Translating propositional formulas

Question

I've been given statements to translate into a propositional formula. The question states that it may also be possible for a statement to have multiple interpretations.

I've having trouble determining whether or not this is the case for the following statement, due to the word "when": "I wear a coat when it's cold"

Let $w$ = "I wear a coat"; $c$ = "It's cold".

This might mean that both are simply happening at the same time, and thus I might interpret it as $(w \land c)$.

However, another possibility could be that "when" acts as a conditional operator, and thus it could be interpreted as $(c \implies w)$, where $c$ is a condition for $w$.

But is this really a proper interpretation of the word "when"?

The term "when" has to do with tense. I would translate the above by (c⟹w), but it does seem somewhat forced. — Doug Spoonwood, Sep 19 '13 at 01:38

score 3 · Accepted Answer · answered Oct 09 '13 at 17:30

In my reading, yes (but please be aware of the fact that English is not my first language).

In this sentence, we want to express that "I wear a coat" is enforced by "It is cold". In natural language, we could also say:

I wear a coat whenever it is cold.

or indeed:

I wear a coat if it is cold.

So, for all intents and purposes, $c \implies w$ is an appropriate translation of our sentence.

Its truth/validity in the sense of a logical system will depend on the properties of this logical system (whether it "recognises time", and perhaps other properties).

Of course, this only applies to the particular use of "when" under discussion. It evidently doesn't work with "When does it get cold?" or even "I take a shower when I come home." Linguistically, it's probably to do with "it's cold" being a condition instead of a time specification.

Translating propositional formulas

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