Matthew B. answered • 04/17/19

Cheap and likes teaching.

Note that P ∨ ~P is a tautology, so it isn't really needed here. Nevertheless,

P ⊃ ~M from 1

~M ⊃ ~C by modus tollens of 2

Notice that 3 is logically equivalent to ~C ⊃ ~L

Hence, by transitivity, P ⊃ ~L

By 4,

(~P ⊃ ~E) • (~E ⊃ ~C)

By transitivity, ~P ⊃ ~C

By 3,

~C ⊃ ~L

By transitivity,

~P ⊃ ~L

Hence, by 5, P ∨ ~P ⊃ ~L ∨ ~L

Finally, ~L ∨ ~L ≡ ~L.

I'll leave it to you to put it into formal notation. The key here was transitivity. When given an "or" statement (i.e. ∨), working out the full implications of both sides can lead to an immediate solution.