Search results for "conditionals"
showing 10 items of 11 documents
Compound conditionals as random quantities and Boolean algebras
2022
Conditionals play a key role in different areas of logic and probabilistic reasoning, and they have been studied and formalised from different angles. In this paper we focus on the de Finetti's notion of conditional as a three-valued object, with betting-based semantics, and its related approach as random quantity as mainly developed by two of the authors. Compound conditionals have been studied in the literature, but not in full generality. In this paper we provide a natural procedure to explicitly attach conditional random quantities to arbitrary compound conditionals that also allows us to compute their previsions. By studying the properties of these random quantities, we show that, in f…
Iterated Conditionals and Characterization of P-Entailment
2021
In this paper we deepen, in the setting of coherence, some results obtained in recent papers on the notion of p-entailment of Adams and its relationship with conjoined and iterated conditionals. We recall that conjoined and iterated conditionals are suitably defined in the framework of conditional random quantities. Given a family \(\mathcal {F}\) of n conditional events \(\{E_{1}|H_{1},\ldots , E_{n}|H_{n}\}\) we denote by \(\mathcal {C}(\mathcal {F})=(E_{1}|H_{1})\wedge \cdots \wedge (E_{n}|H_{n})\) the conjunction of the conditional events in \(\mathcal F\). We introduce the iterated conditional \(\mathcal {C}(\mathcal {F}_{2})|\mathcal {C}(\mathcal {F}_{1})\), where \(\mathcal {F}_{1}\)…
Obligations and Conditionals
2015
The paper considers two kinds of medieval obligational disputations (positio, rei veritas) and the medieval genre of sophismata in relation to the kinds of inferences accepted in them. The main texts discussed are the anonymous Obligationes parisienses from the early 13th century and Richard Kilvington’s Sophismata from the early 14th century. Four different kinds of warranted transition from an antecedent to a consequent become apparent in the medieval discussions: (1) the strong logical validity of basic propositional logic, (2) analytic validity based on conceptual containment, (3) merely semantic impossibility of the antecedent being true without the consequent, and (4) intuitively true…
Algebraic aspects and coherence conditions for conjoined and disjoined conditionals
2019
We deepen the study of conjoined and disjoined conditional events in the setting of coherence. These objects, differently from other approaches, are defined in the framework of conditional random quantities. We show that some well known properties, valid in the case of unconditional events, still hold in our approach to logical operations among conditional events. In particular we prove a decomposition formula and a related additive property. Then, we introduce the set of conditional constituents generated by $n$ conditional events and we show that they satisfy the basic properties valid in the case of unconditional events. We obtain a generalized inclusion-exclusion formula and we prove a …
Canonical Extensions of Conditional Probabilities and Compound Conditionals
2022
In this paper we show that the probability of conjunctions and disjunctions of conditionals in a recently introduced framework of Boolean algebras of conditionals are in full agreement with the corresponding operations of conditionals as defined in the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and specified as suitable random quantities. We do this by first proving that the canonical extension of a full conditional probability on a finite algebra of events to the corresponding algebra of conditionals is compatible with taking subalgebras of events.
Iterated Conditionals, Trivalent Logics, and Conditional Random Quantities
2022
We consider some notions of iterated conditionals by checking the validity of some desirable basic logical and probabilistic properties, which are valid for simple conditionals. We consider de Finetti’s notion of conditional as a three-valued object and as a conditional random quantity in the betting framework. We recall the notions of conjunction and disjunction among conditionals in selected trivalent logics. Then, we analyze the two notions of iterated conditional introduced by Calabrese and de Finetti, respectively. We show that the compound probability theorem and other basic properties are not preserved by these objects, by also computing some probability propagation rules. Then, for …
Connexive Logic, Probabilistic Default Reasoning, and Compound Conditionals
2023
We present two approaches to investigate the validity of connexive principles and related formulas and properties within coherence-based probability logic. Connexive logic emerged from the intuition that conditionals of the form if not-A, then A, should not hold, since the conditional’s antecedent not-A contradicts its consequent A. Our approaches cover this intuition by observing that the only coherent probability assessment on the conditional event A | not-A is p(A | not-A) = 0. In the first approach we investigate connexive principles within coherence-based probabilistic default reasoning, by interpreting defaults and negated defaults in terms of suitable probabilistic constraints on con…
Logical Operations among Conditional Events: theoretical aspects and applications
2019
We generalize the notions of conjunction and disjunction of two conditional events to the case of $n$ conditional events. These notions are defined, in the setting of coherence, by means of suitable conditional random quantities with values in the interval $[0,1]$. We also define the notion of negation, by verifying De Morgan's Laws. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals and we show that some well known properties which are satisfied by conjunctions and disjunctions of unconditional events are also satisfied by conjunctions and disjunction of conditional events. We also examine in detail the coherence of the prevision a…
Interpreting Connexive Principles in Coherence-Based Probability Logic
2021
We present probabilistic approaches to check the validity of selected connexive principles within the setting of coherence. Connexive logics emerged from the intuition that conditionals of the form If \(\mathord {\thicksim }A\), then A, should not hold, since the conditional’s antecedent \(\mathord {\thicksim }A\) contradicts its consequent A. Our approach covers this intuition by observing that for an event A the only coherent probability assessment on the conditional event \(A|\bar{A}\) is \(p(A|\bar{A})=0\). Moreover, connexive logics aim to capture the intuition that conditionals should express some “connection” between the antecedent and the consequent or, in terms of inferences, valid…
Probabilistic inference and syllogisms
2014
Traditionally, syllogisms are arguments with two premises and one conclusion which are constructed by propositions of the form “All S are P ” and “At least one S is P ” and their respective negated versions. We will discuss probabilistic notions of the existential import and the basic sentences type. We will develop an intuitively plausible version of the syllogisms that is able to deal with uncertainty, exceptions and nonmonotonicity. We will develop a new semantics for categorical syllogisms that is based on subjective probability. Specifically, we propose de Finetti’s principle of coherence and its generalization to lower and upper conditional probabilities as the fundamental corner ston…