In mathematics when a "P implies Q" statement is true it means that every time P is true, Q is true also. What about everyday usage? For example consider the statement: "If it is raining, then I am inside the house." Assume that is true that it is raining and that I am inside the house. I think no one will argue that this holds in general. So this conditional is only true for the time we make the statement? But how outside of mathematics can someone know when to understand such if-then statements as temporary statements? Also when we speak about our habits we usually say "When I watch TV, I eat popcorn." Does the speaker mean that he does this every single time?

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    The if-then of ordinary language is called indicative conditional. Its use can not be distilled into any compact set of rules, as much of it is context specific and intuitive, people are trained to use it instead. One learns to understand if-then and other sentences when they master the language, usually as a child. People get quite adept at figuring out implicit stipulations of conversational statements and interpreting what others mean depending on context. Life is not math, and formalizing natural language is not practically possible. – Conifold Apr 4 at 1:07
  • @Conifold , there is a project of Relevance logic that does try to make a better go at understanding conditionals! We intuitively want to say in implication that the information that P would suffice for Q; Relevance logic is about constructing a formal semantics where this seemingly mathematical idea can ground a working logic of inference. (It is non-classical, of course!) – Sofie Selnes Apr 4 at 4:21
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    @SofieSelnes There is also strict conditional of Lewis, suppositional and other non-truth-functional versions. The problem is that they all attempt to fit a box to a cloud. Each captures some aspect of the indicative conditional, and mismatches other aspects, because the goal is to find neat rules of logical form, whereas ordinary use is patchy and decidedly content specific. Toulmin's model of domain specific warrants and rebuttals is closer, but even it does not capture everything. – Conifold Apr 4 at 4:46
  • I suppose I just think it’s worth pushing back against the impossibility of formalising natural language. If we take that comment at face value then we’re sidelining the good work being done in linguistics and semantics which is helping clarify the concepts at work when we talk to one another in conditional terms. – Sofie Selnes Apr 4 at 5:01
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    @SofieSelnes I have the opposite concern about raising unrealistic expectations that are then used as a standard to discredit the entire enterprise, when models inevitably fall short. We see this dynamic in many areas, with epidemiological models most recently. I think, in the end, it is better to disclose the limitations upfront. The impossibility of formalising natural languages is fairly obvious, math is a wrong standard. But that does not mean that the work on its formalizable aspects, and modeling its non-formal aspects as well, is not insightful and worthwhile, as you rightly point out. – Conifold Apr 4 at 5:45

In the world of formal, mathematical logic, statements are treated as either true or false. In the world of natural language, all kinds of statements are uncertain or ambiguous. The same thing applies to conditional statements, "if A then B", which may likewise be true, false, or uncertain, ambiguous, and equivocal. The attempts to extend formal mathematical logic to these kinds of statements have not been generally successful.

The three valued logic of Lukasiewicz is possibly the most direct attempt to deal with uncertainty. This has not been generally accepted in part because important logical laws fail, although this failure can be traced to the behavior of conditional statements with the third logical value. This has much in common with the fuzzy logic of Zadeh, and again, the case of doubtful conditionals causes difficulties.

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  • In 1965, Lofti Zadeh proposed fuzzy logic which has been embraced in engineering circles as alternative to binary logic by providing an infinitely graded approach to truth. – J D Apr 8 at 15:17
  • @Confutus Thanks for the answer. Is it pleonasm to say "If P is true, then Q is true" instead of "Ιf P, then Q"? – ado sar Apr 8 at 21:19
  • " If it is raining, then I am inside the house", as it is used in ordinary language, implies a quantification over time :

for all time t , if it is raining at time t, then I am inside the house at time t.

In other words and equivalently :

there is no time t such that (1) it is raining at t, and (2) I am not inside the house at time t.

  • Suppose I live in a country where it never rains, there is no time t such that it is raining at t. So, a fortiori, there is no time t such that (1) it is raining at t, and (2) I am not inside the house at time t. So, the sentence is true.

  • Suppose I live in a country when it rains 364 days out of 365. On the non-raining day, I am outside. The sentence is still true : there is no time t such that (1) it is raining at t, and (2) I am not inside the house at time t.

  • So, the natural language understanding of " if... then " is not always opposed to the mathematical one.

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