Here's a little situation I want math to resolve for me :

  1. If I study, I make the exam ,
  2. If I do not play tennis, I study ,
  3. I didn't make the exam

Can I conclude that was playing tennis ?

Trying to put this into the symbology of inference logic and propositional classic logic :

$P1 : \text{study} \implies \text{exam}$

$P2 : (\text{tennis}\, \vee \text{study}) \wedge (\neg \text{tennis} \implies \text{study})$ (disjunctive syllogism)

$p3 : \neg \text{exam}$

My reasoning :

Step 1 : the contrapositive of $P1$ is $P1' : \neg \text{exam} \implies \neg \text{study}$ ;

Step 2 : By Modus Tollens ( $ [(P \implies Q) \wedge \neg Q] \implies \neg P$) we have : $(\text{study} \implies \text{exam}) \wedge (\neg \text{exam} \implies \neg \text{study})$

Step 3 : should we suppose : $\neg \text{tennis} \wedge \neg \text{study}$, then $\neg ( \text{tennis} \vee \text{study})$, then (by $P2$) $\text{tennis}$ or otherwise the $P1$ would fall since $\neg \text{study}$ and $\neg (False \implies False)$.

Step 4 : reductio ad absurdum from step $(3)$, we have $(\text{tennis} \vee \text{study})$, henceforth, in $P2$, $\neg \text{tennis}$ or else $false \implies false$.

So, have I been playing tennis or is my inferential logic bad ?

  • 4
    $\begingroup$ The title should be more informative. $\endgroup$ – Paracosmiste Dec 25 '13 at 15:38
  • $\begingroup$ Would the person that "minused" the question care to say why ? That would be nice ! $\endgroup$ – Gloserio Dec 25 '13 at 16:11
  • $\begingroup$ I'm not the downvoter. $\endgroup$ – Paracosmiste Dec 25 '13 at 16:26
  • $\begingroup$ I am not accusing either, and I've just asked the question to see what I can avoid next time :) $\endgroup$ – Gloserio Dec 25 '13 at 16:28

Yes, indeed, we can easily arrive at the conclusion that you played tennis: and the repeated use of modus tollens, alone (plus one invocation of double negation) gets you that conclusion.

Our premises, in "natural language":

1. If I study, I make the exam ,
2. If I do not play tennis, I study ,
3. I didn't make the exam


$S:\; $ I study.

$E:\; $ I make the exam.

$P:\; $ I play tennis.

Then our premises translate to:

$(1): S \rightarrow E$.

$(2): \lnot P \rightarrow S.$

$(3): \lnot E.$

$(4)\quad \lnot S$ follows from $(1), (3)$ by modus tollens.

$(5)\quad \lnot \lnot P$ follows from $(2), (4)$ by modus tollens.

$\therefore (6) \quad P$, by $(5)$ and double negation.

Hence you can conclude you played tennis.

  • $\begingroup$ As @Matt Brenneman did, you translated the second statement to : $\neg P \implies S$ while I've translated it to : $(P \vee S) \wedge \neg S \implies P$, which I thought was safer. Admitting your translation, I would totally agree with your reasoning, but admitting mine, would we come still to the conclusion that I play tennis ? $\endgroup$ – Gloserio Dec 25 '13 at 16:19
  • 1
    $\begingroup$ Yes, absolutely you would! $\lnot P \rightarrow S \equiv \lnot \lnot P \lor S\equiv P \lor S$. Then since we have $\lnot S$, too, you can conclude $P$. $\endgroup$ – Namaste Dec 25 '13 at 16:36
  • $\begingroup$ I know you love logic and for this reason this answer is excellent;-)+1 $\endgroup$ – user63181 Dec 25 '13 at 16:51
  • $\begingroup$ @amWhy : thank you, now it's clear ! $\endgroup$ – Gloserio Dec 25 '13 at 18:16
  • $\begingroup$ You're welcome, @Gloserio! $\endgroup$ – Namaste Dec 25 '13 at 18:23

You made a mistake in your step 3, because considering only P2, the term $\neg ( tennis \vee study)$ does not imply $tennis \vee study$.

This may sound counter-intuitive to your introduction.

The reason is, that your P2 is an arguable translation of statement 2. It is not equivalent to "if I don't play tennis, I study". Rather it states "if I don't play tennis and if I study, I study", which is a tautology. You can see this by drawing a truth-table for P2.

Also note that $\neg exam, \neg tennis, \neg study$ satisfies P1,P2 and P3.

So I would replace P2 by $$\neg tennis \rightarrow study.$$ This also repairs your Step3.

  • $\begingroup$ +1, you're probably true, that why I've asked this question, because I felt as if my $P2$ was somewhat redundant. Thanks for pointing it out ! $\endgroup$ – Gloserio Dec 25 '13 at 16:26

Yes. It just reduces down to look at the contrapositives of your statements.
Statement 1 is logically equivalent to : ~(make exam) implies ~study.
Statement 2 is logically equivalent to: ~study implies (play tennis).
So the truth of ~(make exam) directly implies you played tennis (use modus ponens twice).

  • $\begingroup$ How is second statement logically equivalent to $\neg study \implies tennis$ ? $\endgroup$ – Gloserio Dec 25 '13 at 15:52
  • $\begingroup$ It is the contrapositive of the statement: "~(play tennis) implies study" $\endgroup$ – Matt Brenneman Dec 25 '13 at 15:54
  • $\begingroup$ So how that you converted natural langage to logic symobols, and in its valid. $\endgroup$ – Gloserio Dec 25 '13 at 16:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.