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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] In the notation of modal logic, this can be written as $\phi \leftrightarrow \Box \phi$ .

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 $\phi \rightarrow \Box \phi$ 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]

References

^ ^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.
^ Schmid, Joseph C. (2022). "The fruitful death of modal collapse arguments". International Journal for Philosophy of Religion. 91 (1): 3–22. doi:10.1007/s11153-021-09804-z. ISSN 0020-7047. S2CID 236427989.
^ 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.
^ Kovač, Srećko (2012), Szatkowski, Miroslaw (ed.), "15. Modal Collapse in Gödel's Ontological Proof", Ontological Proofs Today, DE GRUYTER, pp. 323–344, doi:10.1515/9783110325881.323, ISBN 978-3-11-032515-7, retrieved 2022-04-28
^ 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. ISSN 1569-7312. S2CID 248106829.

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[:0-1] 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". 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", Ontological Proofs Today, DE GRUYTER, pp. 323–344, doi:10.1515/9783110325881.323, ISBN 978-3-11-032515-7, retrieved 2022-04-28

[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. ISSN 1569-7312. S2CID 248106829.

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 {{Short description|Concept in modal logic}}
-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 truth]]s, or to put it another way, that "everything exists necessarily".<ref name=":0">{{Cite journal |last=Tomaszewski |first=Christopher |date=2019-04-01 |title=Collapsing the modal collapse argument: On an invalid argument against divine simplicity |url=https://academic.oup.com/analysis/article/79/2/275/5062919 |journal=Analysis |language=en |volume=79 |issue=2 |pages=275–284 |doi=10.1093/analys/any052 |issn=0003-2638}}</ref><ref>{{Cite journal |last=Schmid |first=Joseph C. |date=February 2022 |title=The fruitful death of modal collapse arguments |url=https://link.springer.com/10.1007/s11153-021-09804-z |journal=International Journal for Philosophy of Religion |language=en |volume=91 |issue=1 |pages=3–22 |doi=10.1007/s11153-021-09804-z |s2cid=236427989 |issn=0020-7047}}</ref> In the notation of modal logic, this can be written as <math>\phi \leftrightarrow \Box \phi</math>.
+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 truth]]s, or to put it another way, that "everything exists necessarily".<ref name=":0">{{Cite journal |last=Tomaszewski |first=Christopher |date=2019 |title=Collapsing the modal collapse argument: On an invalid argument against divine simplicity |url=https://academic.oup.com/analysis/article/79/2/275/5062919 |journal=Analysis |language=en |volume=79 |issue=2 |pages=275–284 |doi=10.1093/analys/any052 |issn=0003-2638}}</ref><ref>{{Cite journal |last=Schmid |first=Joseph C. |date=2022 |title=The fruitful death of modal collapse arguments |url=https://link.springer.com/10.1007/s11153-021-09804-z |journal=International Journal for Philosophy of Religion |language=en |volume=91 |issue=1 |pages=3–22 |doi=10.1007/s11153-021-09804-z |s2cid=236427989 |issn=0020-7047}}</ref> In the notation of modal logic, this can be written as <math>\phi \leftrightarrow \Box \phi</math>.
-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]].<ref name=":0" /><ref>{{Cite book |last1=Benzmüller |first1=Christoph |last2=Paleo |first2=B. W. |title=The Square of Opposition: A Cornerstone of Thought |date=2016 |chapter=The Ontological Modal Collapse as a Collapse of the Square of Opposition |series=Studies in Universal Logic |pages=307–313 |chapter-url=https://www.semanticscholar.org/paper/The-Ontological-Modal-Collapse-as-a-Collapse-of-the-Benzm%C3%BCller-Paleo/34575c54ff23f69620138a1039bcaefd9f3ef330 |doi=10.1007/978-3-319-45062-9_18|isbn=978-3-319-45061-2 |s2cid=42006078 }}</ref> For example, [[Gödel's ontological proof]] contains <math>\phi \rightarrow \Box \phi</math> as a theorem, which combined with the axioms of [[S5 (modal logic)|system S5]] leads to modal collapse.<ref>{{Citation |last=Kovač |first=Srećko |title=15. Modal Collapse in Gödel's Ontological Proof |date=2012-12-31 |url=https://www.degruyter.com/document/doi/10.1515/9783110325881.323/html |work=Ontological Proofs Today |pages=323–344 |editor-last=Szatkowski |editor-first=Miroslaw |publisher=DE GRUYTER |doi=10.1515/9783110325881.323 |isbn=978-3-11-032515-7 |access-date=2022-04-28}}</ref>  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.<ref>{{Cite journal |last1=Pedersen |first1=Daniel J. |last2=Lilley |first2=Christopher |date=2022-04-08 |title=Divine Simplicity, God's Freedom, and the Supposed Problem of Modal Collapse |url=https://brill.com/view/journals/jrt/16/1-2/article-p127_8.xml |journal=Journal of Reformed Theology |volume=16 |issue=1–2 |pages=127–147 |doi=10.1163/15697312-bja10028 |s2cid=248106829 |issn=1569-7312}}</ref>
+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]].<ref name=":0" /><ref>{{Cite book |last1=Benzmüller |first1=Christoph |last2=Paleo |first2=B. W. |title=The Square of Opposition: A Cornerstone of Thought |date=2016 |chapter=The Ontological Modal Collapse as a Collapse of the Square of Opposition |series=Studies in Universal Logic |pages=307–313 |chapter-url=https://www.semanticscholar.org/paper/The-Ontological-Modal-Collapse-as-a-Collapse-of-the-Benzm%C3%BCller-Paleo/34575c54ff23f69620138a1039bcaefd9f3ef330 |doi=10.1007/978-3-319-45062-9_18|isbn=978-3-319-45061-2 |s2cid=42006078 }}</ref> For example, [[Gödel's ontological proof]] contains <math>\phi \rightarrow \Box \phi</math> as a theorem, which combined with the axioms of [[S5 (modal logic)|system S5]] leads to modal collapse.<ref>{{Citation |last=Kovač |first=Srećko |title=15. Modal Collapse in Gödel's Ontological Proof |date=2012 |url=https://www.degruyter.com/document/doi/10.1515/9783110325881.323/html |work=Ontological Proofs Today |pages=323–344 |editor-last=Szatkowski |editor-first=Miroslaw |publisher=DE GRUYTER |doi=10.1515/9783110325881.323 |isbn=978-3-11-032515-7 |access-date=2022-04-28}}</ref>  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.<ref>{{Cite journal |last1=Pedersen |first1=Daniel J. |last2=Lilley |first2=Christopher |date=2022 |title=Divine Simplicity, God's Freedom, and the Supposed Problem of Modal Collapse |url=https://brill.com/view/journals/jrt/16/1-2/article-p127_8.xml |journal=Journal of Reformed Theology |volume=16 |issue=1–2 |pages=127–147 |doi=10.1163/15697312-bja10028 |s2cid=248106829 |issn=1569-7312}}</ref>
 == References ==