Given a 1-ary sentence operator , we describe L - another 1-ary operator - as as a left inverse of in a given logic if in that logic every formula is provably equivalent to L. Similarly R is a right inverse of if is always provably equivalent to R. We investigate the behaviour of left and right inverses for taken as the operator of various normal modal logics, paying particular attention to the conditions under which these logics are conservatively extended (...) by the addition of such inverses, as well as to the question of when, in such extensions, the inverses behave as normal modal operators in their own right. (shrink)

We investigate, for several modal logics but concentrating on KT, KD45, S4 and S5, the set of formulas B for which ${\square B}$ is provably equivalent to ${\square A}$ for a selected formula A (such as p, a sentence letter). In the exceptional case in which a modal logic is closed under the (‘cancellation’) rule taking us from ${\square C \leftrightarrow \square D}$ to ${C \leftrightarrow D}$ , there is only one formula B, to within equivalence, in this inverse image, (...) as we shall call it, of ${\square A}$ (relative to the logic concerned); for logics for which the intended reading of “ ${\square}$ ” is epistemic or doxastic, failure to be closed under this rule indicates that from the proposition expressed by a knowledge- or belief-attribution, the propositional object of the attitude in question cannot be recovered: arguably, a somewhat disconcerting situation. More generally, the inverse image of ${\square A}$ may comprise a range of non-equivalent formulas, all those provably implied by one fixed formula and provably implying another—though we shall see that for several choices of logic and of the formula A, there is not even such an ‘interval characterization’ of the inverse image (of ${\square A}$ ) to be found. (shrink)

We show that the modal prepositional logicILM (interpretability logic with Montagna's principle), which has been shown sound and complete as the interpretability logic of Peano arithmetic PA (by Berarducci and Savrukov), is sound and complete as the logic ofπ 1-conservativity over eachbE 1-sound axiomatized theory containingI⌆ 1 (PA with induction restricted tobE 1-formulas). Furthermore, we extend this result to a systemILMR obtained fromILM by adding witness comparisons in the style of Guaspari's and Solovay's logicR (this will be done in a (...) separate continuation of the present paper). (shrink)

It is shown that the propositional modal logic IRM (interpretability logic with Montagna's principle and with witness comparisons in the style of Guaspari's and Solovay's logicR) is sound and complete as the logic ofII 1-conservativity over each∑ 1-sound axiomatized theory containingI∑ 1. The exact statement of the result uses the notion of standard proof predicate. This paper is an immediate continuation of our paper [HM]. Knowledge of [HM] is presupposed. We define a modal logic, called IRM, which includes both ILM (...) andR (the logic of [GS]) and prove an arithmetical completeness theorem in the style of [GS], thus showing that IRM is the logic ofII 1-conservativity with witness comparisons. The reader is recommended to have Smoryński's book [Sm] at his/her disposal. (shrink)

This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order (...) arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368–387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338–355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287–304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy. (shrink)

This paper investigates the conditions under which diagonal sentences can be taken to constitute paradigmatic cases of self-reference. We put forward well-motivated constraints on the diagonal operator and the coding apparatus which separate paradigmatic self-referential sentences, for instance obtained via Gödel’s diagonalization method, from accidental diagonal sentences. In particular, we show that these constraints successfully exclude refutable Henkin sentences, as constructed by Kreisel.

In this paper the PA-completeness of modal logic is studied by syntactical and constructive methods. The main results are theorems on the structure of the PA-proofs of suitable arithmetical interpretationsS of a modal sequentS, which allow the transformation of PA-proofs ofS into proof-trees similar to modal proof-trees. As an application of such theorems, a proof of Solovay's theorem on arithmetical completeness of the modal system G is presented for the class of modal sequents of Boolean combinations of formulas of the (...) form p i,m i=0, 1, 2, ... The paper is the preliminary step for a forthcoming global syntactical resolution of the PA-completeness problem for modal logic. (shrink)

This paper is the first of a series of three articles that present the syntactic proof of the PA-completeness of the modal system G, by introducing suitable proof-theoretic objects, which also have an independent interest. We start from the syntactic PA-completeness of modal system GL-LIN, previously obtained in [7], [8], and so we assume to be working on modal sequents S which are GL-LIN-theorems. If S is not a G-theorem we define here a notion of syntactic metric d(S, G): we (...) calculate a canonical characteristic fomula H of S (char(S)) so that ⊢G ∼ H → (∼S) and ⊢GL-LIN ∼ H, and the complexity σ of ∼ H gives the distance d(S, G) of S from G. Then, in order to produce the whole completeness proof as an induction on this d(S, G), we introduce the tree-interpretation of a modal sequent Q into PA, that sends the letters of Q into PA-formulas describing the properties of a GL-LIN-proof P of Q: It is also a d(*, G)-metric linked interpretation, since it will be applied to a proof-tree T of ∼ H with H = char(S) and σ(∼ H) = d(S, G). (shrink)

In this paper are studied the properties of the proofs in PRA of provability logic sentences, i.e. of formulas which are Boolean combinations of formulas of the form PIPRA, where h is the Gödel-number of a sentence in PRA. The main result is a Normal Form Theorem on the proof-trees of provability logic sequents, which states that it is possible to split the proof into an arithmetical part, which contains only atomic formulas and has an essentially intuitionistic character, and into (...) a logical part, which is merely instrumental. Moreover, the induction rules which occur in the arithmetical part are implicit. Some applications of the Normal Form Theorem are shown in order to obtain some syntactical results on the PRA-completeness of modal logic. In particular a completeness theorem for Boolean combinations of modalities is given. (shrink)

The paper shows how we can add a truth predicate to arithmetic (or formalized syntactic theory), and keep the usual truth schema Tr( ) ↔ A (understood as the conjunction of Tr( ) → A and A → Tr( )). We also keep the full intersubstitutivity of Tr(>A>)) with A in all contexts, even inside of an →. Keeping these things requires a weakening of classical logic; I suggest a logic based on the strong Kleene truth tables, but with → (...) as an additional connective, and where the effect of classical logic is preserved in the arithmetic or formal syntax itself. Section 1 is an introduction to the problem and some of the difficulties that must be faced, in particular as to the logic of the →; Section 2 gives a construction of an arithmetically standard model of a truth theory; Section 3 investigates the logical laws that result from this; and Section 4 provides some philosophical commentary. (shrink)

The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally described by Skyrms, Anderson, and Solovay. In this article, some background is explained to clarify Égré’s solutions, all three of which hinge on (...) intricacies of provability logic and its arithmetical interpretations. To check whether Égré’s solutions are satisfactory, we use the criteria for solutions to paradoxes defined by Susan Haack and we propose some refinements of them. This article aims to describe to what extent the knower paradox can be solved using provability logic and to what extent the solutions proposed in the literature satisfy Haack’s criteria. Finally, the article offers some reflections on the relation between knowledge, proof, and provability, as inspired by the knower paradox and its solutions. (shrink)

It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for (...) selecting beliefs. The significance of this question for assessing "intensional" results like Godel's Second Theorem, and their bearing on Hilbert's Program are discussed. (shrink)

-/- Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. As a modal logic, provability logic has been studied since the early seventies, and has had important applications in the foundations of mathematics. -/- From a philosophical point of view, provability logic is interesting because (...) the concept of provability in a fixed theory of arithmetic has a unique and non-problematic meaning, other than concepts like necessity and knowledge studied in modal and epistemic logic. Furthermore, provability logic provides tools to study the notion of self-reference. (shrink)

Niklas Luhmann verwendet in seiner soziologischen Systemtheorie offenbar etwas, das er den Büchern des englischen Mathematikers George Spencer Brown entnimmt. Dessen Formenkalkül ist für Luhmann, wie Günther Schulte treffend bemerkt, “Mädchen für alles, mit dem er nicht nur in der Lage ist Teezukochen, sondern auch Auto oder Straßenbahn zu fahren”. Der erste Blick in Spencer Browns Laws of Form vermittelt einen anderen Eindruck: nichts scheinen sie mit soziologischer Systemtheorie zu tun zu haben. Der vorliegende Text bearbeitet hieran anknüpfend eine recht (...) bescheidene Frage, die sich gleichwohl jedem Luhmann-Leser schon einmal gestellt haben dürfte: Was wollen die Laws of Form und was will Luhmann mit ihnen? Als Antwort ergibt sich, nach Zurückverfolgung der relevanten Fußnoten, eine gute und eine schlechte Nachricht. Die schlechte Nachricht ist, daß die Lektüre der Laws of Form offenbar niemandem wirklich weiterhelfen kann, auch Luhmann selbst nicht. Die gute ist folglich, daß dem Luhmann-Leser die Notwendigkeit erspart bleibt, einen so dunklen, weil sparsamen Kalkül zu verstehen. Das meiste nämlich, was Luhmann den Laws of Form angeblich entnimmt, steht auf den zweiten Blick nicht darin. Er wird es also ohnehin durch andere Texte begründen müssen. (shrink)

In researching presuppositions dealing with logic and dynamic of belief we distinguish two related parts. The first part refers to presuppositions and logic, which is not necessarily involved with intentional operators. We are primarily concerned with classical, free and presuppositonal logic. Here, we practice a well known Strawson’s approach to the problem of presupposition in relation to classical logic. Further on in this work, free logic is used, especially Van Fraassen’s research of the role of presupposition in supervaluations logical systems. (...) At the end of the first part, presuppositional logic, advocated by S.K. Thomason, is taken into consideration. The second part refers to the presuppositions in relation to the logic of the dynamics of belief. Here the logic of belief change is taken into consideration and other epistemic notions with immanent mechanism for the presentation of the dynamics. Three representative and dominant approaches are evaluated. First, we deal with new, less classical, situation semantics. Besides Strawson’s theory, the second theory is the theory of the belief change, developed by Alchourron, Gärdenfors, and Makinson (AGM theory). At the end, the oldest, universal, and dominant approach is used, recognized as Hintikka’s approach to the analysis of epistemic notions. (shrink)