S5 (modal logic)
In logic and philosophy, S5 is one of five systems of modal logic proposed by
The axioms of S5
The following makes use of the modal operators ("necessarily") and ("possibly").
S5 is characterized by the axioms:
- K: ;
- T: ,
and either:
- 5: ;
- or both of the following:[3]
- 4: , and
- B: .
The (5) axiom restricts the accessibility relation of the
Kripke semantics
In terms of Kripke semantics, S5 is characterized by frames where the accessibility relation is an equivalence relation: it is reflexive, transitive, and symmetric.
Determining the satisfiability of an S5 formula is an
Applications
S5 is useful because it avoids superfluous iteration of qualifiers of different kinds. For example, under S5, if X is necessarily, possibly, necessarily, possibly true, then X is possibly true. Unbolded qualifiers before the final "possibly" are pruned in S5. While this is useful for keeping propositions reasonably short, it also might appear counter-intuitive in that, under S5, if something is possibly necessary, then it is necessary.
Alvin Plantinga has argued that this feature of S5 is not, in fact, counter-intuitive. To justify, he reasons that if X is possibly necessary, it is necessary in at least one possible world; hence it is necessary in all possible worlds and thus is true in all possible worlds. Such reasoning underpins 'modal' formulations of the ontological argument.
S5 is equivalent to the adjunction .[4]
Leibniz proposed an ontological argument for the existence of God using this axiom. In his words, "If a necessary being is possible, it follows that it exists actually".[5]
S5 is also the modal system for the metaphysics of saint Thomas Aquinas and in particular for the Five Ways.[6]
However, these applications require that each operator is in a serial arrangement of a single modality.[7] Under multimodal logic, e.g., "X is possibly (in epistemic modality, per one's data) necessary (in alethic modality)," it no longer follows that X being necessary in at least one epistemically possible world means it is necessary in all epistemically possible worlds. This aligns with the intuition that proposing a certain necessary entity does not mean it is real.
See also
References
- ISBN 0-521-22476-4
- ISBN 0-415-12599-5
- ISBN 9780444500557.
- ^ "Steve Awodey. Category Theory. Chapter 10. Monads. 10.4 Comonads and Coalgebras" (PDF).
- The Stanford Encyclopedia of Philosophy(Spring 2020 ed.), Metaphysics Research Lab, Stanford University, retrieved 2022-06-03
- ^ Gianfranco Basti (2017). Logica III: logica filosofica e filosofia formale- Parte I: la riscoperta moderna della logica formale [Logics III: philosophical Logic and formal philosophy - Part I: the modern rediscovery of the formal logic] (PDF) (in Italian). Rome. pp. 106, 108. Archived from the original (PPT) on 2022-10-07.
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: CS1 maint: location missing publisher (link) - ISBN 978-1-4020-8589-5.
External links
- http://home.utah.edu/~nahaj/logic/structures/systems/s5.html
- Modal Logic at the Stanford Encyclopedia of Philosophy