Sunday, 21 April 2013

The Probability of a Carnap Sentence

In the simplest, "logical empiricist"-style, framework for the formalization of scientific theories, we have 1-sorted language $L_{O,T}$, where the vocabulary has been partitioned into O-predicates and T-predicates (it's easy to include constants and function symbols if one wishes; but it's simpler to omit them). And scientific theories are formulated in $L_{O,T}$. The language obtained by deleting the T-predicates can be denoted $L_{O}$ and is called the observational sublanguage of $L_{O,T}$.

Suppose that $\Theta(\vec{O}, \vec{T})$ is a single axiom for a finitely axiomatized theory in $L_{O,T}$, where $\vec{O}$ is a sequence of O-predicates and $\vec{T}$ is a sequence of T-predicates. Then the Ramsey sentence of $\Theta$ is defined by:
$\Re(\Theta) := \exists \vec{X} \Theta(\vec{O}, \vec{X})$, 
where $\vec{X} = (X_1, \dots)$ is a sequence of second-order variables matching the arities of the predicates $T_1, \dots$ in $\vec{T}$. So, the theoretical predicates have been replaced by second-order variables, and existentially quantified.

Nothing has been said about the meanings of the O-predicates and T-predicates. In principle, one could simply assume some $L_{O,T}$-interpretation $\mathcal{I}$, and let $(L_{O,T}, \mathcal{I})$ be the corresponding fully interpreted language. However, the logical empiricists---the first group of thinkers aiming to apply the newly emerging methods of mathematical logic to the formalization of scientific theories---did not adopt this approach, Instead, largely because of their empiricist metasemantics, they assumed only an $L_{O}$-interpretation $\mathcal{I}^{\circ}$, and consequently $(L_{O,T}, \mathcal{I}^{\circ})$ is then a partially interpreted language.

Because the language is partially interpreted, for each O-predicate $O_i$, there is now a meaning, $(O_i)^{\mathcal{I}^{\circ}}$. How then do the T-predicates get their meanings? Certainly not by explicit definition in terms of O-predicates! In a sense, the new underlying idea is that the meanings of T-terms are not pinned down uniquely and independently of theory, but rather implicitly defined by theories themselves. The basic way of implementing this view of meaning is to consider the Carnap sentence of the theory $\Theta$, i.e.,
$\Re(\Theta) \to \Theta$
and to insist that this sentence is analytic --- true in virtue of meaning.

As Hannes Leitgeb has pointed out in the talk I mentioned in the post "The Probability of a Ramsey Sentence" yesterday, it now seems reasonable, to assign probability 1 to the Carnap sentence. After all, if $\phi$ is analytically true, surely its probability should be 1, whether or not probability is understood subjectively or not. So, we assume that we have some probability function $Pr(.)$ defined over $L_{O,T}$-sentences.

What can we say about connections between the probabilities of theories and their ramsifications? Well, as explained in tre previous post, if the Carnap sentence has probability 1, i.e.,
$Pr(\Re(\Theta) \to \Theta) = 1$
then we can show that,
$Pr(\Re(\Theta)) = Pr(\Theta)$.
On the other hand, suppose that the Carnap sentence has probability slightly lower than 1. E.g., suppose that,
$Pr(\Re(\Theta) \to \Theta) = 1 - \epsilon$
for some small parameter $\epsilon$. In this case, it follows that
$Pr(\Re(\Theta)) = Pr(\Theta) + \epsilon$.
Proof: By the Lemma in the previous post,
$Pr(\Theta) + Pr(\Theta \to \Re(\Theta)) = Pr(\Re(\Theta)) + Pr(\Re(\Theta) \to \Theta)$.
But $Pr(\Theta \to \Re(\Theta)) = 1$, because $\Theta \vdash \Re(\Theta)$ (assuming second-order logic). So,
$Pr(\Theta) + 1 = Pr(\Re(\Theta)) + 1 - \epsilon$.
So, $Pr(\Re(\Theta)) = Pr(\Theta) + \epsilon$, as required. QED.

So, if the Carnap sentence for a theory has a probability lower than 1 by some amount, then the Ramsey sentence for the theory has a higher probability than the theory does, by that same amount. This makes sense intuitively, because the Ramsey sentence is, in a number of senses, weaker than the theory itself (unless it happens to be inconsistent, of course).

1 comment:

  1. I have hints of a response to the Carnap sentence in my book, the Dimensional Philosopher's Toolkit.

    Essentially, there is no sentential requirement of the integral between R and theta. In other words, if R has a spatial meaning Phi, there is no guarantee that that spatial meaning is 'located' on the variable theta. This is what must have been familiar to earlier analytic philosophers about assumed consequences. Further, if we treat system R as corresponding instead to an integral Psi, reflecting instead an empirical meaning for R (instead of a philosophical one, however logical), it appears then that Theta should have precedent over R, for the truth of the theory is defined empirically rather than rationally, as would be familiar to those who have read Hume.

    In my theories of categorical deduction from geometric categories I try to treat theories such as R-Phi as dependent on an integral condition of abstracta. In other words, clarification must be made as to whether the analyst wants 'clear' or 'ugly' results. This is similar to what has already been encountered in quantum mechanics, but applies to a much broader swath of information. If I assume clear, then only perfected data makes sense. In that case, it is possible to look for data of certain types, without which even clear theories of specialized significance may be inadequate. Two of these categories are complexity and perfection. Three others are exclusivity, polarity, and analogy. As it turns out, one other is history.

    Perhaps some of this would be considered useful to those wishing to go deep into theories of R(theta) = theta.

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