A Natural Semantics for the Pullback of Fiber Bundles of Structures

Abstract We remark that forcing in fiber bundles of structures of first order languages is not a compatible semantics with the pullback (of fiber bundles). Motivated by a combination of epistemology and geometry, we describe a semantics which behaves well with respect to the pullback. This new semantics uses parallel transport in its definition and allows to introduce two different types of extensions for the formulas: vertical and horizontal extensions.