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In a multiple conclusion argument from A to B, C, any warrant we have for A does not necessarily transmit to B or C: the only conclusion we are warranted to draw is the disjunction B or C, so it seems for an analysis of consequence in terms of warrant we need to understand some logical vocabulary (in this case, disjunction) in order to understand the consequence relation.
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The unenforceable determinant in the case of disjunction is ⟨F, F, F⟩, dictating that when the disjuncts have the values listed in the first two positions, the disjunction has the value listed in the final position.
For example, in the case of disjunction, let A and B each be a defining condition of A∨B as expressed by the primitive rules A → A∨B and B → A∨B.
It is embedded in the very foundations of his system and all other relations (with the exception of different cases of disjunction, Bolzano 1837, § 160), deducibility included, are special cases of it.
In this case, the conjunction, disjunction, and negation of Dunn/Belnap's 4-valued system are componentwise definable by conjunction, disjunction, or negation, respectively, of classical logic, i.e. this 4-valued system is a product of two copies of classical two-valued logic.
In this case, the multiplicative disjunction and conjunction can become non-commutative.
Thus, to defend a conjunction, a player must be able to defend any of the conjuncts, while in the case of a disjunction, it is sufficient to be able to defend one of the disjuncts.
According to Łukasiewicz' trivalent logic: if F x q and F x)~q are two future contingents, i.e., if they are both undetermined, then the same will be the case for the disjunction of the two statements, F x q ∨ F x)~q.
Priest's qualification to the LCC is that it applies only to atomic sentences and their negations: otherwise we would have to admit the case where a disjunction \(p \vee q\) held right up to a limit in virtue of \(p\) holding at the rational points and \ q\) holding at the irrational points: this would be capricious behaviour in which we can make no sense of the past determining the future.
In the limit, this would approximate a fossil record so poor that it could not be used to place constraints on the ages of nodes, in which case almost any disjunction between continents could be interpreted as having occurred within the window of Gondwanan events.
We may be able to specify the relevant term only in principle in cases where the disjunction is infinitely long.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com