The logical (conditional, causal, or concessive) senses have not been in the focus of research when the semantics of prepositions is studied. My dissertation (Roch, 2018) addresses the context-dependence of these relations by annotating factuality and modality markers that presumably play a role in the sentential context.
Example (1.) illustrates the conditional sense of German bei with the modal auxiliary können (‘can’) in the non-factual main clause.
[…] denn bei rascher Annahme kann diese wichtige Richtlinie umgehend in Kraft treten.
because by rapid adoption can.AUX this important directive immediately enter.into.force.INF
[…] because, if it is adopted rapidly, this directive will be able to enter into force very soon. (Europarl Chapter 10)
The annotations for three German prepositions (bei, mit, ohne) are mined in search for usage patterns by employing multiple correspondence analysis. This multivariate statistical method explores the visual interpretation of corresponding sense categories and annotated factuality and modality features in low-dimensional maps.
MCA map for bei
In the map for _bei_, the first principal axis separates conditional and concessive senses from spatial, temporal and circumstancial ones. At the same time the first axis divides non- and counterfactuality from factuality, as well as present and future tense from past tenses. The conditional sense in the fourth quadrant corresponds with German subjunctive verbal mood features (Konjunktiv I and II), with all types of readings of modal auxiliary verbs (epistemic, root, and ambiguous) and non-factuality. Both concessive senses in the first quadrant are close in the same space with sentential negation, counterfactuality, and focus particles. In sum, the explored hypothesis that the senses are accompanied by certain modality and factuality devices, is reasonable and can be visualized with a dimensionality reduction technique.
References
2018
doctoral thesis
Kontextabhängigkeit konditionaler Interpretationen von Präpositionalphrasen