Definition of opacity

Quine in Word and Object gives a criterion of purely referential position namely that it must be subject to the substitutivity of identity.¹ Other positions are opaque.

See http://dennisdarland.com/philosophy/naming.html (I define ‘belief’)

And http://dennisdarland.com/philosophy/identity_dd.html

Why does opacity matter?

Quine does not permit quantification into opaque contexts. And quantifying (being the value of a variable) is Quine’s criterion for being – ontological status. So the ontological status of mental objects depends on opacity.

Intensional contexts

There are traditionally both extensional and intensional contexts in propositions. Extensional contexts are truth functional. E.g. “p & q” depends truth functionally on “p” and “q”. It is extensional. But S believes “p” does not depend truth functionally on “p”. S may believe either p or ~p independently of whether p or ~p.

Applying my definition of belief to intensions

Review my definition at:

http://dennisdarland.com/philosophy/naming.html

So back to applying the definition of belief to intension:

S believes “p” at time t does not involve “p”.

E.g. when p = R(a,b,c) it means

(
belief_rS,t,w,x,y,z)

And

Symbol_1r(S,t,w,R)
And
Symbol_0r(S,t,x,a)
And
Symbol_0r(St,y,b)
And
Symbol_0r(S,t,z,c)
}

Which does not involve R(a,b,c)

But, as intensional functions of functions are defined in Principia Mathematica (pp. 72,73), belief is still an intensional function of the function R