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Implication and information
Published in Alfredo Pereira, William Alfred Pickering, Ricardo Ribeiro Gudwin, Systems, Self-Organization and Information, 2018
Marcos Antonio Alves, Itala M. Loffredo D’Ottaviano
The main results obtained from the elements presented in this work are: Material implication and probabilistic implication are not equivalent and may have different results in the same situation;Material implication, when treated in terms of Sℙ, does not always have the same results as when treated in terms of SV. There are some results that are specific to Sℙ, given their inability to be stated in terms of SV; and that there are some results shared by the two semantic perspectives. Furthermore, some results are characteristic of one perspective or another, due to the nature of the value (probability or truth) attributed to the formulae. When we deal only with extreme values (0 or 1, corresponding to truth values F and T), Sℙ and SV share relations relative to the value of a formula and the value of its constituents.In SV, there is the possibility of the absence of the relation of content between the elements of a sentence, even when it comes to valid formulae, that is, one whose value is always true. We show that this characteristic also remains in Sℙ, paying particular attention to probabilistic material implication.We show that in a large number of cases there is no relation between the reasonableness of an implication and the connection of content between its constituents. This same result can be proved for conjunction, negation, and biimplication. The material implication, from the probabilistic point of view, is also subject to the paradoxes of material implication, when evaluated in SV.We show that probabilistic material implication does not capture the intuitive notion of implication when it is understood as a classical causality relation. This notion presupposes some kind of connection between the cause (antecedent) and the effect (consequent). However, it can be said that, just as with veritative-functional material implication, the implication does not express, nor does it intend to, the notion of causality.We show that the material implication does not capture the notion of information as suggested in the MTC. Certain sentences have a very different informational value than they should present. One of the main reasons for this is the lack of a dependence relation between antecedent and consequent in material implication. As a result, CPL itself does not capture such a notion of information. A logical system with a probabilistic implication can perhaps do so more appropriately.
Infectious and transparent emotivism
Published in Journal of Applied Non-Classical Logics, 2022
Notice that both transparent and non-transparent interpretations of SL-conditionals imply the paradoxes of material implication. That is, and ). On the other hand, WK-conditionals do not imply them. That is, and ). A WK-countermodel for the former is a valuation where and ; for the latter, it is a valuation where and .
From KLM-style conditionals to defeasible modalities, and back
Published in Journal of Applied Non-Classical Logics, 2018
Katarina Britz, Ivan Varzinczak
The exigence of avoiding the paradoxes of material implication was one of the main motivations for the introduction of conditional logics by Stalnaker and Lewis. Our integration of defeasible implication with our language of defeasible modalities is rather a natural progression of work in the non-monotonic reasoning tradition.