LMPL_interpretations

LMPL_interpretations - Working with LMPL Interpretations...

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Working with LMPL Interpretations Philosophy 12A June 17, 2010 1 Working with Given LMPL Interpretations Consider the following ( given ) LMPL interpretation I 1 : ( I 1 ) F G H I J α + + - + - β - - - + + γ + - - - + In other words, the interpretation I 1 has the following features: D = { α,β,γ } , Ext (F) = { α,γ } , Ext (G) = { α } , Ext (H) = ∅ (where, is the null set ), Ext (I) = { α,β } , and Ext (J) = { β,γ } . Question : What are the I 1 -truth-values of ? Ja ( x)[Jx (Gx Fx)] Fc Ic ± ( x)Gx ( y)(Fy Gy) ( x)(Jx Hx) ( y)( x)[Gy (Jx (Ix Fx))] Solutions : Ja ’ is true on I 1 . This is because ‘ Ja ’ is false on I 1 , since α Ext (J) . Fc Ic ’ is false on I 1 . This is because its antecedent Fc ’ is true on I 1 , since γ Ext (F) ; but its consequent Ic ’ is false on I 1 , since γ Ext (I) . ( x)(Jx Hx) ’ is true on I 1 . The instances of ‘ ( x)(Jx Hx) ’ on I 1 are: ( i ) ‘ Ja Ha ’, ( ii ) Jb Hb ’, and ( iii ) ‘ Jc Hc ’. Instances ( ii ) and ( iii ) are false on I 1 ( why ?). But, instance ( i ) is true on I 1 , because ‘ Ja ’ and ‘ Ha ’ are both false on I 1 , since α Ext (J) and α Ext (H) . 1 ( x)[Jx (Gx Fx)] ’ is false on I 1 . The instances of ‘ ( x)[Jx (Gx Fx)] ’ on I 1 are as follows: ( i ) ‘ Ja (Ga Fa) ’, ( ii ) ‘ Jb (Gb Fb) ’, and ( iii ) ‘ Jc (Gc Fc) ’. Instances ( i ) and ( iii ) are true on I 1 ( why ?). But, instance ( ii ) is false on I 1 . This is because ‘ Jb ’ is true on I 1 , since β Ext (J) ; but ‘ Gb Fb is false on I 1 , since β Ext (G) and β Ext (F) . 2
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LMPL_interpretations - Working with LMPL Interpretations...

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