1.0 (Boolean theory),,, (boolean), : (theorem), (antitheorem),,,,, T,, T,,,, T (high voltage), (low voltage), (power) (ground) (boolean value), (nullary boolean operators),,,, (not) (), x(negation) ; (truth table) : T T, T, ;, T,,, ( ),... =,,,, x y(conjunction), xy (conjunct) x y(disjunction),(disjunct)x y (implication), x(antecedent), y(consequent) x y, x, yx=y(equation), 3
(left side), (right side)xy(unequation),, T T; T, ;, T, T T T T T T T T T T T T T T = T T T T, T T, T,, T,; T T, : ( T) T (T T),,, 9, 10,,, T T =, = 16,,,,, :,, 256,, (conditional composition), if x then y else z: TTT TT T T T TT T T if then else T T T T, n 2 2n, x y, xy(variables),, x y, ( ( T))x, ( T)y, : ( ( T )) ( T) (substitution), (instantiation),,,,, ( T )x, 4
, Ty,, x x T T, T,, x yt T T,,, (Number Theory), 1+12, 1+1=2,, ;, (unclassified), 1/0=5,,,,, (consistent);,, (inconsistent), (complete);,, (incomplete) 1.0.0,,, (Axiom):, ; (antiaxiom), (Evaluation):, (Completion):,, (Consistency):,, (Instance):, (instance) (axiom),, (antiaxiom) T, T,,, 5
,,,, T T = T, T T = T, : (T), (T T)=(T T = T) T x, x, ; x,,, x x, 1/0=5 1/0=5,,,,,,,,, expression 0, expression 0 expression l, expression l? expression l,, expression 0 expression l,,, expression l (detachment) (modus ponens), expression,, expression, expression, expression: expression expression,,,, 1+1=2,,, ----------------------------------------------------------------------------------------------------------------------------------------- (T ),,,, (law),,, 2,,, 1.0.1, 6
: a b c a b c, a b,,,,, :,, ( first part second part ),,, first part = second part (proof) : expression 0 0 = expression 1 1 = expression 2 2 = expression 3 3 expression 0 = expression 1 expression 1 = expression 2 expression 2 = expression 3 0 expression 0 = expression 1; 1 expression 1 = expression 2, = expression 0 = expression 3 (laws of portation) a b c a b c = a (b c) (Material Implication) = (a b) c (Duality) = a b c = a b c = a (b c),, :, (Duality) 7
, (a b) a b;,,,,, =,, (a b c = a (b c)), = ( (a b) c = a ( b c)) = ( a b c = a b c) = = T,, T ( a " b = a! b) = a = = ( a " b = ( a! b = a)) = T T!!,,, expression 0 = expression 1,,, = expression 2 -------------------------------------------------------------------------------------------------------------------------------------- 1.0.2, Conflation (a b) (c d) a c b d a c b d = (a c b) (a c d) = ((a b) (c b)) ((a d) (c d)) = (a b) (a d) (a b) (c d) (c b) (a d) (c b) (c d) (a b) (c d) =, a c b d (a b) (c d), (generalization) 8
a! a ( a! b) " ( b! a) = ( a = b) ( a! b) " ( b! c)! ( a! c)! a! b abastrongerb(weaker)! T a! b! c! a " c! b a b c! a c! b a c! a b c! b (monotonic) a! b! ( b! c)! ( a! c) (antimonotonic) a a a! b a b a! b a b a! b a b a! b a b if a then b else c b c 2(k) ( a " ( a! b)) a " a! b a! b a ( a! b) a " ( a! b) ( a " ( a! b)) ( a " ( a! b))! ( a! a) = T ( a # ( a " b))! T ( a " ( a! b)) ( a " ( a! b)) TTT T=!! =! a " ( a! b) " a! a =! (proof by contradition) a $ ( a # b)!" ( a " ( a! b)) T ------------------------------------------------------------------------------------------------------------------------------------ 1.0. 3, : 9
assumption ( expression 0 = expression 1 = expression 2 = expression 3 expression 0 = expression 1 assumption, expression1 = expression2,,,,, expression0 = expression3, assumption (expression0 = expression3) expression0, assumption expression0 assumption assumption, if then else if possibility then ( possibility) else (, possibility) something, something else, if possibility then something else something else, (Case Idempotent Law),, exp ression 0! exp ression1 = exp ression 0! exp ression2 exp ression1 exp ression2 exp ression0 exp ression0 exp ression0 exp ression1 exp ression2 exp ression 0! exp ression1 exp ression 0! exp ression2 exp ression 0! exp ression1 = exp ression 2! exp ression1 exp ression1 exp ression 0! exp ression1 = exp ression 2! exp ression3 exp ression0 exp ression 1 = exp ression3 exp ression1 exp ression 0 = exp ression2 a! a a a T a! a a a 10
= a! T a a T a! a a a = T! a a! a = T! T context exp ression 0! exp ression1 exp ression0 exp ression1 exp ression 0! exp ression1 exp ression1 exp ression0 exp ression 0! exp ression1 exp ression0 exp ression1 exp ression 0! exp ression1 exp ression1 exp ression0 exp ression 0! exp ression1 exp ression0 exp ression1 exp ression 0! exp ression1 exp ression1 exp ression0 exp ression 0! exp ression1 exp ression0 exp ression1 exp ression 0! exp ression1 exp ression1 exp ression0 if exp ression 0 then exp ression1else exp ression2, exp ression1 exp ression0 if exp ression0 then exp ression1else exp ression2 exp ression2 exp ression0 2(k) ( a " ( a! b)) ( a " ( a! b)) a ( a! b) = ( a! ( T! b))! = ( a! T = ( a "!)! =! = T --------------------------------------------------------------------------------------------------------------------------------------------------- 1.0.4,, (),,,, (),,,,, 11
,, and, also, but, yet, however moreover They're red, ripe, and juicy, but not sweet red ripe juicy sweet or, They are either small or rotten. small rotten, small rotten; Either we eat them or we preserve them., eat preserve, if, =, If it rains, we ll stay home., rain home If it snows, we can go skiing. and if it doesn t, we can t snow=ski --------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------- 1.1 (number theory),,, +x x x x+y x y x y x / y x xy xy xy xy x y xy if a then x else y x,y, a x < y x y x > y x y x = y x y xy xy xy xy xy, xy xy, xy,,, x (extremes) #! < x "! + x =! (absorption) 12
,, 1/0=5 0<( 1) 1/2 -------------------------------------------------------------------------------------------------------------------------------------------------------- 1.2 (character)`a A, `1 1, `, ``, succ(), pred(), = < > if then else ------------------------------------------------------------------------------------------------------------------------------------------------- =,, if then else, (Generic),,, x=x,,, 5=5 <,, >,,, ----------------------------------------------------------------------- --------------------------------------------------------------------------- (bunch) (set)(string)(list),(function) (predicate)(relation)(specification)(program),,, 13