## Archive for August, 2010

### Beth’s Definability Theorem

August 31, 2010

We’ll set this up in the most general way and later see how Hellman’s account in terms of $\phi$ and $\psi$ applies.

Letting $\textup{L}$ be a language with $p$ a $k$-ary relation symbol not appearing in the relation-symbol set of $\textup{L}$, $\textup{L}^{+}$ is the language that extends $\textup{L}$ and which includes $p$. $\textup{T}^{+}$ is a theory in $\textup{L}^{+}$.

We say that $p$ is explicitly definable if, and only if, there is an $\textup{L}$ formula $\theta$ with free variables $x_{0}\dots x_{k-1}$ such that $\textup{T}^{+} \models p x_{0}\dots x_{k-1} \leftrightarrow \theta$.  And we say that $p$ is implicitly definable if, and only if, for any $\textup{L}$- structure $\mathfrak{A}$ and any $\textup{P}, \textup{Q} \subseteq \ \vert \mathfrak{A}\vert^{k}$ (i.e., the universe of the structure), if both $(\mathfrak{A}, \textup{P})$, and $(\mathfrak{A}, \textup{Q})$ are models of $\textup{T}^{+}$, then $\textup{P} = \textup{Q}$.

Beth’s definability theorem says that if $\textup{L}$ a language with $p$ a $k$-ary relation symbol not appearing in the relation-symbol set of $\textup{L}$, $\textup{L}^{+}$ is the language that extends $\textup{L}$ and which includes $p$ and $\textup{T}^{+}$ is a theory in $\textup{L}^{+}$, then $p$ is explicitly definable if, and only if, $p$ is implicitly definable.

To prove that explicit definability implies implicit definability is straightforward. Assume $\textup{T}^{+} \models p x_{0}\dots x_{k-1} \leftrightarrow \theta$.  Show for $\textup{P}, \textup{Q} \subseteq \ \vert \mathfrak{A}\vert^{k}$, if $(\mathfrak{A}, \textup{P}), (\mathfrak{A}, \textup{Q}) \in \textup{M} od (\textup{T}^{+}) \rightarrow \textup{P} = \textup{Q}$.  Since we’re assuming $\textup{T}^{+} \models p x_{0}\dots x_{k-1} \leftrightarrow \theta$, $\textup{P}$ is definable over $\mathfrak{A}$ (i.e., for $k$-ary $p$, $\textup{P} = \phi^{\mathfrak{A}}$, for $\phi$ with $k$ free variables.  Let $\phi = \theta$.  So $\textup{P} = \theta$;  but $\textup{Q} \subseteq \ \vert \mathfrak{A}\vert^{k}$ and $(\mathfrak{A},\textup{Q})$ models $\textup{T}^{+}$. So $\textup{P} = \theta^{\mathfrak{A}}$ $\square$.

To prove that implicit definability implies explicit definability takes just a bit longer, but it’s simple; using the Compactness Theorem  and the Craig Interpolation Theorem.

Next time I’ll turn to how Hellman deals with the problems posed by this theorem.

### Reduction in α-Structures and Physicalist Materialism

August 27, 2010

Equipped with a notion of definability, Hellman gives an account of reducibility.  Reducibility holds when all the terms of the vocabulary being reduced are definable in the reducing vocabulary.

Letting $\phi$ be the vocabulary of mathematical physics, $\psi$ the vocabulary by means of which all other truths may be stated and $\alpha$ as the set of structures representing scientific possibility, we have:

Physical Reductionism (PR): In $\alpha$-structures, $\phi$ reduces $\psi$ if, and only if, $(\forall \textup{P})(\textup{P} \in \psi \rightarrow \textup{P}$ is definable in terms of $\phi$ in $\alpha$ -structures$)$. And $\phi$ effectively reduces $\psi$ if every term in $\psi$ is definable in a recursively enumerable set of definitions.

An interesting thing to note here is Hellman’s footnote 17 regarding the extension of reduction beyond linguistic primitives to sentences and laws stated in the language reduced.  If in $\alpha$-structures, $\phi$ reduces $\psi$, then each law containing $\psi$ terms, even bridge laws, is definitionally equivalent to a law stated in purely $\phi$ terms.  This, of course, shows the incompatibility of PR with views on emergence, like the view that while the theory of evolution does not increase the number of physical entities in the universe, it does introduce lawlike regularities not expressible in terms of physical law.  It will be interesting to see what the story will be about the compatibility of Physicalist Materialism (next paragraph) and emergence theses.

Now, Hellman goes on to claim that while assumptions about the mathematical-physical determination of all truths and, possibly, reference are regulative principles of scientific theory construction –the assumption in general that all terms and all theories are reducible to mathematical-physical terms is without foundation and likely false.  The physicalism that he endorses instead is made up of PE ($(\forall x)(\exists \alpha)(x\in \textup{R}(\alpha))$, for $\textup{R}(\alpha)$ a rank in the hierarchy), and principles of physical determination, Det-T and Det-R. This he calls Physicalist Materialism.  Like PE is independent of PRPE is also independent from physical determination.

It’s important here to highlight just what physicalist materialism is:  Determination has traditionally been thought to imply reductionism.  Anti-reductionism has been traditionally held to be incompatible with determination theses. Physicalist materialism denies both of these claims; it is simultaneous support for principles of determination of truth and reference and anti-reductionism.

My next set of notes will be on Hellman’s evaluation of the effects of Beth’s definability theorem on physicalist materialism.

### A Note on Definability

August 26, 2010

Since we’re talking about definability I wanted to say that in general, for any language $\textup{L}$, any structure $\mathfrak{A}$ in $\textup{L}$ and any $\textup{L}$-formula $\phi$ with $n$ free variables, $x_{0}, \dots, x_{n-1}$, the $n$-ary relation (or predicate, since predicates represent relations) over the universe $\textup{A}$ of $\mathfrak{A}$, $\phi^{\mathfrak{A}} : = \{( a_{0}, \dots, a_{n-1}) \in \textup{A}^{n} : \models_\mathfrak{A}\phi [a_{0}, \dots, a_{n-1}\}$, is the relation defined over $\mathfrak{A}$ by $\phi$.

In Hellman’s terminology, $\phi^{\mathfrak{A}}$ is $\textup{A}$, and he is saying that a predicate $\textup{P}$ is definable in a family of structures $\alpha$ if, and only if for every $\mathfrak{A}$ in $\alpha$, $\textup{P} = \phi^{\mathfrak{A}}$.

With the next update I’ll move into Hellman’s account of reducibility.

### φ-Definability in α structures

August 25, 2010

At the start of §2.2 Reduction and Determination, Hellman introduces the notion of definability:

An $n$-place predicate $\textup{P}$ is definable in terms of a vocabulary $\phi$ in $\alpha$ structures if, and only if, there is a finite sentence $\textup{A}$ containing no nonlogical terms not in $\phi$ and with occurrences of $n$ distinct variables, $x_{1},\dots, x_{n}$, such that every structure in $\alpha$ models $(\forall x_{1})\dots (\forall x_{n}) (\textup{P}x_{1} \dots x_{n} \leftrightarrow \textup{A})$.

$\textup{P}$ is not just coextensive with a term in the $\phi$ vocabulary.  What the claim of definability says is that there is a term, $\textup{A}$, in the $\phi$ vocabulary such that the $n$-place predicate $\textup{P}$ is coextensive with it in every structure in $\alpha$.

Hellman notes that to say that a term is definable in a given vocabulary is not to say that the term is synonymous with a term in that vocabulary.  While the notion of definability can be spelled out explicitly, the notion of synonymy cannot.  Also, the coextensiveness at issue here involves definability of terms in $\phi$ terms over the structures in $\alpha$ and since each structure $\mathfrak{A}$ in $\alpha$ is a model of the laws of science,  Hellman claims that definability is a kind of lawlike coextensiveness between terms terms.

From here, Hellman will go on to discuss reducibility that holds when all the terms in the vocabulary being reduced are definable in the reducing vocabulary.

### Determination of Reference

August 24, 2010

We’ve set out the determination of truth:

Det-T: In structures $\alpha$, $\phi$ truth determines $\psi$ truth iff $(\forall m)(\forall m') ((m, m' \in \alpha \ \wedge \ m \vert \phi \equiv m' \vert \phi) \rightarrow m \vert \psi \equiv m' \vert \psi)$.

Now, the fact that two elementary equivalent structures are indiscernible in point of reference if each term has the same reference in each leads to a determination of reference principle similar to Det-T.

Det-R: In structures $\alpha$, $\phi$ reference determines $\psi$ reference iff $(\forall m)(\forall m')((m, m' \in \alpha \ \wedge \ m \vert \phi = \ m' \vert \phi) \rightarrow m \vert \psi = m' \vert \psi)$

Det-R says that if two $\alpha$ structures agree in what they assign to the $\phi$ terms, then they agree on what they assign to the $\psi$ terms. In terms of the structure of the models we’re discussing, if the set $\alpha$ is closed under automorphism, then Det-R is the condition that any bijection between the domain of m and m’ that is a $\phi$-isomorphism is a $\psi$-isomorphism.

And regarding the connection between Det-T and Det-R, Hellman points out that model-theoretically they are independent principles.  We have the set $\alpha$, the $\phi$ and the $\psi$ such that in $\alpha$ structures, $\phi$ reference determines $\psi$ reference, but $\phi$ truth does not determine $\psi$ truth.

Hellman continues: if $\alpha$ is the set of structures representing scientific possibility,  $\phi$ the vocabulary of mathematical physics and $\psi$ the (broader) vocabulary in which truths can be stated, then Det-T and Det-R are principles of physical determination.  Since we want $\alpha$ to represent scientific possibility, every structure in $\alpha$ must (at least) model all of the laws of science. But if having each member of $\alpha$ model all of the laws of science is a sufficient condition for $\alpha$ representing scientific possibility then, if T represents all scientific theory, the term ‘$\alpha$‘ could be uniformly replaced in every instance with ‘{m : m models T}’.

Hellman points out that since it seems possible to formulate T in language with only a finite number of non-logical symbols, T has models which go against the principles of physical determination.  Such models must be excluded if we are to hold onto these principles. In §2.2 Reduction and Determination, Hellman will set out ways to exclude these models while articulating more precisely the notion of scientific possibility.

### Determination of Truth

August 19, 2010

§2.1 is about determination.  Hellman begins by introducing the determination of truth about one set of facts by another.  If a collection of facts, A, determines a separate collection of facts, B, then the truths about B cannot change without a change in the truths about A. The task is to spell this out model-theoretically to be able to evaluate the connection between determination and reduction.

We have at our disposal a family of languages where the interpretation of terms appearing in more than one language remains fixed across languages.  We have two sets of non-logical symbols, $\phi$ and $\psi$ and a set $\alpha$ of structures that represent what is scientifically possible.

In elementary model theory two models m and m’ are elementary equivalent (m $\equiv$ m’) if they make the same sentences true.  Also, the reduct of a model m to a given vocabulary, L (m $\vert$ L) is the structure obtained form m by excluding the interpretation of all the terms not appearing in L.

Set out the following: In structures $\alpha$, $\phi$ truth determines $\psi$ truth iff $(\forall m)(\forall m') ((m, m' \in \alpha \ \wedge \ m \vert \phi \equiv m' \vert \phi) \rightarrow m \vert \psi \equiv m' \vert \psi)$.  This says that once you have fixed the $\phi$ facts, the $\psi$ facts are also fixed -or that once a complete description of things has been given in $\phi$ terms, then there is only one correct way to describe things in $\psi$ terms.

This notion of determination of truth becomes more engaging when $\alpha$ is a subset of the models of a theory T, composed of lawlike truths and both $\phi$ and $\psi$ are each subsets of the language used to articulate T and $\psi \not\subset \phi$.  If T contains sentences with indispensable occurrences of both $\phi$ and $\psi$, then T serves to connect the $\phi$ and $\psi$ terms.  Such sentences link the determining phenomena with the determined phenomena.  If all the models of T are elementarily equivalent or if all of the models of T are isomorphic then the determination is trivial.

Having set out the determination of truth, Hellman will then introduce the determination of reference before moving onto evaluating reduction.

### Evaluating PE and IPI

August 17, 2010

§1.2 is shorter but makes a very important point about PE and IPI, reductionism and dualism. Let physical reductionism be the claim that for the theory, formulated in a suitable language, that contains all the lawlike truths of science (including physical science), every scientific predicate is definable in physical terms.  That is to say, for every n-place predicate P, one can derive using only the laws of science, a formula of this form:

$(\forall x_{1})\dots (\forall x_{n}) (\textup{P}x_{1} \dots x_{n} \leftrightarrow \textup{A})$

Here, A is a finite sentence that contains only physical vocabulary as the nonlogical terms and n distinct variables, $x_{1} \dots x_{n}$. These equivalences are provable within scientific theory and are logical consequences of its laws. What may come as a shock to those who want to avoid mysterious dualism by adhering to such a strong form of reductionism is that even this reductionism is compatible with ontological dualism.

Hellman sets out the following basic theory to make this point. Let $\Sigma$ be the theory that contains only two one-place predicates, P and Q and these non-logical axioms:

$(\exists x)(\exists y) (x \not= y \wedge (\forall z) (z = x \vee z = y))$
$(\exists x) (\textup{P}x \wedge (\forall y) (\textup{P}y \rightarrow y = x))$
$(\exists x) (\textup{Q}x \wedge (\forall y) (\textup{Q}y \rightarrow y = x))$
$(\forall x) (\textup{P}x \vee \textup{Q}x)$

All that $\Sigma$ says is that there are just two objects and that just one of them is a P and just one of them is Q and everything else is either a P or a Q.  It follows from $\Sigma$ that Q is definable in terms of P, but $\Sigma$ gives no assurance that every object is exhausted by things that are P –as a matter of course, every interpretation where the axioms of $\Sigma$ are true partitions the domain into disjoint subsets of P and Q type things.  And dualism here is a minimal case, since the reasoning can be carried over to any finite number of predicates.

This is a quick and elegant way of showing that reductionism is no shelter from dualism.

What Hellman concludes here is that while PE is necessary for physicalism, neither it nor IPI are sufficient for these reasons: PE does is silent about the scope and power of physical laws, while in IPI, quantification is restricted to the actual world and consequently neither can be used to voice the physicalist thesis that all phenomena are determined by physical phenomena.

We’ll see in §2 what the precise connection is between determination, definability and reduction.

### Wrapping up §1.1 of “Physicalist Materialism”

August 14, 2010

Now, before moving on to §1.2, we follow Hellman in noting that IPI an IPI’ don’t imply PE, but they do imply that there can be no more than one entity apart from the sum of all basic physicl entities.  IPI and IPI’ are stronger than PE, since everything may be exhausted by mathematic0-physical entities, but physical language may not be powerful enough to tell among nonidenticals.  In any case, like I said in the previous post, the upshot is that while IPI and IPI’ make use of the expressive power of the physical language, none of the principles, PE, IPI and IPI’ imply reductionism or even accidental extensional equivalence between the φ and the ψ. And, regardless of its complexity, no physical predicate covers the the extension of any biological or psychological predicate.

To summarize, in §1.1, Hellman has accomplished the following:

1. He has set out the physicalist thesis:”Everything is accounted for by mathematico-physical things satisfying predicates in Γ”.

2. He shows us how to build up a physicalist ontology using Γ.

3. Given the physicalist ontology, he defines the principle of Physical Exhaustion (PE): (∀x)(∃α) (x ∈ R(α)), where R(α) is a rank in the hierarchy.

4. Then he introduces the principle of the Identity of Physical Indiscernibles (IPI): (∀u)(∀v)((∀φ)(φuφv) → (u = v), where φ ranges over physical predicates and u and v range over arbitrary n-tuples of physical objects.

5.  And follows up by introducing IPI’: (∀ψ)(∀u)(∀v)(∃φ)(ψu & ¬ψv φu & ¬φv), which says that for any non-physical predicate and any distinction that it makes, there is a physical predicate that makes that distinction as well.

6. He points out that PE, IPI and IPI’ do not imply reductionism or even accidental coextension between the physical and non-physical predicates.

### Physicalist Principles: No Difference Without a Physical Difference

August 13, 2010

Hellman then introduces:

Identity of Physical Indiscernibles (IPI): (∀u)(∀v)((∀φ)(φuφv) → (u = v), where φ ranges over physical predicates and u and v range over arbitrary n-tuples of physical objects.

IPI says that if  u falls under a φ-predicate if and only if v does, then u and v are identical.  Letting ψ range over predicates not in Γ (i.e., non-physical predicates used to describe phenomena in any branch of science), and by contraposition of Leibniz’s laws,  IPI is equivalent to:

IPI’: (∀ψ)(∀u)(∀v)(∃φ)(ψu & ¬ψv φu & ¬φv).

IPI’ says that for any non-physical predicate and any distinction that it makes, there is a physical predicate that makes that distinction as well.

Now, in a footnote Hellman explains that IPI’ is different from IPI”, where the existential quantifier (∃φ) comes before (∀u)(∀v).  IPI” says that for any ψ there is a φ that distinguishes just what ψ does. IPI” implies (∀ψ)(∃φ)(∀u)(ψu φu), in cases where ψ is neither universal nor empty, which says that every nonphysical predicate has the same extension as a physical predicate and is a  limited form of reductionism.  IPI’ is a lot weaker than IPI”, and doesn’t imply even limited reductionism.

The upshot will be that none of the principles, PE, IPI and IPI’ imply reductionism or even accidental extensional equivalence between the φ and the ψ.