Stuck on proof and need help to solve it

Question

1. This is not homework
2. I do it for fun.

I am using the SD system.

Derive $L\supset T$

1. $L\supset\left ( C\vee T \right ).$ Assume
2. $\left ( \sim L\vee B \right )\wedge\left ( \sim B\vee\sim C \right ).$ Assume

As we see from the image it is valid in SL truth tree form

There does not seem to be an easy way to derive it. I figure the the easiest method is to derive a contradiction, but how to do it after the initial set up ?

4. $\sim\left ( L\supset T \right ).$ Assume

For "$Q$" and "$\sim Q$" from #2, we could use $B$ and $\sim B,$ but how to separate them so they could be used ?

Any help would greatly appreaciated.

Answer 1

A proof by contradiction for this one will be more work than doing a conditional proof, i.e. start a subproof, assume $L$, try to get to $T$, end the subproof, and perform a Conditional Introduction to get $L \supset T$

The rest of this proof is doing the Disjunctive Syllogism pattern over and over. So, if you have the Disjunctive Syllogism rule from SD+ this should actually be quite straightforward:

\begin{array}{lll} 1. & L \supset (C \lor T) & Assumption\\ 2. & (\sim L \lor B) \& (\sim B \lor \sim C) & Assumption\\ 3. & \quad L & Assumption\\ 4. & \quad \sim L \lor B & \& \ Elimination \ Premise \ 2\\ 5. & \quad \sim \sim L & Double \ Negation \ 3\\ 6. & \quad B & Disjunctive \ Syllogism \ 4, 5\\ 13. & \quad ...\\ ? & \quad T\\ ?+1 & L \supset T & \supset \ Introduction \ 3-?\\ \end{array}

In SD this will be a lot more tedious., but I'll show you how to do one of these. In particular, once you have assumed $L$ as part of your conditional proof, you should be able to get to $B$ using $\sim L \lor B$:

\begin{array}{lll} 1. & L \supset (C \lor T) & Assumption\\ 2. & (\sim L \lor B) \& (\sim B \lor \sim C) & Assumption\\ 3. & \quad L & Assumption\\ 4. & \quad \sim L \lor B & \& \ Elimination \ Premise \ 2\\ 5. & \quad \quad \sim L & Assumption\\ 6. & \quad \quad \quad \sim B & Assumption\\ 7. & \quad \quad \quad L & Reiteration \ 3\\ 8. & \quad \quad \quad \sim L & Reiteration \ 5\\ 9. & \quad \quad B & \sim \ Elimination \ 6-8\\ 10. & \quad \quad B \ & Assumption\\ 11. & \quad \quad B \ & Reiteration \ 10\\ 12. & \quad B & \lor \ Elimination \ 4, 5-9,10-11\\ 13. & \quad ...\\ ? & \quad T\\ ?+1 & L \supset T & \supset \ Introduction \ 3-?\\ \end{array}

Answer 2

Hint: Start with your given assumptions. Assume L. Use $P\to Q ~\equiv \neg P \lor Q$ to make a chain of implications to obtain T from L.