Internalizing Tarski

Take standard first-order semantics.  Consider a model $M$ of a certain theory $T$. Now suppose that we wish to include a subset a $\Psi \subset M$ to represent first-order formulas $\phi$. We denote the element corresponding to a formula $\phi$ by  $\bar{\phi} \in \Psi$.  Suppose we wanted all subsets of $M$ to also have internal representation via elements set $\Omega \subset M$. Given $X \subset M$ let the representative be denoted by $\bar{X} \in \Omega$.  Then we wish to have a predicate $m(x,y)$ such that

\[    M \Vdash  m(\bar{X},\overline{\phi(x)})  \text{  iff }  \{z \in M:   z \in  \mathcal{I}\phi(x)\} = X \]

where $\mathcal{I}$ is the Tarskian interpretation in $M$. We will investigate later if this can lead to a paradox or limitation result.

Now suppose that in FOL we had an operator $[\,]$ which transforms a formula $\phi$ into a constant $[\phi]$: the free variables in $\phi$ this time (unlike in Bealer's system) are not free in $[\phi]$, it is a constant (or closed term).  Suppose we had a predicate $Sub(a,b)$ which holds precisely when $a = [\psi(x)]$ for some $\psi(x)$ with single free variable $x$ and $b = [\psi([\psi(x)])]$.  This expresses the reflection-into-self of $a$.

Now suppose we had a truth-like-predicate $T(a)$ which holds precisely when $a = [\phi]$ for some sentence $\phi$  which is, for instance, provable in a deductive system $D$ or true in a given class of models $M$, etc.

We assume we have the basic properties: i) $T([\phi]) \rightarrow \phi$ and ii)  $T([\neg \phi])$ iff $ \neg T([\phi])$.

Consider now the formula $G(x) \equiv  \neg\exists y. Sub(x,y) \,\&\, T(y)$ and let $Sub([G(x)], g)$.

Suppose that $T(g)$. Then by hypothesis $Sub([G(x)], g)$  and so  by definition of $G(x)$, $\neg G([G(x)])$ and hence by i) and ii) we get  $\neg T([G([G(x)])])$, that is, $\neg T(g)$, a contradiction. 

Note how $g$ involves a further reflection-into-self mediated by negation. The result is thus a limitation on predicates satisfying i) and ii).  It seems very likely that the above argument could be transposed to Bealer's system.

In Bealer's system if we postulate $y\Delta [\phi(x)]_x \equiv \phi(y)$ then we can obtain  a Russell-type paradox (without relying on practically any other axiom).

A very interesting kind of 'limit theorem' involves how formal systems (such as $HA^\omega$) cannot represent the totality of total recursive functions.

Also the curious fact that we can define the set of indefinable real numbers.

This reflection-into-self, can it be given a geometric embodiment (if we wish to consider the connection between geometry and logic as in topos theory) ?

The downward Löwenheim-Skolem theorem is not the same kind of limit result. We consider that it is formalized within ZF(C) itself. If we do not accept this (i.e. the self-reflectivity of ZF(C)) then we are done. Otherwise we see that ZF(C) reflected-into-itself is aware that it is not really set theory, but only a countable reflection of set theory. This is a positive result. A limited system is  itself 'aware' of its own limitation.

The logic of Kant's Critique A70-76 in modern form

Consider Kant's table of the 'forms of the understanding'  obtained by abstracting judgments from their content, expounded in A70-76 (B95-101) and divided into quantity, quality, relation and modality.

Judgments are defined inductively.

Let $C_1$ and $C_2$ be concepts. Then $\mathcal{Q}C_1\epsilon C_2$, $\mathcal{Q}C_1\bar{\epsilon} C_2$ and $\mathcal{Q}C_1\epsilon \bar{C_2}$ are  judgments where $\mathcal{Q}\in \{U,P,S\}$.

Let $J_1$ and $J_2$ be a judgements. Then $J_1 \Rightarrow J_2$ is a judgment.

Let $J_1,...,J_n$ be judgements. Then $J_1 \vee...\vee J_n$ is a judgment.

Let $J$ be a judgement. Then $\lozenge J$ and $\square J$ are judgments.

Here $\bar{C}$ is term-negation. As in the infinite judgment 'the soul is not-mortal'.

But Kant distinguishes between judgment and the content of a judgment, in particular the proposition. Also considering Kant's example,  disjunction seems to be term-based:

Let $C_1,...,C_n$ be concepts such that $\bigcup_i C_i = \mathcal{U}$ (or rather, their extensions satisfy this) and $C$ a concept. Then $C\epsilon C_1\vee...\vee C\epsilon C_n$ is a judgment. So we can use Frege's 'vertical line', ancestor of our $\vdash$ and reformulate the syntax of the logic as follows:

Judgments are defined inductively.

Let $C_1$ and $C_2$ be concepts. Then $\mathcal{Q}C_1\epsilon C_2$, $\mathcal{Q}C_1\bar{\epsilon} C_2$ and $\mathcal{Q}C_1\epsilon \bar{C_2}$ are are propositions where $\mathcal{Q}\in \{U,P,S\}$.

Let $P_1$ and $P_2$ be propositions. Then $P_1 \Rightarrow P_2$ is proposition.

Let $C_1,...,C_n$ be concepts such that $\bigcup_i C_i = \mathcal{U}$ and $C$ a concept. Then $C\epsilon C_1\vee...\vee C\epsilon C_n$ is a proposition.

Let $P$ be a proposition. Then $\lozenge P$, $\vdash P$ and $\square P$ are judgments. 

See also the important remark in B141 where Kant confirms the above presentation. Also B11. Analytic judgments are $=_I$ in CIL whilst synthetic judgments are $=_N$, the are 'extensional' in Kant's own terminology !

Can Kant even express his 'analogies of experience' - which surely must be judgments ! - in such a logic ? The original version of the second analogy in A was: for everything that happens there is something which succeeds it, according to a rule. Alles, was geschiet (anhebt zu sein) setzt etwas voraus, worauf es nach einer Regel folgt, $\forall x. Hap(x) \rightarrow \exists y. RegSuc(x,y)$. 

According to Bobzien and Shogry Stoic logic could handle this.

What is a system ?

We cannot leave out time to begin with. Nor possibility, multiple times.  A first attempt would be as follows. A system involves an input $I$ space, an state space $S$ and an output space $O$. We consider that the outputs are completely determined by states, that is, there is  function $\phi: S \rightarrow O$.  Let $T$ be time which has a total order $<$. Given a set $X$ let $X^T$ be the set of all maps $f : T \rightarrow X$.  Then a system  $\mathfrak{S}$ over $(I,S,O,\phi)$ is a function $\mathcal{S}: I^T \rightarrow \mathcal{P}S^T$.

$I$ has a distinguished element $\bot \in I$ which represents doing nothing. We write $f \triangle g$ for $f,g \in X^T$ if there is a $t \in T$ such that $ t' < t$ implies that $f(t') = g(t')$.

A link function is a function $\upsilon : O \rightarrow I$.  With a link function we can define the plugging of one system into another.  The link function can also be considered as $O \rightarrow I^n$ expressing simultaneous different outputs.

$I$ can be given the structure of a category so we can define composition, etc.  We consider the possibility of multiple input and multiple output (and composition, plugging in of systems, including feedback). Thus there is an abstract cartesian product for a more general version of $I$ (and $O, \phi$, etc.).

We define $X^T_<$ to be the set of all functions $f : \downarrow t \rightarrow X$ where $\downarrow t =\{ t' \in T: t' < t\}$ for some $t$.  Then another definition of system is a function $\mathcal{S}: I^T_< \rightarrow \mathcal{P}S^T_<$ which respects $t$.

We can consider a category $\mathcal{I}^T_<$ consisting of sets  $I^{T^{t}_<}$ of functions $\downarrow t \rightarrow I$  for different  values of $t$. If $t_1 < t_2$ then we have natural restriction map.  This is a category. Likewise we can define categories $\mathcal{S}^T_<$ and $\mathcal{O}^T_<$.  We can also define the category  whose elements, for a given $t$, are sets of elements of $I^{T^{t}_<}$. A system is then a functor

\[ \mathcal{S}:  \mathcal{I}^T_< \rightarrow P\mathcal{S}^T_<\]

$S$ is itself a moduli space of structures, networks. Thus $S = S_s \times S_d$ where the first component is a configuration and the second the global state. 

Note that $I$ is more general than 'information', its can also include matter and energy as in biosystems.

In general our system will be a composition of other systems. We will need a calculus of dynamic reconfiguration , merging, separation, etc.

This second definition of system is better; it is the only one accessible to us, for we cannot observe infinite input,output or state histories.  

Perhaps one of the most fundamental division is between systems having an origin (and perhaps an end) and those that are not postulated as having an origin. Thus we need a special state $\bot \in S$ of non-being.  Then we can ask: does the system come into being because of a certain input (does this even make sense) or not ? We can conceive a kind of output from another system which is a self-replication or construction or generation of the new system.  The state at the first instance $t_0$ in which the system has come to be, is its initial state $s_0 \in S$.  Thus a generated system is given by a map (functor between categories) $\mathcal{S}: I^{T^{t_0}_<} \rightarrow \mathcal{P}S^{T^{t_0}_<}$ which respects $t$.

Note that the output or state-change is not necessarily to be considered instantaneous. Our framework is general enough to capture delay, feedback and a great variety of different kinds of causality, quasi-causality, indeterminism, etc.  It is however important to be able to give the images  $\mathcal{S}(s)$ in $\mathcal{P}S^{T^{t_0}_<}$  the structure of a $\sigma$-algebra and a measure (in particular complex-valued).

Thus a quantum system associates to each input path $p$ from an initial time $t_0$ to a time $t$ a  space $\Omega_p$ of possible state paths from time $t_0$ to $t$ together with a (complex-valued) probability measure on $\Omega_p$. We assume that all the $\Omega_p$ are endowed with a $\sigma$-algebra structure in a coherence way (perhaps as induced by such a structure on the space of all state paths from $t_0$ to $t$).

If $T$ is the usual real line, then we can consider input sequences which are almost everywhere $\bot$ or only have a finite number of non-$\bot$ inputs. In this special case the framework above reduces to the classical Feynman formalization.