Preserving truth

What does it mean that argument is valid when conclusion is true whenever the premises are true?

Suppose you have a function, a+b. You say that a+b > 0 whenever a > 0 and b >0, i.e.

(a > 0), (b > 0) ⊢ (a+b > 0)

 

The point of premises is to constrain the function domain to the area where the conclusion holds.If function is binary, say f = a \/ b, we may say

a ⊢ f

The constrained area (red region) can be better seen on the Karnaugh map,

It is apparent that the OR-function is tautology in the red region, where A = True.We can say the same by

⊢ a → f

 which says that it is true that whenever a is true, f holds.We can also define function entirely in the premise

a \/ b ⊢ 1

Here the premise is exactly support-wide so that the function is constant 1 in that region.

We can go another direction and add one constraint at a time. For instance, we can write function a /\ b can be written a ⊢ b or b ⊢ a or b → a (I do not make difference between premises and assumptions and wikipedia on consequents takes the same stance, saying that "The intuitive meaning of the sequent Г ⊢ Σ is that under the assumption of Γ the conclusion of Σ is provable.") as well as b → a → 1 (see curring and partial application). 

We have just transformed a function of two variables into a function of zero variables, which is constant 1 under valuation a = 1, b = 1.