If necessary truth is unavailable to empirical science yet empirical science yields genuine demonstrative knowledge, necessity cannot be a universal condition on demonstrative principles.
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knowledge(Distinguished from mere true belief, which may be the product of indoctrination and need not exercise deliberative capacities.)
Justified true belief — true belief that has been arrived at through the exercise of deliberative capacities, including comparison of and deliberation among alternatives.
necessary truth(Mill's empiricist reinterpretation of modal concepts)
A proposition whose denial seems inconceivable, explained by Mill not as a metaphysical fact but as a result of psychological association making the proposition deeply ingrained.