Timeline for Why are true and false the only truth values used in mathematics?
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| Apr 9, 2018 at 12:04 | comment | added | Discrete lizard | Yes, of course you can do that. This definition, however, contradicts modern stochastics. In modern stochastics, probability measures P are functions from a sigma algebra S of measurable events to the unit interval. There is no function over a 'propositional language'. That there apparently is another definition of 'probability' (there are many such!) within logic doesn't imply that stochastics, a field within mathematics, needs to be consistent with this interpretation! I still don't understand why you seem to be claiming that your own thoughts need to be accepted from within stochastics! | |
| Apr 9, 2018 at 11:46 | comment | added | Jo Wehler | plato.stanford.edu/entries/logic-probability Probabilistic semantics thus replaces the valuations v : L→{0,1} of classical propositional logic with probability functions P : L→ ℝ, which take values in the real unit interval [0,1]. The classical truth values of true (1) and false (0) can thus be regarded as the endpoints of the unit interval [0,1], and likewise, valuations v : L→{0,1} can be regarded as degenerate probability functions P : L→ [0,1]. In this sense, classical logic is a special case of probability logic, .... | |
| Apr 9, 2018 at 11:12 | comment | added | Discrete lizard | 1) This contradicts the interpretation(s) in mainstream stochastics. 2) The concept of a (probability) measure is a very different mathematical object than a logical variable or operations on such variables. 3) The mainstream generalisation you want is known as 'fuzzy logic', see here for the differences between this and probability. | |
| Apr 9, 2018 at 10:52 | comment | added | Jo Wehler | What's wrong in your opinion with considering probability a generalization(!) of the two discrete truth values? | |
| Apr 9, 2018 at 10:09 | comment | added | Discrete lizard | "Probability does not only capture our uncertainty of the event". Nevertheless, there are only 2 truth values in mainstream stochastics. I don't think anyone's opinion on what stochastics should be matters. Therefore, I think this answer is misleading at best. | |
| Apr 9, 2018 at 10:00 | comment | added | Jo Wehler | Thanks for pointing out the wrong axiom. - I consider probability a generalization of the two discrete truth values. Probability does not only capture our uncertainty of the event. In quantum mechanics probability often captures a property inherent to the event, independent from our knowledge. | |
| Apr 9, 2018 at 9:57 | history | edited | Jo Wehler | CC BY-SA 3.0 |
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| Apr 9, 2018 at 9:49 | comment | added | Discrete lizard | I'm not sure if stochastics is that relevant here. Even in stochastics, each event either occurs or doesn't. There are still only 2 truth values. The only thing that is on an interval is our uncertainty of the event. Also, your axiom is wrong. Not only do the events need to be disjoint, they also need to independent. | |
| Apr 9, 2018 at 6:52 | history | answered | Jo Wehler | CC BY-SA 3.0 |