Can I deduce anything from statement?

Question

Statement: If you study, then you can get a diploma. If you don't study then you can still get a diploma.

$S \to D$
0 1 0
0 1 1
1 0 0
1 1 1
$\text{not}(S)\to D$
1 0 0
1 1 1
0 1 0
0 1 1

When I use truth table I get conflict(not all 1's), but I'm not sure that this means I can deduce anything from that statement.

Please correct me if I'm wrong.

Answer 1

You can deduce (in classical propositional logic) that 'you can get a diploma' is true.

Proof. Keeping in line with your notation we have that $S \implies D$ and $\neg S \implies D$. Now the Constructive Dilemma is the following rule $$ (( p \implies q ) \wedge (r \implies s) \wedge (p \vee r)) \vdash q \vee s $$ Setting $p := S, r := \neg S$ and $q := s := D$ this yields $$ (( S \implies D ) \wedge ( \neg S \implies D) \wedge (S \vee \neg S)) \vdash D \vee D. $$

Now, by Tertium non datur (Law of Excluded Middle), $\vdash S \vee \neg S$ and hence $$ (( S \implies D ) \wedge ( \neg S \implies D)) \vdash D \vee D. $$ Since $\vdash (D \vee D) \iff D$, this may be further simplified to $$ (( S \implies D ) \wedge ( \neg S \implies D)) \vdash D. $$

Answer 2

I see that you are quite focused on "getting all 1s". Please know that it is a common mistake to think that getting all 1s is good and not getting all 1s is bad: whether this is good or bad all depends on what you are trying to do. In particular, though, if a statement 'gets all 1s', then this certainly does not mean that anything follows from it. In fact, very little follows from a statement with all 1s, and in fact in general the opposite is true: the fewer 1s a statement has, the more things follow from it!

It is important to understand why this is so (and indeed, how truth tables show this). Take the following statement: 'tomorrow it will rain or not'. Now, do you learn anything from this statement? Do you say: "Ah, thank you for that statement, that really tells me something!" No, you don't. In fact, you say: "that doesn't tell me anything at all! of course it is either going to rain or not". OK, so one way of putting this is that this statement conatins no information. However, look what happens when we put the statement on the truth-table: we get all 1s! This is what tautologies like 'it rains or it doesn't' are like: they are always true ... and of course they are always true, because they in effect don't say anything at all, and if you don't say anything at all, you can't be wrong! So you see, from a statement with all 1s you typically can't infer anything (except other tautologies).

Ok, let's look at the two statements you have. Notice that they are not always true ... This means that they do in fact say something of interest, because they can be wrong. In fact, there are only two rows in which both statements are true: rows 2 and 4. Ok, and now look at statement D: statement D is also true in those two rows. This means that statement D is implied by the two statements. If, however, the two statements would have been true in row 1 or 3, D would not have been implied, since D is not true in those rows. So again, you see that the fewer 1s you have, the more likely it is that something follows from it.

The extreme case of having few 1s is of course having no 1s at all, i.e. A statement that has all 0s. This is called a contradiction. Indeed, the contradiction is the 'maximally informative' or 'maximally strong' statement: in effect it is saying that everything is true: it says that pigs can fly, but also that pigs can't fly, that 1+ 1 equals 2, but also that 1+1 equals 3, etc etc. Indeed, everything follows from a contradiction! ... Which actually isn't all that useful either, if you think about it. Yes, everything follows from it, but if you then ask yourself: "OK, but what is actually true?", you don't get much help from a contradiction, because it will simply say that everything is true ...

Answer 3

For any conditional statement $p\implies q$, its contrapositive $\lnot q\implies \lnot p$ is always an indirect deduction. This is valid for each statement or theorem you face in Mathematics or real world.

In your case, the contrapositive of the first statement is "If you could not get diploma then you didn't study"