If synthetic a priori truths (e.g., '7+5=12') are necessary but not grounded in contradiction, the Principle of Contradiction cannot be the sole governing principle of necessary truths.
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(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.
synthetic a priori truths(as a category of knowledge Kant distinguished)
Statements that are both (1) true based on how the world actually is, not just from definitions, AND (2) knowable through pure reasoning without needing to observe anything. For example, 'a straight line is the shortest distance between two points' is something we can know is true just by thinking about it.