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?
2The 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.– ConifoldApr 4, 2020 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 SelnesApr 4, 2020 at 4:21
1@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.– ConifoldApr 4, 2020 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 SelnesApr 4, 2020 at 5:01
3@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.– ConifoldApr 4, 2020 at 5:45
I watched some YT video on Modal Logic. And in it, if I remember correctly Lewis said when we mean p→q we actually mean □(p→q), where box is the necessity operator.
1This site doesn't have MathJax, so I replaced the MathJax symbols with their text equivalents. Dec 25, 2022 at 7:09
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.
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 DApr 8, 2020 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 sarApr 8, 2020 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.
In natural language there is ofcourse need of necessary provision that the speaker is just lieing.
Putting that aside provision must also be made for Unreliable Narrator. May be the speaker is thinking about only particular data points and not have in mind other data that is counter to his assertion.
Third possibility is, he is emphasizing. He is not saying that there is no counter data, he has that in mind, he is just not giving it importance. May be he think that counter data is too little to be of any significance.
Interpret "When it rains I am at home" as "I think that enough time during rainings I am at home to have it as a general rule" or "You are very much likely to find me at home during raining".
In short, its understood by both speaker and listener that what is meant is a general rule, not that there are no exceptions.
If speaker meant there are no exceptions he would have to say "I am always at home when it rains" or more accurately "Never it happen that I am outside home when it rains".
See how lawyers question the witnesses. They explicitly ask for exception: "Doctor, do no patient ever in this disease was able to walk?", "There sure must be days when rain caught you on your way home".
Existence of exceptions are never questioned in daily speak, except when they are specifically denied by the speaker.
There is an important difference between formal systems of logic and everyday speech. In formal systems, we define statements as having specific and unambiguous meanings, which they always hold. In everyday speech we use words that are open to interpretation, we use them loosely and inconsistently, and what a speaker might intend to mean (assuming they know what they mean) is not necessarily what is understood by a listener.
Given the above, logic in everyday speech is necessarily fuzzy. Take the example you gave, which I might word as 'if it is raining then I am inside the house'. Clearly that is not always true. You might be in your car, for example, or away on holiday. Or you might have some vitally important reason for staying outside and getting wet. Or heavy rain might have started while you were at the far end of the garden, so you decided to shelter under a tree instead. And the word 'raining' is ambiguous- it might mean a torrential downpour or such a light precipitation that it would not drive you indoors or keep you there.
Clearly then the statement 'if it is raining then I am in the house' is a vague generalisation, and short-hand for something more like 'If I am at home and if it is raining heavily, then I will tend to stay in the house unless I have a special reason to go outside'. In everyday speech we don't usually take a lot of trouble to express ourselves precisely, and we expect, consciously or otherwise, that the people with whom we are conversing will not take our words literally but make a common sense interpretation of them.
Clearly also there are degrees of vagueness and generalisation, and there can be cases of everyday speech in which if-then statements are used with the force of formal logic, or nearly so. For example, you might say 'if you give me half of your money then you will still have half left- such a sentence has very little scope for misinterpretation.
You might try to classify all the degrees of freedom inherent in the fuzziness of everyday speech, but there could be no absolutely correct way to achieve it and no hope of achieving it comprehensively, since the scope for words to mean different things to different people is virtually boundless. Indeed, even when two people profess to mean the same thing by a particular word, in many cases you will have no way of being certain that they do, especially where the word describes an abstract idea.