Which Implications are True in Mathematical Logic?

[Terraist]

Hi

This is a question from my self study of ch.2 of Alfred Tarski's Introduction to Logic

Which of the following implications are true from the perspective of mathematical logic?

a) If a number x (assuming x is an integer) is divisible by 2 or by 6, then it is divisible by 12

b) if 18 is divisible by 3 and by 4, then 18 is divisible by 6

Both statements are obviously false from the standpoint of ordinary language. However, as far as my reasoning for b) goes, it is a logically true statement, because the antecedent (18 divisible by 3 and 4) is false, thus the truth table values of the statement will be either FF or FT, leading to overall true meaning of the statement.

I find a) to be more confusing. It is clearly untrue (18 and 6 are counterexamples) yet looking at the antecedent and consequent separately, can lead me to believe that the statement can be true from a logical perspective. For instance if it's true that X is divisible by 2 or 6, but false that it is divisible by 12, then the statement is false. On the other hand, if x is divisible by 12 then the statement is true. Am I mired in confusion here?

My question is, am I correct in trying to view this kind of problem in terms of truth tables? Sometimes the truth value of the antecedent and consequent can be confusing or ambiguous, even though the overall meaning of the statement may appear obvious.

 

[Simon Bridge]

The question concerns the implications.
The trick will be figuring if you are expected to restrict yourself just to logic or if the statements must also be true in arithmetic... or in what way.

Implication: "if A fulfills conditions X then statement Y is true."

But A does not fulfill the conditions! What does this say about statement Y?

Statement (a) uses "or" while statement (b) uses "and".
What does the logical "or" do?

 

[Terraist]

The question concerns the implications.
The trick will be figuring if you are expected to restrict yourself just to logic or if the statements must also be true in arithmetic... or in what way.

Implication: "if A fulfills conditions X then statement Y is true."

But A does not fulfill the conditions! What does this say about statement Y?

Then it appears that statement Y is false? I'm getting really confused. As far as I understood from the book, under a material implication conventionally used in logic, the truth of the statement depends solely on the exclusive truth value of each part (i.e. the truth tables). So it doesn't matter if there's a formal connection between the two. For instance "the sky is green implies that a duck makes a quack" is true from a logical perspective, presumably because the fact that a duck quacks is true regardless of the truth value of the antecedent. With a mathematical statement, I assume the same thing applies too, and the meaning of the implication statement is irrelevant to whatever connection of the two terms. If my understanding is wrong then I'm in need of a little bit of clarification!

Statement (a) uses "or" while statement (b) uses "and".
What does the logical "or" do?

The "or" in that statement is non-exclusive which means that x can be divisible by either 2 or 6 (or both) for the antecedent to be true. But then there are numbers like 6 or 18 which aren't divisible by 12. So I suppose it's a necessary but not a sufficient condition for the consequent. But would that make the overall statement true or false?

 

[Byzantine]

I believe the first statement is false, because you can come up with a number divisible by 2 or by 6 that is not divisible by 12. To disprove a statement all you need a single counterexample, and in this case 6 should work as proof enough of a number that satisfies the conditions does not satisfy the conclusion.

The second case is true, because if the conditions are false then the overall statement is always true, regardless of if it makes any sense. You need true conditions and a false conclusion for it to be false.

Or at least that is how I remember it.

 

[Simon Bridge]

Then it appears that statement Y is false?

My understanding is that would be a logical fallacy. The statement actually tells you nothing about the truth of statement Y.

If the antecedent is false, then the inference provides no further information.

I'm getting really confused. As far as I understood from the book, under a material implication conventionally used in logic, the truth of the statement depends solely on the exclusive truth value of each part (i.e. the truth tables). So it doesn't matter if there's a formal connection between the two. For instance "the sky is green implies that a duck makes a quack" is true from a logical perspective, presumably because the fact that a duck quacks is true regardless of the truth value of the antecedent.

Then what is the point of the "if the sky is green..." part?

With a mathematical statement, I assume the same thing applies too, and the meaning of the implication statement is irrelevant to whatever connection of the two terms. If my understanding is wrong then I'm in need of a little bit of clarification!

What it the point of having an anticedent in the first place?

In the kind of statement above, 18 is clearly divisible by 6 ... so the entire statement, by that logic, will be true even though the anticedent is false.

The "or" in that statement is non-exclusive which means that x can be divisible by either 2 or 6 (or both) for the antecedent to be true. But then there are numbers like 6 or 18 which aren't divisible by 12. So I suppose it's a necessary but not a sufficient condition for the consequent. But would that make the overall statement true or false?

Here X satisfies A OR B, therefore Y is true.
However, we have a situation where X satisfies A, but Y is false.
What does that tell you about the truth of the logical inference?

I think you need to go over the previous chapter again.

 

[Bipolarity]

I asked a very similar question on this a while ago. I would encourage a read on it.
https://www.physicsforums.com/showthread.php?t=662112

BiP

 

[Simon Bridge]

That's an informative thread yep - will probably shed light on Tarski.

 

[Terraist]

I believe the first statement is false, because you can come up with a number divisible by 2 or by 6 that is not divisible by 12. To disprove a statement all you need a single counterexample, and in this case 6 should work as proof enough of a number that satisfies the conditions does not satisfy the conclusion.

The second case is true, because if the conditions are false then the overall statement is always true, regardless of if it makes any sense. You need true conditions and a false conclusion for it to be false.

Or at least that is how I remember it.

Thanks! Your response is the one that made the most sense to me. My main confusion was in how to tell whether a condition was true or not, but yours and Simon's explanations made the formulation much clearer.

With that in mind I would answer the question in the following way:

a) If a number x (assuming x is an integer) is divisible by 2 or by 6, then it is divisible by 12

If x being divisible by 2 or 6 was true, then it being also divisible by 12 is clearly false, so the statement is false.

b) if 18 is divisible by 3 and by 4, then 18 is divisible by 6

18 divisible by 3 and 4 is false, but the overall implication is true regardless of 18 being divisible by 6.

I'm also looking over the thread that was linked to, it's very informative.

 

[RonAGove]

Hi

This is a question from my self study of ch.2 of Alfred Tarski's Introduction to Logic

Which of the following implications are true from the perspective of mathematical logic?

a) If a number x (assuming x is an integer) is divisible by 2 or by 6, then it is divisible by 12

b) if 18 is divisible by 3 and by 4, then 18 is divisible by 6

Both statements are obviously false from the standpoint of ordinary language. However, as far as my reasoning for b) goes, it is a logically true statement, because the antecedent (18 divisible by 3 and 4) is false, thus the truth table values of the statement will be either FF or FT, leading to overall true meaning of the statement.

I find a) to be more confusing. It is clearly untrue (18 and 6 are counterexamples) yet looking at the antecedent and consequent separately, can lead me to believe that the statement can be true from a logical perspective. For instance if it's true that X is divisible by 2 or 6, but false that it is divisible by 12, then the statement is false. On the other hand, if x is divisible by 12 then the statement is true. Am I mired in confusion here?

My question is, am I correct in trying to view this kind of problem in terms of truth tables? Sometimes the truth value of the antecedent and consequent can be confusing or ambiguous, even though the overall meaning of the statement may appear obvious.

Truth table s apply
a is false because as you not x=6 is a counter example
b is true because the hypothesis is false. A=>b is true whenever A is False or B is true as can be seen fro a truth table