Central to the claim that science is the only useful epistemic method is the connection between private, individual knowledge and public knowledge. The scientific method is simple: Find the simplest explanation that accounts for the facts. There are four components to the scientific method: Facts, explanations, accounts and simplicity.
An explanation is any instance of manipulating symbols using some sort of deterministic method; ordinary logic is an instance of a deterministic symbol manipulation method. If you start with some set of axioms, and apply inference rules in a particular order, you will always reach the same conclusion.
An account (or an explanation that accounts for) is an explanation that differentiates between valid and invalid conclusions. Again, ordinary logic in an instance of account: we interpret a conclusion reached by applying inference rules to axioms as valid, and the inverse of that conclusion to be therefore invalid. Since we can derive "2+2=4", that conclusion is valid, and "it is not the case that 2+2=4", its inverse, is invalid.
Just having an explanatory method which generates accounts is not sufficient. We must relate the validity of the conclusions to some facts to discuss the notions of truth and falsity. Without such a relation, the sentence "2+2=4" is no more meaningful than "@=@+$". Arithmetic is equally "valid" even if we manipulate the axioms without any reference at all to our understanding of counting actual things in the real world; Arithmetic is true precisely to the extent that it provides an explanatory account of facts about how we actually count (some) things in the real world.
Simplicity just consists of counting the number of irreducible premises, axioms or stipulations, plus the number of inference rules required to determine the account.
I leave the definitions of "explanation" and "account" minimally defined precisely not to presuppose all of propositional calculus and other forms of mathematics and logical symbol manipulation. We have to find that propositional calculus is deterministic and does distinguish between valid and invalid. We can compare alternative methods on their simplicity. We also find that we can generate agreement with the facts by just by choosing axioms; we do not need to alter the fundamental method. For this reason, we usually take propositional calculus for granted except under esoteric circumstances.
The scientific method is fundamentally different from the deductivist method. The deductivist method says: Find those statements which are derivable from true premises.
I compare the two methods in more detail in my first article on the scientific method, The Failure of Deductivism. Briefly, we seem to have an insuperable problem finding true axioms from which we can derive anything useful. Premises simple enough to use deductively seem impossible to generate foundationally (without deduction), and statements we can generate foundationally (especially statements about perception) are too complex to deduce anything interesting from.
Probably the most important difference between the deductive and scientific methods are that the deductive method is truth-finding, and the scientific method is falsity-finding. We can never be sure that some explanation accounts for the facts not yet in evidence, and we can never be sure that the explanation really is the simplest. We can, however, be sure when some explanation fails to account for some facts that are in evidence. We must simply bite this bullet to employ the scientific method. (Besides the foundational problem, the deductivist must bite his own bullet: He cannot be sure that the inverse of his theorems are themselves non-theorems. Even with a perfect axiomatic foundation the deductivist can never be sure of falsity.)
The facts are statements uncontroversially accepted as true. This is a general definition; the meanings of "uncontroversial" and "accepted as true" depend on the level we are employing the scientific method.
Philosophers often blithely assert that the scientific method requires the importation of a considerable amount of metaphysical baggage, notably metaphysical realism, the presumption of consistency, the presumption of the reliability of the senses, and the a priori intelligibility of language. Science, or so these philosophers assert, is a complex socially constructed language game, and has little or nothing to do with the fundamental philosophical issues of epistemology and ontology. I believe such philosophers are mistaken, and they are mistaken because an enormous amount of fundamental scientific work has already been performed by evolution: We human beings come to the task of philosophy with cognitive tools that seem a priori but are actually posterior to at least five hundred million years of evolution. (The first nervous systems appear at least in the Cambrian, if not the late Proterozoic era; if we admit some degree of symbolic representation of the outside world to bacteria, we can go back four billion years.)
The justification for the scientific method as a fundamental epistemic method rests on four pillars: It is metaphysically sparse; we can, in theory, build up to our present-day panoply of knowledge by nothing more than conscious employment of the scientific method; we can show that the social construct of institutional science depends essentially on our sparse formulation of the scientific method; and we can explain our actual history by relating the scientific method directly to the process of evolution.
I will address each of these pillars in my next series of posts.