2

I'm stuck with a logic problem like this

I eat ice cream if I am sad.

I am not sad.

Therefore I am not eating ice cream.

Is this conclusion logical? The first sentence can be understood both like "ice cream $\implies$ sad" and vice versa. He stated that he is not sad but does that not mean that he is not eating ice cream ? I'm confused.

Community
  • 1
Gummy
  • 31
  • so you are saying that this statement is not logical then – Gummy Oct 19 '14 at 17:22
  • 2
    "I eat ice cream if I am sad” does not say that I eat ice cream "only if" I am sad. Maybe I eat ice cream every day. Maybe when I'm sad I eat ice cream because I'm sad, and when I'm happy I eat ice cream because I'm happy. – MJD Oct 19 '14 at 17:29

3 Answers3

4

In logic, there is a difference between implication ($a \implies b$) and equivalence ($a \iff b$). The word "if" (without "only") usually means the first one:

"I am sad" $\implies$ "I eat ice cream"

If "I am sad" is false, we cannot logically say anything about "I eat ice cream". It may be true or false. This makes sense, because being sad is not the only reason for eating ice cream.

See also: Denying the antecedent

Sebastian Negraszus
  • 2,045
  • so what if it has the word "only" in it ? Does it means that "SAD imples ICE CREAM" ? – Gummy Oct 19 '14 at 17:42
  • @Gummy Yes, "only" would make them equivalent and imply each other – Sebastian Negraszus Oct 19 '14 at 17:56
  • how can "only" imply each other ? Since you cant say:" I am sad only if I eat ice-cram" but you have to say:"I eat ice cream only if I am sad". "SAD" is the necessary condition and "Ice cream" is the sufficient condition ? So SAD must implies Ice cream ? NO ? Please I suck and I know. – Gummy Oct 19 '14 at 18:07
  • @Gummy In a logical equivalence, there are no necessary or sufficient conditions. – Sebastian Negraszus Oct 19 '14 at 18:13
  • but i dont think they are logically equivalent to each other – Gummy Oct 19 '14 at 18:16
3

$P\implies Q$ does not necessarily mean $Q \implies P$

But what does this mean exactly?

Well think of it like this.

All Carrots are vegetables

but this doesn't mean that

All vegetables are Carrots

Ali Caglayan
  • 5,726
0

Argument:

"I eat ice cream if I am sad. I am not sad.Therefore I am not eating ice cream."

Whenever you are stuck in an argument, try represent it in proper symbolism:

Glossary:

$E \equiv $ 'I eat ice cream'

$S \equiv $ 'I am sad'

Then we can represent your above argument like this:

  1. $S \to E$
  2. $\neg S$

$\therefore \neg E$

Now is this a valid argument? (Recall the definition of a valid argument!) For instance, Let the statements (1) and (2) be true. Can $\neg E$ be false?

Bruno Bentzen
  • 3,362
  • no i dont think ¬E can be false – Gummy Oct 21 '14 at 13:42
  • @Gummy What about have a look at its truth table? $$\begin{array} {|c|} \hline S & E & S\to Q & \neg S & \neg E\ \hline 1 & 1 & 1& 0 & 0\ \hline 1 & 0 & 0 &0 &1\ \hline 0 & 1 & \color{blue}{1} &\color{blue}{1} &\color{red}{0}\ \hline 0 & 0 & \color{blue}{1} &\color{blue}{1} &\color{blue}{1}\ \hline \end{array}$$ – Bruno Bentzen Oct 21 '14 at 14:02
  • yeah I understand this, but how do you know when an argument is valid or not. Sorry, my 1st language is not Eng, so I might a have a little problem at understanding. So yes not E can be false – Gummy Oct 21 '14 at 14:27
  • @Gummy Roughly, an argument is valid if and only if (iff) it's not be the case that it's premises are true and the conclusion false. So now what can we say about this argument? And don't worry about your English, take your time :) – Bruno Bentzen Oct 21 '14 at 14:46
  • so this argument is not valid since we statement (1) and (2) are premises and they are both true and the conclusion of ¬E is false .Thus this argument is not valid ? – Gummy Oct 21 '14 at 14:51
  • @Gummy Perfect! Good work! So, in your words, this conclusion is not "logical" (= it does not follow from the premises). – Bruno Bentzen Oct 21 '14 at 14:56
  • yes !, thank you but I got it wrong in the assessed worksheet. Anyway it's all good. Thank you so much for the patient and help. – Gummy Oct 21 '14 at 15:03