# Logicians do it with Models

A self-guided jaunt in philosophical logic

## Archive for September, 2010

### Determination and Definability: Newtonian Mechanics & Statistical Dynamics

September 30, 2010

The last determination-without-reduction example I’ll talk about from Hellman is the one he gives regarding classical particle mechanics and statistical mechanics.   A process is reversible, if it could proceed forward in time as well as backwards (think of playing, in reverse, a video of a model of the orbits of the planets around the sun).  A process is irreversible when it can only proceed in one direction of time and would violate physical law proceeding in the opposite time direction (think of playing, in reverse, a video of a gas escaping a bottle).

Newtonian laws governing motion are time-symmetric and reversible: forward motions of Newtonian systems are on par with motions backwards in time.  Statistical mechanics, on the other hand, attempts to explain irreversible behavior of the higher-level observable phenomena of thermodynamics, like temperature, diffusion, pressure and entropy.  The macroscopic properties of thermodynamics are defined in terms of the phase quantities of Newtonian mechanics with the addition of a measure theoretic probability density function as well as some assumptions a-priori about distribution (e.g., equiprobability of equal volumes of phase space).  For macroscopic properties like entropy, more complex probabilistic concepts come into play: dividing phase space into cells and adding to the mechanical motions the periodical average of the cells, then entropy increases and the distribution density tends uniformly to equilibrium.  This is what the Ehrenfests (Paul and Tanya) called coarse-graining and it’s a method for converting a probability density in phase space into a piece-wise constant function by density averaging in cells in phase space. Coarse-grained densities are needed to avoid paradoxical results concerning how irreversible processes of thermodynamics arise from completely reversible mechanical interactions.

The question Hellman poses is: can the higher-level concepts of thermodynamics be explicitly defined in the language of Newtonian mechanics?  Determination, at least, holds: having fixed any two closed particle systems that are identical at the level of Newtonian mechanics, their higher level behavior (studied by thermodynamics) will be identical.  Each of these systems will be represented by the same trajectory in phase space.  Hellman gives the example of one system entering higher entropy regions at a given time, then the same entropy regions will be entered by the other system.

Definiability on the other hand is more difficult to establish, since the language of classical Newtonian mechanics is not as mathematically robust as that of statistical dynamics.  And ultimately, definability in this case requires a significant change to the language of mechanics: additional vocabulary for measure theory or even set theory to speak of mathematical objects more generally.

We’ve covered some examples of determination without reduction: (1) No explicit definition of the truth predicate $\mathsf{Tr}(x)$ in $\textup{L}$ in spite of  $\textup{L}$-truth and $\textup{L}$-reference determining $\mathsf{Tr}$-truth and $\mathsf{Tr}$-reference, respectively, in $\alpha$-structures; (2) in $\alpha^{\textup{C}}$-structures, $\textup{L}$-reference determines $\textup{DF}arith$-reference, but within this same class of structures, $\textup{L} + \textup{G}(x)$ does not inductively define $\textup{DF}arith$; (3) The mechanical properties of two fixed, closed particle systems determine their macro-level thermodynamic properties without thereby establishing definability, as the language of Newtonian mechanics would have to undergo significant changes incorporating the language, at least, of measure theory.

Next, I’m going to cover an important distinction Hellman makes between the ontological and and ideological status of  properties, relations, attributes, etc. beyond predicates and sets.  After that I’ll go into a detailed, critical, evaluation of some of the anti-materialist claims David Chamers makes in The Conscious Mind: In Search of a Fundamental Theory.

### Hellman’s Second Definability Example

September 27, 2010

Tarski’s theorem shows that the set $\mathsf{Th} (\Omega)^\#$ of code numbers of sentences true in $\Omega$ is not definable.  This has negative consequences for the prospects of aritmetically defining a truth predicate for arithmetic. Nevertheless, Tarski showed how to define truth in terms of satisfaction and by giving an inductive definition of satisfaction beginning with atomic sentences and on up with sentences of higher and higher complexity in terms of the satisfaction of their parts. While a stronger set theory or higher-order logic is still required to convert the inductive definition into an explicit one, Hellman investigates how Det-T and Det-R measure up against this weaker type of definability.

Addison’s theorem establishes that the class of arithmetically definable sets of numbers is itself not an arithmetically definable class of sets.  This means that in the language, $\textup{L}$, of arithmetic extended by a one place predicate $\textup{G}(x)$, no formula $\textup{S}$ is true in $\Omega$ when $\textup{G}(x)$ is assigned a set $\textup{A}$ of numbers such that $\textup{A}$ is definable over $\Omega$.  The proof is involved and is clinched by contradiction on the existence of a generic arithmetical set (I may, or may not, get around to explaining what this is in the next post or so, since it involves explanation of the technique of forcing).

The example turns on this: Addison’s theorem shows that the predicate $\textup{DF}arith$ = ‘set of numbers definable in arithmetic’ is not inductively definable in  arithmetic. Nevertheless, $\textup{DF}arith$ is determined by the primitive predicates of $\textup{L}$.  Set out the following: $\alpha$ is the set of standard $\omega$-models of arithmetic.  Now extend each model $m \in \alpha$ by adding the class $\textup{C}$ of all sets $\textup{X}$ of natural numbers from the domain of $m$ such that $\textup{X}$ is in the extension in $m$ of a formula $\textup{B}(x)$ of $\textup{L}$ with one free variable.  Let $\alpha^{\textup{C}}$ be the class of $\alpha$-structures extended in this way.

So, $\alpha^{\textup{C}}$ contains all the standard models of arithmetic that have standard interpretations of $\textup{DF}arith$. This means that in $\alpha^{\textup{C}}$-structures, $\textup{L}$ reference determines $\textup{DF}arith$ reference but within this same class of structures, $\textup{L} + \textup{G}(x)$ does not inductively define $\textup{DF}arith$.  In spite of this lack of definiability,  Det-R still holds since (and this is all up to isomorphism) any two structures that assign the same interpretations to the primitives of $\textup{L}$ must also assign the same extension to the well-formed-formulas of $\textup{L}$ with only one free variable.  So, up to isomorphism, the same sets of natural numbers are assigned to the distinguished elements of $\textup{C}$

The next example from Hellman is not mathematical, but from classical particle mechanics.  After that I will go into Hellman’s clarification on the difference between the ontological and ideological status of attributes, properties and relations before moving into constructive work on the mental.

### Fun With Robinson Arithmetic

September 17, 2010

in yesterday’s notes I mentioned Robinson Arithmetic ($\mathsf{Q}$) as a subsystem of the axiom system $\textup{T}$ in Hellman’s example. Just because it’s so much fun, let’s talk about $\mathsf{Q}$.  The axioms are listed below.  The dot notation distinguishes between the symbol used and the relation it represents (this notation was used earlier without explanation here and here). $\dot{\mathsf{S}}x$ indicates the successor function that returns the successor of $x$.

## Robinson Arithmetic

$(\textup{S1}) \ \forall x (\neg \dot{0} \ \dot{=} \ \dot{\mathsf{S}}x)$
$(\textup{S2}) \ \forall x \forall y (\dot{\mathsf{S}}x \ \dot{=} \ \dot{\mathsf{S}} y \rightarrow x \dot{=} y)$
$(\textup{L1}) \ \forall x (\neg x \dot{<} \dot{0})$
$(\textup{L2}) \ \forall x \forall y [x \dot{<} \dot{\mathsf{S}}y \leftrightarrow (x \dot{<} y \vee x \dot{=} y)]$
$(\textup{L3}) \ \forall x \forall y [(x \dot{<} y) \vee (x \dot{=} y) \vee (y \dot{<} x])$
$(\textup{A1}) \ \forall x (x \dot{+} \dot{0} \dot{=} x)$
$(\textup{A2}) \ \forall x \forall y [x \dot{+} \dot{\mathsf{S}}y \dot{=} \dot{\mathsf{S}}(x \dot{+} y)]$
$(\textup{M1}) \ \forall x (x \dot{\times} \dot{0} \dot{=} \dot {0})$
$(\textup{M2}) \ \forall x \forall y [(x \dot{\times} \dot{\mathsf{S}}y) \dot{=} (x \dot{\times} y) \dot{+} x]$

These axioms are from Hinman and correspond to what Boolos, Burgess and Jeffrey (in chapter 16 of Computability and Logic) call “minimal arithmetic”. Since the Boolos et al book is so widely studied, here is a map linking both axiomatizations.

$(\textup{S1}) \mapsto 1$
$(\textup{S2}) \mapsto 2$
$(\textup{L1}) \mapsto 7$
$(\textup{L2}) \mapsto 8$
$(\textup{L3}) \mapsto 9$
$(\textup{A1}) \mapsto 3$
$(\textup{A2}) \mapsto 4$
$(\textup{M1}) \mapsto 5$
$(\textup{M2}) \mapsto 6$

This is going to be as confusing as keeping track of names in One Hundred Years of Solitude. Boolos et al compare $\mathsf{Q}$/minimal arithmetic to the system $\mathsf{R}$, which has historically been called “Robinson Arithmetic”.  $\mathsf{R}$ differs from $\mathsf{Q}$ in that it contains,

$(\textup{Q0}) \ \forall x [x \dot{=} \dot{0} \vee (\exists y) (y \dot{=} \dot{\mathsf{S}}y)]$

And it replaces $\textup{L1}$-$\textup{L3}$ with,

$(\textup{Q10}) \ \forall x \forall y [x \dot{<} y \leftrightarrow \exists z (\dot{\mathsf{S}}z \dot{+} x \dot{=} y)$

In their comparison Boolos et al conclude that $\mathsf{Q}$ and $\mathsf{R}$ have a lot in common (e.g., some of the same theorems are provable), but in some cases $\mathsf{R}$ is stronger than $\mathsf{Q}$ –e.g., in the non-standard ordinal model of $\mathsf{Q}$ some simple laws (like $\dot{1} \dot{+} x \dot{=} x \dot{+} \dot{1}$) fail to hold, in addition to $(\textup{Q0})$ and $(\textup{Q10})$.  At the same time, howerver, $\mathsf{Q}$ is in some cases stronger than $\mathsf{R}$.  For example, there is a model (with domain $\omega$ and non standard elements $a$ and $b$ and natural interpretations of $\dot{0}$, $\dot{+}$, $\dot{\times}$, $\dot{\mathsf{S}}x$ of $\mathsf{R}$ where basic laws like $x \dot{<} \dot{\mathsf{S}}x$ fail to hold.

While it’s easier to represent all recursive functions in $\mathsf{Q}$ is than it is to do so in $\mathsf{Q}$ than in $\mathsf{R}$, any one of these will do for Hellman’s example.  What’s interesting is that in theories like $\mathsf{Q}$ and $\mathsf{R}$ which lack an induction schema (and thus fall just short of Peano Arithmetic) the truth predicate is undefinable.

Hopefully in the next update I’ll be able to get to Hellman’s second definability example.

### Hellman’s First Definability Example

September 15, 2010

What Tarski’s Theorem shows is that interpreted formal languages that are interesting (i.e., with enough expressive machinery to represent arithmetic or fragments thereof) cannot contain a predicate whose extension is the set of code numbers (e.g., $\mathsf{Th}(\Omega)^\#$) of sentences true in the interpretation.  The extension of any proposed truth predicate in such a system escapes the definitional machinery of the system.  Of course, the truth predicate for first-order arithmetic can be defined with appeal to  a stronger system, like second-order arithmetic, in the case of the Peano Axioms, etc.

Hellman’s first example is the following.  It is a corollary of Tarski’s theorem that a theory in the language, $\textup{L}$, of arithmetic (e.g., an axiom system $\textup{T}$ containing Robinson Arithmetic ($\mathsf{Q}$)) with symbols for zero, successor, addition, and multiplication, when extended with a one place predicate, $\mathsf{Tr}(x)$ (read “true in arithmetic” such that for each closed sentence $\textup{S}$ in $\textup{L}$ a new axiom of the form $\ulcorner \mathsf{Tr}(n) \leftrightarrow \textup{S}\urcorner$ (where $n$ is the numeral for a code number for the sentence $\textup{S}$), the resulting theory $\textup{T}^{*}$ contains no explicit definition of  $\mathsf{Tr}(x)$ in $\textup{L}$.

Connecting this to our ongoing discussion of determination of truth and reference in special collections of models, suppose that $\alpha$ is the class of standard $\omega$-models of $\textup{T}^{*}$.  Then we have:

• In $\alpha$-structures $\textup{L}$-truth determines $\mathsf{Tr}$-truth.
• In $\alpha$-structures $\textup{L}$-reference determines $\mathsf{Tr}$-reference

Which means that once you have the arithmetical truths in the class $\alpha$, then so are the ‘true-in-arithmetic’ truths and the same goes for the reference of the vocabularies.  To avoid collapsing to reductionism via Beth’s theorem, note that there is no first-order theory (like those under discussion) in a language with finitely many non-logical symbols has as it’s models just the models in in $\alpha$.

If you extend $\alpha$ to $\alpha^{*}$ containing all models of $\textup{T}^{*}$, then you do get reductionism, since determination of reference in $\alpha^{*}$ amounts to implicit definability in $\textup{T}^{*}$ -thus showing that there exist non-standard models of arithmetic.

This is a good example because it is clear, based on popular, well established results and firmly shows how determination of truth and reference in one core theory carry over to it’s extension, without thereby reducing the extension to the core.

In the next update I’ll discuss Hellman’s second definability example.

Posted in Robinson Arithmetic, Supervenience and Determination, Tarski's Theorem, Undefinability of Truth | Leave a Comment »

### Tarski’s Theorem

September 14, 2010

Today we’re going to follow Hinman’s proof of Tarski’s theorem. The proof is by contradiction, and employs diagonalization. First, assuming clauses (i)-(iii) on definability, a universal $\textup{U}$-section for the class of $\Omega$-definable sets is established and it’s diagonal is shown to not be definable over $\Omega$.

A $\textup{U}$-section is a set defined for any set $\textup{A}$ and any relation $\textup{U} \subseteq \textup{A} \times \textup{A}$ such that for each $a \in \textup{A}$, $\textup{A}_{a} := \{b : \textup{U}(a, b)\}$.  $\textup{U}$ is universal for a class $\mathcal{C} \subseteq \wp (\textup{A})$ if, and only if, every member of the class $\mathcal{C}$ is a $\textup{U}$-section.  The diagonal set of $\textup{U}$ is $\textup{D}_{\textup{U}} := \{a : \textup{U}(a, a)\}$.

Second, assuming the definability of $\mathsf{Th}(\Omega)^{\#}$, there is a formula $\phi (y)$ such that for all $p$, $p \in \mathsf{Th}(\Omega)^{\#} \Longleftrightarrow \phi (\dot{p}) \in \mathsf{Th}(\Omega)$.  But since the function $\mathsf{Sb}$ is effectively computable, it is definable  and so, $\mathsf{Sb}(m) = p \Longleftrightarrow \psi (\dot{m}, \dot{p}) \in \mathsf{Th}(\Omega)$, for some $\psi (x, y)$.  But this means that,

$m \in \textup{D}_{\textup{U}} \Longleftrightarrow \exists p [\phi(\dot{p}) \in \mathsf{Th}(\Omega)$ and $\psi(\dot{m}, \dot{p}) \in \mathsf{Th}(\Omega)]$

$m \in \textup{D}_{\textup{U}} \Longleftrightarrow \exists y [\phi(y) \wedge \psi(\dot{m}, y)] \in \mathsf{Th}(\Omega)$,

which means that $\textup{D}_{\textup{U}}$ is definable.  This is a contradiction.  So, $\mathsf{Th}(\Omega)^{\#}$ is not definable and not effectively countable.  Nor is $\mathsf{Th}(\Omega)$ effectively countable.

In the next update (probably on a plane!) I’ll discuss this theorem and get into Hellman’s example.

### Statement of Tarski’s Theorem

September 13, 2010

The theorem showing that the theory determined by the standard model of arithmetic, $\mathsf{Th} (\Omega)$, is undecidable and consequently not decidably axiomatizable (this means that $\mathsf{Th} (\Omega) \not= \mathsf{Th} (\Gamma)$, for some $\Gamma$ consisting of axioms of arithmetic, like those of Peano Arithmetic or one of it’s extensions) can be made stronger by showing that $\mathsf{Th}(\Omega)^{\#} := \{p: \chi_{p} \in \mathsf{Th}(\Omega)\}$,  the set of indices of sentences that are true in $\mathsf{Th} (\Omega)$, is not definable over $\Omega$, which means that $\mathsf{Th}(\Omega)$ is not effectively enumerable.  This is known as Tarski’s Theorem on the undefinability of truth.

Now set up the (provably effectively computable) function $\mathsf{Sb}(m):= \#(\chi_{m}(\dot{m}))$, which returns the least number $m$ such that $\chi_{m}$ is a formula of $\textup{L}_{\Omega}$.

In the next update I’ll discuss the proof of Tarski’s Theorem and move into the corollary that Hellman uses to give an example of determination without reduction.

### Assumptions On Definability Over The Standard Model Ω, Part 2

September 11, 2010

So we set out the basic assumptions about definability over $\Omega$ that factor into the proof of Tarski’s theorem.  Let’s go over each one to get clear on all the definitions.

(i) Every effectively computable function $\textup{F}: \omega \rightarrow \omega$ is definable over $\Omega$.

A function $\textup{F}: \textup{X} \rightarrow \textup{Z}$ is effectively computable if, and only if, there is an effective procedure such that, for any $x \in \textup{X}$ the procedure calculates the value $\textup{F}(x)$.  The claim is that every such function defined on the natural numbers is definable in the standard model.

(ii) Every decidable set $\textup{X} \subseteq \omega$ is definable over $\Omega$.

A subset $\textup{A} \subseteq \textup{X}$ is decidable if, and only if, the property $\mathcal{P}$ defined as $(\mathcal{P}(x): \Longleftrightarrow x \in \textup{A})$ is decidable in $\textup{X}$.  And a property $\mathcal{P}$ defined over a set $\textup{X}$ is decidable in $\textup{X}$ if, and only if, there is an effective (or decision) procedure for deciding, for any $x \in \textup{X}$ whether or not $\mathcal{P}(x)$ holds.  Here the claim is straightforward, every such set of natural numbers is definable in the standard model.

(iii) Every effectively countable set $\textup{X} \subseteq \omega$ is definable over $\Omega$

Finally, a set $\textup{X}$ is effectively countable if, and only if, $\textup{X}$ is empty or there is an effectively computable function $\textup{F}: \omega \rightarrow \textup{X}$ that counts $\textup{X}$ (i.e., $\textup{X}$ is the image of $\textup{F}$, $\textup{X} = \mathsf{Im}(\textup{F}) := \{\textup{F}(n): n \in \omega\}$). So, what’s being assumed here is that the effectively countable subsets of the natural numbers are definable in the standard model.

Just a few more definitions and we can get into Tarski’s theorem.  In the next update I’ll define the set of indices for the theorems of $\Omega$ as well as a diagonalization function that will play a role in the proof.

Posted in Metatheory of First Order Logic, Models of Arithmetic, Undefinability of Truth | Leave a Comment »

### Assumptions On Definability Over The Standard Model Ω, Part 1

September 10, 2010

In the late 30′s Tarski proved that arithmetical truth cannot be defined in arithmetic.  In the next few updates I’m going to be discussing Tarski’s Undefinability theorem and will follows chapter 4 of Hinman’s Fundamentals.  Check this earlier note if you want to get clear on the definability we’re talking about.

Below are some of the basic assumptions about definability and the standard model of arithmetic that will factor into the proof of Tarski’s theorem.

(i) Every effectively computable function $\textup{F}: \omega \rightarrow \omega$ is definable over $\Omega$.

(ii) Every decidable set $\textup{X} \subseteq \omega$ is definable over $\Omega$.

(iii) Every effectively countable set $\textup{X} \subseteq \omega$ is definable over $\Omega$.

In the next update I’ll break down each of these assumptions and hopefully move into the theorem itself.  The point of all this is just to be able to get through Hellman’s example of determination without reduction using a corollary of the undefinability theorem.

Posted in Metatheory of First Order Logic, Models of Arithmetic, Undefinability of Truth | Leave a Comment »

### Hellman’s Physicalist Materialism

September 8, 2010

I’m moving onto Hellman’s “Physicalist Materialism”, a paper whose aim is to apply the physicalist materialist position he developed in “Physicalism” to problems in philosophy and philosophy of science.  I want to focus on just three parts of this paper. The first is the examples Hellman gives of determination without reduction. The second is the distinction he clarifies between the ontological and ideological status of attributes, properties and relations.  The third is the section on the mental.  I really want to get into the section on theoretical equivalence, but I will only do so if I can somehow fit it into the evaluation of the anti-materialism of Chalmers.

### “Physicalism” Concluding Summary

September 7, 2010

It took me a good while to get through this paper –more than I expected –but here we are.

What has Hellman accomplished?

First he showed us how to build the ontological principle of physical exhaustion, PE, $(\forall x)(\exists \alpha) (x \in \textup{R}(\alpha))$. PE allows us to say that everything is exhausted by the physical without (embarrassingly) implying that everything is in the extension of a basic physical predicate.

Then he introduced the identity of physical indiscernables, IPI, and IPI’, $(\forall u)(\forall v)((\forall \phi) (\phi u \leftrightarrow \phi v) \rightarrow (u = v)$ and $(\forall \psi) (\forall u) (\forall v) (\exists \phi) (\psi u \wedge \lnot \psi v \rightarrow \phi u \wedge \lnot \phi v)$, respectively.  IPI says that if two objects have the same physical properties, then they are the same thing, while IPI’ says that no two objects are distinct with respect to a $\psi$ property without being distinct with respect to a $\phi$ property.

These three principles neither independently nor in conjunction imply or require reduction to the physical, but they are also too weak to express the physicalist thesis that physical phenomena determine all phenomena.  He then turns his attention to principles of determination. In addition to PE and IPI/IPI’ Hellman introduces the determination of truth, Det-T: in $\alpha$ structures $\phi$ truth determines $\psi$ truth if, and only if, $(\forall m)(\forall m')((m, m' \in \alpha \wedge m \vert \equiv m' \vert \phi) \rightarrow m \vert \psi \equiv m' \vert \psi)$, and Det-R: in $\alpha$ structures $\phi$ reference determines $\psi$ reference if, and only if, $(\forall m)(\forall m') ((m, m' \in \alpha \wedge m \vert \phi = m' \vert \phi) \rightarrow m \vert \psi = m' \vert \psi)$.

Det-T says that once you have given a complete description of things in $\phi$ terms, there is only one correct way to describe them in $\psi$ terms.  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.

Given a notion of definability, Hellman is able to state the thesis of physical reduction, PE: 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$)$.

Given that assumptions about the mathematical-physical determination of all truths and, maybe, 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 probably false. Hellman’s physicalist materialism is instead composed of PE, DET-T and DET-R, with PE independent of PR and the determination principles.

Because there are non-standard models of the laws of science, our formal systems do not model scientific possibility in a way that permits the move from physical determination to physical reduction –thus Beth’s definibility theorem poses no threat to physicalist materialism.  And any way you cut it, the link between theories as syntactic entities and reductionism doesn’t carry over to determination of reference, ruling out even accidental co-extensiveness between terms.

This sets up the theoretical background for an evaluation of the applications of physicalist materialism across disciplines and problems in philosophy of science, mind and social theory.  In the next updates I  will be covering Hellman’s 1977 “Physicalist Materialism” (Noûs) as well as giving a detailed, critical, evaluation of some of the anti-Materialist claims David Chamers makes in the early chapters of his book The Conscious Mind: In Search of a Fundamental Theory.
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