Analyzing the Truth of Jack Winning a Contest

[EternusVia]

Homework Statement

Suppose someone says to you that the following statement is true: “If
Jack is younger than his father, then Jack will not lose the contest.” Did Jack
win the contest? Why or why not? Explain.

Homework Equations

Truth table from textbook:

A--------B--------A implies B
True----True-------True
True----False------False
False---True-------True
False---False------True

The Attempt at a Solution

A = Jack is younger than his father.
B = Jack will not lose the contest = Jack will win the contest (assuming you can't draw)

A is true because one is ALWAYS younger than his/her father.
B is true because A is true.

Therefore A -> B is also true and Jack did win the contest.

This question is probably really easy, but I was wondering if anyone could validate my answer? I'm really new to proofs and this kind of logical thought.

Thanks!

 

[Ray Vickson]

Homework Statement

Suppose someone says to you that the following statement is true: “If
Jack is younger than his father, then Jack will not lose the contest.” Did Jack
win the contest? Why or why not? Explain.

Homework Equations

Truth table from textbook:

A--------B--------A implies B
True----True-------True
True----False------False
False---True-------True
False---False------True

The Attempt at a Solution

A = Jack is younger than his father.
B = Jack will not lose the contest = Jack will win the contest (assuming you can't draw)

A is true because one is ALWAYS younger than his/her father.
B is true because A is true.

Therefore A -> B is also true and Jack did win the contest.

This question is probably really easy, but I was wondering if anyone could validate my answer? I'm really new to proofs and this kind of logical thought.

Thanks!

'Implication' can be tricky, and this is one such case. Obviously, Jack can win the contest or he can lose the contest, so the implication is not really making a predictive statement. In fact, implications of the form A -> B can connect two totally unrelated statements or concepts, such as "If my eyes are blue then Kansas produced a lot of corn last year".

All you can say is that if Jack did win the contest the implication was true (but irrelevant), and if he lost the contest the implication was false (but, again, irrelevant). The point is that the implication is, itself, a logical statement and can thus be true or false.

Implication need not signal causation; for example (after the late E.T. Jaynes): "If it is raining at 10:00 then there were clouds at 9:59." This implication is true, but the rain at 10:00 did not "cause" the clouds at 9:59.

 

[BvU]

Oh, this is a nice one. If the 'someone' speaks the truth, then the following is true:
B. If Jack is younger than his father, then Jack will not lose the contest.

Furthermore the following is always true (don't ask me why...)
A. Jack is younger than his father.

So A is true; if A is true, then Jack will not lose the contest. Conclusion: Jack will not lose the contest.

No more, no less. You already exclude a draw. You don't mention the case where the contest hasn't taken place yet.

What on Earth do you mean with EternusVia ? Via is female, so is Vita.

 

[EternusVia]

Oh, this is a nice one. If the 'someone' speaks the truth, then the following is true:
B. If Jack is younger than his father, then Jack will not lose the contest.

Furthermore the following is always true (don't ask me why...)
A. Jack is younger than his father.

So A is true; if A is true, then Jack will not lose the contest. Conclusion: Jack will not lose the contest.

No more, no less. You already exclude a draw. You don't mention the case where the contest hasn't taken place yet.

What on Earth do you mean with EternusVia ? Via is female, so is Vita.

Thanks for the help. And yes, I am aware of the grammatical shortcomings of my username. I have little to no knowledge of Latin and made up the name a few years ago XD

 

[EternusVia]

'Implication' can be tricky, and this is one such case. Obviously, Jack can win the contest or he can lose the contest, so the implication is not really making a predictive statement. In fact, implications of the form A -> B can connect two totally unrelated statements or concepts, such as "If my eyes are blue then Kansas produced a lot of corn last year".

All you can say is that if Jack did win the contest the implication was true (but irrelevant), and if he lost the contest the implication was false (but, again, irrelevant). The point is that the implication is, itself, a logical statement and can thus be true or false.

Implication need not signal causation; for example (after the late E.T. Jaynes): "If it is raining at 10:00 then there were clouds at 9:59." This implication is true, but the rain at 10:00 did not "cause" the clouds at 9:59.

Good points. Thank you!