Transitivity

A third stimulus connects the relation

Training A-to-B and B-to-C creates a two-link baseline. The transitivity probe presents A and tests selection of C.

If A-to-C was directly taught, prompted, or revealed during practice, later A-to-C performance cannot show initial emergence under that preparation.

Direction matters

Transitivity in the conventional notation is A-to-C. A C-to-A test combines transitivity with symmetry and is often called an equivalence test.

Report which direction was probed. “The learner passed transitivity” can hide whether the test was A-C, C-A, or a pooled set.

Baseline mastery is a prerequisite

Weak A-B or B-C performance makes an A-C failure hard to interpret. Set and verify a mastery criterion for each trained relation before derived probes.

Recheck baseline around probe blocks when the design calls for it. Avoid excessive rechecks that expose the same stimuli so often that history changes substantially.

A fictional transitivity example

Nadia learns three sets connecting abstract symbols, printed route names, and destinations. A-to-B and B-to-C relations reach the predefined mastery criterion. A-to-C receives no direct training.

On twelve initial A-to-C probes, Nadia responds correctly in 9 of 12. Results by set are 4/4, 3/4, and 2/4. The third set deserves separate analysis.

On C-to-A probes, she responds correctly in 6 of 12. That lower combined reverse result should not be merged with A-to-C transitivity.

Keep probe consequences neutral

Differential feedback can teach A-C during assessment. Define the initial probe block, consequence arrangement, trial count, and criterion in advance.

If a relation is taught after weak probes, begin a new phase. Later correct responses show acquisition after intervention, not initial derivation.

Control positions and familiar knowledge

Balance comparison locations and use neutral presentation. Inspect materials for unique colors, borders, wear, or other cues.

Document prior learning. Real words, pictures, and familiar categories may already support A-C outside the study. Abstract stimuli offer control, while practical stimuli may answer a more meaningful applied question.

Nodal distance can affect performance

A and C are separated by one intermediate node, B. Larger networks can place more nodes between tested stimuli. Performance may vary with nodal distance, training structure, and relation type.

State the network rather than comparing raw accuracy from tasks with different numbers of links.

Transitivity sits within a broader literature

A review of derived relations in applied research categorized trained and emergent relations, procedures, generalization, and maintenance. The literature includes diverse participants, modalities, and tasks.

Another derived-relations review describes how small trained networks can support new matching. Neither source makes one procedure universally suitable.

Accessibility is part of validity

Ensure that the person can perceive samples and comparisons, scan the array, and respond through an effective method. Provide needed AAC, sensory, motor, and language supports.

Record adaptations and system failures. An inaccessible comparison array weakens interpretation of both correct and incorrect responses.

Practical use requires another measure

Transitive responding can support efficient instruction when a few taught relations yield useful new ones. Measure whether the person uses those relations in communication, education, work, or another selected context.

The BACB Test Content Outline covers emergent relations and generative performance. It is examination content rather than proof that a particular relational program is clinically indicated.

Build a network and exposure audit

Draw each stimulus as a node and every directly trained direction as an arrow. Use a different line for each reserved probe. The diagram should show that A-C is absent from teaching while A-B and B-C are mastered.

Beside the network, log every exposure from assessment, instruction, ordinary materials, and prior sessions. A supposedly derived destination relation may already appear on classroom signs or a familiar app. Preserve that context and narrow the claim when needed.

Report opportunities by relation

Create one row per A-C relation with the number of initial probes, correct responses, latency, omissions, and any feedback. A pooled total can hide one consistently weak set. If relation sets have different array sizes or nodal distances, avoid ranking them from percentage alone.

After initial probes, decide whether the practical goal calls for direct teaching. Scientific interest in emergence should never delay useful, accessible instruction. Start a new phase and retain the initial probe record.

Archive the network diagram, stimulus files, trial order, and phase-specific data together for future independent review.