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

Compactness and Löwenheim-Skolem theorems in extensions of first-order logic

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Enrique Casanovas Ruiz-Fornells[en] Lindström’s theorem characterizes first-order logic as the most expressive among those that satisfy the countable Compactness and downward Löwenheim-Skolem theorems. Given the importance of this results in model theory, Lindström’s theorem justifies, to some extent, the privileged position of first-order logic in contemporary mathematics. Even though Lindström’s theorem gives a negative answer to the problem of finding a proper extension of first-order logic satisfying the same model-theoretical properties, the study of these extensions has been of great importance during the second half of the XX. century: logicians were trying to find systems that kept a balance between expressive power and rich model-theoretical properties. The goal of this essay is to prove Lindström’s theorem, along with its prerequisites, and to give weaker versions of the Compactness and Löwenheim-Skolem theorems for the logic L ( Q 1 ) (first-order logic with the quantifier "there exist uncountably many"), which we present as an example of extended logic with good model-theoretical properties

Is English consequence compact?

Abstract: By mimicking the standard definition for a formal language, we define what it is for a natural language to be compact. We set out a valid English argument none of whose finite subarguments is valid. We consider one by one objections to the argument's logical validity and then dismiss them. The conclusion is that English—and any other language with the capacity to express the argument—is not compact. This rules out a large class of logics as the correct foundational one, for example any sound and complete logic, and in particular first‐order logic. The correct foundational logic is not compact

Realisability for Infinitary Intuitionistic Set Theory

We introduce a realisability semantics for infinitary intuitionistic set theory that employs Ordinal Turing Machines (OTMs) as realisers. We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary intuitionistic logic, and that all axioms of infinitary Kripke-Platek set theory are realised. As an application of our technique, we show that the propositional admissible rules of (finitary) intuitionistic Kripke-Platek set theory are exactly the admissible rules of intuitionistic propositional logic

Ontologies étalées

The notion of Mathematics as Ontology (as defined by Badiou in his work) is brought into question from a working mathematician's perspective. Notions of independence in set theory and model theory are contrasted with the original equation Mathematics=Ontology. The author builds an extension of mathematical ontology from set theory to a foliated, étale, setting

Contraction, Infinitary Quantifiers, and Omega Paradoxes

Our main goal is to investigate whether the infinitary rules for the quantifiers endorsed by Elia Zardini in a recent paper are plausible. First, we will argue that they are problematic in several ways, especially due to their infinitary features. Secondly, we will show that even if these worries are somehow dealt with, there is another serious issue with them. They produce a truth-theoretic paradox that does not involve the structural rules of contraction.Fil: Da Re, Bruno. Instituto de Investigaciones Filosóficas - Sadaf; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Rosenblatt, Lucas Daniel. Instituto de Investigaciones Filosóficas - Sadaf; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Gödel Mathematics Versus Hilbert Mathematics. II Logicism and Hilbert Mathematics, the Identification of Logic and Set Theory, and Gödel’s 'Completeness Paper' (1930)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined