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|Field II 145
Dialethism/Priest/Paradoxa/Field: (Priest 1998): Thesis: the sentence of the liar as well as its negation are both assertable (and also their conjunction). The rules of the logic are weakened (> stronger/weaker), so that not every assertion can be asserted by this.
Most attractive variant: builds on Kleene's trivalent logic.
Trivalent logic/Kleene/Priest/Field: Priest assumes here that the valid inferences are those that guarantee "correct assertion". But an assertion is only correct if it has one of the two highest truth values in the truth value table.
Curry paradox: is thus excluded, since the only conditional in this language is the material conditional.
Material conditional/Field: the material conditional is defined by ~ and v. It does not fully support the modus ponens in the logic of Kleene/Priest.
Liar/KleeneVsPriest: (and other "deviant" sentences): have truth-value gaps. But there are no agglomerations of truth values.
Deviating Sentence: E.g. Liar sentence, has no truth-value agglomerations but truth-value gaps.
Liar/PriestVsKleene: (and other deviating sentences): have, conversely, truth-value agglomerations and no gaps.
Problem/Kleene: here one cannot establish an equivalence between "p" and "p" is true! For to assert a truth-value gap in a sentence "A" would be to assert: "~ [true ("A") v true ("~A")]" and this should be equivalent to "~ (A v ~ A)". But one sentence of this form can never be legitimate in Kleene.
Truth-value gap/logical form/Field: to assert a truth-value gap in a sentence "A" would mean to assert: "~ [true ("A") v true ("~ A")]" and this should be equivalent to "~ (A v ~ A)".
Solution/Priest: if "A" is a deviating sentence, this is then a correct assertion in Priest. Also the assertion of the absence of a truth-value agglomeration in a sentence "A" would be the assertion "~ [(true ("A") u true ("A)"]" which should be equivalent to "~(a u ~A)". Kleene cannot claim this absence for deviant sentences, Priest can do this.
Beyond the Limits of Thought Oxford 2001
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980