Show that q→r based on the given information

Question

I'm a bit unsure about how to approach this problem. Any help is appreciated.

Given the following: $$\begin{align} (p \wedge t) &\to (r \vee s) \\ q &\to (u \wedge t) \\ u &\to p \\ ¬s& \end{align}$$ Show that $q \to r$.

Answer 1

Show that $q\to r$.

What you require is a "conditional proof".   Assume $q$, derive $r$, thereby deducing that $q\to r$.

To get you started, (presuming you are familiar with Fitch style proof displays) your proof should look somewhat like this ...:

$\def\fitch#1#2{~~\begin{array}{|l}#1\\\hline #2\end{array}} \fitch{~1.~q\to (u\land t)\\~2.~u\to p\\~3.~(p\land t)\to (r\lor s)\\~4.~\neg s}{\fitch{~5.~q}{~6.~u\wedge t\quad: 1,5,\to\textsf{elimination (modus ponens)}\\~~~\vdots\\12. ~r}\\13.~q\to r}$

Fill in the missing details.

Answer 2

(1) q. P(presuppositions)

(2) q→(u∧t). P

(3) u∧t. (1),(2),I

(4) u. (3),I

(5) t. (3),I

(6) u→p. P

(7) p. (4),(6),I

(8) p∧t. (5),(7),I

(9) (p∧t)→(r∨s). P

(10)r∨s. (8),(9),I

(11)￢s. P

(12) r. (10),(11),I

(13) q→r. CP