Search results for "Mathematics - Logic"
showing 6 items of 16 documents
Probabilistic entailment and iterated conditionals
2018
In this paper we exploit the notions of conjoined and iterated conditionals, which are defined in the setting of coherence by means of suitable conditional random quantities with values in the interval $[0,1]$. We examine the iterated conditional $(B|K)|(A|H)$, by showing that $A|H$ p-entails $B|K$ if and only if $(B|K)|(A|H) = 1$. Then, we show that a p-consistent family $\mathcal{F}=\{E_1|H_1,E_2|H_2\}$ p-entails a conditional event $E_3|H_3$ if and only if $E_3|H_3=1$, or $(E_3|H_3)|QC(\mathcal{S})=1$ for some nonempty subset $\mathcal{S}$ of $\mathcal{F}$, where $QC(\mathcal{S})$ is the quasi conjunction of the conditional events in $\mathcal{S}$. Then, we examine the inference rules $A…
Selective versions of chain condition-type properties
2015
We study selective and game-theoretic versions of properties like the ccc, weak Lindel\"ofness and separability, giving various characterizations of them and exploring connections between these properties and some classical cardinal invariants of the continuum.
Probabilistic squares and hexagons of opposition under coherence
2017
Various semantics for studying the square of opposition and the hexagon of opposition have been proposed recently. We interpret sentences by imprecise (set-valued) probability assessments on a finite sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square and of the hexagon in terms of acceptability. Then, we show how to construct probabilistic versions of the square and of the hexagon of opposition by forming suitable tripartitions of the set of all coherent assessments on a finite sequence of conditional events. Finally, as an application, we present new versions of the square and of the…
Noetherian type in topological products
2010
The cardinal invariant "Noetherian type" of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_\omega$ with the countable box topology, $(2^{\aleph_\omega})_\delta$, is shown in Section 3 to be closely related …
The Syllogistic with Unity
2011
We extend the language of the classical syllogisms with the sentence-forms “At most 1 p is a q” and “More than 1 p is a q”. We show that the resulting logic does not admit a finite set of syllogism-like rules whose associated derivation relation is sound and complete, even when reductio ad absurdum is allowed.
A Completeness Proof for a Regular Predicate Logic with Undefined Truth Value
2023
We provide a sound and complete proof system for an extension of Kleene's ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of "is defined" is extended to terms and formulas via a straightforward recursive algorithm. The "is defined" formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of the formula, its negation, and the negation of its "is defined" formula is true on the constructed model. Many other ternary logics in the literature can be reduced to ours. Partial functions …