Understanding Conditional Proposition - Logical Operators

 I was having a problem with understandin 'if-then' statements' truth tables and validities, so what better way to understand anything by teaching.

Always imagine if-then statements as those flyer ads you see on local trains.

Let's imagine you see an advertisement as - 'If you eat our magical pill, then you will get 6 pack abs'

This compound statement can be broken down into its two statements, 'Eat our magical pill' & 'you will get 6 pack abs', if we denote each simple statement as p and q respectively, then the whole compound statement can be represented as 'if p, then q'.

Now to test the validity of the compound statement, let us consider the truth table of conditional statements.

The truth table of conditional statement states: 

If p is true and q is true, then 'if p, then q' is true.

If p is true and q is false, then 'if p, then q' is false.

If p is false, then regardless of the truth functionality of q, (doesn't matter if q is True or False) 'if p, then q' will be not false, i.e it will be true always.

Now let us test to see how this truth table is working by going back to the compound statement and seeing how each scenario can play out. To do this, let us assume that before taking the pill, you went to your lawyer friend and ask him what if the magic pill does not work? Can I file a lawsuit against the fraud advertisement then? The layer friend will tell you that you can file the case and win the case, only if you can prove that the advertisement's claim is false, i.e. you have to go to the court and convince the CEO of the company that their statement 'if you eat our magical pill then you will get 6 pack abs' is false.

Scenario 1) You ate the magical pill and you actually 6 pack abs, thus making both p and q true, i.e you ate the magical pill (p) and you got 6 pack abs (q), so the whole compound statement gets true. So now you have no reason to complain and will not go to the CEO to challenge the advertisement as their statement is proved true by you yourself.

Scenario 2) You did eat the magical pill (p) but you did not get 6 pack abs (q), thus making p true, but q false. As you can see, you did eat the pill (p is true) but did not get the 6 pack abs (q is false). So now you can challenge the CEO that their advertisement is false, thus proving that the compound statement is false.

Scenario 3) You did not eat the magical pill (p) but....wait a minute, you did not eat the magical pill! Now it doesn't even matter what the outcome is, you now simply can't challenge the CEO that their claim is false, because the CEO can simply say that 'You didn't even try our product, how can you question the validity of our claim then?'. The CEO can say that since you cannot prove our claim is false, then we will declare it true, and you can't do anything about it because you haven't even tried the pill. 

Thus if p is false in a conditional statement, it doesn't matter what q is, the statement can always be claimed 'not false' (as you cannot prove that it is false), thus making it true (because in conditional logic, things are either true or false, nothing in between, so as here it is not false, so it can be called true.)

Now let us take another more abstract example to solidify the concept.

Compound statement ---> 'If 1+1 = 3, then monkeys can fly.'

Let us break down '1+1=3' as p, and 'monkeys can fly' as q. Now let us try to answer the question that is the compound statement is true or false, i.e. you have to find out that if the compound statement is true or not. 

Let us allot the truth function to individual sentences of p and q. We know that '1+1=3' can not be true, i.e it is false. Thus we established p is false.

Now, 'monkeys can fly', we also know monkeys can't fly, thus it is a false statement, therefore q is also proved false.

Now by truth table of implications, we know F + F = T. Thus the claim, 'If 1+1 = 3, then monkeys can fly' is true. 

(If this is confusing, imagine someone challenging you to prove the compound statement false, how will you do it? You might reason him that 1+1 is not 3 then how can the statement be true? He can reply back that that is exactly his point, in his statement he is saying 'if 1+1=3', he is not talking of any other scenario, then it doesn't even matter what alternate scenario you bring to him. 

Let us take one last example. Let's assume you gave a bet to your friend that, 'If you let me slap your face twice (p), then I will pay you 1000rs. (q)' . Let's assume your friend agrees to the bet on the condition that if you do not stay true to your words then he will slap you twice instead. Now let's see how you can use this 'if-then' statement to your advantage. Remember, he can slap you back only if he proves your statement false. So you know that your statement always has to be true now, for which you need to prove p as false, because if p is false, then the conditional statement always is true. So to prove p as false, you can either choose between not slapping him even once OR slap him just once, both the cases making 'me slapping your face twice (p)' as false (because you did not slap his face twice, you slapped him only once). In this way, you can slap your friend, not give him the 1000rs while also not getting slapped yourself. 

Understood? Good. Ask any questions below if you have any.