Inferential logic in a simple-life situation.

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 ?

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

KEY:

$S:\;$ 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.

• 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 ? – Gloserio Dec 25 '13 at 16:19
• 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 $P$. – Namaste Dec 25 '13 at 16:36
• I know you love logic and for this reason this answer is excellent;-)+1 – user63181 Dec 25 '13 at 16:51
• @amWhy : thank you, now it's clear ! – Gloserio Dec 25 '13 at 18:16
• You're welcome, @Gloserio! – 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.

• +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 ! – 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).

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