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Suppose I were to say "2+2=4 is probably true". Would that be incorrect, since it is necessarily true? I believe "probably true" means "there is a greater than 50% chance it is true". By this definition, statements that are true with 100% probability are automatically probably true. So, my question is, are necessarily true statements also probably true?

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  • Something might be 'probably true' from an empirical approach, and 'necessarily true' from a mathematical proof. Until someone finds a flaw in either: a deprecation of an observation procedure, or an advance in mathematic theory which reveals an exception. Commented Nov 23, 2022 at 20:38
  • How sure are you about the definition? In common usage I would think 'probably' means 'above some likelihood threshold, but not certain', that is to say, probably and certainly are mutually exclusive. I thought this was true for inductive logic as well, but happy to be proven wrong Commented Nov 24, 2022 at 5:48
  • The usual approach in Modal Logic is that necessity implies possibility. Mapping them to probability means that True is 100% true while "probably true" means some level between 50 and 100... Commented Nov 24, 2022 at 7:39
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    Arguably, this is simply a question of the pragmatics of language. When we say "probably true" we might mean "merely probable but not certain", or we might mean "at least more probable than not". Such things have to be resolved by reference to the context in which the statement is uttered.
    – Bumble
    Commented Nov 24, 2022 at 14:27

2 Answers 2

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If something is necessarily true, is it probably true?

I would say yes. Start with the question’s contrapositive: If P is not probably true, then P is not necessarily true. This if-then statement appears to be true (to this writer, at least). Because the contrapositive is the equivalent of the original, the two have the same truth value.

So my answer to the question is yes.

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Much of philosophy starts from a "logic" reference, but the assumptions behind this can be misleading. Our world is CONTINGENT, there is nothing necessary about it. What rules or principles apply to aspects of our world are established thru investigation, not by reasoning.

2+2 equaling 4 is a prerequisite of arithmetic, but arithmetic does not necessarily apply to all aspects of our world. Examples -- arithmetic works very well most of the time with wooden blocks, at least so long as the timescale is less than a decade. It works far less well with rabbits, particularly with timescales of a decade. Rabbits do not live for a decade, and they multiply like -- rabbits. It also works far less well with things like raindrops -- which often combine to form bigger raindrops.

Arithmetic also works very poorly with probabilities, which anyone who takes a probabilities and statistics class will discover.

Arithmetic would also work very poorly with wooden blocks if one were placing them in a burning fireplace.

One can characterize the circumstances where arithmetic works well with our world, and others where it does not. Stability and conservation of the units one is applying it to are necessary (no dead rabbits, born baby rabbits, burned blocks, or merged drops). Linearity is also needed (statistics is not linear). Much of our universe is linear. And large portions of it are mostly stable for reasonable periods of time like minutes and hours.

Does this make arithmetic "probably true"? Your question presumed that our universe and arithmetic are in the same logic space, and they are not. Arithmetic is a math system, which is necessarily "true" if one specifies postulates, and a logic method of implementing them. But no math or logic system necessarily applies to our world.

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