# Soggy Predicates

**Warning: I suggest to either directly use the subset notation (see below) or introduce a SoggyLink to turn any predicate into a soggy predicate where the domain of that soggy predicate is and only the powerset of the domain the given predicate. Automatically extending the domain of a predicate to its powerset leads to bad foundations.**

A Simple Observation Grounded predicate, or Soggy predicate, is an uncertain predicate F so that: For each x, the number F(x) lies in [0,1] and can be interpreted as the average degree to which an arbitrary element of some set O of observations has property F. (Here we assume that the degree to which a specific observation has a property F is itself a number in [0,1])

(Note that the set of observations O need not be the observations actually made by the AI system whose memory contains the predicate — it may be observations made by some other hypothesized entities, etc.)

In this case we have a clear interpretation for

EvaluationLink <s> PredicateNode F Atom x

So we can say

PredicateNode F <t>

means the average degree to which an arbitrary element y of the AI system’s “default set of observations” satisfies F (i.e. the average over this default observation-set of F(y))…

Or, for an observation-set C, we can say

ContextLink <t1> ConceptNode C PredicateNode F

where t1 is the average over F(y), where y is counted in the average with a weight proportional to the degree to which y is in C.

So then

PredicateNode F <t>

means, conceptually,

ContextLink <t> >default context< PredicateNode F

Next, we can then define

EvaluationLink <s> PredicateNode F Atom x

as being equivalent to

MemberLink <s> Atom x SatisfyingSet PredicateNode F

Basically, this is just defining the membership function of the fuzzy set

SatisfyingSet PredicateNode F

in a particular way.

We can then convert this to

Subset <s> Atom x SatisfyingSet PredicateNode F

because of the way F was originally defined.

This becomes a bit subtle to interpret in the case that the argument of F is a list.

For example,

EvaluationLink <s> PredicateNode “eat” ListLink ConceptNode “cat” ConceptNode “mouse”

converts to

SubsetLink <s> ListLink ConceptNode “cat” ConceptNode “mouse” SatisfyingSet ConceptNode “eat”

which means that s is the average degree to which an observation (in the default overall observation-set) involving “eat”, also involves the pair (eater = cat, eatee=mouse).

## Interpretation in terms of Multisets

One way to phrase the above interpretation is to introduce the multiset interpretation of fuzzy membership degrees from

https://pdfs.semanticscholar.org/c18f/ba07ba1dbb99700391df30f6ca8798e5df9f.pdf

and to associate F with a multiset **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M_F}**
, in which an entity **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x}**
has **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(F,x)}**
copies, where this number of copies is then proportional to the degree to which **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x}**
has property **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F}**
. If we assume the degrees of membership of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F}**
are of the form **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k \over N}**
, then **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(F,x) = m}**
means **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(x)}**
has degree **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m \over N}**
.

Suppose **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F}**
is Soggy, and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x}**
has been observed **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M}**
times, and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p}**
of these observations have fully manifested property **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M-p}**
observations have manifested

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(F,x) = N * {p \over M} }**

More generally we have

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(F,x) = N * { \sum_{i=1}^M d(F,x_i) \over M } }**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i}**
is the i’th observation of x, and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d(F,x_i)}**
is the degree to which **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i}**
manifests

This multiset interpretation has the advantage of

- presenting fuzzy degrees as probabilities (across multisets rather than ordinary sets
- providing a formal justification for the use of fuzzy set (min/max) algebra