Question

Complete the formal proof of (~Q→~R)v(R&~Q) from no premises. The empty premise line is not numbered. Hint: there are longer ways of doing the proof that require the 5-step plan in the middle somewhere, but we require to you find the shortcuts and do it in 11 lines.

Use -> for arrow, # for contradiction; justify subproof assumptions with Assume; always drop outer parentheses; no spaces in PROP.

Answer 1

To complete the formal proof of (~Q→~R)v(R&~Q) from no premises, you can use the disjunctive syllogism.

Step-by-step explanation:

To complete the formal proof of (~Q→~R)v(R&~Q) from no premises, you can use the disjunctive syllogism. Here's a step-by-step proof:

1. Assume ~(~Q→~R) as a subproof assumption.
2. Use disjunction introduction to derive (~Q→~R)v(R&~Q).
3. Assume ~(R&~Q) as a subproof assumption.
4. Use disjunction introduction to derive (~Q→~R)v(R&~Q).
5. Assume ~Q as a subproof assumption.
6. Use reductio ad absurdum on the assumption ~Q to derive Q.
7. Use conjunction introduction to derive (Q & ~Q).
8. Use contradiction elimination to derive ~(~Q→~R).
9. Use disjunction introduction to derive (~Q→~R)v(R&~Q).

Answer 2

The question requires constructing a formal proof of (~Q→~R)v(R&~Q) using logical principles like Reductio Ad Absurdum and disjunctive syllogisms, within an 11-line constraint and without any given premises.

Step-by-step explanation:

The student is asking for help in constructing a formal proof in propositional logic. This specific proof requires finding a concise argument path that arrives at the conclusion (~Q→~R)v(R&~Q) without any given premises, using only 11 lines. A hint suggests that finding shortcuts is essential, rather than using the longer 5-step plan typically associated with logical proofs. The proof likely employs logical principles such as Reductio Ad Absurdum (RAA), disjunctive syllogisms, and rules for introducing and eliminating logical operators.