Theory: Set of statements that is supposed to explain a set of statements of supposed observable fact. To be minimally adequate, the theory must be consistent and deductively entail what it is supposed to explain.
There are, of course, other possible stipulative definitions of what a theory is, but normally these comprise that a theory is a set of statements.
The criterion of minimal adequacy is added to dismiss inconsistent theories and nondeductive theories. The motivations are as follows.
That inconsistency is an undesirable property for a theory is based on the fact that in standard logic anything whatsoever follows from an inconsistent set of statements.
That nondeductiveness is an undesirable property for a theory is based on the fact that in standard logic a theory that is supposed to explain whatever it explains in a nondeductive manner does not permit a deductive step of what a theory is to what it is supposed to explain, nor indeed a deductive step to the falsehood of the theory if it has a false prediction.
The consistency requirement removes all manner of theories that deductively entail consequences known to be false, and the deductiveness criterion removes all manner of stories that may be suggestive but don't really explain deductively.
The relations between a theory, its predictions, and the observations it explains can be sketched as follows:
Here explanation is a deduction from a theory, as prediction is, while expectation is a deduction from a prediction, and test a deduction about a prediction based on an observation. Abduction and induction are principles of inference.
It should be noted also, since this is often missed, that any testable theory goes beyond the known facts, for if it does not it cannot be tested.
The six relations of inference indicated by arrows in the above picture may be somewhat more fully explained as follows:
An abduction is the inference of a Theory to account for a (presumed) Observation. This inference is normally not deductively valid, and indeed a theory cannot be tested independently if it does not deductively imply statements that go beyond the known evidence. A Theory is a set of statements that has been inferred to account for (a) presumed observation(s) and has been assigned some probability or degree of belief (if only tentatively, in some cases).
An explanation is the inference of a (presumed) observation from a theory. This inference must be deductively valid to be a proper explanation. The theory that is presumed may be any theory one has, and as just indicated good testable theories always go beyond the known evidence (and therefore never can be deduced from the evidence).
A prediction is the inference of a presumed statement of fact from a Theory. This presumed statement of fact is called Prediction (capital P), and a proper Prediction is not known to correspond to observational fact when it is made. It should follow deductively from a Theory.
An induction is the inference of a new probability or degree of belief for a Theory when a new Observation verifies or falsifies an earlier Prediction. Inductions follow deductively from the fact that a new Observation that verifies or falsifies an earlier Prediction has been made together with the rules and assumptions of probability theory.
An expectation is the inference of a presumed Observation from a Prediction. The presumed Observation should not have been made or refuted when expected, and the degree of its expectation depends on the probability of or degree of belief in the Theory that allowed the Prediction.
A test is the inference that a Prediction is true or false from the fact that the expectation has been found to be true or false by observation.
Note that in the above outline of only theories are nondeductively inferred, though it also should be added that inductions can be inferred deductively only with the help of the assumption of probability theory and some further assumption.
