Logic Laws - Material Implication - Conditional Laws

[jpanknin]

I'm going through a Set Theory book and one of the logic laws listed is what this book calls the "Conditional Laws" (see below). In Google searches I've also found it called the Material Implication.

The rule is: [math](P \implies Q) \iff (\neg P \lor Q)[/math]
I was trying to understand this equivalence and put together the tables below using a simple example of [imath]x = 2 \implies x^{2} = 4[/imath]. They all make sense except for the one on the bottom right highlighted in red. I get that in this case not P is true and Q is false, so that "not P or Q" would be true. But how can it be true if x = 2 that x^2 does not = 4? This line highlighted in red is supposed to be true so that it's logically equivalent to the table on the left, but I don't see how that can be the case. If x = 2 then x^2 does in fact = 4.

 

[Dr.Peterson]

But how can it be true if x = 2 that x^2 does not = 4? This line highlighted in red is supposed to be true so that it's logically equivalent to the table on the left, but I don't see how that can be the case. If x = 2 then x^2 does in fact = 4.

I don't like the tables you made, which are confusing. It looks as if in each row you were changing what P and Q are, but evidently you just mean, for example, that if [imath]x\ne2[/imath], then [imath]P[/imath] is false, so [imath]\neg P[/imath] is true. But the table doesn't make that clear. So you seem to have confused yourself. (Did you get this style of table from somewhere, or make it up yourself?)

As I understand it, that last row should represent the case that P is false (so that [imath]x\ne2[/imath]) and Q is false (so that [imath]x^2\ne4[/imath]). That's entirely possible: Let [imath]x=3[/imath]. And if that happened, then, as you note, the statement [imath]\neg P \lor Q[/imath] would be true, since the first statement, [imath]\neg P[/imath] would be true, and that is all you need for an "or" to be true. So your last column is wrong.

It is not true in that line that [imath]x=2[/imath]! That's what is NOT true. That's where your table confused you.

I would have expected you to complain about the second row, not the fourth. Each case in a table like this is hypothetical; you aren't claiming these things are true or false as claimed, but looking at what would happen if the truth values were as stated.

Looking at the other rows,

• the first says that if [imath]x=2[/imath] and [imath]x^2=4[/imath] (which is possible), then the statement is true, because although [imath]\neg P[/imath] is false, [imath]Q[/imath] is true;
• the second row says that if [imath]x=2[/imath] and [imath]x^2\ne4[/imath] (which is impossible), then the statement would be false, because both [imath]\neg P[/imath] and [imath]Q[/imath] are false;
• the third says that if [imath]x\ne2[/imath] and [imath]x^2=4[/imath] (which is possible, if [imath]x=-2[/imath]), then the statement is true, because both parts are true.

But it doesn't matter which can actually happen; this is all "what if". Logic doesn't care what's real, only what would be true if it were.

The usual concern I see is on the other side, misunderstanding the meaning of the conditional in the last row. But apparently you have no trouble there.

 

[jpanknin]

I don't like the tables you made, which are confusing. It looks as if in each row you were changing what P and Q are, but evidently you just mean, for example, that if [imath]x\ne2[/imath], then [imath]P[/imath] is false, so [imath]\neg P[/imath] is true. But the table doesn't make that clear. So you seem to have confused yourself. (Did you get this style of table from somewhere, or make it up yourself?)

As I understand it, that last row should represent the case that P is false (so that [imath]x\ne2[/imath]) and Q is false (so that [imath]x^2\ne4[/imath]). That's entirely possible: Let [imath]x=3[/imath]. And if that happened, then, as you note, the statement [imath]\neg P \lor Q[/imath] would be true, since the first statement, [imath]\neg P[/imath] would be true, and that is all you need for an "or" to be true. So your last column is wrong.

It is not true in that line that [imath]x=2[/imath]! That's what is NOT true. That's where your table confused you.

I would have expected you to complain about the second row, not the fourth. Each case in a table like this is hypothetical; you aren't claiming these things are true or false as claimed, but looking at what would happen if the truth values were as stated.

Looking at the other rows,

• the first says that if [imath]x=2[/imath] and [imath]x^2=4[/imath] (which is possible), then the statement is true, because although [imath]\neg P[/imath] is false, [imath]Q[/imath] is true;
• the second row says that if [imath]x=2[/imath] and [imath]x^2\ne4[/imath] (which is impossible), then the statement would be false, because both [imath]\neg P[/imath] and [imath]Q[/imath] are false;
• the third says that if [imath]x\ne2[/imath] and [imath]x^2=4[/imath] (which is possible, if [imath]x=-2[/imath]), then the statement is true, because both parts are true.

But it doesn't matter which can actually happen; this is all "what if". Logic doesn't care what's real, only what would be true if it were.

The usual concern I see is on the other side, misunderstanding the meaning of the conditional in the last row. But apparently you have no trouble there.

The top tables are pretty standard. For the bottom tables, I tried to come up with an example where I could replace the T and F with the truth/falsity of the actual expression. Given how long it took me to figure that out, I'm very disappointed you don't like them

I spent a while looking at this and am starting to think that the table approach isn't the best way to make sense of this intuitively. From what the expression says, If P implies Q is true, then either [imath]x \neq 2[/imath] or [imath]x^2 = 4[/imath]. LIkewise, if [imath]x^2 = 4[/imath] or [imath]x \neq 2[/imath], then P implies Q will be true.

I'm trying to wrap my head around how this would be used in everyday conversation. When I do that, I usually understand the formal version much better. Kind of like "If it rains, then the grass is wet." For the contrapositive, "If the grass is not wet, then it did not rain" makes sense. Then I understand the formal notation and the notation makes sense intuitively.

 

[Dr.Peterson]

The top tables are pretty standard. For the bottom tables, I tried to come up with an example where I could replace the T and F with the truth/falsity of the actual expression. Given how long it took me to figure that out, I'm very disappointed you don't like them

Yes, I was referring to the bottom tables.

If you want to put an example in a table, it might be better to add a column or two on the left stating the situation (which might be more or less what you have under P and Q, and just put T and F into the other columns, just as in the top tables.

Or possibly you could color each cell to remind you whether the statement at the top of the column is true or false:

​

This way, at least you know that everything green is being assumed to be true, and what's red is assumed to be false. But it still isn't quite clear what each heading means, and what is true.

I'm trying to wrap my head around how this would be used in everyday conversation. When I do that, I usually understand the formal version much better. Kind of like "If it rains, then the grass is wet." For the contrapositive, "If the grass is not wet, then it did not rain" makes sense. Then I understand the formal notation and the notation makes sense intuitively.

Sometimes you have to be very careful, because formal logic doesn't map well onto ordinary language. The tense of verbs may have to change (as you did here), or the English may imply a causal connection that is not intended in the logic, for example.

 

[jpanknin]

Got it. Need to think on this one for a while. Very much appreciate the help.