are tnie th en the conclusion is true. N othing is being said about why the sentences in
volved in the argum ent are true, or about truth as such at all. Also, the objection rests on
a sim p lified c o n c e p tio n o f tru th ’s role in logical c o n se q u e n c e - w h ed ier we are m onists or pluralists. A gain it is useful to take as starting p o in t a (classical) two-valued logic, where
logical c o n se q u e n c e . If we are w orking with this c o n c e p tio n , th en there seem s to be one d o m ain invariant n o tio n o f truth involved. But n o tice, that’s g o t n o th in g to d o with the q u estion o f c o n se q u e n c e as such. T h e sen se in w hich logical c o n se q u e n c e involves a
“im iq u e truth p r ed ica te” is ju st the sam e sen se in w hich tn ith value ascription does.
If we take a step back from the pluralism issue, it is n o ticea b le that th ere is som eth in g o d d ab ou t T a p p o let’s two questions;
• WTiat predicate is u sed in an a cco u n t o f valid inferences?
• V\1iy is that p red icate n o t the only o n e we need?
W hen we m ove from p rep o sitio n a l to first order logic we m ust refine our n o tio n o f logical c o n se q u e n c e . In first ord er logic we d o n ’t co n cer n ourselves m u ch with tn ith simpliciter.
Instead we focu s o n a relativized n o tio n o f truth: truth in a model (relative to an assignm ent o f values to the variables). A m o d e l M consists o f a d om ain - a non-em [)ty set o f objects - and an in terp retation fu n ctio n that m aps nam es to objects, on e-p lace predicates to sets
o f objects and m any-place predicates to tuples o f objects. It’s alm ost the default view that the m ost accurate a cco u n t o f logical c o n se q u e n c e is given in term s o f truth preservation across m odels:
A form ula
A
is logical c o n se q u e n c e o f a set o f form ulas F iff for every 5 G F and every m o d e lM.,
ifv{B, Ai )
= i th env{A, Ai ) =
i .If th at’s the right a cco u n t we w ould have an answer to the first question: the predicate required is “truth in a m o d e l” (or equivalently “truth relative to a m o d e l”). But now the
se c o n d q u estio n seem s like a N o n sequitur. It d o e s n ’t follow that we d o n ’t n e e d another o n e truth p red icate b ecau se w e ’ve got truth in a m od el. In fact, it’s obvious that truth in
a m o d el isn ’t directly relevant for the pragm atic tasks that fall ou tsid e logic.
To return to the d istin ctio n from §3.1, truth in a m o d el (relative to an a.ssignment) is
required to specify in g red ie n t values o f se n ten ces. W liat’s special about the m o d e l theo retic d efin itio n o f logical c o n se q u e n c e is that it d o e s n ’t em p lo y a n o tio n that is relevant for c o n n e c tin g the tech n ical d efin itio n with ordinary assessm ents o f assertion and infer
en ce . For that p u rp ose, we n e e d to recover so m e n o tio n o f truth that plays the role o f
a s s e r to r ic v a lu e o f s e n te n c e s . In th is s e ttin g , all w e c a r e a b o u t w h e n w e assess t h e a c c u r a c y o f a s s e r tio n s is t r u t h in a n in te n d ed m o d e l. T h is w o u ld h e t h e m o d e l t h a t r e p r e s e n t s th e c o r r e c t e x t e n s i o n o f th e s i n g u l a r te r m s a n d p r e d ic a te s in t h e a c tu a l w o rld . B u t p r e s e r v a tio n o f tru th in the in te n d ed m odel is n o t s u f f ic ie n t f o r lo g ic a l c o n s e q u e n c e ; t h a t ’s m e r e c o n t i n g e n t t r u t h p r e s e r v a tio n . S o , if w e a r e i n t e r e s t e d in a m o d e l t h e o r e t i c n o t i o n o f c o n s e q u e n c e s , T a p p o l e t ’s q u e s t i o n s a r e m is g u id e d .
L e t ’s c o m e a t t h e is s u e f r o m a d i f f e r e n t a n g le . I f w e c o n s i d e r a g a in a m a n y -v a lu e d s e m a n t ic s w ith th e s e t o f t n i t h v a lu e s (o, i}, w e a c tu a lly n e e d to give a n a c c o u n t o f lo g ic a l c o n s e q u e n c e in te r m s o f a s s e r to r ic v a lu e s r a t h e r t h a n i n g r e d i e n t v a lu e s. W lia t m a t t e r s in m a n y -v a lu e d s e m a n t ic s is p r e s e r v a tio n o f designated value:
A n a r g u m e n t (F ,
0
) is v a lid iff f o r e v e iy s e n t e n c e il> E T a n d e v e ry v a lu a tio n v , if v { ip ) £ V t h e n v (0
) € T>.T h is d e f i n i t i o n is s h a r e d a c r o s s m a n y -v a lu e d s e m a n t ic t h e o r i e s . T h e r e is n o t one s p e c ific a c c o u n t o f v a lid a r g t u n e n t h e r e b e c a u s e it d e p e n d s o n w h a t v a lu e ( s ) w e c o i m t as d e s ig n a t e d . F o r c x a iiq ) le , s u p p o s e w e a c c e p t th e f o llo w in g a c c o u n t o f t h e c o n n e c tiv e s ;
0 1 1 1 2 2 1 0 Va 0 i 1 0 0 0 0 1 2 ° 2 2 1 0 i 1 0 21 1 0 1 1 1 1 1 1 1 2 2 2 1 0 2L 1 V v 0 21 1 0 0 21 1 1 1 1 2 2 2 1 1 1 1
H a v in g s e ttle d th e t r u t h f i u i c t i o n a l c h a r a c t e r o f t h e c o n n e c t iv e s d o e s n ’t s e ttle w h a t lo g i c a l c o n s e q u e n c e r e l a t i o n is in play. F o r e x a m p le , if w e say t h a t th e d e s i g n a t e d v a lu e s is {^, i) , w e a r riv e a t a p a r a c o n s i s t e n t lo g ic w h e r e ex fa ls a quodlibet d o e s n ’t h o l d , b u t th e law o f t h e e x c l u d e d m i d d l e d o e s ( P r ie s t, 2 0 0 6 ) . O n t h e o t h e r h a n d , if w e say th a t 1 is th e o n ly d e s i g n a t e d v a lu e , w e a r r iv e a t p a r a c o m p l e t e lo g ic t h a t v a lid a te s ex fa ls o quodlibet b u t n o t t h e law o f th e e x c l u d e d m id d l e . It t h e r e f o r e m a tte r s a lo t h o w w e s e ttle th e a s s e r to r ic val u e s o f s e n te n c e s w ith in th is c o n t e x t. I f w e a r e s ta r t in g f r o m a m a n y -v a lu e d s e m a n t ic s th e s it u a ti o n is th e m i r r o r im a g e o f t h e m o d e l t h e o r e t i c a c c o i u u . N o w i t ’s t h e a s s e r to r ic (i.e . d e s i g n a t e d vs u n d e s i g n a t e d ) v a lu e s t h a t a r e r e le v a n t f o r d e f i n i n g lo g ic a l c o n s e q u e n c e s a n d n o t t h e i n g r e d i e n t v a lu e s (i.e . t r u t h v a lu e s ) . W e still n e e d tw o d i f f e r e n t p r e d ic a te s : o n e f o r h a n d l i n g lo g ic a l c o n s e q u e n c e a n d a s s e r tio n , a n d o n e f o r h a n d l i n g lo g ic a l c o n n e c tiv e s.
T h is p o in ts a g a in in th e d ire c tio n we s h o u ld g o in o r d e r to m a k e se n se o f p lu ra l ism. We hav e to se lect so m e a p p r o p r ia te c o m b in a tio n o f in g r e d ie n t a n d a sse rto ric values th a t a re p lu ralistically a c c e p ta b le a n d u se th o se to m a k e sen se o f th e tru th -p re s e rv in g c h a r a c te r o f th e c o n s e q u e n c e re la tio n . B ut th e r e is n o th in g in p r in c ip le th a t m akes th e p lu ra lis t’s p o s itio n d if f e r e n t fro m th o se w ho invoke a n o tio n o f c o n s e q u e n c e m o re c o m p le x th a n th e o n e u s e d in tw o-valued p ro p o s itio n a l logic. T a p p o le t’s a r g u m e n t seem s to re st o n a c o n fu s io n a b o u t w h a t is a t stak e in any a c c o u n t o f c o n s e q u e n c e .
3.3
Expressive D evice
In §3-1-1 I said p lu ra lism s h o u ld in c o r p o r a te a m o n a d ic tr u th p r e d ic a te . O n e o f th e re a so n s m e n tio n e d was th a t d o in g so m ak es it p o ssib le fo r th e p h u a lis t to a c c o u n t fo r th e ro le o f tr u th as a n ex p ressiv e device. T h is was d iscu ssed in §2.1, b u t i t ’s well w o rth c o n s id e rin g th e im p lic a tio n s th is has fo r h o w we s h o u ld m ak e se n se o f p lu ralism .
I l e r e ’s a te llin g e x a m p le . B efo re e m b a rk in g o n his P ersia c a m p a ig n , A le x a n d e r visited Pythia, I'h e O ra c le o f D e lp h i. U p o n e n te r in g th e te m p le h e says,
E v e ry th in g T h e O ra c le says is tru e .
This is a case o f b lin d e n d o r s e m e n t. W Tiatever else we c a n say a b o u t th e s itu a tio n is th a t “trvie” u s e d in this seiU en ce c a n n o t b e d o m a in in d e x e d . F o r o tir p u rp o s e s it’s b e st to re p r e s e n t th is as q u a n tify in g o v er p ro p o sitio n s: