I am a little bit frustrated in how we use hypothetical reasoning in everyday life. Many times we make "if-then" statements. For example, if i get ill ,then i cant go to work and if i cant go to work , then i cant get money. But i have a problem in understanding the andecent . It says in the case i get ill but does this says anything else about the real world ? i mean it isnt sure that i wont get money because maybe a rich man give me tons of dollars . So when we do hypothetical reasoning we take as granted the world how we live now and modify some conditions ? Also the above example is a material implication or an example of hypothetical reasoning ?
Real conditionals often, perhaps usually, do not behave like material implications. Because of that, some common rules that apply to material implication do not always work with real conditionals. Such rules include hypothetical syllogism, contraposition and strengthening of the antecedent (monotonocity). Often when we express a conditional we have in mind some background circumstances that hold by default but might have exceptions. For example:
- If Alice spends lots of money on luxury goods she'll become poor.
- If Alice wins the lottery she will spend lots of money on luxury goods. But not:
- If Alice wins the lottery she'll become poor.
This is an example of hypothetical syllogism failing. Another, discussed by Ernest Adams, is
- If President Brown's party loses the election he will resign after the election.
- If President Brown dies before the election, his party will lose the election. But not:
- If President Brown dies before the election, he will resign after the election.
In each case, there is a shift in the assumed default circumstances between the two premises and this suffices to prevent the conclusion being valid. These are similar to your example. By default, it is reasonable to suppose that if you can't work you won't get money, because this is the normal way you get money. But there are obviously exceptions. As you say, someone might give you the money, or you might win a bet on the horses, or a rich uncle might leave you an inheritance. In practice, it is infeasible to list all the circumstances needed to turn a real conditional into a true set of necessary and sufficient conditions, so we don't bother and make do with default assumptions. We then take it as read that that these defaults might turn out false.
Ernest Adams' probability logic copes well with this kind of situation. It may be highly probable that C given B, and highly probable that B given A, but it does not always follow that it is highly probable that C given A.
In natural language, material implication works only for pairs of logical propositions that are both unambiguously either true or false at some instant in time. Like mathematics as a whole, material implication has nothing to do with causality or the passage of time.
P implies Q means only that it is not the case that both P is true and Q is false. This is often given as definition in textbooks, but it can be derived from other well-accepted principles of logic.
An argument is considered valid if the conclusion cannot be false when all of the premises are true.
A hypothetical syllogism is a valid argument. If both the premises are true, then the conclusion must be.
The premises, B if A, and C if B, logically entails the conclusion that C if A.
Of course, there is no guarantee what the conclusion will be should any of the premises be unjustified.
If you cannot justify the premises, then a valid argument will have an unsound conclusion.