The phrase "accounts of validity" is correct and usable in written English.
It can be used in contexts discussing the legitimacy or credibility of various claims, studies, or arguments.
Example: "The researchers provided several accounts of validity to support their findings in the study."
Alternatives: "reports of validity" or "narratives of validity".
Most, if not all, of these accounts of validity can be thought of as variant ways of giving formal precision to the idea that necessity is truth in every possible world or conceivable state of affairs.
Validity can also be defined for various modal predicate logics by combining the definition of LPC-validity given earlier (see above Validity in LPC) with the relevant accounts of validity for modal systems, but a modal logic based on LPC is, like LPC itself, an undecidable system.
For a number of axiomatic modal systems, however, no satisfactory account of validity has been devised.
The issues raised by the above examples no doubt differ significantly, but they all suggest a threefold rather than a twofold division of propositions and hence the possibility of a logic in which the variables may take any of three values (say 1, 1/2, and 0), with a consequent revision of the standard PC account of validity.
Most medieval logicians didn't, however, abandon the account of validity based on containment or something similar.
With validity then defined in the usual way as preservation of truth (simpliciter), the account of validity coincides with classical validity.
Beall (2000) argued that the account of validity used in multi-valued logics gives pluralists the resources to deal with the problem of mixed inferences.
In other words, substitutivity of terms as captured by (ST) is not a key element of Boethius' account of validity — neither terminologically nor conceptually.
However truth tables cannot be used to provide an account of validity in modal logics because there are no truth tables for expressions such as 'it is necessary that'it'is is obligatory that', and the like.
One could define validity over all worlds (of all interpretations); however, while this would yield a 'suitable conditional' in the going sense, the conditional would be even weaker than the (very) weak (but nonetheless suitable) conditional generated from the normal-worlds-only account of validity.
Adopting a designated-value account of validity, pluralists can simply take F1, …, Fn to be the relevant designated values and define an inference as valid just in case the conclusion is designated if each premise is designated (i.e., one of F1, …, Fn).
