PHI 134 California Baptist University Frame Definability Modal Logic Problems Modal Logic problems 2 questions with small parts. Not a paper but solutions

PHI 134 California Baptist University Frame Definability Modal Logic Problems Modal Logic problems 2 questions with small parts. Not a paper but solutions or explanation. I can provide you with the textbook for reference PHI134 Assignment # 2
Completeness and Filtrations
This assignment consists of two questions, and is worth 25% of your final grade. It is due on
Wednesday the 3rd of June at 11:59pm.
Question 1: (15 marks) In this question we will walk through, step by step, the argument showing that if Σ contains the modal formula ♦A → ♦A, then RΣ is convergent in the sense that
If RΣ Γ ∆ and RΣ Γ ∆∗ then there exists a Θ such that RΣ ∆Θ and RΣ ∆∗ Θ
To do this we need to show that given (i) −1 Γ ⊆ ∆ and (ii) ♦∆∗ ⊆ Γ , that there exists
a complete and consistent set Θ ∈ W Σ such that RΣ ∆Θ and RΣ ∆∗ Θ. Consider the set
Θ− where:
Θ− = −1 ∆ ∪ −1 ∆∗
If this set Θ− is consistent, then there must be a complete Σ-consistent set extending
it, and we can let that set be our Θ.
Part (i):
Show that if Θ is a complete consistent set extending Θ− that we
will have both RΣ ∆Θ and RΣ ∆∗ Θ
All that remains to be done, then, is to show that Θ− is consistent. Suppose, for a
contradiction, that it isn’t. Then we must have that Θ− `Σ ⊥. So we must have
D1 , . . . , Dn ∈ ∆ and S1 , . . . , Sm ∈ ∆∗ for which
D1 ∧ . . . ∧ Dn , S1 ∧ . . . ∧ Sm `Σ ⊥
So by the deduction theorem (Proposition 3.36(4)) we have S1 ∧ . . . ∧ Sm `Σ (D1 ∧
. . . ∧ Dn ) → ⊥, and so by (Proposition 3.36(5)) and the fact that (A → ⊥) → ¬A is
a tautological instance we have
S1 ∧ . . . ∧ Sm `Σ ¬(D1 ∧ . . . ∧ Dn )
So by Lemma 4.6 it follows that
(S1 ∧ . . . ∧ Sm ) `Σ ¬(D1 ∧ . . . ∧ Dn )
Part (ii):
From the above it follows that:
1. (D1 ∧ . . . ∧ Dn ) ∈ ∆
2. (S1 ∧ . . . ∧ Sm ) ∈ ∆∗
3. ¬(D1 ∧ . . . ∧ Dn ) ∈ ∆∗
Explain why.
1
So, as ¬(D1 ∧. . .∧Dn ) ∈ ∆∗ it follows that we must have ♦¬(D1 ∧. . .∧Dn ) ∈
Γ , and so by the fact that ♦A → ♦A ∈ Σ, that ♦¬(D1 ∧ . . . ∧ Dn ) ∈ Γ . So by
the fact that −1 Γ ⊆ ∆ it follows that we must have ♦¬(D1 ∧ . . . ∧ Dn ) ∈ ∆. But (as
you will show) this means that ∆ is inconsistent.
Part (iii):
Explain why the fact that (D1 ∧ . . . ∧ Dn ) ∈ ∆ and ♦¬(D1 ∧
. . . ∧ Dn ) ∈ ∆ means that ∆ is inconsistent, ideally by providing an appropriate derivation.
But ∆, being by hypothesis a world in W Σ must be consistent. And so by reductio
ad absurdem it follows that Θ− must also be consistent. So by Lindenbaum’s Lemma
there is a complete Σ-consistent set Θ extending it in W Σ , and the result follows by
Part (i) above.
Question 2 (10 marks): This question is about filtrations. Consider the following model
M2 = hW2 , R2 , V2 i, where:
• W2 = {0} ∪ {na, nb|n ∈ Z+ } ∪ {n∗ , n† |n > 1 and n odd}1
• R2 = {h0, 1ai, h0, 1bi} ∪ {hna, mai|m = n + 1} ∪ {hnb, mbi|m = n + 1} ∪
{hna, m∗ i|n even and m = n + 1} ∪ {hnb, m† i|n even and m = n + 1}
• V2 (pi ) = ∅ for all i
Here’s a picture:
3∗
5∗
1a
2a
3a
4a
5a
…
1b
2b
3b
4b
5b
…
0
3†
5†
In this question we will be looking at what filtrations of this model through the
subformulas of ♦⊥ look like. We’ll go through this in stages.
1. Give the set Γ of subformulas of ♦⊥.
2. Let M∗2 be a filtration of M2 through the set of formulas Γ from the previous
question. Give W2∗ by describing which worlds in W2 are ≡Γ . Which members
of Γ are true in each equivalence class [w]≡Γ in W2∗ ?
1 Z+
is the set of all positive integers 1, 2, 3, . . .
2
3. What is the finest filtration of M2 through Γ ?
4. What is the coarsest filtration of M2 through Γ ?
3

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