Tweedledum and Tweedledee: Symmetry in Behavior Analysis

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Tweedledum and Tweedledee: Symmetry in Behavior Analysis. M. Jackson Marr Georgia Institute of Technology mm27@prism.gatech.edu. Maxwell’s Equations. Symmetry between electricity and magnetism: Explained the nature of light as electromagnetic waves moving at speed c.

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## Tweedledum and Tweedledee: Symmetry in Behavior Analysis

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### Tweedledum and Tweedledee:Symmetry in Behavior Analysis

M. Jackson Marr

Georgia Institute of Technology

mm27@prism.gatech.edu

Maxwell’s Equations
• Symmetry between electricity and magnetism: Explained the nature of light as electromagnetic waves moving at speed c.
Einstein’s Principles of Relativity
• (1) Laws of physics independent of states of motion.
• (2) Constancy of the speed of light, independent of states of motion.
• One implication: Equivalence of mass and energy.

E = mc2

SYMMETRY AND CONSERVATION LAWS
• Emmy Noether’s Theorem: For every symmetry exhibited by a physical law, there is a corresponding observable quantity that is conserved.
• Momentum: translations in space (including rotations).
• Energy: translations in time.
• Many other examples of symmetry/conservation, e.g., in particle physics (e.g., CPT—charge, parity, time—all together.).
SYMMETRIES IN BEHAVIOR ANALYSIS

Positive and negative reinforcement

Positive and negative punishment

Reinforcement and punishment

Discrimination and generalization

Shaping dynamics

Three-term contingency

Equivalence and n-term contingency

Schedule performance

Generally, invariances in behavior-analytic principles (contingencies, species, intra- and inter-individual, and socio-cultural behaviors)

MEN ACT UPON THE WORLD, AND CHANGE IT, AND ARE CHANGED IN TURN BY THE CONSEQUENCES OF THEIR ACTION.B. F. Skinner, Verbal Behavior, 1957, p. 1

O-rules,

functional relations

B = f (r)

r:

feedback

B:

output

E-rules,

feedback functions

r = g (B)

Figure 1. The behavior-environment feedback system

Operant Conditioning

SD

SR

Ro

SD: discriminative stimulus

Ro: operant class

SR: reinforcer

Three-term Contingency

F: force

m: mass

dv/dt: acceleration

Newton’s Second Law of Motion

The Equivalence Relation:

Ordered pairs of all elements that participate in the reinforcement contingency

EQUIVALENCE RELATION

GIVEN SETS: A, B, & C

A = A; B = B; C = C (reflexivity or matching)

AND

A = B → B = A; B = C → C = B (symmetry)

AND

IF A = B and B = C, then A = C (transitivity)

### If f (x) = c xα

then,

log f (x) = log c + αlog x.

This is a linear function on a log-log scale, with slope  (the scaling factor).

Scaling in IRT>t Schedules

IRT>t Scheduled Value

MATCHING LAW
• R1 / R2 = r1 / r2 Herrnstein
• R1 / R2 = b (r1 / r2)a Baum
HERRNSTEIN’S HYPERBOLA

Conservation of Behavior:

Exclusive choice is conserved; either you do one thing or another, each action with differential consequences (r and r0), and total behavior in the situation is constant (k).

Symmetry, as wide or as narrow as you define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.

Hermann Weyl, Symmetry (1952, p. 5)