252 research outputs found
Consequences of a Goedel's misjudgment
The fundamental aim of the paper is to correct an harmful way to interpret a
Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the
Goedel's fault is rather venial, its misreading has produced and continues to
produce dangerous fruits, as to apply the incompleteness Theorems to the full
second-order Arithmetic and to deduce the semantic incompleteness of its
language by these same Theorems. The first three paragraphs are introductory
and serve to define the languages inherently semantic and its properties, to
discuss the consequences of the expression order used in a language and some
question about the semantic completeness: in particular is highlighted the fact
that a non-formal theory may be semantically complete despite using a language
semantically incomplete. Finally, an alternative interpretation of the Goedel's
unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic
incompleteness, categoricity, arithmetic, second-order languages, paradoxesComment: English version, 19 pages. Fixed and improved terminolog
Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism
This essay examines the philosophical significance of -logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of -logical validity can then be countenanced within a coalgebraic logic, and -logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of -logical validity correspond to those of second-order logical consequence, -logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets
Suszko's Problem: Mixed Consequence and Compositionality
Suszko's problem is the problem of finding the minimal number of truth values
needed to semantically characterize a syntactic consequence relation. Suszko
proved that every Tarskian consequence relation can be characterized using only
two truth values. Malinowski showed that this number can equal three if some of
Tarski's structural constraints are relaxed. By so doing, Malinowski introduced
a case of so-called mixed consequence, allowing the notion of a designated
value to vary between the premises and the conclusions of an argument. In this
paper we give a more systematic perspective on Suszko's problem and on mixed
consequence. First, we prove general representation theorems relating
structural properties of a consequence relation to their semantic
interpretation, uncovering the semantic counterpart of substitution-invariance,
and establishing that (intersective) mixed consequence is fundamentally the
semantic counterpart of the structural property of monotonicity. We use those
to derive maximum-rank results proved recently in a different setting by French
and Ripley, as well as by Blasio, Marcos and Wansing, for logics with various
structural properties (reflexivity, transitivity, none, or both). We strengthen
these results into exact rank results for non-permeable logics (roughly, those
which distinguish the role of premises and conclusions). We discuss the
underlying notion of rank, and the associated reduction proposed independently
by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve
compositionality in general, meaning that the resulting semantics is no longer
truth-functional. We propose a modification of that notion of reduction,
allowing us to prove that over compact logics with what we call regular
connectives, rank results are maintained even if we request the preservation of
truth-functionality and additional semantic properties.Comment: Keywords: Suszko's thesis; truth value; logical consequence; mixed
consequence; compositionality; truth-functionality; many-valued logic;
algebraic logic; substructural logics; regular connective
Hybrid type theory: a quartet in four movements
This paper sings a song -a song created by bringing together the work of four great names in the history of logic: Hans Reichenbach, Arthur Prior, Richard Montague, and Leon Henkin. Although the work of the first three of these authors have previously been combined, adding the ideas of Leon Henkin is the addition required to make the combination work at the logical level. But the present paper does not focus on the underlying technicalities (these can be found in Areces, Blackburn, Huertas, and Manzano [to appear]) rather it focusses on the underlying instruments, and the way they work together. We hope the reader will be tempted to sing along
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
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