5 research outputs found
Probability Semantics for Aristotelian Syllogisms
We present a coherence-based probability semantics for (categorical)
Aristotelian syllogisms. For framing the Aristotelian syllogisms as
probabilistic inferences, we interpret basic syllogistic sentence types A, E,
I, O by suitable precise and imprecise conditional probability assessments.
Then, we define validity of probabilistic inferences and probabilistic notions
of the existential import which is required, for the validity of the
syllogisms. Based on a generalization of de Finetti's fundamental theorem to
conditional probability, we investigate the coherent probability propagation
rules of argument forms of the syllogistic Figures I, II, and III,
respectively. These results allow to show, for all three Figures, that each
traditionally valid syllogism is also valid in our coherence-based probability
semantics. Moreover, we interpret the basic syllogistic sentence types by
suitable defaults and negated defaults. Thereby, we build a knowledge bridge
from our probability semantics of Aristotelian syllogisms to nonmonotonic
reasoning. Finally, we show how the proposed semantics can be used to analyze
syllogisms involving generalized quantifiers
Probabilistic inferences from conjoined to iterated conditionals
There is wide support in logic, philosophy, and psychology for the hypothesis that the probability of the indicative conditional of natural language, , is the conditional probability of given , . We identify a conditional which is such that with de Finetti's conditional event, . An objection to making this identification in the past was that it appeared unclear how to form compounds and iterations of conditional events. In this paper, we illustrate how to overcome this objection with a probabilistic analysis, based on coherence, of these compounds and iterations. We interpret the compounds and iterations as conditional random quantities which, given some logical dependencies, may reduce to conditional events.
We show how the inference to from and can be extended to compounds and iterations of both conditional events and biconditional events.
Moreover, we determine the respective uncertainty propagation rules.
Finally, we make some comments on extending our analysis to counterfactuals
Square of Opposition Under Coherence
Various semantics for studying the square of opposition have been proposed recently. So far, only (Gilio et al., 2016) studied a probabilistic version of the square where the sentences were interpreted by (negated) defaults. We extend this work by interpreting sentences by imprecise (set-valued) probability assessments on a sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square in terms of acceptability and show how to construct probabilistic versions of the square of opposition by forming suitable tripartitions. Finally, as an application, we present a new square involving generalized quantifiers