Study of a self-transparent believer (2) Logic not classical

The beliefs that a believer can express are the contents of the beliefs s/he has. In terms of the semantics sketched in the previous post, the contents of the agent’s beliefs in world x is the agent’s belief set. Its members are the propositions which are true in every world doxastically possible relative to x, the set R(x), which is the set { y in W: xRy}.

My interest is in the logic of expression of belief, that is to say, in what I called the doxastic consequence relation. That relation will be determined by what is characteristic of belief sets. So let’s begin by asking what is in a belief set.

I will use the same letters referring to sentences and to propositions. Actually, whenever “p” stands for a sentence it is in red ink, and otherwise in green ink, but the colors don’t show up here — I’m sure the reader will have no difficulty recognizing what is meant in each case.

Four characteristics of belief sets

Consider the belief set X of the agent in world x.

T1. if Bp is in X, then p is in X

For if Bp is in X then BBp is true in x, and by the relevant axiom of LSTB, Bp is true in x. Therefore p is true in all members of R(x), and hence belongs to X.

T2. if p is in X, then Bp is in X

For if p is in X then Bp is true in x, hence by the relevant axiom of LSTB, BBp is true in x, and so Bp is in X.

T3. There is a sentence p such that neither (if Bp then p) nor (if p then Bp) belongs to all belief sets

We can give a single example to show that ~B(if Bp then p) and ~B(if p then Bp) are satisfiable, for simple (atomic) sentences p. For example, there is a model structure <W, R> in which world x is such that R(x) = W. So B(if Bp then p) is true in x if and only if (if Bp then p) is true in all worlds. But beliefs are generally false in many worlds where they are held — typically there will be another world y such that p is false in y but true in all members of R(y). Similarly, (if p then Bp) is not likely to be true in all worlds, in fact, it is rarely the case, even logically speaking, that an agent believes everything that is true.

T4. Neither (Bp & ~p) nor (~Bp & p) belongs to any belief set.

For suppose that (Bp & ~p) is true in all members of R(x). Then both Bp and ~p are true in all members of R(x). So BBp and B(~p) are true in x. By the relevant axiom of LSTB, Bp and B(~p) are both true in x, which implies that both p and ~p are true in all members of R(x). By axiom A4, R(x) is not empty, so there is a world in which p and ~p are both true, which is impossible.

Equally, suppose that (~Bp & p) belongs to belief set X. Then p is in X, hence by T2. above, Bp is in X. So both ~Bp and Bp are in X, but axiom A4 guaranteed that a belief set is not inconsistent.

Another way to think of this is as follows. The examples to look at here are cases in which p is true in some worlds and also false in some worlds that are doxastically possible. Then neither p nor Bp belongs to the belief set, and hence neither does any conjunction in which one is a conjunct.

Characteristics of the doxastic consequence relation

Whatever is the logic of this consequence relation — let us call it LDOX — it is clear from T1. – T4. that it is very non-classical. For example, T1 means that p is a doxastic consequence of Bp, while T3 shows that (if Bp then p) is not a theorem:

TT1. Bp —> p, but it is not the case that —>(if Bp then p)

The analogue to the classical ‘deduction theorem’ does not hold.

Note well, though, that the ‘deduction theorem’ presents not an ‘ordinary rule’ like modus ponens, but a ‘metarule‘: it does not tell you something like ‘ Y entails p’ but rather something like ‘if Y entails p then Z entails q’.

Here is another failure of a metarule: the rule of Reductio ad Absurdum:

1. (Bp & ~p) assumption
2. Bp from 1
3. p from 2, because Bp —> p
4. ~p from 1
5. ~(Bp & ~p) from 1-4 by reductio
6. (if Bp then p) from 5

Since we know that 6 is not a theorem of LDOX, due to fact T3, this is not deduction in LDOX — the mistake is that Reductio ad Absurdum is not an admissible rule for LDOX.

Nevertheless, on the positive side, we can note these two facts:

TT2. If p is theorem of LSTB then it is a theorem of LDOX

TT3. Modus ponens is an admissible rule of / for LDOX

The first is clear because LSTB has the rule that if p is a theorem then so is Bp, and thus in that case p belongs to every belief set. And we have also seen that what is believed is closed under the consequence relation of LSTB. We will just have to be very careful not to rely on metarules which are admissible for LSTB!

The next task we have is very clear, and challenging: formulate logic LDOX, and prove its soundness and completeness in the intended respect.