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 
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.
of code numbers of sentences true in 
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 
, no formula 
is true in 
of numbers such that 
-models of arithmetic. Now extend each model 
by adding the class 
of all sets 
of natural numbers from the domain of 
such that 
of 
) as a subsystem of the axiom system 
in Hellman’s example. Just because it’s so much fun, let’s talk about 
indicates the successor function that returns the successor of 
.
![(\textup{L2}) \ \forall x \forall y [x \dot{<} \dot{\mathsf{S}}y \leftrightarrow (x \dot{<} y \vee x \dot{=} y)]](http://s0.wp.com/latex.php?latex=%28%5Ctextup%7BL2%7D%29+%5C+%5Cforall+x+%5Cforall+y+%5Bx+%5Cdot%7B%3C%7D+%5Cdot%7B%5Cmathsf%7BS%7D%7Dy+%5Cleftrightarrow+%28x+%5Cdot%7B%3C%7D+y+%5Cvee+x+%5Cdot%7B%3D%7D+y%29%5D&bg=ffffff&fg=333333&s=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])](http://s0.wp.com/latex.php?latex=%28%5Ctextup%7BL3%7D%29+%5C+%5Cforall+x+%5Cforall+y+%5B%28x+%5Cdot%7B%3C%7D+y%29+%5Cvee+%28x+%5Cdot%7B%3D%7D+y%29+%5Cvee+%28y+%5Cdot%7B%3C%7D+x%5D%29&bg=ffffff&fg=333333&s=0 "(\textup{L3}) \ \forall x \forall y [(x \dot{<} y) \vee (x \dot{=} y) \vee (y \dot{<} x])")
![(\textup{A2}) \ \forall x \forall y [x \dot{+} \dot{\mathsf{S}}y \dot{=} \dot{\mathsf{S}}(x \dot{+} y)]](http://s0.wp.com/latex.php?latex=%28%5Ctextup%7BA2%7D%29+%5C+%5Cforall+x+%5Cforall+y+%5Bx+%5Cdot%7B%2B%7D+%5Cdot%7B%5Cmathsf%7BS%7D%7Dy+%5Cdot%7B%3D%7D+%5Cdot%7B%5Cmathsf%7BS%7D%7D%28x+%5Cdot%7B%2B%7D+y%29%5D&bg=ffffff&fg=333333&s=0 "(\textup{A2}) \ \forall x \forall y [x \dot{+} \dot{\mathsf{S}}y \dot{=} \dot{\mathsf{S}}(x \dot{+} y)]")
![(\textup{M2}) \ \forall x \forall y [(x \dot{\times} \dot{\mathsf{S}}y) \dot{=} (x \dot{\times} y) \dot{+} x]](http://s0.wp.com/latex.php?latex=%28%5Ctextup%7BM2%7D%29+%5C+%5Cforall+x+%5Cforall+y+%5B%28x+%5Cdot%7B%5Ctimes%7D+%5Cdot%7B%5Cmathsf%7BS%7D%7Dy%29+%5Cdot%7B%3D%7D+%28x+%5Cdot%7B%5Ctimes%7D+y%29+%5Cdot%7B%2B%7D+x%5D&bg=ffffff&fg=333333&s=0 "(\textup{M2}) \ \forall x \forall y [(x \dot{\times} \dot{\mathsf{S}}y) \dot{=} (x \dot{\times} y) \dot{+} x]")
, which has historically been called “Robinson Arithmetic”. ![(\textup{Q0}) \ \forall x [x \dot{=} \dot{0} \vee (\exists y) (y \dot{=} \dot{\mathsf{S}}y)]](http://s0.wp.com/latex.php?latex=%28%5Ctextup%7BQ0%7D%29+%5C+%5Cforall+x+%5Bx+%5Cdot%7B%3D%7D+%5Cdot%7B0%7D+%5Cvee+%28%5Cexists+y%29+%28y+%5Cdot%7B%3D%7D+%5Cdot%7B%5Cmathsf%7BS%7D%7Dy%29%5D&bg=ffffff&fg=333333&s=0 "(\textup{Q0}) \ \forall x [x \dot{=} \dot{0} \vee (\exists y) (y \dot{=} \dot{\mathsf{S}}y)]")
-
with,
) fail to hold, in addition to 
and 
. At the same time, howerver, 
and 
and natural interpretations of 
, 
, 
, 
fail to hold.
) 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.
(where 
is the numeral for a code number for the sentence 
contains no explicit definition of 
containing all models of 
-section for the class of 
such that for each 
, 
. 
if, and only if, every member of the class 
is a 
.
, there is a formula 
such that for all 
, 
. But since the function 
is effectively computable, it is definable and so, 
, for some 
. But this means that,
and ![\psi(\dot{m}, \dot{p}) \in \mathsf{Th}(\Omega)]](http://s0.wp.com/latex.php?latex=%5Cpsi%28%5Cdot%7Bm%7D%2C+%5Cdot%7Bp%7D%29+%5Cin+%5Cmathsf%7BTh%7D%28%5COmega%29%5D&bg=ffffff&fg=333333&s=0 "\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)](http://s0.wp.com/latex.php?latex=m+%5Cin+%5Ctextup%7BD%7D_%7B%5Ctextup%7BU%7D%7D+%5CLongleftrightarrow+%5Cexists+y+%5B%5Cphi%28y%29+%5Cwedge+%5Cpsi%28%5Cdot%7Bm%7D%2C+y%29%5D+%5Cin+%5Cmathsf%7BTh%7D%28%5COmega%29&bg=ffffff&fg=333333&s=0 "m \in \textup{D}_{\textup{U}} \Longleftrightarrow \exists y [\phi(y) \wedge \psi(\dot{m}, y)] \in \mathsf{Th}(\Omega)")
,
is definable. This is a contradiction. So, 
effectively countable.
, is undecidable and consequently not decidably axiomatizable (this means that 
, for some 
consisting of axioms of arithmetic, like those of Peano Arithmetic or one of it’s extensions) can be made stronger by showing that 
, the set of indices of sentences that are true in 
, which returns the least number 
is a formula of 
.
is definable over 
is effectively computable if, and only if, there is an effective procedure such that, for any 
the procedure calculates the value 
. The claim is that every such function defined on the natural numbers is 
is definable over 
is decidable if, and only if, the property 
defined as 
is decidable in 
holds. Here the claim is straightforward, every such set of natural numbers is 
that counts 
, 
). So, what’s being assumed here is that the effectively countable subsets of the natural numbers are definable in the standard model.
. 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.
and 
, 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 
property without being distinct with respect to a 
property.
, and Det-R: in 
.
is definable in terms of 
.
