Presupposition in logic

Question

Let S be a presupposition for Q. For example:

1. S: I used to smoke
2. Q: I quit smoking

As far as I know, Q has a truth value if and only if S is true. However, my understanding of the concept of presupposition contradicts this, so I feel that I'm wrong—but where and how?

I feel that if S is false, Q has the truth value False. Because for quitting smoking I have to had been smoking before. Starting is the necessary condition for quitting. And If I didn't satisfy this condition, then I just logically can't quit smoking. Therefore I just didn't quit. That is, Q is false, which means that it does have a truth value.

Answer 1

I am Posting this Answer , at the risk of getting Downvoted , to aid the new OP , who should include more Context/Details to the Question Post , Eg Where was the Question from , What OP thinks is the wrong/right way , Why OP wants to look into this.

Background :

Essentially , this is a matter of Semantics , Linguistics , Philosophy & Natural Languages , not generally covered in regular Mathematical Logic texts.

Linguistics & Philosophy texts cover this in Detail.
Eg :
https://plato.stanford.edu/entries/presupposition/ lists 150+ Articles.

In terms of Mathematical Logic , there are some treatments in some theories by some authors , though it is not covered in texts like "Logic and Structure - Dirk Van Dalen" & "Logic for Mathematicians - Yuri I. Manin"

OP Query :

(1) The main Objection (out of many other Objections) is that this will not satisfy PEM the Principle of Excluded Middle , which says that Either "a Proposition $X$ is true" or "it is not true & $\lnot X$ is true" : there is no other alternative & Both can not be true.

In OP Case :
PEM says that : if we take "I quit smoking" is true , then "I did not quit smoking" is not true , while if we take "I quit smoking" is not true , then "I did not quit smoking" is true.

But the OP Way of thinking makes Both true , which Contradicts PEM.

$S$ : I used to smoke
$Q_1$ : I quit smoking
$Q_2$ : I did not quit smoking

PEM : If we take $Q_1$ is not true , $\lnot Q_1$ must be true.
PEM : If we take $Q_2$ is true , $\lnot Q_1$ must be not true.
OP Way : Both $Q_1$ & $\lnot Q_1$ are true , Both $Q_2$ & $\lnot Q_2$ are not true.

Conclusion : It is not whether $Q$ & $Q_1$ & $Q_2$ can have truth value , it is whether that truth value is Consistent.

(2) One other Issue to look into is that the truth value of $Q$ (which is not valid by itself) can always make $S$ true , whether that is the Intention or not.
Eg : "Did the Army stop murdering Civilians ?" , "Did the gangster continue to beat the neighbours ?" , "Did the Minister know that the assistant was involved in Corruption ?"
The Questions can be made into Propositions "The Army stopped murdering Civilians" & "The gangster continued to beat the neighbours" & "The Minister knew that the assistant was involved in Corruption"
No matter whether these are true or not , that will automatically make the Presumptions true :
"The Army was earlier murdering Civilians" & "The gangster was earlier beating the neighbours" & "The assistant was involved in Corruption"

Hence , Mathematical Logic will not generally treat these as Atomic Propositions , in general. There are some authors who treat these as Compound Propositions , though that is rare.

Answer 2

1. I used to smoke - S
2. I quit smoking - Q

But I feel that, even if S is false, Q still can have truth value.

Since you want the loaded statement "I quit smoking" to formally have a truth value regardless of the truth value of S, I suggest the clearer and actually consistent alternative interpretation

1. S: I used to smoke
2. Q: I still smoke

so that "I quit smoking" is formalised as S ∧ ¬Q instead.