Logical deductions are infallibly certain as long as no assumptions are made and the starting premise is verifiably true. The only way they can be false is if they are true, since even logic being false would mean that some things necessarily follow from others--for example, it would follow from logic being false that logic would not provide absolute certainty. If it does not follow from a verifiable idea that another idea is true, then anyone who believes otherwise has accepted something based on an assumption, not on purely rational deduction. "Inductive" reasoning, which extrapolates unverifiable patterns from individual events or units, does not establish any grander truth than the premise it starts with even if someone uses it "correctly."
Many people still openly assume that inductive reasoning is philosophically valid, or at least they act exactly like they would if they held this belief, which can only be held by an intentional or unintentional assumption. The mere mention of the logical fact that one cannot know if patterns in our sensory experiences--experiences that can seem very stable, at that--will continue is enough to confuse or even frustrate them. Of course, when they appeal to the supposed knowability of a pattern's continued occurrence in the future, they must appeal to ignorance, emotion, authority, or popularity. They have no way whatsoever to prove that something like gravitation or electricity will persist. Furthermore, if they do persist, these blind advocates of inductive reasoning have no way to prove that they will continue without some major change.
There is no way to prove exactly how probable it is that a car will turn on when the keys are inserted and twisted, how probable it is that the sun will rise tomorrow, or how probable it is that Jesus rose from the dead in terms of mathematical percentages. There is not even a way to know every variable and relevant piece of evidence! In the case of the former two examples, the sharp distinction between correlation and causation means no amount of repetition proves the same events will occur after that which has always preceded them, and in the case of the latter, there is no way to even know exactly how many unbiased historical texts support the resurrection of Jesus. How many honest or dishonest documentations of his life may have been lost?
To think of probabilities of this sort in mathematical terms is asinine. Every probabilistic claim is either true or false, and only whether the claim is true or false is up in the air without a way to even know if 20%, 76%, or some other percentage of the available evidence slants in favor of one possibility or the other. What can be known is that one of the two exclusive possibilities regarding whether a given claim/idea is true or false must be correct, that certain things follow or do not follow from either possibility, that there is or is not accessible evidence for one possibility or the other, and that the inherent truth of logical facts is unaffected by unknown probabilities. None of this makes it 90% probable that the sun will rise tomorrow (or makes any other specific percentage valid).
Nonetheless, we are left with genuine evidence that the sun will rise again in the form of memories of the pattern of sunset and later sunrise, even if memory and sensory perceptions fall short of proving anything other than recalled experiences. One can still reflect on memories of the sun rising and setting; one can still analyze those memories and the concepts of things like the sun's rising and setting. Genuine probabilistic patterns are supported by evidence. There is nevertheless also evidence that points to key disruptions of certain probabilistic patterns, such as the position hat Jesus resurrected. For multiple reasons, including the absolute certainty and therefore non-probabilistic nature of sound logical deductions and the possibility or probability of abnormal events, probabilism is not the height of epistemology.
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