We can refer to the "basic" of Aristotelian science :
The notion of "valid argument" is as old as "formal logic" itself; see Aristotle's logic :
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 [emphasis added]. (Prior Analytics I.2, 24b18-20)
The core of this definition is the notion of “resulting of necessity” . This corresponds to a modern notion of logical consequence: X results of necessity from Y and Z if it would be impossible for X to be false when Y and Z are true. We could therefore take this to be a general definition of “valid argument”.
In a nutshell, we can say that according to Aristotle, science, i.e. true knowledge, is what we can deduce from true premises with a valid argument :
For this reason, science requires more than mere deduction. Altogether, then, the currency of science is demonstration (apodeixis), where a demonstration is a deduction with premises revealing the causal structures of the world, set forth so as to capture what is necessary and to reveal what is better known and more intelligible by nature.
Gettier Problems arise from Gettier's critique of the Justified True Belief account of knowledge; according to this view,
A subject S knows that a proposition P is true if and only if:
(i) P is true, and
(ii) S believes that P is true, and
(iii) S is justified in believing that P is true.
Gettier's Case II is :
Smith has a justified belief that "Jones owns a Ford". Smith therefore (justifiably) concludes (by the rule of disjunction introduction) that "Jones owns a Ford, or Brown is in Barcelona", even though Smith has no knowledge whatsoever about the location of Brown.
In fact, Jones does not own a Ford, but by sheer coincidence, Brown really is in Barcelona. Thus, Smith had a belief that was true and justified, but not knowledge.
The justified true belief came about, as the result of entailment (the inference licensed by disjunction introduction) from justified false beliefs that "Jones owns a Ford".
The issue is that, according to JTB theory, the conclusion : "Jones owns a Ford, or Brown is in Barcelona" appear to be knowledge because it is true (item (i)) : it's a fact that Brown is in Barcelona, and it is believed by Smith (item (ii)) and is justified by a correct use of inference rules (item (iii)) :
Smith, when he believes, holds a justified true belief. However, is Smith's belief an instance of knowledge? Intuitively, Smith's belief cannot be nowledge; it is merely lucky that it is true.
In fact it is not "real" knowledge, as we can see applying A's point of view : it is deduced from a false premise.
But the issue regards the "belief" involved in the definition : today we know that A's point of view regarding the possibility of achieving "absolutely" certain first principles to be used as starting points for science and knowledge is difficult to maintain.
JTB is a tentative account of knowledge which try to address this issue.