and complete with respect to all of them. Two further sequent calculi and \ calculus) will be considered, which serve the same purpose and which are obtained by applying general methods to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the \ and the \ calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the \ and the \ calculus, we also hint at its philosophical significance. (shrink)

The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth predicates (...) for the infinitary languages $\mathcal {L}_{\text {Ord},\omega }$, allowing any class parameter A; to the existence of truth predicates for the language $\mathcal {L}_{\text {Ord},\text {Ord}}$ ; to the existence of $\text {Ord}$ -iterated truth predicates for first-order set theory $\mathcal {L}_{\omega,\omega }$ ; to the assertion that every separative class partial order ${\mathbb {P}}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text {Ord}+1$. Unlike set forcing, if every class forcing notion ${\mathbb {P}}$ has a forcing relation merely for atomic formulas, then every such ${\mathbb {P}}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between $\text {GBC}$ and Kelley–Morse set theory $\text {KM}$. (shrink)

In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic, an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we (...) will show that a calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof. (shrink)

ABSTRACT The Gödel–Dummett logic and Łukasiewicz one are two main many-valued logics used by the fuzzy logic community. Our goal is a quantitative comparison of these two. In this paper, we will mostly consider the 3-valued Gödel–Dummett logic as well as the 3-valued Łukasiewicz one. We shall concentrate on their implicational-negation fragments which are limited to formulas formed with a fixed finite number of variables. First, we investigate the proportion of the number of true formulas of (...) a certain length n to the number of all formulas of such length built with exactly one variable. Then, we investigate such proportion for satisfiable formulas. Second, we generalise our investigation on formulas written with $ k\ge 1 $ k ≥ 1 variables. The primary goal of the paper is the research on the asymptotic behaviour of these fractions when the length n tends to infinity. If such limits exists, they are real numbers between 0 and 1, which are called the density of truth or the density of SAT. To compare the density of truth and the density of satisfiable formulas for both fragments of 3-valued Gödel–Dummett's and Łukasiewicz's logics we use the powerful theory of analytic combinatorics. This paper is a natural continuation of the previous brief conference note by Kostrzycka and Zaionc (2020) as well as enriched with some previous results from Kostrzycka and Zaionc (2003). In the conference note we computed analytically the density of truth and the density of SAT (with a determined precision) for 3-valued Łukasiewicz's logic restricted to a language with only one variable. In Kostrzycka and Zaionc (2003) we computed the same values for exactly the same fragment of the 3-valued Gödel–Dummett logic. This paper answers the more general questions of the existence of density of truth and density of SAT for both many-valued logics with an arbitrary finite number of variables. Therefore this paper gives an an interesting picture of two main families of finite-valued fuzzy logics problems treated quantitatively. This picture is taken from the perspective of classical logic. It shows that unexpectedly there is quantitatively a little distance between these two approaches. (shrink)

We show that the elementary theory of the recursively enumerable tt-degrees has the same computational complexity as true first-order arithmetic. As auxiliary results, we prove theorems about exact pairs and initial segments in the tt-degrees.

The Gödel–Dummett logic and Łukasiewicz one are two main many-valued logics used by the fuzzy logic community. Our goal is a quantitative comparison of these two. In this paper, we will mostly consider the 3-valued Gödel–Dummett logic as well as the 3-valued Łukasiewicz one. We shall concentrate on their implicational-negation fragments which are limited to formulas formed with a fixed finite number of variables. First, we investigate the proportion of the number of true formulas of a (...) certain length n to the number of all formulas of such length built with exactly one variable. Then, we investigate such proportion for satisfiable formulas. Second, we generalise our investigation on formulas written with k≥1 variables. The primary goal of the paper is the research on the asymptotic behaviour of these fractions when the length n tends to infinity. If such limits exists, they are real numbers between 0 and 1, which are called the density of truth or the density of SAT. To compare the density of truth and the density of satisfiable formulas for both fragments of 3-valued Gödel–Dummett's and Łukasiewicz's logics we use the powerful theory of analytic combinatorics. This paper is a natural continuation of the previous brief conference note by Kostrzycka and Zaionc (Citation2020) as well as enriched with some previous results from Kostrzycka and Zaionc (Citation2003). In the conference note we computed analytically the density of truth and the density of SAT (with a determined precision) for 3-valued Łukasiewicz's logic restricted to a language with only one variable. In Kostrzycka and Zaionc (Citation2003) we computed the same values for exactly the same fragment of the 3-valued Gödel–Dummett logic. This paper answers the more general questions of the existence of density of truth and density of SAT for both many-valued logics with an arbitrary finite number of variables. Therefore this paper gives an an interesting picture of two main families of finite-valued fuzzy logics problems treated quantitatively. This picture is taken from the perspective of classical logic. It shows that unexpectedly there is quantitatively a little distance between these two approaches. (shrink)

The author's motivation for constructing the calculi of this paper\nis so that time and tense can be "discussed together in the same\nlanguage" (p. 282). Two types of enriched propositional caluli for\ntense logic are considered, both containing ordinary propositional\nvariables for which any proposition may be substituted. One type\nalso contains "clock-propositional" variables, a,b,c, etc., for\nwhich only clock-propositional variables may be substituted and that\ncorrespond to instants or moments in the semantics. The other type\nalso contains "history-propositional" variables, u,v,w, etc., for\nwhich only history-propositional variables (...) may be substituted and\nthat correspond to maximally linearly ordered subsets of instants.\nThis second type is for tense logics in which time is not linear\nand where at any moment there may exist alternative future courses\nof events. Quantifiers are included in both types but only for clock-\nand history-propositional variables. However, these quantifiers are\ngiven an essentially substitutional interpretation. In addition,\nbesides each clock-propositional variable being true at exactly one\ninstant, the semantics requires that for each instant there be at\nleast one clock-propositional variable true at that instant, which\nin effect amounts to having each instant of the model "denoted" in\nthe formalism by some clock-propositional variable. Thus it is not\nsurprising that quantification over clock-propositional variables\nturns out to have a "well behaved first-order semantics" (p. 284).\n{One axiom relating history- with clock-propositions seems to need\nrevision: CKTauTbuAUabUba. Since the clock-propositional variables\nmight be true at the same instant the consequent should include an\nalternative, namely, LCab.} Besides the standard truth-functional\nand tense operators G,H,F,P, the author includes L for "it is\nalways the case that". The model-theoretic earlier-than relation\nordering instants and the semantic truth-at (an instant) relation\nare formalized by Uab and Taα (for a wff α) which\nare defined respectively as LCbPa (whenever the clock-proposition\nb is the case the clock-proposition a was the case) and LCaα\n(α is the case whenever the clock-proposition a is the case).\nBecause the future tense operators are definable in terms of U\nand T and the latter are definable in terms of L and P, the\ndual of H, the author notes that his completeness proofs can be\nrestricted to L and H as primitive and with the future tense\noperators used for other interpretations, especially Prior, A.'s\nindeterminist "it will be the case that", where time is not linear.\nThe author concludes with some rather technical observations showing\nthat "in ordinary tense logics the instants are nonstandard elements\nof the models. This provides", according to the author, "a semantic\npressure to build the instants into tense logic, so that they will\nappear as standard elements in the models" (p. 283). (shrink)

Applying first-order logic to derive the consequences of laws that are only approximately true of empirical phenomena involves idealization of a kind that is akin to applying arithmetic to calculate the area of a rectangular surface from approximate measures of the lengths of its sides. Errors in the data, in the exactness of the lengths in one case and in the exactness of the laws in the other, are in some measure transmitted to the consequences deduced from them, (...) and the aim of a theory of idealization is to describe this process. The present paper makes a start on this in the case of applied first-order logic, and relates it to Plato's picture of a world or model of 'appearances' in which laws are only approximately true, but which to some extent resembles an ideal world or model in which they are exactly true. (shrink)

Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: "kiss me and hug me" is the conjunction of "kiss me" with "hug me". This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied—what more (...) is there to say? Much more, I argue. "If you love me, kiss me", a conditional imperative, mixes a declarative antecedent ("you love me") with an imperative consequent ("kiss me"); it is satisfied if you love and kiss me, violated if you love but don't kiss me, and avoided if you don't love me. So we need a logic of three -valued imperatives which mixes declaratives with imperatives. I develop such a logic. (shrink)

In many real-life applications of logic it is useful to interpret a particular sentence as true together with its negation. If we are talking about classical logic, this situation would force all other sentences to be equally interpreted as true. Paraconsistent logics are exactly those logics that escape this explosive effect of the presence of inconsistencies and allow for sensible reasoning still to take effect. To provide reasonably intuitive semantics for paraconsistent logics has traditionally proven (...) to be a challenge. Possible-translations semantics can meet that challenge by allowing for each interpretation to be composed of multiple scenarios. Using that idea, a logic with a complex semantic behavior can be understood as an appropriate combination of ingredient logics with simpler semantic behaviors into which the original logic is given a collection of translations preserving its soundness. Completeness is then achieved through the judicious choice of the admissible translating mappings. The present note provides interpretation by way of possible-translations semantics for a group of fundamental paraconsistent logics extending the positive fragment of classical propositional logic. The logics PI, Cmin, mbC, bC, mCi and Ci, among others, are all initially presented through their non-truth-functional bivaluation semantics and sequent versions and then split by way of possible-translations semantics based on 3-valued ingredients. (shrink)

Suppose that a sign at the entrance of a hotel reads: “Don’t enter these premises unless you are accompanied by a registered guest”. You see someone who is about to enter, and you tell her: “Don’t enter these premises if you are an unaccompanied registered guest”. She asks why, and you reply: “It follows from what the sign says”. It seems that you made a valid inference from an imperative premise to an imperative conclusion. But it also seems that imperatives (...) cannot be true or false, so what does it mean to say that your inference is valid? It cannot mean that the truth of its premise guarantees the truth of its conclusion. One is thus faced with what is known as “Jørgensen’s dilemma” (Ross 1941: 55-6): it seems that imperative logic cannot exist because logic deals only with entities that, unlike imperatives, can be true or false, but it also seems that imperative logic must exist. It must exist not only because inferences with imperatives can be valid, but also because imperatives (like “Enter” and “Don’t enter”) can be inconsistent with each other, and also because one can apply logical operations to imperatives: “Don’t enter” is the negation of “Enter”, and “Sing or dance” is the disjunction of “Sing” and “Dance”. A standard reaction to this dilemma consists in basing imperative logic on analogues of truth and falsity. For example, the imperative “Don’t enter” is satisfied if you don’t enter and is violated if you enter, and one might say that an inference from an imperative premise to an imperative conclusion is valid exactly if the satisfaction (rather than the truth) of the premise guarantees the satisfaction of the conclusion. But before getting into the details, more needs to be said on what exactly imperatives are. (shrink)

Free logics are a family of first-order logics which came about as a result of examining the existence assumptions of classical logic (Hintikka _The Journal of Philosophy_, _56_, 125–137 1959 ; Lambert _Notre Dame Journal of Formal Logic_, _8_, 133–144 1967, 1997, 2001 ). What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential (...) import. Free logics reject the claim that names need to denote in (ii). _Positive_ free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true, _negative_ free logic treats them as uniformly false, and _neutral_ free logic as taking a third value. There has been a renewed interest in analyzing proof theory of free logic in recent years, based on intuitionistic logic in Maffezioli and Orlandelli (_Bulletin of the Section of Logic_, _48_(2), 137–158 2019 ) as well as classical logic in Pavlović and Gratzl (_Journal of Philosophical Logic_, _50_, 117–148 2021 ), there for the positive and negative variants. While the latter streamlines the presentation of free logics and offers a more unified approach to the variants under consideration, it does not cover neutral free logic, since there is some lack of both clear formal intuitions on the semantic status of formulas with empty names, as well as a satisfying account of the conditional in this context. We discuss extending the results to this third major variant of free logics. We present a series of G3 sequent calculi adapted from Fjellstad (_Studia Logica_, _105_(1), 93–119 2017, _Journal of Applied Non-Classical Logics_, _30_(3), 272–289 2020 ), which possess all the desired structural properties of a good proof system, including admissibility of contraction and all versions of the cut rule. At the same time, we maintain the unified approach to free logics and moreover argue that greater clarity of intuitions is achieved once neutral free logic is conceptualized as consisting of two sub-varieties. (shrink)

Although it is frequently said that logic is a purely formal discipline lacking any content for special philosophical subdisciplines, I argue in this essay that the concepts of predication, and of the properties of objects presupposed by standard first-order logic are sufficient to address many of the traditional problems of ontology. The concept of an object's having a property is extended to provide an intensional definition of the existence of an object as the object's possessing a maximally consistent (...) property combination, consisting, for any property or its complement. Nonexistent objects by the proposed definition are those that either lack both some property and its complement, or have both a property and its complement included in its corresponding property combination. The definition in turn makes it possible to offer solutions based on purely logical concepts of such longstanding metaphysical problems as why there is something rather than nothing, and why there exists exactly one logically contingent actual world, itself a maximally consistent combination of true predication instances or facts. Additionally, the ontology upholds an argument in support of modal actualism and against modal realism, treating all logically possible worlds other than the actual world as predicationally incomplete nonexistent objects or semantic fictions. (shrink)

With their book Logical Pluralism, Jc Beall and Greg Restall have elaborated on their previous statements on logical pluralism. Their view of logical pluralism is centred on ways of understanding logical consequence. The essay tries to come to grips with their doctrine of logical pluralism by highlighting some points that might be made clearer, and questioning the force of some of Beall’s and Restall’s central arguments. In that connection seven problems for their approach are put forth: (1) The Informal Common (...) Core Problem, (2) The Formal Common Core Problem, (3) The Superior Judge Problem, (4) The Problem of the Conditional, (6) The Problem of an Unsettled Concept of Consequence, (7) The Methodological Problem, (8) The General Logical Form Problem. It seems that the case for logical pluralism is far from clear. It is even unclear what exactly logical pluralism is and where is stops. It is also unclear if logical pluralism could be stated as it is, if it were true. So far universalism seems to be the better position to take. (shrink)

A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬BOXp, read as “p is true but you do not know p.” (...) Not only is the Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful. In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas. (shrink)

A nonempty sequence T1,...,Tn of theories is tolerant, if there are consistent theories T 1 + ,..., T n + such that for each 1 i n, T i + is an extension of Ti in the same language and, if i n, T i + interprets T i+1 + . We consider a propositional language with the modality , the arity of which is not fixed, and axiomatically define in this language the decidable logics TOL and TOL. It is (...) shown that TOL (resp. TOL) yields exactly the schemata of PA-provable (resp. true) arithmetical sentences, if (A1,..., An) is understood as (a formalization of) PA+A1, ..., PA+An is tolerant. (shrink)

It is a paradox, whose importance familiarity fails to diminish, that the most highly developed and useful scientific theories are ostensibly expressed in terms of objects never encountered in experience. The line traced by a draughtsman, no matter how accurate, is seen beneath the microscope as a kind of corrugated trench, far removed from the ideal line of pure geometry. And the “point-planet” of astronomy, the “perfect gas” of thermodynamics, or the “pure species” of genetics are equally remote from exact (...) realization. Indeed the unintelligibility at the atomic or sub-atomic level of the notion of a rigidly demarcated boundary shows that such objects not merely are not but could not be encountered. While the mathematician constructs a theory in terms of “perfect” objects, the experimental scientist observes objects of which the properties demanded by theory are and can, in the very nature of measurement, be only approximately true. As Duhem remarks, mathematical deduction is not useful to the physicist if interpreted rigorously. It is necessary to know that its validity is unaltered when the premise and conclusion are only “approximately true.” But the indeterminacy thus introduced, it is necessary to add in criticism, will invalidate the deduction unless the permissible limits of variation are specified. To do so, however, replaces the original mathematical deduction by a more complicated mathematical theory in respect of whose interpretation the same problem arises, and whose exact nature is in any case unknown. (shrink)

Imperatives cannot be true, but they can be obeyed or binding: `Surrender!' is obeyed if you surrender and is binding if you have a reason to surrender. A pure declarative argument — whose premisses and conclusion are declaratives — is valid exactly if, necessarily, its conclusion is true if the conjunction of its premisses is true; similarly, I suggest, a pure imperative argument — whose premisses and conclusion are imperatives — is obedience-valid (alternatively: bindingness-valid) exactly (...) if, necessarily, its conclusion is obeyed (alternatively: binding) if the conjunction of its premisses is. I argue that there are two kinds of bindingness, and that a vacillation between two corresponding variants of bindingness-validity largely explains conflicting intuitions concerning the validity of some pure imperative arguments. I prove that for each of those two variants of bindingness-validity there is an equivalent variant of obedience-validity. Finally, I address alternative accounts of pure imperative inference. (shrink)

Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that the domain of interpretation is not empty, every name denotes exactly one object in the domain and the quantifiers have existential import. Free logics usually reject the claim that names need to denote in, and of the systems considered in this paper, the positive free (...) class='Hi'>logic concedes that some atomic formulas containing non-denoting names are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject, are likewise considered. These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut, while streamlining free logics in a way no other approach has. We then present a simple and unified system of abstract semantics, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different logics. The final part of this paper is dedicated to extending the system with modalities by using a labeled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework. (shrink)

Besides pure declarative arguments, whose premises and conclusions are declaratives, and pure imperative arguments, whose premises and conclusions are imperatives, there are mixed-premise arguments, whose premises include both imperatives and declaratives, and cross-species arguments, whose premises are declaratives and whose conclusions are imperatives or vice versa. I propose a general definition of argument validity: an argument is valid exactly if, necessarily, every fact that sustains its premises also sustains its conclusion, where a fact sustains an imperative exactly if (...) it favors the satisfaction over the violation proposition of the imperative, and a fact sustains a declarative exactly if, necessarily, the declarative is true if the fact exists. I argue that this definition yields as special cases the standard definition of validity for pure declarative arguments and my previously defended definition of validity for pure imperative arguments, and that it yields intuitively acceptable results for mixed-premise and cross-species arguments. (shrink)

We study the system IFP of intuitionistic fixed point logic, an extension of intuitionistic first-order logic by strictly positive inductive and coinductive definitions. We define a realizability interpretation of IFP and use it to extract computational content from proofs about abstract structures specified by arbitrary classically true disjunction free formulas. The interpretation is shown to be sound with respect to a domain-theoretic denotational semantics and a corresponding lazy operational semantics of a functional language for extracted programs. We (...) also show how extracted programs can be translated into Haskell. As an application we extract a program converting the signed digit representation of real numbers to infinite Gray code from a proof of inclusion of the corresponding coinductive predicates. (shrink)

Logical monism is the claim that there is a single correct logic, the 'one true logic' of our title. The view has evident appeal, as it reflects assumptions made in ordinary reasoning as well as in mathematics, the sciences, and the law. In all these spheres, we tend to believe that there aredeterminate facts about the validity of arguments. Despite its evident appeal, however, logical monism must meet two challenges. The first is the challenge from logical pluralism, (...) according to which there is more than one correct logic. The second challenge is to determine which form of logicalmonism is the correct one. One True Logic is the first monograph to explicitly articulate a version of logical monism and defend it against the first challenge. It provides a critical overview of the monism vs pluralism debate and argues for the former. It also responds to the second challenge by defending a particularmonism, based on a highly infinitary logic. It breaks new ground on a number of fronts and unifies disparate discussions in the philosophical and logical literature. In particular, it generalises the Tarski-Sher criterion of logicality, provides a novel defence of this generalisation, offers a clearnew argument for the logicality of infinitary logic and replies to recent pluralist arguments. (shrink)

Besides pure declarative arguments, whose premises and conclusions are declaratives (“you sinned shamelessly; so you sinned”), and pure imperative arguments, whose premises and conclusions are imperatives (“repent quickly; so repent”), there are mixed-premise arguments, whose premises include both imperatives and declaratives (“if you sinned, repent; you sinned; so repent”), and cross-species arguments, whose premises are declaratives and whose conclusions are imperatives (“you must repent; so repent”) or vice versa (“repent; so you can repent”). I propose a general definition of argument (...) validity: an argument is valid exactly if, necessarily, every fact that sustains its premises also sustains its conclusion, where a fact sustains an imperative exactly if it favors the satisfaction over the violation of the imperative, and a fact sustains a declarative exactly if, necessarily, the declarative is true if the fact exists. I argue that this definition yields as special cases the standard definition of validity for pure declarative arguments and my previously defended definition of validity for pure imperative arguments, and that it yields intuitively acceptable results for mixed-premise and cross-species arguments. (shrink)

We consider the modality “ $\varphi $ is true in every $\sigma $ -centered forcing extension,” denoted $\square \varphi $, and its dual “ $\varphi $ is true in some $\sigma $ -centered forcing extension,” denoted $\lozenge \varphi $, which give rise to the notion of a principle of $\sigma $ -centered forcing. We prove that if ZFC is consistent, then the modal logic of $\sigma $ -centered forcing, i.e., the ZFC-provable principles of $\sigma $ -centered forcing, (...) is exactly $\mathsf {S4.2}$. We also generalize this result to other related classes of forcing. (shrink)

Several accounts of logical truth are compared and shown to define distinct concepts. Nevertheless, conditions are given under which they happen to declare exactly the same sentences logically true. These conditions involve the variety of objects in the domain, the richness of the language, and the logical resources available. It is argued that the class of sentences declared logically true by each of the accounts depends on particularities of the actual world.

For the given logical calculus we investigate the proportion of the number of true formulas of a certain length n to the number of all formulas of such length. We are especially interested in asymptotic behavior of this fraction when n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which we may call the density of truth for the investigated logic. In this paper we apply this approach to (...) the intuitionistic logic of one variable with implication and negation. The result is obtained by reducing the problem to the same one of Dummett's intermediate linear logic of one variable (see [2]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more than 93%) of classical prepositional calculus. Despite using strictly mathematical means to solve all discussed problems, this paper in fact, may have a philosophical impact on understanding how much the phenomenon of truth is sporadic or frequent in random mathematics sentences. (shrink)

In the theory of meaning, it is common to contrast truth-conditional theories of meaning with theories which identify the meaning of an expression with its use. One rather exact version of the somewhat vague use-theoretic picture is the view that the standard rules of inference determine the meanings of logical constants. Often this idea also functions as a paradigm for more general use-theoretic approaches to meaning. In particular, the idea plays a key role in the anti-realist program of Dummett and (...) his followers. In the theory of truth, a key distinction now is made between substantial theories and minimalist or deflationist views. According to the former, truth is a genuine substantial property of the truth-bearers, whereas according to the latter, truth does not have any deeper essence, but all that can be said about truth is contained in T-sentences (sentences having the form: ‘P’ is true if and only if P). There is no necessary analytic connection between the above theories of meaning and truth, but they have nevertheless some connections. Realists often favour some kind of truth-conditional theory of meaning and a substantial theory of truth (in particular, the correspondence theory). Minimalists and deflationists on truth characteristically advocate the use theory of meaning (e.g. Horwich). Semantical anti-realism (e.g. Dummett, Prawitz) forms an interesting middle case: its starting point is the use theory of meaning, but it usually accepts a substantial view on truth, namely that truth is to be equated with verifiability or warranted assertability. When truth is so understood, it is also possible to accept the idea that meaning is closely related to truth-conditions, and hence the conflict between use theories and truth-conditional theories in a sense disappears in this view. (shrink)

Prima facie, there are two kinds of expression used in English to make reference to time: those involving explicit mention of time and temporal ordering relations, and tenses involving no such explicit reference. Taking as a criterion of adequacy the unification of both these aspects, a systematization is proposed (owing much to Reichenbach) which provides a characterization of tenses. The theory is not based on the notion of a proposition with variable truth value which formed the cornerstone of Arthur Prior’s (...) work. His conception receives a good deal of criticism, ultimately on the ground that relativising to exactly one time is inadequate. We don’t know in advance whether or not no time of perhaps more than one time is relevant, and where this is the case, the analysis cannot be reduced to propositions true at just one time. (shrink)

Predicate modal formulas with non-modalized quantifiers (call them Q-formulas) are considered as schemata of arithmetical formulas, where is interpreted as the provability predicate of some fixed correct extension T of arithmetic. A method of constructing 1) non-provable in T and 2) false arithmetical examples for Q-formulas by Kripke-like countermodels of certain type is given. Assuming the means of T to be strong enough to solve the (undecidable) problem of derivability in QGL, the Q-fragment of the predicate version of the (...) class='Hi'>logic GL, we prove the recursive enumerability of the sets of Q-formulas all arithmetical examples of which are: 1) T-provable, 2) true. In. particular, the first one is shown to be exactly QGL and the second one to be exactly the Q-fragment of the predicate version of Solovay's logic S. (shrink)

Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the standard (...) readings, for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this. (shrink)

This is a paper about the constituents of arguments. It argues that several different kinds of truth-bearer may be taken to compose arguments, but that none of the obvious candidates—sentences, propositions, sentence/truth-value pairs etc.—make sense of logic as it is actually practiced. The paper goes on to argue that by answering the question in different ways, we can generate different logics, thus ensuring a kind of logical pluralism that is different from that of J. Beall and Greg Restall.

This expository paper on Aristotle’s prototype underlying logic is intended for a broad audience that includes non-specialists. It requires as background a discussion of Aristotle’s demonstrative logic. Demonstrative logic or apodictics is the study of demonstration as opposed to persuasion. It is the subject of Aristotle’s two-volume Analytics, as its first sentence says. Many of Aristotle’s examples are geometrical. A typical geometrical demonstration requires a theorem that is to be demonstrated, known premises from which the theorem is (...) to be deduced, and a deductive logic by which the steps of the deduction proceed. Every demonstration produces knowledge of its conclusion for every person who comprehends the demonstration. Aristotle presented a general truth-and-consequence theory of demonstration meant to apply to all demonstrations: a demonstration is an extended argumentation that begins with premises known to be truths and that involves a chain of reasoning showing by deductively evident steps that its conclusion is a consequence of its premises. In short, a demonstration is a deduction whose premises are known to be true. Aristotle’s general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining theory of deduction was meant to apply to all deductions: any deduction that is not immediately evident is an extended argumentation that involves a chaining of immediately evident steps that shows its final conclusion to follow logically from its premises. His deductions, both direct and indirect, were rule-based and not tautology-based. The idea of tautology-based deduction, which dominated modern logic in the early years of the 1900s, is nowhere to be found in Analytics. Rule-based deduction was rediscovered by modern logicians. To illustrate his general theory of deduction, Aristotle presented a prototype: an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. With reference only to propositions of the four so-called categorical forms, he painstakingly worked out exactly what those immediately evident deductive steps are and how they are chained to complete deductions. In his specialized prototype theory, Aristotle explained how to deduce from a given categorical premise set, no matter how large, any categorical conclusion implied by the given set. He did not extend this treatment to non-categorical deductions, thus setting a program for future logicians. The prototype, categorical syllogistic, was seen by Boole as a “first approximation” to a comprehensive logic. Today, however it appears more as the first of the dozens of logics already created and as the first exemplification of a family that continues to expand. (shrink)

Williamson argues for the contention that substructural logics are ‘ill-suited to acting as background logics for science’. That contention, if true, would be very important, but it is refutable, given what is already known about certain substructural logics. Classical Core Logic is a substructural logic, for it eschews the structural rules of Thinning and Cut and has Reflexivity as its only structural rule. Yet it suffices for classical mathematics, and it furnishes all the proofs and disproofs one (...) needs for the hypothetico-deductive method in science. We explain exactly what Classical Core Logic is, why it is a substructural logic par excellence, and what the basic requirements would be for a logic to be ‘suited to acting as [a] background logic for science’. We also explain how Classical Core Logic meets all these requirements. We end by examining Williamson’s argument in order to expose where its error lies. (shrink)

Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects $\S e_1, \S e_2$, are identical if, and only if, an equivalence relation $Eq_\S$ holds between the entities $e_1, e_2$) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? (...) In this paper, we investigate the matter by deriving the axioms of second-order Peano Arithmetic from Frege's Basic Law V (the extension of $F$ is identical with the extension of $G$ if, and only if, $F$ and $G$ are extensionally equivalent) in the presence of a relevant higher-order logic. The results are interesting. Not only must we take on logic as true, and not only must we apply our logic to abstraction principles, but also we have to apply our theory of abstraction back to the logic in order to arrive at arithmetic. Thus, what Abstractionism gives us is not simply what we get from abstraction via logic, but also what we get from logic via abstraction. (shrink)

One True Logic is a rare contribution to the most fundamental issues in the philosophy of logic. The book pushes a remarkably clear and uncompromising monistic.

Since the late nineteenth century ‘Scientific Philosophy’ has become a label ascribed to many research programs. German theoretical philosophy of the early twentieth century was dominated by three different trends—Phenomenology, Neo-Kantianism, and Logical Empiricism: Each trend claimed to represent the ‘Scientific Philosophy’. In this context it is astonishing that we know almost nothing about the relationships between these schools. It is true, all of them rejected the speculative metaphysics found, for example, in German Idealism, but knowledge about other connections (...) is scarce. Starting with the question what exactly ‘Scientific Philosophy’ means and how these meanings relate to each other, a thorough investigation of the connections between all three trends has yet to take place.The main goal of the conference was, therefore, to better understand the relationships of these schools through their notions of ‘Scientific Philosophy’. In this context the works of Edmund Husserl (for Ph .. (shrink)

Logical pluralism is the claim that different accounts of validity can be equally correct. Beall and Restall have recently defended this position. Validity is a matter of truth-preservation over cases, they say: the conclusion should be true in every case in which the premises are true. Each logic specifies a class of cases, but differs over which cases should be considered. I show that this account of logic is incoherent. Validity indeed is truth-preservation, provided this is (...) properly understood. Once understood, there is one true logic, relevance logic. The source of Beall and Restall’s error is a recent habit of using a classical metalanguage to analyse non-classical logics generally, including relevance logic. (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 Hard Truths, Elijah Millgram maintains that analytic philosophy rests on a mistake. 1 It is committed to bivalence – the contention that every truth bearer is either true or false. As a result of this commitment, its views about logic and metaphysics are profoundly misguided. He believes that rather than restricting ourselves to two truth values, we should recognize a plethora of partial truths – sentences, beliefs and opinions that are partly true or true in (...) a way. These are located on a multidimensional continuum between truth and falsity. Millgram never says exactly what partial truth is. The closest he comes is ‘Partial truth is the not fully articulated standard, or family of standards to which we hold [utterances, inscriptions and thoughts] in partial truth inferences’ , where partial truth inferences are those used in cases where ‘there is a recognized mismatch between representation and world – but not in any way that requires changing the representations’ . A critical issue then is what constitutes such a mismatch. I will argue that Millgram’s reasons for recognizing partial truths rest on untenable conceptions of logic, truth and language. Perhaps there are partial truths; perhaps recognizing them would enhance our logic or metaphysics. But Millgram’s arguments fail to show it. 1. Logic Recognizing that there can be some truth in sentences that as a whole are false is not unprecedented. Analyses by Ullian and Goodman 2 and, more recently, Yablo 3 reveal how a false sentence or proposition can be true about a particular item. ‘The dog barks and the cat speaks French’ is true about the dog, even though the conjunction as a whole is false. This does not undermine classical logic. Rather, such analyses reveal that truth about a particular subject matter is more fine grained than truth simpliciter. ‘True …. (shrink)

A class of Kripke models is modally definable if there is a set of modal formulas such that the class consists exactly of models on which every formula from that set is globally true. In this paper, a class is also considered definable if there is a set of formulas such that it consists exactly of models in which every formula from that set is satisfiable. The notion of modal definability is then generalized by combining these two. (...) For thus obtained types of modal definability on the level of Kripke models, we give characterization theorems in the usual form, in terms of algebraic closure conditions. As some consequences of these, various preservation results are presented. Also, some characterizations are strengthened by replacing closure under ultraproducts with closure under ultrapowers. (shrink)