  First-order logic is typically presented as the study of deduction in a
setting with elementary quantification. In this paper, we take another vantage
point and conceptualize first-order logic as a linear space that encodes
"plausibility". Whereas a deductive perspective emphasizes how (i.e., process),
a space perspective emphasizes where (i.e., location). We explore several
consequences that a shift in perspective to "signals in space" has for
first-order logic, including (1) a notion of proof based on orthogonal
decomposition, (2) a method for assigning probabilities to sentences that
reflects logical uncertainty, and (3) a "models as boundary" principle that
relates the models of a theory to its "size".
