Suppose the following statement. "If I kick the ball then the ball will hit the wall." Can this sentence have a truth value? I mean the time that I kick the ball, it hasn't reached the wall so the consequent is false, therefore the conditional false. What if I state it differently, that is: "If I kick the ball then the ball will hit the wall after 5 seconds." Can this statement now be true? It bothers me because in logic or mathematics time is irrelevant for the conditionals but in everyday life such statements make sense.
As Logikal suggests, there are several implicit premises which would make the claim more precise.
- I kicked the ball
- The kick was in the direction of the wall
- The kick has enough force to reach the wall
- There is nothing obstructing the path between point A and B which would prevent the ball hitting the wall
:. The ball will hit the wall
If and only if any one of the premises is false, then conclusion is false. And that's a fact.
Both of your statements can be true. However, in propositional logic the question is whether the truth of these sentences is solely dependent on the truth of its subsets. And that is not the case here.
- "If I kick the ball, then the ball will hit the wall" can be true or false independently of whether "I kick the ball" is true and whether "the ball will hit the wall" is true.
- "If I kick the ball then the ball will hit the wall after 5 seconds" can be true or false independently of whether its subsets are true or false.
Yet, a sentence like "The ball will hit the wall or the ball will not hit the ball" is logically true. No matter whether its subsets are true or false, it will always be true.
Strictly speaking, it is just false.
Is it true that if I kick the ball, then the ball will hit the wall?
Is it true that if I kick the ball (now and in the proper way), then the ball will (necessarily) hit the wall?
We don't assess such conditionals in the same way as we do a conditional such as for example If x = 2, then x > 0.
We insert context assumptions, and a probable modality: (Given the specific situation) if I kick the ball (now and in the proper way), then the ball will (with good probability) hit the wall.
Even then, the conditional retains a predictive value. Broadly, it only really suggests that there is a good physical possibility of kicking the ball so that it hits the wall, rather than asserting that if I actually kicked the ball, then it would definitely hit the wall.
This is still deductive logic, but our assumptions when we make such a statement would probably be too difficult to all identify exactly.
So, in the end, it seems to be a deduction entirely based on intuition, somewhat like you would say there is a tree just because you see a tree.
The deduction is that, based on empirical considerations, it is reasonable to believe the conclusion is true, rather than it is actually true. And, we don't bother to articulate the caveat because, broadly, nobody needs to be explained.
But what we mean and what we understand is not the same as what is actually said.
It is easy to pretend we are sure the ball will hit the wall as long as we don't do it, but no one would reasonably offer a billion dollars as security just in the supposedly very unlikely case that they missed.
Words are cheap.
You could rephrase it as :
if there is a time t such that I kick the ball at t, then there is a time t' such that t'> t and the ball hit the wall at t'?
This sentence has definite truth conditions, manely it is false if there is a t-time as defined but no t' time, and true otherwise.
But it would be difficult to test its truth value : since no condition is imposed on the lenght of the time interval t'-t , one could always say : there is a time t' in the future that will satisfy the definition of t' .
What I want to say is that it would not be possible to falsify the claim ( if no condition is imposed as to the length of the time interval t'-t).