# 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.