Global supervenience, Halvorson, and logic

All students of analytic philosophy should read the part of Hans Halvorson’s new book, The Logic in Philosophy of Science, about definability and Beth’s theorem, and appreciate its relevance to discussions of reduction, supervenience, and the possibility of non-reductive physicalism.

I am very much in agreement with the sentiments I read as expressed in this part.  But it does seem to me that Hans does not do justice to the idea of global supervenience in his note about Petrie (Halvorson pp. 203-204). Petrie defines:

“let A and B be sets of properties. We say that A globally supervenes on B just in case worlds which are indiscernible with regard to B are also indiscernible with regard to A.” (B. Petrie, Philosophy and Philosophical Research 1987)

Like Hans I want at once to paraphrase Petrie’s talk about properties in the formal mode, switching to the context of first-order logic, rather than immerse myself in analytic metaphysics.  And my paraphrase would be this:

Let Σ¹ and Σ² be two disjoint signatures, and Σ⁺ their union, and let us assume that the models of Σ⁺ are just those that are ‘allowed’ in a specific context (not ruled out by some background theory formulated in Σ⁺).  Perhaps the first signature is the vocabulary of folk psychology and the second that of physics.  Informally then:

Global supervenience of Σ¹ discourse on Σ² discourse:   for any text F formulated in Σ¹, if F is true then the world could not be different in that respect unless something formulated in Σ² would also have a different truth-value.

The models of Σ⁺ can be partitioned into the family Q⁽²⁾ = {MOD(T): T a maximal consistent set of sentences in signature Σ²}.  Suppose that F, a text formulated in Σ¹, is true in model M and false in model M’.  Then by global supervenience as defined above, these two models cannot belong to the same member of Q⁽²⁾.  So if M does lie in MOD(T), a member of Q⁽²⁾, then all of MOD(T) is part of MOD(F).  And similarly, if M’, where F is false, is in MOD(T’) then MOD(T’) is entirely disjoint from MOD(F).  This means that MOD(F) is the union of cells in the partition in question, that is, the union of {MOD(T) in Q⁽²⁾: MOD(T) is included in MOD(F)}.

In other words, if we can think of union as the infinitary form of disjunction, then as far as truth-value is concerned, any text F in Σ¹ is a disjunction of texts in Σ².  But by this word “infinitary” hangs a tale:  the ‘reduction’ or ‘translation’ of the former discourse into ‘disjunctions’ of the latter is, in general, not a humanly graspable translation.  It is not, in general, sentence by sentence, paragraph by paragraph, or recursively definable set by recursively definable set … it is, as Pascal phrased it, at the far end of infinity.  And if the matching is between indefinable sets of sentences in the two signatures, it may well exist in Cantor’s paradise or its category-theoretic equivalent, but asserting that it in support physicalism makes that a thesis designed to be irrefutable – and what value has that?  So, I see this as supporting rather than detracting from Hans’ negative reaction to the physicalism story.