Defining premise and conclusion Trying to better understand the concept of premise and conclusion.
In classic propositional logic, I think it usually takes on the following form: $p \to q$, where $p$ is the premise and $q$ is the conclusion.
But doesn't this depend on our semantic interpretation of the $\to$ operator? And the very concept of "premise" and "conclusion" being defined and only making sense in this particular logical framework?
Or is "premise" and "conclusion" an even stronger claim we can define on the metalogical level, e.g. $p \vdash q$ or something to this effect, that would apply to any logic system?
 A: An argument is a linguistic "object":

In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (both spellings are acceptable) intended to determine the degree of truth of another statement, the conclusion. The logical form of an argument in a natural language can be represented in a symbolic formal language.

The concept of valid (deductive) argument has been defined firstly by Aristotle :

A deduction is speech (logos) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so. (Prior Analytics, I.2, 24b18–20)

Each of the “things supposed” is a premise (protasis) of the argument, and what “results of necessity” is the conclusion (sumperasma).
The key discovery of Aristotle is that, in order to assess the validity of an argument, we have to consider its Logical Form.
In order to do this, is useful to "formalize" an argument using variable (i.e. reducing the linguistic argument to its "schematic" structure); see Syllogism :

Major premise: All $M$ are $P$.
Minor premise: All $S$ are $M$.

Conclusion: All $S$ are $P$.


Modern mathematical logic has improved "formalization" using the modern mathematical symbols developed for algebra.
Propositional logic is useful because in it we can have a simplified model of language: it proxy statements of natural language with propositional symbols (or variables). Thus, propositional logic provides a simple model for deductive arguments.
In propositional logic we define a formal counterpart of entailment (or: logical consequence) : $Γ⊨φ$.
The symbol reads : "formula $φ$ is a logical (or: tautological, in the case of propositional logic) consequence of the set of formulas $Γ$" and it is defined in terms of semantical concept: truth assignments (or interpretations).
The semantical concepts are related to the syntactical ones: setting up the logical calculus, we introduce rules of inference that allow us to infer  a formula (the conclusion) from an initial set of formulas (the premises).
With them we define the relation of derivability, defined as follows: "$Γ ⊢ φ$ iif there is a derivation with conclusion $φ$ and with all hypotheses (or assumptions) in $Γ$."
A derivation, in turn, is a finite sequence of applications of rules of inference.
The two sides: semantical and syntactical, are linked by the property of soundness and completeness.

In propositional logic, $p \to q$ is a formula: it is a conditional with $p$ as antecedent and $q$ as consequent.
$p → q,p ⊢ q$ is the formal counterpart of a valid argument (modus ponens), where $p → q$ and $p$ are the premises and $q$ is the conclusion.
