Why false implies true

This is an often overlooked tool in writing good proofs. If it sounds silly when read aloud in this way the chances are high that something is wrong.

For example, the base case in the inductive proof above could be read as:. Why would that need to be implied by some ugly formula? Ah, it would not need to be. Perhaps the implication is not meaningful. For many problems the mistaken proof is nevertheless a good way to begin thinking about a correct proof. This is because by assuming what is to be proved one has something to begin with. To prove the result it is often a matter of only reversing the scratch work.

Notice also that we have included many more words and punctuation to the proof which makes it both readable and also helps us to understand the logic of the proof more easily. As the left hand side equals the right hand side, the base case is settled. I am having a hard time understanding why two false statements in a conditional makes it true.

I tried to use different statements to create a truth table but I get stuck on the same concept. I tried a sentence like "If a polygon is a square, then the sides are equal. Being true or false does not even apply. Could you explain the reason just in the logic way? Because when we say "work the same as subsets", we must prove it does work the same When I trust the set theory, I must trust logic first.

So I want to understand the reason just on the way logic goes. But the conditional statement is not meant to suggest that even though many examples given in texts look that way. It is a mathematical definition, not everyday reasoning Second, this conditional statement is in a sense just something mathematicians define for their own purposes, not something that necessarily agrees with the natural-language use of the phrase.

And what we need in logic at least in traditional two-valued logic, as opposed to a logic that might include an "undetermined" value is for every statement to be either true or false. Here I gave a familiar answer: Traditionally, mathematicians subscribe to a sort of "innocent until proven guilty" rule: we can't say something is false just because there is no evidence ; instead, when there is no evidence of truth or falsity, we say it is true.

This is what lies behind the related facts about sets: we say that the null set is a subset of any set because there is no evidence that it is not--there is no element in the null set that is NOT an element of the other set!

This is what works in describing logical arguments Ultimately, I realized, the reason for the choice comes from our application, not from the real world. One application of symbolic logic is to validate arguments, and here, if the truth value of a conditional statement were not defined as it is, then a valid argument would fail the test: Let's consider an argument like this: I have a cold. If I have a cold, then my nose is running. Now the truth table accurately reflects the validity of the argument.

And that, I think, is why we make this definition: it works. Why does it work? Because we want to say an argument is valid when the conclusion follows from the premise: if A is really true, then B had better be true. There's the meaning behind that " innocent until proven guilty " idea. An example In , I answered another question, which gave me a chance to fill out a couple areas: Logic Statement False Implies True I am well familiar with the linguistic arguments which clarifies this somehow confusing concept.

By layman definition, logic is something that allows "naturally" the consequence to flow from a premise. The dilemma of this concept, though I use it myself to prove some propositions like the empty set is a subset of every set, is that it kills the "natural" connectedness inherent in logic.

Unfortunately, the very definition of logic itself is so intuitive and vague in the same way the set or sanity is defined, otherwise undefined! Even though I trained myself to live with this concept and I use it in my formal proofs, I try to avoid using it as much as possible. It is like employing proof by contradiction. I would rather prove directly. He responded with a comment that overstated the conclusion: I found it rather interesting that there is indeed some philosophical underlying mode of thinking built into the definition of "P implies Q," that is, if there is no evidence that something is false, then we must assume that it is true.

Viewed 37k times. Rodrigo de Azevedo YoussefDir YoussefDir 1 1 gold badge 2 2 silver badges 6 6 bronze badges. Maybe what bothers me is that we can link any two unrelated issues by an implication if the sufficient condition is false. Do we see this in real life?

Add a comment. Active Oldest Votes. Marc Paul Marc Paul 2, 1 1 gold badge 12 12 silver badges 19 19 bronze badges. And I've seen a lot Let P be the statement If the policeman sees you speeding, then you will have to pay a fine. GEdgar GEdgar If the policeman see's you speeding false , then you will have to pay a fine false.

This is still a true statement. It is also true that if the police see's you speeding false , then you will have to pay a fine true , is also a true statement.

The "q" part could be either true or false, hence both forms are true. Is this valid thinking? Suhail Suhail 2, 10 10 silver badges 17 17 bronze badges. Pin J. Pin