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Schematic variables in first-order logic are usually trivially eliminable in second-order logic, because a schematic variable is often a placeholder for any property or relation over the individuals of the theory. This is the case with the schemata of Induction and Replacement mentioned above. Higher-order logic allows quantified variables to ...
An example of second-order conditioning. In classical conditioning, second-order conditioning or higher-order conditioning is a form of learning in which a stimulus is first made meaningful or consequential for an organism through an initial step of learning, and then that stimulus is used as a basis for learning about some new stimulus.
In Henkin semantics, a separate domain is included in each interpretation for each higher-order type. Thus, for example, quantifiers over sets of individuals may range over only a subset of the powerset of the set of individuals. HOL with these semantics is equivalent to many-sorted first-order logic, rather than being stronger than first-order ...
As a result, second-order logic has greater expressive power than first-order logic. For example, there is no way in first-order logic to identify the set of all cubes and tetrahedrons. But the existence of this set can be asserted in second-order logic as: ∃P ∀x (Px ↔ (Cube(x) ∨ Tet(x))). We can then assert properties of this set.
Use ordinary first-order logic, but add a new unary predicate "Set", where "Set(t)" means informally "t is a set". Use ordinary first-order logic, and instead of adding a new predicate to the language, treat "Set(t)" as an abbreviation for "∃y t∈y" Some first-order set theories include: Weak theories lacking powersets:
Second-order logic can be extended by a transitive closure operator in the same way as first-order logic, resulting in SO[TC]. The TC operator can now also take second-order variables as argument. SO[TC] characterises PSPACE. Since ordering can be referenced in second-order logic, this characterisation does not presuppose ordered structures. [20]
This section describes second-order arithmetic with first-order semantics. Thus a model of the language of second-order arithmetic consists of a set M (which forms the range of individual variables) together with a constant 0 (an element of M), a function S from M to M, two binary operations + and · on M, a binary relation < on M, and a ...
A second-order stimulus is a type of visual stimulus employed in psychophysics where objects are distinguished from their backgrounds through variations in contrast or texture. In contrast, a stimulus distinguished by differences in luminance is termed a first-order stimulus.
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