The 'a Priori' Method

Peirce present four methods for fixing beliefs ["Fixation of Belief," 1876].  One of these, the a priori method:

Let the action of natural preferences be unimpeded, then, and under their influence let men, conversing together and regarding matters in different lights, gradually develop beliefs in harmony with natural causes. … Systems of this sort have not usually rested upon any observed facts, at least not in any great degree.  They have been chiefly adopted because their fundamental propositions seemed "agreeable to reason."[CP 5.382]

gets short shrift in leading up to the scientific method and its concern with truth and falsity, but it is hard to see just how "conversing together" is necessarily a priori in this undervalued sense.

It would be a priori in this sense if we are conversing about things like logic, mathematics, or even metaphysics, since then there is little or no reference to "observed facts" and not much concern with truth or falsity.  It would also a priori in this sense if we are conversing analytically about and within the linguistic system of a science.

The ultimate goal of a positive science is the development of a "theory" or "hypothesis" that yields valid and meaningful (i.e., not truistic) predictions about phenomena not yet observed.  Such a theory is, in general, a complex intermixture of two elements.  In part, it is a "language" designed to promote "systematic and organized methods of reasoning." ["The Methodology of Positive Economics," p. 6]

But such sciences are also:

In part, it is a body of substantive hypotheses designed to abstract essential features of complex reality. [Ibid.]

and if our conversation were to wander over into this second part, it would be based upon "observed facts," and it would also, ostensibly, be concerned with truth and falsity.
 
But let's say we're "conversing together" in a sports book, because we want to bet on which team is going to win the NBA Playoffs this year.  Our conversation would clearly be based on observed facts — it could be full of detailed statistics and intricate arguments — and there would be a definite concern with truth or falsity — which team is going to win (actually, which team's odds of winning are better than the payoffs being given).  This conversing might be a priori in the sense we don't know which team will actually win, and we can't know until it's too late, but it doesn't come down to whatever is "agreeable to reason."  And, as a method, it makes up a lot more of our day-to-day reasoning, including virtually all major decisions, than does experimental science.