Halvorson, H. (2012), What Scientific Theories Could not Be

The PDF of the paper can be found here. My comment and understanding of the paper can be found in the essay Structure on Models and what follows are some other, mostly related, remarks.

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Here I collect brief notes on particular topics connected or of particular interest for my First Bachelor Thesis Concept. On the same topic, see Halvorson, H. (2013), The semantic view, if plausible, is syntactic. For other papers by the same author, visit Hans Halvorson

Identity Crisis for Theories

In this part of the paper Halvorson presents three possible identity criteria for theories in the semantic view: for $mod(T):=\{\mathcal{M}$ is a model $:T\models \mathcal{M}\}$.

• $T{\sim }_E{T}^{\prime }:=$ exists $f:mod(T)\to mod(T^{\prime })$ s.t. $f$ is bij.
• $T{\sim }_P{T}^{\prime }:=$ exists $f:mod(T)\to mod(T^{\prime })$ s.t. $f$ is bij. and ${\forall }_{\mathcal{M}\in mod(T)}\mathcal{M}\cong f(\mathcal{M})$.
• $T{\sim }_I{T}^{\prime }:=mod(T)=mod(T^{\prime })$ These criteria, I believe, assume an intuition on $mod(T)$, namely that it is no more than a set. For instance, if $mod(T)$ had some structure, like a partial order or the one of a group, one would define some equivalence relation as isomorphisms of such structures. For this reason I am interested in finding some further structure to $mod(T)$ so that some alternative equivalence relation as an isomorphism of such structures can be defined.

Partial Order

Following the usual construction of a maximally order set through the procedure of the Lindenbaum’s Lemma (see 7. Soundness and Completeness I, whose proof gives a practical grasp to the algorithm), one notices that for ${\mathcal{P}}_T:=\{A\subseteq \mathcal{L}:T\subseteq A\wedge A/\models \perp \}$, $({\mathcal{P}}_T,\subseteq )$ is a partial order whose minimal element is $T$ and maximal elements of each chain are in $mod(T)$. Following the previous reasonings one could now consider two other identity criteria for theories:

• $T{\sim }_{P{O}_1}{T}^{\prime }\iff ({\mathcal{P}}_T,\subseteq )\cong ({\mathcal{P}}_{T^{\prime }},\subseteq )$
• $T{\sim }_{P{O}_2}{T}^{\prime }\iff ({\mathcal{P}}_T,\subseteq )\cong ({\mathcal{P}}_{T^{\prime }},\subseteq )$ for $f$ isom., ${\forall }_{\mathcal{M}\in mod(T)}\mathcal{M}\cong f(\mathcal{M})$

One might also consider: $({\mathcal{P}}_T,\subseteq )\cong ({\mathcal{P}}_{T^{\prime }},\subseteq )$ for $f$ isom., ${\forall }_{A\in {\mathcal{P}}_T}A\cong f(A)$. Though clearly $A\cong A^{\prime }$ for $A,A^{\prime }\in \mathcal{P}$ is not defined since $A,A^{\prime }$ are only consistent sets, i.e. theories, but not structures. One could instead define:

• $T{\sim }_{P{O}_3}{T}^{\prime }\iff ({\mathcal{P}}_T,\subseteq )\cong ({\mathcal{P}}_{T^{\prime }},\subseteq )$ for $f$ isom., ${\forall }_{A\in mod(T)}A{\sim }_{P{O}_2}f(A)$ It might be the case though that $T{\sim }_{P{O}_3}{T}^{\prime }\iff T{\sim }_{P{O}_2}{T}^{\prime }$.

It would be then interesting to see whether ${\sim }_{P{O}_1}$, ${\sim }_{P{O}_2}$ and ${\sim }_{P{O}_3}$ are equivalent to one of the previously mentioned relation or if they are subject to the same critique.

Special Identities

Halvorson considers only those identity criteria that can be applied on all theories in full generality, just like the method I described above. Though, one might be willing, for some other reason, to consider only some theories in particular and define some equivalence relation not subject to Halvorson’s critique. I do not dive into the numerous possibilities here but want instead consider one precise method that goes in this direction which is strictly related to my First Bachelor Thesis Concept in particular to: Algebra on Worlds.

The thesis I want to argue for in part of my thesis is that, thanks to some structure on generators of ultrafilters, I am able to transfer such a structure up to ultraproducts (i.e. some models). For the sake of this comment, I will assume that there is some algebra $\mathcal{A}$ on ultraproducts of $(A_n{)}_{n\in \mathbb{N}}$ a sequence of structures. More precisely I assume that there are some operations (of any arity) $R_1,...,R_n$ on $\mathcal{X}:=\{{\prod }_{\mathcal{U}}{A}_n:\mathcal{U}$ is ultrafil.$\}$ s.t. for any $X,Y\in \mathcal{X}$ we have $X{R}_{i}Y\in \mathcal{X}$.

Now, in case $mod(T)=\mathcal{X}$,

• $T{\sim }_{{\Pi }_1}{T}^{\prime }\iff mod(T)\cong mod(T^{\prime })$ for $mod(T)=\{{\prod }_{\mathcal{U}}{A}_n:\mathcal{U}$ is ultrafil.$\}$ and operations $R_1,...,R_n$.

The case $mod(T)=\mathcal{X}$ rightfully seems not to be a probable one, though, since in $\mathcal{X}$ we range on all $\mathcal{U}$ and keep $A_n$ fixed, we might set $A_n:=mod(T)$, hence consider: ${\mathcal{X}}_T:=\{{\prod }_{\mathcal{U}}mod(T):\mathcal{U}$ is ultrafil.$\}$ and ${\mathcal{A}}_T$ its algebra. A relevant question here is: does $mod(T)\subseteq {\mathcal{X}}_T$ hold for any theory?

If we have $mod(T)\subseteq {\mathcal{X}}_T$ we can then define:

• $T{\sim }_{{\Pi }_2}{T}^{\prime }\iff {\mathcal{A}}_T\cong {\mathcal{A}}_{T^{\prime }}$ for is ultrafil..

Now back to the question, does $mod(T)\subseteq {\mathcal{X}}_T$ hold for any theory? By Łos Theorem, for $\mathcal{U}$ ultrafil. and $A\equiv A^{\prime }:={\forall }_{\phi \in \mathcal{L}}A\models \phi \iff A^{\prime }\models \phi$, we have ${\forall }_{u\in \mathcal{U}}n\in u\Rightarrow {\prod }_{\mathcal{U}}{A}_n\equiv A_n$ and since ${\forall }_{n\in \mathbb{N}}{\exists }_{\mathcal{U}\subseteq \mathcal{P}(\omega )}(\mathcal{U}$ is ultrafil. $\wedge {\forall }_{u\in \mathcal{U}}n\in u)$, then ${\forall }_{n\in \mathbb{N}}{\exists }_{X\in \mathcal{X}}{A}_n\equiv X$ hence, up to elem. equivalence, $(A_n{)}_{n\in \mathbb{N}}\subseteq \{{\prod }_{\mathcal{U}}{A}_n:\mathcal{U}$ is ultrafil.$\}$. To get to a formal proof I should prove: (i) ${\forall }_{n\in \mathbb{N}}{\exists }_{\mathcal{U}\subseteq \mathcal{P}(\omega )}(\mathcal{U}$ is ultrafil. $\wedge {\forall }_{u\in \mathcal{U}}n\in u)$ and (ii) ${\forall }_{u\in \mathcal{U}}n\in u\Rightarrow {\prod }_{\mathcal{U}}{A}_n\equiv A_n$.

Technical Part

Here I summarise some of the results and definition that Halvorson states from page 191. $\mathcal{L}(T)$ is the language of the theory $T$.

• for $T$, $T^{\prime }$ theories, $F:\mathcal{L}(T)\to \mathcal{L}(T^{\prime })$ preserving var. and arity of constants s.t. for each $\phi ,\psi \in T$, $\phi ⊢\psi \Rightarrow F(\phi )⊢F(\psi )$ p. 191 (Interpretation)
• $F:T\to T’$, $G:T’\to T$ interpretations s.t. ${\forall }_{\phi \in \mathcal{L}(T)}(T⊢G\circ F(\phi )\leftrightarrow \phi )\wedge {\forall }_{\psi \in \mathcal{L}(T^{\prime })}(T^{\prime })⊢F\circ G(\psi )\leftrightarrow \psi )$, ($G$ is weak inverse)
  • I don't see how concatenation of the functions can work here
• $F,G$ interp. s.t. $G$ weak inv. of $F$ ($T$, $T^{\prime }$ definitionally equivalent)
• $T=\{{\exists }_{=1}x(x=x)\}$, $T^{\prime }=T\cup \{i\in \mathbb{N}:Q_0(x)⊢Q_i(x)\}$
  • |$mod(T)|=|mod(T^{\prime })|$
  • there is a bij. $f:mod(T)\to mod(T^{\prime })$ s.t. $f(m)\cong m$. (p. 192bottom)
• $F,F^{\ast }$ interp. s.t. $F^{\ast }$ weak inv. of $F$, ${\exists }_{m^{\prime }\in mod(T^{\prime })}|m^{\prime }|\ne |F^{\ast }(m^{\prime })|$.
  • hence def.equiv. theories don’t have isom. models. (?)