## Does intelligence imply contradiction?

##### Patrizio Frosini

Received 19 June 2002; reveiced in revised form 30 July 2007

### Introduction

#### Idea

Is contradiction accidental, or is it the necessary companion of intelligence?
Thesis: Contradiction can be seen as a tool to increase intelligence in biological systems.

#### Why is this relevant?

A mathematically sound answer to this question could give new perspectives on artificial intelligence. Think for instance about modern loss functions as

• Contrastive
• Triplet
• Capsules

### A brief history of contradiction

#### Mathematics and computer science

• Second theorem
• Many-valued systems

#### Psychology

• Cognitive dissonance
• Time-inconsistent agent

### Intuition

A contradiction is an observed phenomenon in which the same entity evolves in two different ways (in different times) from the same initial state.

### Framework

#### What shall be defined

A contradiction is an observed phenomenon in which the same entity evolves in two different ways (in different times) from the same initial state.

#### Counter-argument

• Different answers are caused by different mind states?
• The question is posed or perceived in a different way?

#### Formalism

$\left(\mbox{Observer}, \mbox{Event}\right)$

### Main theorem

Any sufficiently intelligent entity must be contradictory

Observation: intelligence needs a context, and an observer in order to be studied.

Thus, a quantum-mechanic flavoured replacement has to be put in place: $$\mbox{particle}\mapsto\mbox{observed particle}$$

### Framework for testing

Entity, environment, intelligence, contradiction can be defined in the fairly general contex of cellular automata Furthermore, cellular automata have the property of universal computation.

###### Wolfram (1994)

Video realised by courtesy of Iratxe Ansa and Igor Bacovich, and the dancers of the wonderful Metamorphosis project Code available here

### Framework and epistemology

Def [Existence]: The existence of an entity depends exclusively on the observer.

Caveat: Here the subjectivity of existence is not to be discussed. We should rather decide to take it into account or not.

###### Mc Ginn (1983). The subjective view

Def [Observer]: Let $\mathcal{P}_{ent}$ and $\mathcal{P}_{ENV}$ be non-empty, finite sets of perceptible states for an entity and an environment, respectively. Assume that it exists a privileged element $0\in\mathcal{P}_{ent}$. An observer is any function $$\square = \left(ps_{ent}, ps_{ENV}\right):\Sigma\rightarrow \mathcal{P}_{ent}\times\mathcal{P}_{ENV}$$

Observations

• The observer acts as an encoder
• $0\in\mathcal{P}_{ent}$ to encode absence
• The two sets need to be finite, because observer are not allowed infinite memory, nor computational capability.

### Framework and epistemology

Def [Existence]: The existence of an entity depends exclusively on the observer.

Caveat: Here the subjectivity of existence is not to be discussed. We should rather decide to take it into account or not.

###### Mc Ginn (1983). The subjective view

Def [Observer]: Let $\mathcal{P}_{ent}$ and $\mathcal{P}_{ENV}$ be non-empty, finite sets of perceptible states for an entity and an environment, respectively. Assume that it exists a privileged element $0\in\mathcal{P}_{ent}$. An observer is any function $$\square = \left(ps_{ent}, ps_{ENV}\right):\Sigma\rightarrow \mathcal{P}_{ent}\times\mathcal{P}_{ENV}$$
Def [Entity & lifetime]: Each maximal sequence of consecutive perceived states in $\mathcal{P}_{ent}\setminus\left\{0\right\}$, say $\left\{ ps_{ent}\left(s_{t+i}\right) \right\}_{i=0}^q$ is an entity. The set $\left\{t, \dots, t+q\right\}$ is the lifetime of the entity.

### Framework and epistemology

Def [Environment]: Let $\varepsilon = \left(ps_{ent}\left(s_{t} \right), \dots, ps_{ent}\left(s_{t+q} \right)\right)$ be an entity. We call environment $\left(ps_{ENV}\left(s_{t} \right), \dots, ps_{ENV}\left(s_{t+q} \right)\right)$

Def [Intelligence]: Given an entity $\varepsilon = \left(ps_{ent}\left(s_{t} \right), \dots, ps_{ent}\left(s_{t+q} \right)\right)$ with respect to the observer $\square$, we define the intelligence $L\left(\varepsilon, \square\right)$ as the entity's lifetime $q$.

###### Minsky (1986); Turing (1950); Stenberg (1990).

Reminder: An entity is contradictory if faced with the same circumstances (as perceived by the observer) in two occasions in time, it exhibits different behaviours.

Def [Contradictory entity]: Let $\varepsilon = \left(ps_{ent}\left(s_{t} \right), \dots, ps_{ent}\left(s_{t+q} \right)\right)$ be an entity for $\square$. If there exists $a,b\in\mathbb{N}$, $a, b\leq q$, such that $$ps_{ENV}\left(s_{t+a} \right) = ps_{ENV}\left(s_{t+b} \right) \mbox{ and } ps_{ent}\left(s_{t+a} \right) \neq ps_{ent}\left(s_{t+b} \right)$$ then $\varepsilon$ is contradictory.

### Framework and epistemology

Def [Environment]: Let $\varepsilon = \left(ps_{ent}\left(s_{t} \right), \dots, ps_{ent}\left(s_{t+q} \right)\right)$ be an entity. We call environment $\left(ps_{ENV}\left(s_{t} \right), \dots, ps_{ENV}\left(s_{t+q} \right)\right)$

Def [Intelligence]: Given an entity $\varepsilon = \left(ps_{ent}\left(s_{t} \right), \dots, ps_{ent}\left(s_{t+q} \right)\right)$ with respect to the observer $\square$, we define the intelligence $L\left(\varepsilon, \square\right)$ as the entity's lifetime $q$.

Def [Contradictory entity]: Let $\varepsilon = \left(ps_{ent}\left(s_{t} \right), \dots, ps_{ent}\left(s_{t+q} \right)\right)$ be an entity for $\square$. If there exists $a,b\in\mathbb{N}$, $a, b\leq q$, such that $$ps_{ENV}\left(s_{t+a} \right) = ps_{ENV}\left(s_{t+b} \right) \mbox{ and } ps_{ent}\left(s_{t+a} \right) \neq ps_{ent}\left(s_{t+b} \right)$$ then $\varepsilon$ is contradictory.

#### Counter-arguments

How does this fit in the classical formulation of contradictory theory?

### Framework and epistemology

Def [Environment]: Let $\varepsilon = \left(ps_{ent}\left(s_{t} \right), \dots, ps_{ent}\left(s_{t+q} \right)\right)$ be an entity. We call environment $\left(ps_{ENV}\left(s_{t} \right), \dots, ps_{ENV}\left(s_{t+q} \right)\right)$

Def [Intelligence]: Given an entity $\varepsilon = \left(ps_{ent}\left(s_{t} \right), \dots, ps_{ent}\left(s_{t+q} \right)\right)$ with respect to the observer $\square$, we define the intelligence $L\left(\varepsilon, \square\right)$ as the entity's lifetime $q$.

Def [Contradictory entity]: Let $\varepsilon = \left(ps_{ent}\left(s_{t} \right), \dots, ps_{ent}\left(s_{t+q} \right)\right)$ be an entity for $\square$. If there exists $a,b\in\mathbb{N}$, $a, b\leq q$, such that $$ps_{ENV}\left(s_{t+a} \right) = ps_{ENV}\left(s_{t+b} \right) \mbox{ and } ps_{ent}\left(s_{t+a} \right) \neq ps_{ent}\left(s_{t+b} \right)$$ then $\varepsilon$ is contradictory.

Def [Deterministic environment]: An environment $\left(ps_{ENV}\left(s_{t} \right), \dots, ps_{ENV}\left(s_{t+q} \right)\right)$ is deterministic if for every $a,b\leq q$ such that $$ps_{ENV}\left(s_{t+a} \right) = ps_{ENV}\left(s_{t+b} \right) \mbox{ and } ps_{ent}\left(s_{t+a} \right) = ps_{ent}\left(s_{t+b} \right),$$ we have $ps_{ENV}\left(s_{t+a+1} \right) = ps_{ENV}\left(s_{t+b+1} \right)$.

### Theorem and applications

Let $\varepsilon$ be an entity with finite lifetime and deterministic environment, with respect to the observer $\square$, for the cellular automaton $\mathcal{C}$. Let $k = |\mathcal{P}_{ent}\times\mathcal{P}_{ENV}|$. If the intelligence (lifetime) of $\varepsilon\geq k$, then $\varepsilon$ must be contradictory.

#### Stockholders and share prices

$P_i = \left(\mbox{up}, \mbox{down}, \dots\right)$. $P_i$ wins the $k$th round if $c_{k+1} > c_{k}$ and $P_{i,k} = \mbox{up}$.

$r_{cc}= 2,\, r_{nn} = 0,$ $r_{nc} = \left\{\begin{matrix}n\mapsto 1,\\ c\mapsto -1\end{matrix}\right.$

$\$ = 10000$,$p\in \{900, 1000, 1100\}$,$t = 7 \mbox{ days}\$.