INTRODUCTION TO MODAL LOGIC 2014 HOMEWORK 5

INTRODUCTION TO MODAL LOGIC 2014
HOMEWORK 5
•
•
•
•
Deadline: November 11 — at the beginning of class.
Electronic submissions can be sent to Giovanni Cina [email protected]
Grading is from 0 to 10 points.
Success!
(1) (2pt) Use the Sahlqvist algorithm to compute the first-order correspondent of the
formula:
(♦p ∧ ♦q) → (♦(p ∧ ♦q) ∨ ♦(p ∧ q) ∨ ♦(q ∧ ♦p)).
(2) (2pt) Prove
(a) `K ϕ → (ψ → ϕ)
(b) `K (♦ϕ ∧ (ϕ → ψ)) → ♦ψ
You may find it helpful to note that the following are propositional tautologies:
•
•
•
•
•
ϕ → (ψ → ϕ)
ϕ → (ϕ ∨ ψ)
(ϕ → ψ) → (¬ψ → ¬ϕ)
(ϕ → (ψ → χ)) ↔ ((ϕ ∧ ψ) → χ)
ϕ → (ψ → (ϕ ∧ ψ))
You can also use Proposition 10 from the notes.
(3) (2pt) Recall that
S5 = K + (p → p) + (p → p) + (p → ♦p).
Show that
`S5 ♦p → ♦p
You can use Proposition 10 from the notes.
(4) (2pt)
(a) Show that if a frame F is a bounded morphic image of a frame G, then Log(G) ⊆
Log(F).
(b) Let C be a non-empty class of frames. Prove that Log(C) is contained in the
logic of a single reflexive point or Log(C) is contained in the logic of a single
irreflexive point.
(c) Let Call be the class of all frames. Prove that for each frame class C its logic
Log(C) contains Log(Call ).
1
2
INTRODUCTION TO MODAL LOGIC 2014 HOMEWORK 5
(5) (2pt) Show that S5 is sound with respect to the class of frames (W, R), where R is
an equivalence relation.
(6) (2pt) (BONUS!) The exercise is only for readers who like syntactical manipulations
(and have a lot of time to spare, this is Exercise 1.6.5 from Blackburn et al.).
GL = K + ((p → p) → p).
Prove that `GL p → p. We already discussed in class how to do this semantically.