1.A fact in and of itself is not an atomic fact, just as a vector is not in and of itself a basis vector. To say that a proposition is a truth function of elementary propositions is thus a mistake. One must say instead that any collection of facts which provide axioms from which the world’s truths can be derived guarantees the truth values of a collection of statements, which we can call propositions.

1.1 The study of the possible mappings between different representations of the same logical space can thus be considered the fundamental research program of mathematical logic.

1.1.1 A deduction is a mapping with the property that if every statement in the mapping’s domain is true, then every statement in the mapping’s range must be true.

1.1.2 We should not view a logical space as merely a collection of statements. To define a logical space, we must also state exactly what qualifies as a deduction. Like Klein did with geometry, this approach naturally allows for the problem of classifying all possible logics.

1.1.2.1 In some sense, this is precisely Klein’s erlangen program. A geometric property, like a logical property, is defined merely by its invariance under a transformation in this approach. The words “geometric” and “logical” are undefined terms.

1.1.2.2 Frege’s program of deriving mathematics from logic, even if sucessful, would not have shown logic to be any more fundamental than any other part of mathematics. We could very well be able to derive logic from algebra as well as algebra from logic. One example of this in the real world is the application of Topos theory to studying logic.

1.1.3 A basis of atomic facts for a logical space combined with rules of deduction determines exactly what statements are in a logical space. Likewise, a basis of atomic facts combined with the entirety of the statements in a logical space gives us restrictions which our notion of deduction must satisfy.

1.1.4 Our experiences give us evidence that a collection of statements about the world is true. The image of these statements under all deductions constitutes precisely what we can know about the world.

1.1.4.1 Different definitions of deduction can lead to the extrapolation of different logical spaces, different worlds, from the same experiences. The determination of which definitions of deduction will lead to the extrapolation of worlds that are not contradicted by our future experiences is one of the tasks of scientific theory.