To distinguish the extensional concept of functions-as-sets from the non-extensional concept of functions-as-rules, the latter is often referred to as an ‘intensional’ function concept, in part because of the ostensibly intensional concept of a rule involved.
On the terminology used in the foundations of mathematics, functions-as-sets are classified as extensional since they use the axiom of extensionality as their criterion of identity, and functions-as-rules are classified as intensional because they rely on the ostensibly intensional concept of a rule.
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Type theories characteristically assign types to entities, distinguishing, for example, between numbers, sets of numbers, functions from numbers to sets of numbers, and sets of such functions.
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Strictly speaking, these functions are not propositional functions because they do not take propositions as values.
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We may extend the vagueness sets for individual agents to include a collection of inductive support functions that cover the range of values for likelihood ratios of evidence claims (as well as cover the ranges of comparative support strengths for hypotheses due to plausibility arguments within b, as represented by ratios of prior probabilities).
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In fact, there were at least two quite distinct motivations for abandoning the view of propositions as sets of worlds and adopting the structured proposition account.
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Consequently, decidable sets are often called “recursive sets”.
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The paradoxes of the theory of sets are resolved by reducing assertions about sets to assertions about propositional functions.
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Almost any (indeed many people believe literally any) fact in mathematics can be construed as a fact about sets.
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Combinatory terms are thought of as functions, and functions are thought to have a domain (a set of possible inputs) and a codomain (a set of possible outputs).
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The construction of propositional functions begins with possible worlds and the assumption that there are sets.
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It may be that artifactual kinds are individuated by function, but many artifactual kinds (such as houses) seem to have multiple functions, and some sets of distinct artifactual kinds (such as bicycles and tricycles) may all have the same function.
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As the incompleteness results in particular teach us, there are sets which are only weakly but not strongly representable (the key example being the set of statements provable in the system).
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In particular, they may have to aggregate individual sets of judgments on multiple, logically connected propositions into collective sets of judgments.
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The need for a specification of infinite sets is crucial in the discussion of the related well-ordering problem.
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Mikayla Kelley (2019) has then gone considerably further and generalized Walsh’s results significantly by describing a wide range of possible inaccuracy measures and characterizing the undominated credence functions defined on sets of propositions of different varieties.
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Some include pizza functions and steamers.
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Better, then, but still in keeping with an extensional set-theoretic approach, would be to identify a type as a function from worlds to sets of objects in that world.
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We have the Halo Essentia Series Bassinest Swivel Sleeper, but there's also the Premiere, Luxe, and Luxe Plus, all of which have upgraded features like audio and vibrations, reminders, and even sheet sets, but the basic structure and safety functions are the same across the board.
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