Derivation Rules and Justification. The Simple Rules mA nB oA & Bn, m &I mA & B nAn &E oBn &E mA B...
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Transcript of Derivation Rules and Justification. The Simple Rules mA nB oA & Bn, m &I mA & B nAn &E oBn &E mA B...
The Simple Rules
m An Bo A & B n, m &I
m A & Bn A n &Eo B n &E
m A Bn Ao B m,n E
m A Bn Bo A m,n Ep B m,o E
m An A B m Io Bp A B o I
m An A m R
&Em A & Bn A n &Eo B n &E
, A & B A
, A & B B
, A, B , A & B AT T T T T TT T F T F TT F T T F FT F F T F F
, A, B , A & B BT T T T T TT T F T F FT F T T F TT F F T F F
&Em A & Bn A n &Eo B n &E
, A & B A
, A & B B
, A, B , A & B AT T T T T TT T F T F TT F T T F FT F F T F F
, A, B , A & B BT T T T T TT T F T F FT F T T F TT F F T F F
&Im An Bo A & B n, m &I
, A, B A & B
, A, B , A, B A & BT T T T T T TT T F T T F FT F T T F T FT F F T F F F
&Im An Bo A & B n, m &I
, A, B A & B
, A, B , A, B A & BT T T T T T TT T F T T F FT F T T F T FT F F T F F F
I, A A B
, B A B
m An A B m Io Bp A B o I
, A, B , A A B T T T T T TT T F T T TT F T T F FT F F T F F
, A, B , B A BT T T T T TT T F T F TT F T T T TT F F T F F
I, A A B
, B A B
m An A B m Io Bp A B o I
, A, B , A A B T T T T T TT T F T T TT F T T F FT F F T F F
, A, B , B A BT T T T T TT T F T F TT F T T T TT F F T F F
Em A Bn Bo A m,n Ep B m,o E
, A B, A B
, B A, A B
, A, B , A B, A BT T T T T T TT T F T F T FT F T T F F TT F F T T F F
, A, B , A B, B AT T T T T T TT T F T F F TT F T T F T FT F F T T F F
Em A Bn Bo A m,n Ep B m,o E
, A B, A B
, B A, A B
, A, B , A B, A BT T T T T T TT T F T F T FT F T T F F TT F F T T F F
, A, B , A B, B AT T T T T T TT T F T F F TT F T T F T FT F F T T F F
The Complex Rulesm A Assumptionn Bo A B m-n I
m A Assumptionn B
o Bp Aq A B m-n, o-p I
m A B
n A Assumptiono C
p B Assumptionq Cr C m, n-o, p-q I
m A Assumptionn Bo Bp A m-o I
m A Assumptionn Bo Bp A m-o E
Im A Assumptionn Bo A B m-n I
If , A B, then A B
, A, B If , A B, then A BT T T T T T T TT T F T T F T FT F T T F T T TT F F T F F T T
Im A Assumptionn Bo A B m-n I
If , A B, then A B
, A, B If , A B, then A BT T T T T T T TT T F T T F T FT F T T F T T TT F F T F F T T
I
If , A B, and , B A, then A B
, A, B If , A B, and , B A, then A BT T T T T T T T T T TT T F T T F T F T T FT F T T F T T T F T FT F F T F F T F F T T
m A Assumptionn B
o Bp Aq A B m-n, o-p I
I
If , A B, and , B A, then A B
, A, B If , A B, and , B A, then A BT T T T T T T T T T TT T F T T F T F T T FT F T T F T T T F T FT F F T F F T F F T T
m A Assumptionn B
o Bp Aq A B m-n, o-p I
Em A B
n A Assumptiono C
p B Assumptionq Cr C m, n-o, p-q I
If , A B, A C, and , A B, B C, then , A B C
Em A B
n A Assumptiono C
p B Assumptionq Cr C m, n-o, p-q I
If , A B, A C, and , A B, B C, then , A B C
, A, B, C If , A B, A C, and , A B, B C, then , A B CT T T T T T T T T T T T T T TT T T F T T T F T T T F T T FT T F T T T T T T T F T T T TT T F F T T T F T T F F T T FT F T T T T F T T T T T T T TT F T F T T F F T T T F T T FT F F T T F F T T F F T T F TT F F F T F F F T F F F T F F
Em A B
n A Assumptiono C
p B Assumptionq Cr C m, n-o, p-q I
If , A B, A C, and , A B, B C, then , A B C
, A, B, C If , A B, A C, and , A B, B C, then , A B CT T T T T T T T T T T T T T TT T T F T T T F T T T F T T FT T F T T T T T T T F T T T TT T F F T T T F T T F F T T FT F T T T T F T T T T T T T TT F T F T T F F T T T F T T FT F F T T F F T T F F T T F TT F F F T F F F T F F F T F F
¬Im A Assumptionn Bo Bp A m-o I
If , A B and , A B, then A
, A, B If , A B and , A B, then AT T T T T T T T F T FT T F T T F T T T T FT F T T F T T F F T TT F F T F F T F T T T
¬Im A Assumptionn Bo Bp A m-o I
If , A B and , A B, then A
, A, B If , A B and , A B, then AT T T T T T T T F T FT T F T T F T T T T FT F T T F T T F F T TT F F T F F T F T T T
¬Em A Assumptionn Bo Bp A m-o E
If , A B and , A B, then A
, A, B If , A B and , A B, then AT T T T T T T T F T FT T F T T F T T T T FT F T T F T T F F T TT F F T F F T F T T T