Study of a self-transparent believer (1)

Beliefs are often, perhaps even typically, false. I will call believers self-transparent, however, exactly if they are accurate in one respect: their beliefs about what they believe are entirely correct. To be precise, an agent is a self-transparent believer iff that agent believes that s/he believes that A if and only if s/he does in fact believe that A.

As will appear somewhere below, my motivation for this has much to do with the issues about Moore’s paradox discussed in previous posts — self-transparency automatically protects against falling into Moore’s paradox — but I will not dwell on it for now.

We can set up a simple language for belief attribution, for this case, along the lines of propositional normal modal logic. With B as modal operator for “The agent believes that” the characteristic logical principle will then be

BBp if and only if Bp

But when I’ve done that, I want to turn the focus to the content of such a believer’s body of beliefs. That is, I want to ask the question:

(*) If X is the set of propositions that this believer believes, what is X like?

Since the semantics will of the possible world type, the question will always pertain to a given world, and the set in question, which I will call a belief set, will be

{p: Bp is true in world w}

and I will make question (*) more precise and definite, by asking for the exact properties of this consequence relation:

Y —> p if and only if every belief set that contains set Y also contains p

which I will call the doxastic consequence relation.

I expect this relation to have some unusual features.

The logical system LSTB

The language has the usual vocabulary of normal modal logic, though with the modal operator symbolized as B. The axioms and rules are those of the minimal normal logic K, plus two axioms that spell out the self-transparency:

R0. If p is a theorem of classical propositional logic then ⊢ p

R1. If ⊢ p then ⊢ Bp

A1. ⊢ if B(if p then q) then (if Bp then Bq)
A2. ⊢ if Bp then BBp

A3. ⊢ if BBp then Bp

A4. ⊢ if Bp then ~B~p

The phrase “if … then” is the material conditional and “⊢ p” is read as “p is a theorem”. There is a derivative logical consequence relation: p is logical consequence of set X of sentences exactly if (if q then p) is a theorem for some sentence q that is the conjunction of sentences which belong to X.

By this definition and axiom A1 it follows that if q follows from p1 …, pk then Bq follows from Bp1, …, Bpk. That is, the notion of belief here is such that it includes all that is believed implicitly, in the sense of being something that follows logically from what is believed.

The addition of A4, familiar from deontic logic, rules out from consideration the agent whose beliefs are inconsistent.

The semantic analysis

A model structure is a couple M = <W, R> where W is a non-empty set, the worlds, and R is a binary relation on W, relative doxastic possibility. I will specify properties of R below.

Intuitively, if x and y are worlds, and xRy, then the beliefs the agent holds in world x do not rule out that y is the actual world, i.e. all the agent’s beliefs in x are true in y.

A valuation over M is an assignment of truth-values T, F to all sentences at each world in M, such that the truth tables for connectives &, ~, v, if … then are obeyed, and in addition:

v(x, Bp) = T iff v(y, p) is true in all worlds y such that xRy.

A sentence is valid iff it receives T from all valuations, at all worlds, over all model structures. Set of sentences X semantically entails sentence p iff p receives truth value T from all valuations over all model structures at all worlds therein.

But R needs to have special properties if the axioms are to present valid sentences. The condition on R to ensure validity for A2 (if Bp then BBp) is well-known from normal modal logic S4: R must be transitive.

Similarly, to ensure validity for A4, the requirement is again familiar: For each possible world there is a doxastically possible world as well: if x is a world there is a world y such that xRy. We can refer to the set of worlds to which x bears R as R(x). Then A4 says in effect that R(x) is never empty. A relation with this property is called serial.

What ensures the validity of A3, which is the converse of A2, is not as familiar. Here we may have recourse to a paper by Frederic Fitch (Journal of Philosophical Logic 1973) that deserves to be seminal. Fitch shows how there is a simple recipe in the traditional calculus of relations which relates modal logic principles with properties of the relative possibility relation. I won’t put it in those traditional terms, now generally unfamiliar, but the required property for axiom A3. is this

R is weakly reflexive: for any worlds x, y, if xRy then there is a world z such that xRz and zRy.

We do not want R to be reflexive, though that would guarantee the validity of A3, but would also validate (if Bp then p). The reasons for calling this property weak reflexivity are two. First, if R is reflexive, so that xRx for all x, then it is weakly reflexive. Secondly, in the above formula we detect a threat of infinite regress, for of course it implies that if xRz then there must be some world u such that xRu and uRz, and so forth and so forth. But the regress stops, and the property holds, if any of those worlds is possible relative to itself. For example if xRy and yRy then y itself can serve as the “middle” world — a dash of reflexivity will do.

We have not yet begun to address the crucial question

(*) If X is the set of propositions that this believer believes, what is X like?

but I stop here for now, just to add an Appendix to show that this semantic analysis is adequate for the logical system LSTB.

APPENDIX

As Fitch already pointed out, the completeness proofs for normal modal logics are easily adapted if some axiom is added and R is required to have the corresponding property defined by his recipe. So I will only show that the one unfamiliar feature, axiom A3. is satisfied.

Suppose that in world x, Bp is false

So there is a world y such that xRy and p is false in y

By weak reflexivity, there is a world z such that xRz and zRy

Since zRy and p is false in y, it follows that Bp is false in z

But then BBp is false in x, since xRz.

There is another way to think about this, which I find helpful. BBp is true in x iff for all y and z, if xRy and yRz then p is true in z. You might say: precisely if p is true in all the worlds ‘doubly-connected’ to x. But then, could Bp still be false in x? Yes, if p is false in some world ‘singly-connected’ to x. But by weak reflexivity, there are no singly-connected worlds. For if xRy then there is also access from x to y via some world z, so that this putatively single connection is actually a double connection.