One-Variable Fragments of First-Order Many-Valued Logics

In this thesis we study one-variable fragments of first-order logics. Such a one-variable fragment consists of those first-order formulas that contain only unary predicates and a single variable. These fragments can be viewed from a modal perspective by replacing the universal and existential quantifier with a box and diamond modality, respectively, and the unary predicates with corresponding propositional variables. Under this correspondence, the one-variable fragment of first-order classical logic famously corresponds to the modal logic S5. This thesis explores some such correspondences between first-order and modal logics. Firstly, we study first-order intuitionistic logics based on linear intuitionistic Kripke frames. We show that their one-variable fragments correspond to particular modal Gödel logics, defined over many-valued S5-Kripke frames. For a large class of these logics, we prove the validity problem to be decidable, even co-NP-complete. Secondly, we investigate the one-variable fragment of first-order Abelian logic, i.e., the first-order logic based on the ordered additive group of the reals. We provide two completeness results with respect to Hilbert-style axiomatizations: one for the one-variable fragment, and one for the one-variable fragment that does not contain any lattice connectives. Both these fragments are proved to be decidable. Finally, we launch a much broader algebraic investigation into one-variable fragments. We turn to the setting of first-order substructural logics (with the rule of exchange). Inspired by work on, among others, monadic Boolean algebras and monadic Heyting algebras, we define monadic commutative pointed residuated lattices as a first (algebraic) investigation into one-variable fragments of this large class of first-order logics. We prove a number of properties for these newly defined algebras, including a characterization in terms of relatively complete subalgebras as well as a characterization of their congruences

The Nature and Logic of Vagueness

The PhD thesis advances a new approach to vagueness as dispersion, comparing it with the main philosophical theories of vagueness in the analytic tradition

What the heck is Logic? Logics-as-formalizations, a nihilistic approach

Logic is about reasoning, or so the story goes. This thesis looks at the concept of logic, what it is, and what claims of correctness of logics amount to. The concept of logic is not a settled matter, and has not been throughout the history of it as a notion. Tools from conceptual analysis aid in this historical venture. Once the unsettledness of logic is established we see the repercussions in current debates in the philosophy of logic. Much of the battle over the ‘one true logic’ is conceptually talking past each other. The theory of logics-as-formalizations is presented as a conceptually open theory of logic which is Carnapian in flavour and grounding. Rudolf Carnap’s notions surrounding ‘external’ and ‘pseudo-questions’ about linguistic frameworks apply to formalizations, thus logics, as well. An account of what formalizations are, a more structured sub-set of modelling, is given to ground the claim that logics are formalizations. Finally, a novel account of correctness, the COFE framework, is developed which allows the notions of logical monism, pluralism and nihilism to be more precisely formulated than they currently are in the discourse

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Worlds and Objects of Epistemic Space : A study of Jaakko Hintikka's modal semantics

This study focuses on meaning and knowledge by assessing a distinctive view regarding their relation, namely the modal view of Jaakko Hintikka. The development of this view has not been previously scrutinized. By paying close attention to the texts of Hintikka, I show that, despite the extensive deployment of mathematical tools, the articulation of the view remained intuitive and vague. The study calls attention to several points at which Hintikka omits relevant details or disregards foundational questions. Attempts are made to articulate Hintikka’s certain ideas in a more specific manner, and new problems that result are identified. The central claim argued for is that Hintikka’s exposition was unsatisfactory in many respects and hence the view, as it stands, falls short in its explanatory scope compared to current theories in the intersection of logic, semantics, and epistemology. However, I argue that, despite its shortcomings, the prospects of the modal view are not exhausted. This is verified by introducing a new interpretation of the framework and by sketching new applications relevant in philosophy of language and in epistemology. It is also pointed out that certain early advances of the view closely resemble, and therefore anticipate, the central tenets of the currently influential two-dimensional approaches in logic and semantics.Tutkimus paneutuu merkityksen ja tiedon käsitteisiin tarkastelemalla Jaakko Hintikan työtä modaalisen semantiikan parissa. Tutkimus osoittaa, että Hintikka jätti modaalisen semantiikan kehitystyössään avoimeksi useita perustavia kysymyksiä ja yksityiskohtia. Tutkimuksessa pyritään artikuloimaan täsmällisemmin joitakin Hintikan näkemyksiä, ja tunnistetaan uusia syntyviä ongelmia. Keskeisenä väitteenä on, että Hintikan teoreettinen työ jäi monilta osin epätyydyttäväksi, ja siten hänen modaalinen näkemyksensä ei yllä selitysvoimaltaan ja sovelluspotentiaaliltaan samalle tasolle kuin nykyiset filosofiset teoriat, jotka operoivat logiikan, semantiikan ja epistemologian risteyskohdissa. Tästä huolimatta tutkimuksessa argumentoidaan, että Hintikan teoreettinen viitekehys tarjoaa myös uusia kiinnostavia näköaloja. Tämä todennetaan tarjoamalla Hintikan viitekehykselle uusi tulkinta, ja soveltamalla sitä uusiin kielifilosofisiin kysymyksiin. Tutkimus nostaa myös esiin kirjallisuudessa ohitetun tosiasian, että Hintikan työ ennakoi tärkeällä tavalla nykyisin vaikutusvaltaisia kaksi-dimensionaalisia lähestymistapoja logiikassa ja semantiikassa

Révision automatique de théories écologiques

À l’origine, ce sont des difficultés en biologie évolutive qui ont motivé cette thèse. Après des décennies à tenter de trouver une théorie basée sur la sélection capable de prédire la diversité génomique, les théoriciens n’ont pas trouvé d’alternatives pratiques à la théorie neutre. Après avoir étudié la relation entre la spéciation et la diversité (Annexes A, B, C), j’ai conclu que l’approche traditionnelle pour construire des théories serait difficile à appliquer au problème de la biodiversité. Prenons par exemple le problème de la diversité génomique, la difficulté n’est pas que l’on ignore les mécanismes impliqués, mais qu’on ne réussit pas à construire de théories capable d’intégrer ces mécanismes. Les techniques en intelligence artificielle, à l’inverse, réussissent souvent à construire des modèles prédictifs efficaces justement là où les théories traditionnelles échouent. Malheureusement, les modèles bâtis par les intelligences arti- ficielles ne sont pas clairs. Un réseau de neurones peut avoir jusqu’à un milliard de paramètres. L’objectif principal de ma thèse est d’étudier des algorithmes capable de réviser les théories écologiques. L’intégration d’idées venant de différentes branches de l’écologie est une tâche difficile. Le premier défi de ma thèse est de trouver sous quelle représentation formelle les théories écologiques doivent être encodées. Contrairement aux mathématiques, nos théories sont rarement exactes. Il y a à la fois de l’incertitude dans les données que l’on collecte, et un flou dans nos théories (on ne s’attend pas à que la théorie de niche fonctionne 100% du temps). Contrairement à la physique, où un petit nombre de forces dominent la théorie, l’écologie a un très grand nombre de théories. Le deuxième défi est de trouver comment ces théories peuvent être révisées automatiquement. Ici, le but est d’avoir la clarté des théories traditionnelles et la capacité des algorithmes en intelligence artificielle de trouver de puissants modèles prédictifs à partir de données. Les tests sont faits sur des données d’interactions d’espèces

Replacing truth

Kevin Scharp proposes an original account of the nature and logic of truth, on which truth is an inconsistent concept that should be replaced for certain theoretical purposes. He argues that truth is best understood as an inconsistent concept; develops an axiomatic theory of truth; and offers a new kind of possible-worlds semantics for this theory

Abstraktní studium úplnosti pro infinitární logiky

V této dizertační práci se zabýváme studiem vlastností úplnosti infinitárních výrokových logik z pohledu abstraktní algebraické logiky. Cílem práce je pochopit, jak lze základní nástroj v důkazech uplnosti, tzv. Lindenbaumovo lemma, zobecnit za hranici finitárních logik. Za tímto účelem studujeme vlastnosti úzce související s Lindenbaumovým lemmatem (a v důsledku také s vlastnostmi úplnosti). Uvidíme, že na základě těchto vlastností lze vystavět novou hierarchii infinitárních výrokových logik. Také se zabýváme studiem těchto vlastností v případě, kdy naše logika má nějaké (případně hodně obecně definované) spojky implikace, disjunkce a negace. Mimo jiné uvidíme, že přítomnost daných spojek může zajist platnost Lindenbaumova lemmatu. Keywords: abstraktní algebraická logika, infinitární logiky, Lindenbau- movo lemma, disjunkce, implikace, negaceIn this thesis we study completeness properties of infinitary propositional logics from the perspective of abstract algebraic logic. The goal is to under- stand how the basic tool in proofs of completeness, the so called Linden- baum lemma, generalizes beyond finitary logics. To this end, we study few properties closely related to the Lindenbaum lemma (and hence to com- pleteness properties). We will see that these properties give rise to a new hierarchy of infinitary propositional logic. We also study these properties in scenarios when a given logic has some (possibly very generally defined) connectives of implication, disjunction, and negation. Among others, we will see that presence of these connectives can ensure provability of the Lin- denbaum lemma. Keywords: abstract algebraic logic, infinitary logics, Lindenbaum lemma, disjunction, implication, negationKatedra logikyDepartment of LogicFaculty of ArtsFilozofická fakult