Ever since its popularisation by Piaget around 60 years ago, transitive reasoning (deductively-inferring A greater than C from premises A greater than B and B greater than C) has been of psychological interest both as a mental phenomenon and as a tool in areas of psychological discourse. However, the focus of interest in it has shifted periodically first from child development, to learning disability, to non-humans and currently to cognitive and clinical neuroscience. Crucially, such shifts have always been plagued by one core question--the question of which of two competing paradigms (extensive-training paradigm versus non-training paradigm) is valid for assessing transitive reasoning as originally conceived in Piagetian research. The continued avoidance of this question potentially undermines several important findings recently reported: Such as about exactly what is involved in deducing transitive inferences, which brain regions are critical for reaching transitive inference, and what links exist between weakened deductive transitivity and mental illnesses like schizophrenia. Here, we offer the view that both of the competing paradigms are indexing transitivity, but each one tends to tap a different aspect of it. Then, we summarise studies from child and adult cognitive psychology, disabilities research, and from cognitive neuroscience. These, together with studies of non-human reasoning, seem to afford a theory of transitive reasoning that has two major components; one deductive but the other associative. It is proposed that only a dual-process theory of transitivity (having analytic versus intuitive routes approximate to deductive versus associative processing respectively) can account both for the variety of findings and the apparently-disparate paradigms. However, fuzzy-trace theory ("Gist" processes and representations), if not already embodying such a dual-process theory, will need to be incorporated into any complete theory. (Contains 2 figures.)