and 1 represents truth. (1) and (2) can be interpreted as inform al characterizations of
the semantics for “a n d ” and “o r”. W'here t; is a function that m aps sentences unto the set
{1,0), we can state them m ore precisely as
v { ^4> A ■0”') = 1 iff v {4>) = '^('0) = 1
v { ^4> V ■0”') = o iff v{(f)) =
= o
.\Iternatively, we can say the same using truth tables:
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f v 0 1 0 0 1 1 1 1 Va 0 1 0 0 0 1 0 1
In th e tw o v a liie d sem antics, th e t n it h vahies play th e ro le o f th e in g re d ie n t values. Every sentences is assigned a value o f the set {1 ,0 }. T o g e th e r w ith a recursive d e fin itio n fo r “ v ” , we can assign tr u th values to c o m p le x sentences. A t th e same tim e , the tr u th values play th e ro le o f asssertoric values. Assum e th a t w h e n we assess th e accuracy o f a d eclara tive u tte ra n c e , we d o so by classifyin g th e m as tru e o r false. T h a t is:
A n a ssertion o f (f) is a ccurate i f f (f) is tru e .
G iven th e re a d in g o f th e set o f tr u th values, we have the d e fin itio n that;
A sentence (f) is tru e ifi' = 1
T h e d is tin c tio n s r e rju ire d fo r th e [lurposes o f sem antics a n d th e purposes o f c h a ra c te riz in g the accuracy o f o u r assertions tu rn o u t to he the same. The o n ly d is tin c tio n we need to appeal to in b o th cases is th a t o f b e in g tru e versus b e in g false. H e n ce , in tw o-valued sem antics, th e tr u th values |o, i | are b o th assertoric a n d in g re d ie n t values.
T h is , how ever, is an a rte fa c t o f im p o v e ris h e d resources. The d is tin c tio n betw een as s e rto ric a n d in g re d ie n t va lu e is b lu rr e d because th e classical tr u th values can d o b o th th in g s in p re p o s itio n a l lo g ic . B u t as D u m m e tt p o in te d o u t, “ [tjh e r e is n o a j) r io r i reason w hy the tw o n o tio n s o f tru th -v a lu e s h o u ld c o in c id e ” . T h e e x a m p le D u m m e tt gives fo r w he n they can co m e a p a rt is m any-valued log ic:
As a te c h n ic a l study, m any-valued lo g ic has been d e v e lo p e d p rim a r ily as a m a th e m a tic a l g e n e ra liz a tio n o f tw o-valued lo g ic , w ith little re g a rd to in tu itiv e in te r p r e ta tio n . T h e tru th -v a lu e s are d iv id e d in to those th a t ra n k as ‘desig n a te d ’ a n d those w h ic h ra n k as ‘ u n d e s ig n a te d ’ : a fo rm u la is d e fin e d as v a lid if, u n d e r each a ss ig n m e n t o f tru th -va lu e s to its sentence-letters, it com es o u t as h a v in g a d e s ig n a te d value. .. T h u s an assertion m ade by u tte rin g a given sentence a m o u n ts to a c la im th a t th a t sentence has a d esig n a te d value: in o rd e re , th e re fo re , to grasp the c o n te n t o f any p a rtic u la r assertion, a ll th a t is nece.ssary is to k n o w th e c o n d itio n fo r the sentence u tte re d to have a des ig n a te d value. We d o n o t n eed, fo r th is p u rp o s e to k n o w a n y th in g a b o u t