Modal collapse

In modal logic, modal collapse is the condition in which every true statement is necessarily true, and vice versa; that is to say, there are no contingent truths, or to put it another way, that "everything exists necessarily" (and likewise if something does not exist, it cannot exist). In the notation of modal logic, this can be written as ϕ ↔ ◻ ϕ {\displaystyle \phi \leftrightarrow \Box \phi } . In the context of philosophy, the term is commonly used in critiques of ontological arguments for the existence of God and the principle of divine simplicity. For example, Gödel's ontological proof contains ϕ → ◻ ϕ {\displaystyle \phi \rightarrow \Box \phi } as a theorem, which combined with the axioms of system S5 leads to modal collapse. Since some regard divine freedom as essential to the nature of God, and modal collapse as negating the concept of free will, this then leads to the breakdown of Gödel's argument. However, it has been shown that a modified version of Gödel's original argument in modal logics K and T avoid modal collapse all the while maintaining validity of the argument.

Concept in modal logic

Not to be confused with Model collapse.

In modal logic, modal collapse is the condition in which every true statement is necessarily true, and vice versa; that is to say, there are no contingent truths, or to put it another way, that "everything exists necessarily"^[1][2] (and likewise if something does not exist, it cannot exist). In the notation of modal logic, this can be written as ϕ ↔ ◻ ϕ {\displaystyle \phi \leftrightarrow \Box \phi } .

In the context of philosophy, the term is commonly used in critiques of ontological arguments for the existence of God and the principle of divine simplicity.^[1][3] For example, Gödel's ontological proof contains ϕ → ◻ ϕ {\displaystyle \phi \rightarrow \Box \phi } as a theorem, which combined with the axioms of system S5 leads to modal collapse.^[4] Since some regard divine freedom as essential to the nature of God, and modal collapse as negating the concept of free will, this then leads to the breakdown of Gödel's argument.^[5] However, it has been shown that a modified version of Gödel's original argument in modal logics K and T avoid modal collapse all the while maintaining validity of the argument.^[6]

References

[edit]

1. ^ ^{a b} Tomaszewski, Christopher (2019). "Collapsing the modal collapse argument: On an invalid argument against divine simplicity". Analysis. 79 (2): 275–284. doi:10.1093/analys/any052. ISSN 0003-2638.
2. ^ Schmid, Joseph C. (2022). "The fruitful death of modal collapse arguments" (PDF). International Journal for Philosophy of Religion. 91 (1): 3–22. doi:10.1007/s11153-021-09804-z. ISSN 0020-7047. S2CID 236427989.
3. ^ Benzmüller, Christoph; Paleo, B. W. (2016). "The Ontological Modal Collapse as a Collapse of the Square of Opposition". The Square of Opposition: A Cornerstone of Thought. Studies in Universal Logic. pp. 307–313. doi:10.1007/978-3-319-45062-9_18. ISBN 978-3-319-45061-2. S2CID 42006078.
4. ^ Kovač, Srećko (2012), Szatkowski, Miroslaw (ed.), "15. Modal Collapse in Gödel's Ontological Proof" (PDF), Ontological Proofs Today, de Gruyter, pp. 323–344, doi:10.1515/9783110325881.323, ISBN 978-3-11-032515-7, retrieved 2022-04-28{{citation}}: CS1 maint: work parameter with ISBN (link)
5. ^ Pedersen, Daniel J.; Lilley, Christopher (2022). "Divine Simplicity, God's Freedom, and the Supposed Problem of Modal Collapse". Journal of Reformed Theology. 16 (1–2): 127–147. doi:10.1163/15697312-bja10028. hdl:2164/18891. ISSN 1569-7312. S2CID 248106829.
6. ^ Benzmüller, Christoph (2020-07-09). "A (Simplified) Supreme Being Necessarily Exists, says the Computer: Computationally Explored Variants of Gödel's Ontological Argument". Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning. 17 (1): 779–789. arXiv:2001.04701. doi:10.24963/kr.2020/80. ISSN 2334-1033.

This article about religious studies is a stub. You can help Wikipedia by adding missing information.

• v
• t
• e

This mathematical logic-related article is a stub. You can help Wikipedia by adding missing information.

• v
• t
• e