IMPORTANT!!

As one member of our class recognized, there is a As one member of our class recognized, there is a major mistake on page 180 of the text where major mistake on page 180 of the text where the rule schemas for SD are laid out.the rule schemas for SD are laid out.

It symbolizes a rule it calls It symbolizes a rule it calls E2 – there is no such E2 – there is no such rule! – asrule! – asP P Q QQQPP

Such a rule is NOT truth preserving and not in SDSuch a rule is NOT truth preserving and not in SD

IMPORTANT!!

There is only one rule for eliminating the There is only one rule for eliminating the horseshoe (horseshoe (E). And it is symbolized properly E). And it is symbolized properly inside the front cover of the text and used inside the front cover of the text and used throughout the chapter.throughout the chapter.

P P Q QPPQQ

Less important…

I did not notice that this new edition has us add I did not notice that this new edition has us add the relevant rule following an auxiliary the relevant rule following an auxiliary assumption that starts a subderivation.assumption that starts a subderivation.

This is useful when you’re trying to go back to fill This is useful when you’re trying to go back to fill in line numbers especially if the derivation in line numbers especially if the derivation contains a lot of subderivations and auxiliary contains a lot of subderivations and auxiliary assumptions. assumptions.

Proving SD notions

Using derivations to prove thatUsing derivations to prove thata sentence of SL is a sentence of SL is a theorem in SDa theorem in SDa sentence a sentence PP is derivable in SDis derivable in SD from a set from a set of of sentences of SLsentences of SLan argument of SL is an argument of SL is valid in SDvalid in SDa set of sentences of SL is a set of sentences of SL is inconsistent in SDinconsistent in SDsentences sentences PP and and QQ are are equivalent in SDequivalent in SD

Show that⊦ A (B A)

A A A/A/II BB A/A/II AA 1 R1 R

B B A A II A A (B (B A) A) II

Can we show that⊦ A (B C)

A A A/A/II---------------------- BB A/A/II ---------- CC

B B C C IIA A (B (B C) C) II

An argument is valid in SD IFF its conclusion is derivable from the set consisting of its premises

Show that the following argument is valid in SD:Show that the following argument is valid in SD:

~A v ~B~A v ~BAA----------------------~B~B

1 1 ~A v ~B~A v ~B AA22 AA AA33 ~A~A A/vEA/vE

~B~B~B~B A/vEA/vE~B~B

~B~B vEvE

1 1 ~A v ~B~A v ~B AA22 AA AA33 ~A~A A/vEA/vE44 B B A/~IA/~I

~B ~B ~I~I~B~B A/vEA/vE~B~B RR

~B~B vEvE

1 1 ~A v ~B ~A v ~B AA22 A A AA33 ~A ~A A/vEA/vE44 B B A/~IA/~I5 A5 A 2R2R66 ~A ~A 3R3R77 ~B ~B 4-6 ~I4-6 ~I88 ~B ~B A/vEA/vE99 ~B ~B 8R8R10 ~B10 ~B 1, 3-7, 8-9 vE1, 3-7, 8-9 vE

There’s more than one way to derive a sentence, but some are easier…

1 ~A v ~B1 ~A v ~B AA2 A2 A AA

~B~B how about ~I?how about ~I?

There’s more than one way to derive a sentence, but some are easier…

1 ~A v ~B1 ~A v ~B AA2 A2 A AA33 BB A/~IA/~I

44 AA 2 R2 R

~A~A ~B~B how about ~I?how about ~I?

1 ~A v ~B1 ~A v ~B AA2 A2 A AA33 BB A/~IA/~I44 AA 2 R2 R5 5 ~A ~A A/vEA/vE66 ~A ~A 5R5R

77 ~B ~B A/vE A/vE 88 A A A/~I A/~I 9 B9 B 3 R3 R1010 ~B ~B 7R 7R 11 ~A11 ~A 8-10 ~I8-10 ~I12 ~A12 ~A 5-6, 7-11 vE5-6, 7-11 vE13 ~B13 ~B 3-12 ~I3-12 ~I

One special case of validity…

Show that the following argument is valid in SD:Show that the following argument is valid in SD:

A A B BA A ~B ~BAA

----------------------M M R R

Special cases…

1 A 1 A B B AA2 A 2 A ~B ~B AA33 A A AA

M M R R

Special cases…

1 A 1 A B B AA2 A 2 A ~B ~B AA33 A A4 ~(M 4 ~(M R) R) A/~EA/~E

BB

~B~B

M M R R ~E~E

Special cases…

1 A 1 A B B AA2 A 2 A ~B ~B AA33 A A4 ~(M 4 ~(M R) R) A/~EA/~E

55 B B 1, 3 1, 3 E E

6 ~B6 ~B 2, 3 2, 3 E E7 M 7 M R R 4-6, ~E4-6, ~E

P and Q are equivalent in SD IFF Q is derivable in SD from {P} and P is derivable in SD from {Q}

Show that the following pair of sentences is Show that the following pair of sentences is equivalent in SD:equivalent in SD:

AA~~A~~A

So we need 2 derivationsSo we need 2 derivations

Demonstrating equivalence

1 A1 A AA2 ~A2 ~A A/~IA/~I3 ~A3 ~A 2R2R44 AA 1R1R5 ~~A5 ~~A 2-4,~I2-4,~I

Demonstrating equivalence

1 ~~A1 ~~A AA

AA

Demonstrating equivalence

1 ~~A1 ~~A AA22 ~A ~A A/~EA/~E33 ~A ~A 2 R2 R44 ~~A ~~A 1 R1 R5 A5 A 2-4 ~E2-4 ~E

Demonstrating that a set is inconsistent in SD

A set A set is inconsistent in SD IFF there is some is inconsistent in SD IFF there is some sentence sentence PP such that both such that both PP and and ~P~P are are derivable from derivable from ..

A set A set is consistent in SD IFF there is no is consistent in SD IFF there is no sentence sentence PP such that both such that both PP and and ~P~P are are derivable from derivable from

Show that {A B, B ~A, A} is inconsistent in SD

1 1 A B A2 B ~A A3 A A4 A 3R

~A

Show that {A B, B ~A, A} is inconsistent in SD

1 1 A B A2 B ~A A3 A A4 A 3R5 B 1, 4 E6 ~A 2, 5 E