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The Semantics of Entailment.

Once upon a time, modal logics “had no semantics”. Bearing a real world G, a set of worlds K, and a relation R of relative possibility between worlds, Saul Kripke beheld this situation and saw that it was formally explicable, and made model structures. It came to pass that soon everyone was making model structures, and some were deontic, and some were temporal, and some were epistemic, according to the conditions on the binary relation R. None of the model structures that Kripke made, nor that Hintikka made, nor that Thomason made, nor that their co-workers and colleagues made, were, however, relevant. This caused great sadness in the city of Pittsburgh, where dwelt the captains of American Industry. The logic industry was there represented by Anderson, Belnap & Sons, discoverers of entailment and scourge of material impliers, strict impliers, and of all that to which their falsehoods and contradictions led. Yea, every year or so Anderson& Belnap turned out a new logic, and they did call it E, or R, or E i , or P W, and they beheld each such logic, and they were called relevant. And these logics were looked upon with favor by many, for they captureth the intuitions, but by many more they were scorned, in that they hadeth no semantics. Word that Anderson & Belnap had made a logic without semantics leaked out. Some thought it wondrous and rejoiced,’ that the One True Logic should make its appearance among us in the Form of Pure Syntax, unencumbered by all that set-theoretical garbage. Others said that relevant logics were Mere Syntax. Surveying the situation Routley, and quite independently Urquhart, found an explication of the key concept of relevant implication. Building on Routley [ 19721 , and with a little help from our friends Dunn and Urquhart