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00925njm 22002777a 4500 dc Aranda Utrero, Víctor author Huertas, Antonia author Manzano, María author Martins, Manuel author 2026 The aim of this paper is to present the philosophical influences and motivations of Hybrid Partial Type Theory (HPTT), a partial type theory combined with a strong version of hybrid logic in which the concepts of existence and denotation emerge naturally. The text is divided into three main parts. Firstly, we present and discuss some formal systems dealing with existence and denotation in both first-order logic and type theory. Then, we move to the framework of modal logic, introducing the distinction between constant and varying domains, as well as the hybrid machinery. Finally, we give a succinct account of the semantics of HPTT and provide new definitions of existence and denotation within the system, together with a reflection on de dicto and de re statements and some validities sharing an explicit philosophical characterization. Aranda, V., Huertas, A., Manzano, M., Martins, M. (2026). On the Philosophy and Mathematics of Hybrid Partial Type Theory. In: Antunes, H., Freire, A.R., Rodrigues, A. (eds) Walter Carnielli on Reasoning, Paraconsistency, and Probability. Outstanding Contributions to Logic, vol 32. Springer, Cham. https://doi.org/10.1007/978-3-032-11027-5_3 978-3-032-11026-8 978-3-032-11029-9 978-3-032-11027-5 10.1007/978-3-032-11027-5_3 https://hdl.handle.net/20.500.14352/137077 https://doi.org/10.1007/978-3-032-11027-5_3 https://link.springer.com/chapter/10.1007/978-3-032-11027-5_3 On the philosophy and mathematics of hybrid partial type theory