Translating Fallacy into Symbolic Logic

Question

I think I've read this, once.

1) If you don't like it here, then go somewhere else, or something like 2) You like it here, or you go somewhere else.

Something about this doesn't seem to make sense, to me.

I tried to translate it into Logic, Symbolic Logic, Predicate Logic to try to see where it doesn't seem to make sense.

I've got

1)

¬L→G

¬L

∴G

2)

L∨G

¬L

∴G

I think, I feel like, can't you not like it somewhere, and stay—(maybe to aim to work on things)? I do not think I can see where it makes no sense, logically. As of now, I have only feeling.

Answer 1

The problem arises from attempting to force an imperative sentence (command) in natural language to be a declarative sentence (factual statement). This is simply not allowed.

In classical first-order logic, all statements are declarative, so we don't have the ability to make imperative statements. If you want to stick to classical logic, you would have no choice but to first translate the imperative into a factual statement, and then express it in logic.

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Original imperative:

If you do not like it here, then go somewhere else.

Factual statement with nearly equivalent meaning but totally different tone:

If you ever decide that you do not like it here, then after that you should go somewhere else.

Same factual statement in a form suitable for translation to classical first-order logic (where $X$ stands for "this place"):

For any time $t$, if you decide at time $t$ "that you do not like $X$ at time $t$", then you should after time $t$ try making it so that you go after time $t$ to some place other than $X$.

(Note that I've interpreted the imperative not as a command. If it is a command then it does not mean "you should try to ..." but rather "I require you to ...").

Classical first-order logic translation with restricted quantification: $\def\imp{\rightarrow}$

$\forall t \in Times\ ( DecideAt(t,you,\ulcorner \neg LikeAt(t,you,X) \urcorner)$

$\quad \imp ShouldTryAfter(t,you,\ulcorner \exists p \in Places\ ( p \ne X \land GoAfter(t,you,p) ) \urcorner) )$.

The "$\ulcorner \urcorner$" are used to quote statements, because stating what you think/feel about a statement is different from the truth of falsehood of the statement.

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As you can see, it's not pretty! But literally that is how much detailed nuance the natural language sentence has in its meaning!

It is now easy to see that this is totally different from the other English sentence that you asked about, which is a very unnatural sentence but one that many people would interpret as a factual statement.

~~~

Original sentence:

You like it here, or you go somewhere else.

One likely interpretation:

Either you do not like it here at the present moment or you would go somewhere else.

Since the original sentence is already quite ambiguous, I will not attempt to translate it into logic.

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However, let me show you what your original imperative is equivalent to, which is derived completely logically:

$\neg \exists t \in Times\ ( DecideAt(t,you,\ulcorner \neg LikeAt(t,you,X) \urcorner)$

$\quad \land \neg ShouldTryAfter(t,you,\ulcorner \exists p \in Places\ ( p \ne X \land GoAfter(t,you,p) ) \urcorner) )$.

(You cannot change the inner quoted stuff, just as "I do not believe that A is true" is not the same as "I believe that A is false"!)

This corresponds to the following English sentence:

There is no time $t$ at which you will both decide that you do not like $X$ and you should not try after time $t$ to go to some place other than $X$.

Indeed, it states the same thing as the imperative, which you can slowly take the time to convince yourself after careful thought.

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So the reason you feel differently about both sentences is that they are in fact saying completely different things. One cannot expect classical logic to capture easily natural language. To alleviate that problem, people have invented all sorts of other logics such as modal logics, named so because they were intended to try capturing modality in natural language, which are expressed in English via modal verbs like "shall", "should", "can", "could", "will", "would", "may", "might", "must", "ought to" and adverbs such as "possibly", "necessarily", "probably", "surely", "mostly", "slightly"...