# Basic question regarding logical implications

1. Jul 21, 2014

### maxpan

Three statements:
1) I will take a shower if the water is warm.
2) I will take a shower unless the water is cold.
3) I will not take a shower unless the water is warm.

How different are their meanings?

First guess was:
1) water is warm => I take a shower
2) water is not cold = water is warm => I take a shower
3) water is warm => I could take a shower

Although, I suspect the second one doesn't mean warm, just not cold (could be cool for example). So it's a statement on its own.

So:
1) water is warm => I take a shower
2) water is not cold => I take a shower
3) water is warm => I could take a shower

2. Jul 21, 2014

### micromass

Staff Emeritus
It seems to me that you're not translating (2) and (3) correctly. For example, (2) also says that if the water is cold, then you will not take a shower (well, it's debatable whether it says this, but to me the sentence does imply it).

And (3) says that if the water if not warm, then you will definitely not take a shower. This is missing from your translations.

3. Jul 21, 2014

### MrAnchovy

Correct

OK, but what if the water IS cold? The statement does imply what happens in this case too.

We wouldn't normally say this - "I could take a shower" means "either I take a shower or I do not take a shower" which doesn't tell us anything. However what if the water is NOT warm? The statement has a more meaningful implication then.

4. Jul 21, 2014

### MrAnchovy

Micromass beat me to it.

5. Jul 21, 2014

### micromass

Staff Emeritus
That makes 2 - 1 then :tongue:

6. Jul 21, 2014

### maxpan

Thanks.

So should it be along the lines of:

1) water is warm => I take a shower
2) water is not cold (warm, cool etc) => I take a shower; otherwise, if the water is cold, I will not take a shower
3) water is warm => I might decide to take a shower; otherwise, if the water is not warm (cold, cool etc), I will definitely not take a shower

7. Jul 22, 2014

### micromass

Staff Emeritus
Can you state this briefly? Use iff.

Like MrAnchovy remarked, a thing like "I might decide" isn't really meaningful. To be fair, you could interpret the third sentence in a lot of way. But I interpret it that if the water is warm, then I definitely will take a shower. It is debatable though.

8. Jul 22, 2014

### MrAnchovy

No I don't think you can conclude anything if the water is warm, I liked your previous interpretation better:

9. Jul 22, 2014

### maxpan

p water warm
w water not cold
q take a shower

1) p $\rightarrow$ q
2) w $\leftrightarrow$ q
3) p $\leftarrow$ q ("if the water is not warm, then you will definitely not take a shower" = if I'm taking a shower, then you can be sure the water is warm, but that doesn't mean that if the water is warm I will definitely take a shower)

Last edited: Jul 22, 2014
10. Jul 22, 2014

### maxpan

11. Jul 22, 2014

### maxpan

So, as I understand,

"If" expresses one of the many criteria for something to be done for certain (although, we don't know how many there are, it might be that this critertion is the only one, or maybe there are multiple). And "only if" merely gives restrictions, as to when something CAN be done, but not necessarily will be done.

"If and only if" is the same, in essence, as "only if" combined with "if", that is — we have a restriction that should be satisfied, and it is indeed satisfied. Hence now what was specified HAS to be done.

only if $\subseteq$ if $=$ if and only if

12. Jul 22, 2014

### MrAnchovy

But unlike "if and only if" I don't think the word "unless" has a consistent translation into symbolic logic.

In the phrase "I will take a shower unless the water is cold", the phrase "I will (outcome) unless (proposition)" introduces a necessary and sufficient condition (that the proposition is false i.e. "the water is not cold") to the outcome "(I will) take a shower".

When we see "I will not take a shower unless the water is warm", we are tempted to parse this in the same way so that the outcome is "(I will) not take a shower". But I do not believe this is linguistically correct; the correct parsing is "I will not (outcome) unless (proposition)", with the subtle difference that the condition (that the proposition is true i.e. "the water is warm") is necessary (for the outcome i.e. "(I will) take a shower") but no longer sufficient.

13. Jul 22, 2014

### maxpan

I re-edited my post a few times :) But you seem to have commented an earlier version.

14. Jul 22, 2014

### MrAnchovy

Of course this has nothing to do with mathematical logic, it is linguistics. In mathematical logic we define things precisely so that when we write A → B ("A implies B") it is clear that this means that if A is true then B is also true, but if A is false we don't know anything about B.

15. Jul 22, 2014

### MrAnchovy

Ah yes, I like the new version better - it illustrates the linguistic difference between "unless" and "not ... unless" well.