ASSIGNMENT
Complete the following exercises (i) using the Word version of this document to type in your answers (this option is recommended; if you adopt it please delete the extra spaces between the problems to enhance readability), OR (ii) using very clear and legible handwriting in the printout of this document and then scanning or taking snapshots of all the pages. The spaces between the problems are for those using the second option. No matter how you do it, .
- Show that the following derivability claim holds in SD or SD+ (20 points):
{[(A É B) ∨ (~ B É A)] É C, (D & C) É ~ D} ⊢ ∼ D
[Do not skip ‘DN’ steps!]
- Show that the following set of SL sentences is inconsistent in SD or SD+ (30 points):
{[∼ K ∨ (L & M)] ≡ N, ~ L⊃ ~ K, ~ (M & N) & ~ (~ M & ~ N), N ⊃ K}
[Do not skip ‘DN’ steps!]
- Show that ‘((X ≡Y) ∨(X ≡ Z)) ∨ (Y ≡ Z)’ is a theorem in SD or SD+ (30 points):
[Do not skip ‘DN’ steps!]
By using the natural deduction format and the style of proof similar to the “Proof Theory Exercise” we went over in class, show that the rule ⊃I is eliminable from SD in favor of these two rules in its place:
n ½~ P
½
½P ⊃ Q n ⊃I-right
n ½Q
½
½P ⊃ Q n ⊃I-left
- Note: you can only use the other 10 SD rules in your schematic Fitch-style demonstration; in particular, you are notallowed to use any SD+ rules, or any other derived results.
- the latter concerns the dispensabilityof a certain rule, while the former concerns its replaceability with two other rules. In that respect, this task is similar to that of the “Proof Theory Exercise.”
