modal logic
Table of Contents
1. Correspondence
axiom \(\phi\) (\(\Box \phi \to \Diamond \phi\)) corresponds to a property \(C\) (seirality, a model restriction) if and only if:
- C satisfied -> \(M,w \models \phi\)
- C not satisfied -> there are some alternative valuation \(V'\) and some world \(w \in W\) such that \((W,R,V'),w \nvDash \phi\)
1.1. Use of Correspondence
if \(\phi\) corresponds to property \(C\), then \(\textbf{K} + \phi\) would be sound and complete with respect to models that satisfy \(C\).
And this stacks.
2. translation
Alethic □φ means: “φ is necessarily true.” Epistemic □φ means: “I know that φ is true.” Doxastic □φ means: “I believe that φ is true.” Temporal □φ means: “At every time in the future, φ will be true.” Deontic □φ means: “φ should be true.” Legal □φ means: “φ is legally required to be true.”
3. validity
Backlinks
faith
faith is a choice of knowledge, or knolwedge of a fixed set of choices(assumptions).
In term of modal logic, faith could be defined in term of beliefs, or beliefs of the valance of propositions, such as “when advesary slap you in the left cheek, [you should] present your right”, “man are born good”.
Such propositions often have vague/hard-to-define meaning/objective or lots of alternative forms which can all be true, such as when advesary slap you in the left, you can [slap back], [slap harder], [call mass evidence], [call authority], [do nothing], [present right], [slap back later when they are vulnerable]… Each could be valid strategy and thus valid “you should” belief; and in term of their objectives, what would the bring about, or what do you want to achieve, or what is the meaning of those action, can be pretty vague, and we may have never think about them – present right, so they see violence is meaningless? slap back later so they will surely be destroyed and be out of your way? call mass evidence to mentain social record of advesary’s violence nature, so the social mechnism do what is right for the society?
If you have faith on a set of such propositions (beliefs, beliefs of valance of propositions, etc.; names of the same concept) \(P\), it is equivallent of saying that, for each \(p \in P\) and each pointed model \(M,w\), \(\models \Box p\) (or \(M,w \models \Box p\)). What world model I’m not too sure about, but at least temporal is a fit.
(Axiom)
in addition to modal logic’s axioms:
- Knowledge is truthful - If a knows that \(\psi\) is true, then \(\phi\) is true, \[ \Box_a \phi \to \phi \]
- I konw what I know (positive introspection) - \[ \Box_a \phi \to \Box_a \Box_a \phi \]
- I konw that I don’t konw (negative introspection) - \[ \neg \Box_a \phi \to \Box_a \neg\Box_a \phi \]