The Barcan Formula: A, B, and C-Semantics

2023-2 Modal Logic (Segment 2)

 

Q) Use tableaux to show the difference between what we have called the A semantics and the B semantics (+ inclusion requirement) and the C-semantics for quantified modal logic. Be sure to explain by reference to the question of the validity of the Carnap-Barcan Formula (BF) and its converse. Be sure to explain the situation with

Ax1 (∀x) (ϕx) ⊃ ϕy, where y is free for x in the wff ϕ.

How does Cresswell (book) address the validity of this wff in a B semantics (p. 275 f). Explain why in a B semantic framework where we don’t have the Inclusion Requirement, we get in trouble with the universal validity of the Rule of Necessitation. How then is universal validity and invalidity defined in a B semantics without inclusion requirement. Show how the inclusion requirement resolves this issue. Explain how we resolved the issue in class by rejecting Ax1 in favor of a quantification theory with:

*Ax1: (∀y) (∀x)(ϕx) ⊃ αy, where y is free for x in the wff ϕ.

Explain how it is that with the quantification theory *QT and its axiom *Ax1, we are in a good position to leave open issues pertaining to the Barcan wff and its converse.

This more full statement of the question might help:

Explain how the B semantics assures the universal validity of (∀x)Ax ⊃ Ay, where y is a free variable.

Recall that the definition of valid in a model in the B semantics is this: If for all w∈W, Vu(A, w) = 1 for all u such that u(y) ∈ Dw for every free variable y in A, then A is valid in <W, R, D, Q, V>.

If for some w∈W, Vu(A, w) = 0 for some u such that u(y) ∈ Dw for all free variable y in A, then A is not valid in <W, R, D, Q, V>.

Explain why on this notion of universal validity, if we drop the inclusion requirement, then L((∀x)Ax ⊃ Ay) is no longer universally valid. Thus the rule of Necessitation would fail. How does Cresswell redefines universal validity to keep the rule of Necessitation and yet drop the inclusion requirement. (I.e., explain the B+ semantics which introduced the “Posterity of w” and also introduces: Vu( ∀xAx, w) =1 if Vp(Ax , w) = 1 for all x-alternatives p of u such that p(x) ∈ D+w. Vu( ∀xAx, w) = 0 otherwise.