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I'm working on some problems that demonstrate some simple implications. The logic seems to be very different from the way I'm used to using it in everyday language. I'm not sure what assumptions I am allowed to make to show that the statement is true or false. For example, can I consider hypothetical worlds in which a statement I know to be false is actually true? Below is an implication that I need to show can or cannot always be true.

  1. Every good boy does fine $\Rightarrow$ Some bad boy doesn't do fine.

    • True, assuming every good boy doesn't do fine.
    • It could be false though, assuming every good boy does fine is true and some bad boy doesn't do fine is false.

    If I'm allowed to assume anything I want how can an implication always be true? What am I missing?

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  • $\begingroup$ With the given information, I think the implication is false for the reason you stated in the second bullet: as far as we know, it is possible that some bad boy does fine. To answer your other question, here is an implication that does hold: "Every good boy does fine $\implies$ every boy that does not do fine is bad" (this is the contrapositive). $\endgroup$ – angryavian Sep 4 '14 at 2:50
  • $\begingroup$ @angryavian In response to your answer to my second question, is that because both of your statements are equivalent? $\endgroup$ – StarCute Sep 4 '14 at 3:20
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    $\begingroup$ Yes, both statements are equivalent in my example, so it is a little bit silly. $\endgroup$ – angryavian Sep 4 '14 at 3:22
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$A \Rightarrow B$ doesn't translate well into English ever. In natural language you assume motivation behind statements, so a casual statement $A \Rightarrow B$ often has a suggested "and $A$ is true". It is also very hard to avoid hangups with quantification (ambiguously assuming the natural language speaker meant some kind of "for all A"). Implication in natural language is nasty.

For a statement $A \Rightarrow B$ I suggest translating it as $(\text{not } A) \text{ or } B$. It will be much easier for you to think through. So

  • Every good boy does fine $\Rightarrow$ Some bad boy doesn't do fine

becomes

  • Not every good boy does fine or some bad boy doesn't do fine.

So if you assume "every good boy doesn't do fine", then the statement is true, since the first half of the "or" is true.

If you assume every good boy does fine and every bad boy doesn't do fine then it is also true, since the second half of the or "some bad boy doesn't do fine" is satisfied.

If would be false in the case of both halves of the "or" being false:

  • There is a good boy who doesn't do fine, and
  • There is no bad boy who doesn't do fine, without the double negation: all bad boys do fine

Aside,

The opposite of "every good boy does fine" isn't "every good boy doesn't do fine". It is "there is a good boy who doesn't do fine". You only need 1 boy to be a counter example to the "every".

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  • $\begingroup$ @Danie1V In response to your explanation of the statements being false. What you said is equivalent to A being true and B being false from the original problem statement correct? $\endgroup$ – StarCute Sep 4 '14 at 3:23
  • $\begingroup$ @StarCute Yes, that is correct. $\endgroup$ – DanielV Sep 4 '14 at 3:23
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    $\begingroup$ I find $A\implies B \equiv \neg[A \land \neg B]$ to be more intuitive. $\endgroup$ – Dan Christensen Sep 4 '14 at 14:39
  • $\begingroup$ @DanChristensen That's very true because it avoids the inclusive/exclusive or issue. I considered it, but from his post, the questioner was already having trouble with propagating negations. Perhaps $(A \land \lnot B) \text{ is false }$ would have worked. $\endgroup$ – DanielV Sep 4 '14 at 22:12
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Implication can be understood as thinking in NECESSARY and SUFFICIENT conditions. A good example is to think about someone who has born in Dallas, Texas.

Proposition P could stand for "Someone who has born in Dallas"

Proposition Q could stand for "Someone who is texan"

Truth table for implication is:

enter image description here

So, in this example is easy to check line by line the validity of truth table above:

  • Who has born in Dallas is therefore texan; So the first implication is TRUE.

  • Who has born in Dallas MUST be texan. So, saying that someone who has born in Dallas, in the state of Texas, is not texan is FALSE;

  • Saying who doesn't born in Dallas but is texan anyway (take someone who has born in Houston, for example) is a TRUE statement;

  • At last, who doesn't born in Dallas AND also is NOT texan is also TRUE.

So, thinking in terms of NECESSARY (to be texan is a necessary condition for who has born in Dallas) and SUFFICIENT (to born in Dallas is a sufficient condition to be texan) conditions ease the process to translate implication to natural language.

Other examples of propositions containing necessary and sufficient conditions are:

  • P = To be a priest / Q = To believe in God;
  • P = To be a Police officer / Q = To carry a gun;
  • P = The number is divisible by 4 / Q = The number is even.
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