# Fuzzy Logic

Recall the definition of an $n$-ary logic system as a homomorphism $\varphi:F[X]\to V$ with $X$ a theory, $F$ a structure, and $V$ a structure of cardinality $n$ such that $F$ and $V$ have the same signature.  Fuzzy logic is hereby defined as the study of $c$-ary logic systems where $c$ is the cardinality of the continuum.  In this sense propositions can be thought of as having valence values in some interval, like $[0,1].$  So classical binary boolean logic is a type of fuzzy logic, where $\mbox{ran}\,\varphi=\{0,1\}.$

Example 1.  Recall in our construction of a utilitarian set we had a set $X$ together with a utility function $u:X\to [-1,1).$  Now suppose $T$ is a set of terms which is also a utilitarian set.  Suppose we define a structure $F$ on $T/\sim_u$ by

$\displaystyle [x]\vee [y]=u'^{-1}\left(\min\{u'[x],u'[y]\}\right)=u'^{-1}\left(\min\{u(x),u(y)\}\right)$

$\displaystyle [x]\wedge [y]=u'^{-1}\left(\max\{u'[x],u'[y]\}\right)=u'^{-1}\left(\max\{u(x),u(y)\}\right).$

This structure has signature $(0,0,2,0,...).$  Then $T/\sim_u$ is also a set of terms, with certain terms in $T$ identified, and if $X$ is a theory/subset in $T/\sim_u,$ then $u':F[X]\to [-1,1)$ is a $c$-ary logic system where $[-1,1)$ has the min and max operations.

In the case of Example 1, if we think of the set $T$ of terms as a set of behaviors, which could be construed as terms of persons (acting as words), then the structure $F[X]$ can be interpreted as equivalence classes of compound behaviors that yield the same utility where logical valence of compound behaviors is simply based on their utility.

Since the valence set in a fuzzy logic system is an interval, let us look at some common structures on $[0,1]$ to discuss some intuitive structures on terms.  We already mentioned min and max functions and a corresponding structure on class utilitarian sets.  $[0,1]$ is closed under binary multiplication.  The corresponding binary connective would thus send two propositions to a proposition whose valence is a product of the original two.  Treating binary boolean logic as a fuzzy logic, propositional conjunction satisfies this condition.

Example 2.  Let $(X,\Sigma,P)$ be a probability space.  Let $[0,1]$ be closed under addition where any sum exceeding $1$ is defined as $1$ and define a unary operation by $r^{-1}=1-r.$ Now note that $\Sigma$ is closed under union and complementation; denote this structure on $\Sigma$ by $F.$  If measurable sets are construed as formulas, select a theory $X$ of disjoint sets.  Then $P:F[X]\to [0,1]$ is a fuzzy logic system.

# Capitalism

Definition 1.  We define a static capital system as a triple $(X,\Sigma,\mu)$ with counting measure $\mu$ where $\mu(X)=m$ and  is called the monetary constant, $\Sigma$ is a collection of subsets of $X$ such that $\mu(\cup_{A\in\Sigma}A)=m,$ elements of which are called owners, and $\mu(A)$ is called the worth of $A$ for an owner $A.$

Note we are not requiring $\Sigma$ to be closed under any operations (i.e. it is not an algebra of sets).  Suppose we have two structures on $X,$ $(X,\Sigma_1,\mu)$ and $(X,\Sigma_2,\mu).$   Let $f:\Sigma_1\to P(\Sigma_2)$ (i.e. a multi-valued map into $\Sigma_2$).  Such a function is called a trade (and may correspondingly be thought of as a change of ownership).  We define the trade utility of a trade $f$ as a map $u_f:\Sigma_1\to\mathbb{Z}$ by

$\displaystyle u_f(A)=\mu\left(\bigcup f(A)\right)-\mu(A).$

Again, $\cup f(A)$ need not be in $\Sigma_2,$ but we can of course still define the counting measure on it.

Definition 2.  A composite trade is a map $g\circ f:\Sigma_1\to P(\Sigma_3)$ where $f:\Sigma_1\to P(\Sigma_2)$ and $g:\Sigma_2\to P(\Sigma_3)$ are trades.

Note that $g\,\circ:P(\Sigma_2)\to P(\Sigma_3)$ since it is defined on the image of $f.$  $g\,\circ$ simply evaluates $g$ on all sets in $f(A).$

Definition 3.  Let $(X,\Sigma_t,\mu)_{t\geq 0}$ be a continuum of static capital systems.  We say $(X,\Sigma_t,\mu)_{t\geq 0}$ is a capital system if

1. for every $t\geq 0$ and $\varepsilon\geq t$ there is a unique trade $f_{t,\varepsilon}:\Sigma_t\to P(\Sigma_{t+\varepsilon});$
2. $f_{t,0}=1$ (i.e. $f_{t,0}(A)=\{A\}$);
3. if $f_{t,\varepsilon_1}$ and $f_{\varepsilon_1,\varepsilon_2}$ are trades such that $\varepsilon_1+\varepsilon_2=\varepsilon,$ then $f_{t,\varepsilon}=f_{\varepsilon_1,\varepsilon_2}\circ f_{t,\varepsilon_1}$ for all $t,\varepsilon_1,\varepsilon_2\geq 0.$

Example 4.  A capital system is in a socialist state at time $t$ if $\mu(A)=\mu(B)$ for all $A,B\in\Sigma_t.$  We may further say $(X,\Sigma_t,\mu)_{t\geq 0}$ is socialist during $T\subseteq[0,\infty)$ if $(X,\Sigma_t,\mu)$ is in a socialist state for all $t\in T.$  A capital system is in a communist state at time $t$ if $\Sigma_t=\{X\}.$ Similarly we have the definition for communist during a set $T\subseteq[0,\infty).$

Note that by this definition a communist state implies a socialist state.  In the above regards, a communist state can be thought of as having a single owner (say, “the people”), and socialist state has owners with equal worth.

Definition 5.  A dynamic capital system is a capital system $(X_t,\Sigma_t,\mu)_{t\geq 0}$ where $(X_t,\Sigma_t,\mu)$ is a static capital system for all $t$ where $\mu(X_t)=m_t$ and $X_t,X_s$ are comparable (in the inclusion sense) for all $s,t\geq 0.$  In particular the function $m:[0,\infty)\to\mathbb{N}$ defined by $m(t)=m_t$ is called the monetary policy.  If $m_t$ is strictly increasing during an interval, we say $(X_t,\Sigma_t,\mu)$ is expansionary during that interval.  Similarly it is  contractionary if it is strictly decreasing on some interval.

Definition 6.  A dynamic capital system $(X_t,\Sigma_t,\mu)_{t\geq 0}$ is rational if $u_{f_{t,\varepsilon}}\geq 0$ for all $t,\varepsilon\geq 0.$

Of course if $\varepsilon$ is $0$ we have $f_{t,0}=1$ and thus the condition is satisfied for this case:

$\displaystyle u_{f_{t,0}}(A)=\mu\left(\bigcup \{A\}\right)-\mu(A)=0.$

So in a rational dynamic capital system we have the inequality

$\displaystyle\mu(A)\leq\mu\left(\bigcup f_{t,\varepsilon}(A)\right)\leq m_{t+\varepsilon}$

with $A\in\Sigma_t.$  If $\lim_{t\to\infty}m_t$ exists and is finite, then the rational dynamic capital system $(X_t,\Sigma_t,\mu)$ is said to have an end game.

# A Survey of Utilitarianism

There is a common misconception about the application of the theory of utilitarianism.  Many attempt to apply it to events that have happened in history.  The purpose of utilitarianism is, really, an attempt to establish a choice function on a set of options.  Since events in history presumably happened precisely because of their sets of antecedents, there are no other choices of events;  so utilitarian analysis of them is trivial.  It can be usefully applied to psychology in the form of decision making.  One must decide which behaviors to commit in a given set of circumstances based upon predicted costs and benefits.  To approach this rigorously, we will work in ZF theory with special functions.

Definition 1.  Let $X$ be a set and $u:X\to [-1,1)$ be a function called a utility function.  The pair $(X,u)$ will be called a utilitarian set.  A sub utilitarian set is a pair $(A,u_A)$ where $A\subseteq X$ and $u_A=u|_A$.

We will also define the trivial utilitarian set as the pair $(\varnothing, u)$ where $u(\,)=0$.

Note that the motivation for closing the codomain on the left is that a behavior bringing death is assumed to be of minimal utility.  This allows us to put a choice function on $X$ iff $u(x)=u(y)\Rightarrow x=y$ (i.e. iff the map is injective).  This choice function is defined by

$\displaystyle C(X)=u^{-1}\left(\min_{x\in X}u(x)\right).$

The only problem is that the choice function will pick the “worst” option, and we want to pick the “best”.  Let us define

$X_1=X-\{C(X)\}$,

$X_{n+1}=X_n-\{C(X_n)\}$.

If $X$ is finite and $u$ is injective, then $X_n=\varnothing$ for all $n\geq m$ for some $m$.  In this case $X_{m-1}$ is a singleton consisting of the “best” element.  If $u$ is not injective, then we may have $C(X)\subseteq X$.  This isn’t a problem for the worst elements;  if worst elements all go to the same value, we can just take them all out.  But there is also the possibility of having best elements with the same utility.

Define a relation $\sim_u$ where $x\sim_u y\Leftrightarrow u(x)=u(y)$.  This is clearly an equivalence relation.  Now consider the set $X/\sim_u$.  We have an induced utility function $u'$ on $X/\sim_u$ defined by $u'([x])=u(x)$.  By definition of the equivalence relation, we have that any induced utility function on $X/\sim_u$ is injective.  Hence for any finite $X$, we have a best element of $X/\sim_u$.  This is just the set of best elements of $X$.

Definition 2.  If $(X,u)$ is a utilitarian set, we call $(X/\sim_u,u')$ the class utilitarian set of $(X,u)$.

One can easily see an isomorphism (in the sense that $u'(x)=u''(\varphi(x))$) between $(X/\sim_u,u')$ and $((X/\sim_u)/\sim_{u'},u'')$ where $\varphi([x])=[[x]]$ and so on.  We thus limit ourselves to utilitarian sets where $u$ is injective as it always will be on the class set.

Now assume the Axiom of Choice (for purposes of ordering elements of $X$, and note this still doesn’t allow us to pick a “best” element of it trivially, since best is defined by the element taking largest value, if it exists).  Let $(X,u)$ be a utilitarian set with $X$ denumerable.  Then injectivity of $u$ allows for a set $\{u(x_n)\}$ to be a bounded strictly monotonically increasing sequence in $[-1,1]$ which in turn contains a convergent subsequence $\{u(x_k)\}$.  Note we cannot say the sequence is bounded in the codomain, but it is of course bounded in $[-1,1]$.

Definition 3.  Let $(X,u)$ be a denumerable utilitarian set with $u$ injective and $\{x_n\}$ an ordering of $X$.  We say $(X,u)$ is a decidable set if

$\displaystyle\max\lim u(x_k)<\sup_{n\in\mathbb{N}} u(x_n)=\sup_{x\in X} u(x)$

where the max is taken over all convergent subseqences $\{u(x_k)\}$ of $\{u(x_n)\}$.  If $(X,u)$ is decidable, then the supremum above can be replaced with a maximum (otherwise its value would have been a limit over which the max on the left was taken, and hence, the maximum of them).  Hence if $(X,u)$ is decidable, we define the best choice as

$\displaystyle B(X)=u^{-1}\left(\max_{x\in X}u(x)\right)$.

A utilitarian set is undecidable if it is not decidable.

Consider the function defined by

$\displaystyle d(x,y)=|u(x)-u(y)|$.

This is a semimetric on an arbitrary $(X,u)$ and a metric when $u$ is injective.  We can make a modification and define the opportunity cost of $x$ with respect to $y$ by

$d_y(x)=u(x)-u(y)$.

If one assumes that behaviors consume resources proportional to the utility acquired, then the above describes a proportional number of resources lost (or gained if negative) by choosing behavior $x$ over $y$.

Definition 4.  A discrete path in $X$ is a sequence in $X$, say $\gamma:\mathbb{N}\to X$.  A path is a continuous map $\gamma:[0,1]\to X$ where $X$ is endowed with its topology induced by $d$.  We can define the marginal utility of a path $\gamma$ at time $n$ and $t$ respectively for the type of path by

$\displaystyle\gamma'(n)=d(\gamma(n+1),\gamma(n))=d(x_{n+1},x_n)$

$\displaystyle\gamma'(t)=\lim_{\varepsilon\to 0}\frac{d(\gamma(t+\varepsilon),\gamma(t))}{\varepsilon}$.

Note the second one may not exist.  If one interprets this by defining a sequence $\{x_n\}$ for a behavior $x$ called “consuming a good” and where $n$ represents the number of units of that good consumed, then marginal utility simply represents the change of utility in consuming one additional unit of that good.

If we can accept that all decision making of individuals (i.e. psychology, where biological and environmental factors determine the utility function) models this theory, then aggregately so does that of  groups of individuals–making this the foundation of social science.

# An Approach to a Legal System with Utilitarian Members

Definition 1.  A static legal system is a collection of individuals together with a collection of laws.  (Mathematically, it’s a set $P$, whose elements are called persons, together with a unary operation $\varnothing\in P$, called the null person, and a collection of infinitary $\{0,1\}$-valued maps $\{l_n:P^\infty\to\{0,1\}\}$, called laws, such that all but finitely many of the terms in the domain are $\varnothing$).

For example, a law $l_n$ defined on persons $p_1,...,p_k$ may evaluate $l_n(p_1,...,p_k,\varnothing,...)=1$ , meaning that law $l_n$ pertaining to individuals $p_1,...,p_k$ is legal (or illegal if it returned value $0$).   To accommodate the influence that individual complexity has in a  legal system, we could make the additional assumption that each person is a static legal system with persons and laws respectively replaced by perceptions and thoughts.  Let $p$ denote the set of perceptions,  the thoughts have the form $\{t_n:p^\infty\to\{0,1\}\}$ such that all by finitely many of the perceptions are null perceptions.  The value the thought takes determines whether or not a behavior is executed.  But what is a behavior?  A behavior executed from a thought is a model that satisfies the perceptions on which that thought was defined.  Hence we could write

$\displaystyle t_n(p_1,...,p_k,\varnothing,...)=1\Rightarrow\left( B(t_n)\vDash\{p_1,...,p_k\}\right)$

where $p_i$ are perceptions.  In simple terms, this just means that you only act in agreement with your perceptions (simple thoughts that are assumed true).  To reconnect back to the original static legal system, we may now say that laws dictate the legality of individual or group behavior.  A person is simply a collection of perceptions together with thought functions.  Hence we may say that a law defined on a thought function $t$ is in turn defined on all behaviors $B(t)$.  We simply set $l(B(t))=l(t)$.

Let us now assume every person behaves in a utilitarian manner.  By this I mean that every person has a function $u_p:\mathcal{B}\to\mathbb{R}$ where $\mathcal{B}$ is the set of legal behaviors of an individual that satisfies the following conditions for all thoughts $t$ and perceptions $\pi_1,...,\pi_k$ such that $B\vDash\{\pi_1,...,\pi_k\}$ (we also omit null perceptions for convenience):

1. $u_p(B)\leq 0\Rightarrow t(\pi_1,...,\pi_k)=0$ and
2. $u_p(B)>0\Rightarrow t(\pi_1,...,\pi_k)=1$.

Hence, assuming legality of the behavior, the above implications are equivalences  (i.e. a legal behavior is committed if and only if it has positive utility).  We could extend the utility functions to illegal behaviors, and the legal utility would be defined as

$\lambda(B)=u(B)+u(B')$

where $B'$ is the cobehavior (or punishment) of $B$.  For legal behaviors $B$, we have $B'=\varnothing$ and hence $\lambda(B)=u(B)$.  But we will assume all committed behaviors are legal for simplicity.

Sometimes groups of individuals will work together as a firm (or corporation) and execute aggregate legal behaviors of positive aggregate utility.  The aggregate behavior and utility are defined as some function of the behavior and utility functions of members of the firm.

Example 2.  Suppose $B=\sum_iB_i$ is a corporate behavior which is a formal sum of behaviors of its members.  Define the aggregate utility $U_F$ of a firm $F$ as

$\displaystyle U_F(B)=\sum_i u_i(B_i)$

where $u_i$ is the utility function of the person committing behavior $B_i$.  Perhaps if a member contributes $n$ behaviors per day, then they may say the utility of a behavior is $S_p/(365n)$ where $S_p$ is the salary of person $p$.  In this case the corporate utility summed over behaviors throughout the year would be their (in economic terms) normal profit (assuming no other expenditures), but it isn’t a very useful value since it does not indicate economic profits, which is really their utility in capitalistic economies.

Definition 3.  A dynamic legal system is collection of static legal systems.  A dynamic legal system $P=\{P_t\}_{t\in\mathbb{R}}$ is continuous if for every static legal system $P_t$ there is some $\varepsilon>0$ such that the cardinalities of sets of laws for all static legal systems $\{P_{t-\varepsilon},...,P_t,...,P_{t+\varepsilon}\}$ differ by at most $1$ (i.e. laws don’t change too fast).

A token economy is a special kind of dynamic legal system, indexed by time, where we define the utility of a behavior to be $1$ if it gives the person $1$ token.  What makes a token economy special is conservation of utility.  This means that

$\displaystyle\sum_{ij} u_i(B_{ij})=0$

where $B_{ij}$ is the $j$th behavior committed by person $p_i$ for all static legal systems $P_t$ (i.e. the sum of token changes at any time is equal to $0$).

In particular it implies that any behavior $B_p$ of person (or firm) $p$ with utility $u_p(B_p)$ has a coperson (or cofirm) $p'$ such that $B_p$ is a cobehavior to person $p'$ and $u_p(B_p)+u_{p'}(B_p)=0$.

Note:  It’s definitely choppy in some areas and could use revision, but I thought I’d throw it out first.

Let me briefly summarize the post.  A legal system is a collection of people, where each person is, for all intensive purposes, a collection of behaviors, together with a collection of laws that govern those behaviors.  We can further assume that behaviors are committed provided that they are of positive utility.  The behaviors that take negative values are cobehaviors (behaviors executed by other persons for whom the utility is positive and equal in magnitude).  In actuality when one purchases something, although they lose money, they typically receive something else useful in return; however, our study is limited strictly to currency, in which case they lose from a purchase.  In this case we can think of the economy of currency/tokens as an approximation to a fluid that changes its density through transactions; whereas the individual who loses/destroys tokens in exchange for a product can simply think of the product as a manifestation of the tokens exchanged for it.