‘Perhaps it will be said, that we may not merely conceive the _meaning_ of a proposition, but we may conceive it to be a _true_ proposition, and that it is our being able to conceive it to be true that gives us ground to think it possible. In answer to this, I beg you would attend very carefully to the meaning of these words—“Conceiving a proposition to be true.” I can put no other meaning upon them but judging it to be true, that is, giving some degree of assent to it. Judgment or assent admits of various degrees, from the slightest suspicion to the most deliberate conviction; from the most modest and diffident assent to the most pertinacious and dogmatical. I conceive there maybe inhabitants in the moon; that is, my judgment leans a little that way. I would not use that expression unless the probability, however small, seemed to be on that side of the question. ‘If this be the meaning of conceiving a proposition to be true, then the meaning of the axiom will be, that a proposition which appears to us to have any degree of probability, however small, must at least be possible. But the axiom, taken in this sense, is surely false. It would be superfluous to give instances of this to a mathematician. ‘If it should be said that conceiving a proposition to be true means, neither barely to understand the meaning of the proposition, nor the giving any degree of assent to it, I would be glad to know what it really does mean. For I am at a loss to know what power of the understanding we mean by the conceiving a proposition to be true, if it is neither simple apprehension, by which we barely understand the meaning of a proposition, nor judgment, by which we assent to the proposition or dissent from it. I know of no power of the understanding intermediate between these two. And if there is none, I think the axiom must be false. ‘There are many propositions which, by the faculties God has given us, we perceive to be not only true, but necessarily true; and the contradictions of these must be impossible. So that our knowledge of what is impossible keeps pace with our knowledge of necessary truth.