By Roy T. Cook
This worthy reference introduces undergraduate and post-graduate scholars to the most difficulties and positions of philosophical good judgment. parts comprise the most important figures, positions, terminology, and debates inside philosophical good judgment in addition to concerns that pertain to similar, overlapping disciplines, equivalent to set concept and the philosophy of arithmetic. Entries are generally cross-referenced for identity in the context of wider debates.
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Additional resources for A Dictionary of Philosophical Logic
Thus, the truth table for choice negation (where N is the third value) is: A T N F ~A F N T See also: Boolean Negation, Bottom, DeMorgan Negation, Exclusion Negation, Falsum CHOICE SEQUENCE see Free Choice Sequence CHOICE SET A choice set for a set S is a set containing exactly one member from each set contained in S. The axiom of choice can be understood as asserting that for each non-empty set of sets S there is a choice set for S. See also: Axiom of Countable Choice, Axiom of Dependent Choice, Choice Function, Global Choice, Zorn’s Lemma CHRONOLOGICAL LOGIC see Temporal Modal Logic CHURCH’S THEOREM Church’s Theorem states that validity in first-order logic is not decidable – that is, there is no decision procedure for determining, of an arbitrary formula from a firstorder language, whether or not it is a logical truth.
To derive the paradox, assume that there was an order-type of all ordinal numbers, that is, an ordinal number corresponding to the standard ordering on the collection of all ordinal numbers. Call this order-type Γ. Then, by the principle above, the order-type of the ordinal numbers less than Γ is Γ. So the order-type of the ordinal numbers less than or equal to Γ, that is, the order-type of all the ordinal numbers, is Γ+1. But this contradicts our assumption that the order-type of all the ordinal numbers is Γ, since Γ ≠ Γ+1.
See also: Kripke Semantics, Kripke Structure, Modality b The second letter of the Hebrew alphabet, b is used to denote a particular type of infinite cardinal number. Subscripted ordinal numbers are used to distinguish, and order, the b’s. b0 is identical b to a 0 , the first infinite cardinal. b1 is identical to 2 0. b2 is identical to b 2 1 … and generally: bi b i+1 = 2 1004 02 pages 001-322:Layout 1 16/2/09 15:11 Page 29 barbara 29 With the b notation in place, we can succinctly express both the continuum hypothesis and the generalized continuum hypothesis.