"Justification" words in symbolic logic I am struggling with the symbolic logic equivalent of "justifications" (e.g. therefore, because, since etc.).  For example:
Case 1
I hate company therefore I stay home
The propositions are:
p: I hate company
q: I stay home

What is the symbolic representation?
p ⊙ q
// Don't know which operator to use.  Is it ↔ because in 
// the narrator's mind the two conditions are very tightly coupled?


Case 2
Since I am sick, I am skipping the party
The propositions are:
p: I am sick
q: I am skipping the party

What is the symbolic representation?
p ⊙ q // Don't know which operator to use.

 A: *

*Both cases are (modus ponens) arguments, and are each symbolised as
$$p;\;\therefore q.\tag{1a}$$
I.e., $\,p$ is true and $(p\implies q);\:$ therefore $q$ is true.


*Note that it is incorrect to symbolise them as either of the
sentences $$p\implies q\tag{2a}$$$$p\rightarrow q,\tag3$$
because
neither $\text{(2a)}$ nor $\text{(3)}$ asserts the respective
premise and conclusion.
[Similarly, the argument
$$\text{I am a chair; therefore I am a furniture piece}\tag{1b}$$
is stronger than—and implicitly contains—the sentence
$$\text{If I am a chair, then I am a furniture piece}.\tag{2b}$$
$\text{(1b)}$ is unsound, while $\text{(2b)}$ is true.]


*Incidentally, my answer to your recent
question
elaborates on the difference between sentences $\text{(2)}$ and
$\text{(3)}.$

OP: “Don't know which operator to use.  Is it $\leftrightarrow$ because in the narrator's mind the two conditions are very tightly coupled?”
Not at all! The $\leftrightarrow$ symbol is called the biconditional (notice that it is bidirectional?), and is a logical operator that is used to check whether the two sentences (on its left and right) that it connects have the same truth value; it is intricately related to the concept of equivalence.
On the other hand, the $\,\therefore\,$ symbol just means ‘therefore’, and indicates a forward-directional chain of reasoning; so, $[p;\;\therefore q]$ and $[q;\;\therefore p]$ have opposite meanings.
A: Your goal of justification can be mapped to semantic consequence (⊨) within model theory, or logical consequence (⊢) within proof theory as the familiar Fitch bar in natural deductive system F. It's a well known fact that natural language conditionals are not always truth functional in the sense that the truth value of "If P, then Q" is determined solely by the truth values of P and Q. Thus semantic analysis of conditionals typically propose alternative interpretations built on foundations such as modal logic, relevance logic, probability theory, and causal models. For example the sentence "Mary is not at home whenever Jack is singing." is not truth functional with all your propositional material conditional calculus which simply is not expressive enough to describe whenever, and thus you cannot exactly translate as Q→P where P=Mary is not at home, Q=Jack is singing. But as an English speaker we intuitively know there's some causal connection between P and Q. And for that you have to employ quantification ∀t(Q(t)→P(t)). You can’t tell whether P logically implies Q just by looking at the truth values of P and Q, even given P→Q is true as a premise within zeroth order propositional logic.
Besides quantification, another approach as mentioned in my above paragraph is employing strict conditional ($\Box (p\rightarrow q)$) or variably strict conditional of modern modal logic with possible world relational semantics, but certainly they're more technically involved than quantified FOL.
In summary what we're actually searching for is causality (even you're talking about justification here), but unfortunately there's no easy single path to true causality as the famous ancient laughing philosopher Democritus hinted long ago. Our quantified inference rules (introduction, elimination, etc) are only a first-order step towards such causality, next step we often need the semantic predicate language for the applied field of any specific problems. For more abstract problems, you may also need higher order logic, maybe up to infinity like a Taylor series expansion.
A: In the sentence "I hate company, therefore I stay home" there's a hidden implication: that "if one hates company, therefore one stays home." In the context of formal logic, a syllogism of this form, where one of the parts is ommited, is called an enthymeme. Both the sentences you provided are examples of enthymemes.
In an explicit syllogism, the above example would look like this...
If one hates company, one stays home.
I hate company.
-------------------------------------
I stay home.

...where the horizontal line is representative of a logical conclusion, and could be substituted by a $\therefore$ (therefore) sign.
When it comes to mathematics, though, one will usually have at his disposal sentences/implications that are true either by definition or by hypothesis, and as such it is customary to link propositions by the aid of a simple "$\implies$" sign when in fact you are making a syllogistic argument.
For example, if you were to tell a mathematician that "$6$ is even, therefore $2$ divides it" he would understand it, but only because to him it is known that, by definition, $n$ is even if and only if $2$ divides $n$. In reality, the argument you are making is the following.
$2 \mid n \iff n \text{ is even.}$ 
$6 \text{ is even.}$ 
$\therefore 2 \mid 6.$
But you would only write
$$6 \text{ is even } \implies 2 \mid 6.$$
