# Understanding hypothetical reasoning and material implication

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?

• If we use the "classical" (truth-functional) model for the propositional connectives, the key feature of the conditional ("if..., then...") is its interaction with modus ponens. Asserting the conditional : "(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)" and assuming the case that I'm ill, we are licensed to use mp (twice) to "detach" the consequent, concluding that I will not be paid. – Mauro ALLEGRANZA Sep 4 '18 at 8:01
• This mechanism is the basic of math deduction: we have an axiom (or already proved theorem) A and we want to prove a new theorem T. If we succeed in deducing the statement "if A, then T", we can use mp and we have a proof of T (in the context of the theory with axiom A). – Mauro ALLEGRANZA Sep 4 '18 at 8:03
• I'd say the statement "if I can't go to work, then I can't get money" is false, so the issue isn't logic. The correct statement would be (assuming you don't get sick days or whatever): "if I can't go to work, then I can't get money FROM WORK (unless they decide to give me money for some reason other than working); however, I could certainly get money from other sources". A specific example would be: if I get ill, I can't go to work. If I get ill, I can get apply for disability. If I get disability, I get money. – user935 Sep 4 '18 at 20:33
• Yes, practical inferences of everyday reasoning use background assumptions and presuppositions that are never explicitly stated, and can not be made fully explicit in principle. Sometimes a catchall phrase ceteris paribus (other things being equal) is used, but often not even that. The conditionals themselves are not only not material, they are not formally valid in any sense.They rely on context specific warrants, see argumentation theory. – Conifold Apr 3 '20 at 7:46

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:

1. If Alice spends lots of money on luxury goods she'll become poor.
2. If Alice wins the lottery she will spend lots of money on luxury goods. But not:
3. If Alice wins the lottery she'll become poor.

This is an example of hypothetical syllogism failing. Another, discussed by Ernest Adams, is

1. If President Brown's party loses the election he will resign after the election.
2. If President Brown dies before the election, his party will lose the election. But not:
3. 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.

• "If Alice spends lots of money on luxury goods [then] she'll become poor." This suggests a causal link between between spending and poverty. As such, it is not a material implication. We might convert it to a material implication as follows: "If Alice spends a lot on luxury goods, then she IS poor." In other words, it is not the case that both "Alice spends a lot on luxury goods" is true and "Alice is poor" is false. – Dan Christensen Sep 5 '18 at 16:40
• @DanChristensen I think you misunderstood Bumble's answer -- you might want to read the first paragraph again. – Eliran Sep 5 '18 at 16:42
• @EliranH Bumble wrote: "Real conditionals OFTEN, PERHAPS USUALLY, do not behave like material implications." (My emphasis) I gave an example of "real conditional" (example 1) that might be easily converted to a material implication. – Dan Christensen Sep 5 '18 at 16:57
• @DanChristensen perhaps it might be converted to a material conditional, but that does not mean that the original was one. – Eliran Sep 5 '18 at 19:48
• @EliranH The is no suggestion of causality or changes over time in the material implication. In this case, the material implication describes the state of things at an instant in time. It may even be a more honest portrayal of reality. – Dan Christensen Sep 5 '18 at 20:52

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.

• You refer to a single context of p implies q. What about the others? (~p v q), ~(p & ~q), (p-->(p & q), and so on. Why did you pick one out of many context which all are identical by truth table. Are you only addressing Mathematical Logic and not the others? – Logikal Sep 5 '18 at 22:44
• @Logikal Since they are all logically equivalent, I picked the one that I thought was easiest to use and understand. – Dan Christensen Sep 6 '18 at 4:30
• You do know that the truth table is equivalent but the contexts are different right? The propositions listed each express something different. The term equivalent does not mean the same thing a identical. Do you acknowledge that there are different logic systems and all logic is not mathematics? Initially only Philosophy used the term Material Implication as in a special context to differentiate the common use of conditionals. Math has no need to use the term Material Implication as the philosophers once did. So why use it in Mathematical Logic? The intention is different & it’s purpose. – Logikal Sep 6 '18 at 4:41
• @Logikal 1. Each of expressions that you list can be derived from any of the others. So, they are each interchangable. I don't see that context has anything to do with it. 2. I don't rule out the possibility of inventing other forms of logic, but I am not convinced that it is necessary. 3. AFAIK material implication is the only form of conditional used by the vast majority of mathematicians. – Dan Christensen Sep 6 '18 at 5:18
• @Logikal Re: "We do not disagree on meaningful propositions." So, we do not disagree on anything significant -- a tempest in a teacup! Thanks. – Dan Christensen Sep 7 '18 at 13:46

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.

Example

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.

1. At different times the truth values of antecedent and consequent may change. But usually we evaluated it at the moment we make the statement.
2. 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.
3. 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.