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 can't go to work and if I can't go to work , then I can't get money. But I have a problem in understanding the antecedent. It says in the case I get ill but does this says anything else about the real world? I mean it isn't sure that I won't 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.
First thanks everyone for commenting and answering in this post. You all helped me come up with my answer. After a lot "headaches" about how I should interpret them in everyday usage I understood that trying to apply logic in natural language is very difficult and sometimes isn't even usefull.
If John is in his bedroom, Marie is in her bedroom.
Using truth tables we can evaluate the truth value of this statement. Some problems I recognize in this statement that are actually fallacies/misuse of the natural language.
- At different times the truth values of antecedent and consequent may change. But usually we evaluated it at the moment we make the statement.
- Someone could prove that this is a true statement based on his assumptions and use that reasoning to infer something. Suppose a friend of John who has observed that every time John goes to his bedroom, Marie goes too. He then make the assumption that this can't be change so knowing that John is in his bedroom suffices for him to conclude that Marie is in her bedroom. If someone asks him "how do you know that there is not case that antecedent is true and consequent is false" he would state the above assumptions.
- The above statement may be also interpreted it as a promise. Two friends bet if the above statement is true or false. The winner take some amount of money. In that case they just bet and they don't try to infer anything.
Actually many of us use both 2 and 3. It is just in our nature to make assumptions and trying to conclude something. But how one should interpret them? Well someone who would like to infer something he can make assumptions and conclude that the above statement is always true that is consequent will never be false if the antecedent is true. "It can't be the case". If someone doesn't care he can view it as a bet. At the end of the day is just a matter of the truth values that make the conditional (antecednt and consequent). Antecedent will be true or false and the same for consequent. Only one of the four possible combinations will happen. Trying to guess which one of the four will happen shouldn't bother us.
Unlike natural language in mathematics implications work very well. A statement like "If P then Q" would be true or false. In such system you can actually prove because the system you work with is very strict. You have axioms and definitions. You use all that "keys" (axioms, definitions, proven theorems) you have to a unlock (prove) a new door (theorem). The assumptions (axioms) are stated explicitly. So what you write or what you say or how you should interpret a statement is very clear.