Solutions to Chapter 19 Additional rules for TFL ‣ Solutions to selected exercices ‣ forall x: Calgary

A. The following proofs are missing their citations (rule and line numbers). Add them wherever they are required:

Line number	Formula	Justification
$1$	$W\rightarrow\neg B$
$2$	$A\wedge W$
$3$	$B\vee(J\wedge K)$
$4$	$W$	$\wedge$ E $2$
$5$	$\neg B$	$\rightarrow$ E $1$ , $4$
$6$	$J\wedge K$	DS $3$ , $5$
$7$	$K$	$\wedge$ E $6$

Line number	Subproof level	Formula	Justification
$1$	0	$L\leftrightarrow\neg O$
$2$	0	$L\vee\neg O$
$3$	open subproof, 1	$\neg L$
$4$	1	$\neg O$	DS $2$ , $3$
$5$	1	$L$	$\leftrightarrow$ E $1$ , $4$
$6$	1	$\bot$	$\neg$ E $5$ , $3$
$7$	close subproof, 0	$\neg\neg L$	$\neg$ I $3$ – $6$
$8$	0	$L$	DNE $7$

Line number	Subproof level	Formula	Justification
$1$	0	$Z\rightarrow(C\wedge\neg N)$
$2$	0	$\neg Z\rightarrow(N\wedge\neg C)$
$3$	open subproof, 1	$\neg(N\vee C)$
$4$	1	$\neg N\wedge\neg C$	DeM $3$
$5$	1	$\neg N$	$\wedge$ E $4$
$6$	1	$\neg C$	$\wedge$ E $4$
$7$	open subproof, 2	$Z$
$8$	2	$C\wedge\neg N$	$\rightarrow$ E $1$ , $7$
$9$	2	$C$	$\wedge$ E $8$
$10$	2	$\bot$	$\neg$ E $9$ , $6$
$11$	close subproof, 1	$\neg Z$	$\neg$ I $7$ – $10$
$12$	1	$N\wedge\neg C$	$\rightarrow$ E $2$ , $11$
$13$	1	$N$	$\wedge$ E $12$
$14$	1	$\bot$	$\neg$ E $13$ , $5$
$15$	close subproof, 0	$\neg\neg(N\vee C)$	$\neg$ I $3$ – $14$
$16$	0	$N\vee C$	DNE $15$

B. Give a proof for each of these arguments:

1.

$E\vee F$ , $F\vee G$ , $\neg F\therefore E\wedge G$

Line number	Formula	Justification
$1$	$E\vee F$
$2$	$F\vee G$
$3$	$\neg F$
$4$	$E$	DS $1$ , $3$
$5$	$G$	DS $2$ , $3$
$6$	$E\wedge G$	$\wedge$ I $4$ , $5$

2.

$M\vee(N\rightarrow M)\therefore\neg M\rightarrow\neg N$

Line number	Subproof level	Formula	Justification
$1$	0	$M\vee(N\rightarrow M)$
$2$	open subproof, 1	$\neg M$
$3$	1	$N\rightarrow M$	DS $1$ , $2$
$4$	1	$\neg N$	MT $3$ , $2$
$5$	close subproof, 0	$\neg M\rightarrow\neg N$	$\rightarrow$ I $2$ – $4$

3.

$(M\vee N)\wedge(O\vee P)$ , $N\rightarrow P$ , $\neg P\therefore M\wedge O$

Line number	Formula	Justification
$1$	$(M\vee N)\wedge(O\vee P)$
$2$	$N\rightarrow P$
$3$	$\neg P$
$4$	$\neg N$	MT $2$ , $3$
$5$	$M\vee N$	$\wedge$ E $1$
$6$	$M$	DS $5$ , $4$
$7$	$O\vee P$	$\wedge$ E $1$
$8$	$O$	DS $7$ , $3$
$9$	$M\wedge O$	$\wedge$ I $6$ , $8$

4.

$(X\wedge Y)\vee(X\wedge Z)$ , $\neg(X\wedge D)$ , $D\vee M$ $\therefore$ $M$

Line number	Subproof level	Formula	Justification
$1$	0	$(X\wedge Y)\vee(X\wedge Z)$
$2$	0	$\neg(X\wedge D)$
$3$	0	$D\vee M$
$4$	open subproof, 1	$X\wedge Y$
$5$	1	$X$	$\wedge$ E $4$
$6$	close subproof, open subproof, 1	$X\wedge Z$
$7$	1	$X$	$\wedge$ E $6$
$8$	close subproof, 0	$X$	$\vee$ E $1$ , $4$ – $5$ , $6$ – $7$
$9$	open subproof, 1	$D$
$10$	1	$X\wedge D$	$\wedge$ I $8$ , $9$
$11$	1	$\bot$	$\neg$ E $10$ , $2$
$12$	close subproof, 0	$\neg D$	$\neg$ I $9$ – $11$
$13$	0	$M$	DS $3$ , $12$