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All-equal and not-all-equal function: their values is 1 when the inputs do (not) all have the same value; Exact-count functions: their value is 1 on input vectors with k ones for a fixed k. One-hot or 1-in-n function: their value is 1 on input vectors with exactly one one; One-cold function: their value is 1 on input vectors with exactly one zero
In first-order logic with equality, counting quantifiers can be defined in terms of ordinary quantifiers, so in this context they are a notational shorthand. However, they are interesting in the context of logics such as two-variable logic with counting that restrict the number of variables in formulas. Also, generalized counting quantifiers ...
Logical equality is a logical operator that compares two truth values, or more generally, two formulas, such that it gives the value True if both arguments have the same truth value, and False if they are different.
The predicate calculus goes a step further than the propositional calculus to an "analysis of the inner structure of propositions" [4] It breaks a simple sentence down into two parts (i) its subject (the object (singular or plural) of discourse) and (ii) a predicate (a verb or possibly verb-clause that asserts a quality or attribute of the object(s)).
The corresponding logical symbols are "", "", [6] and , [10] and sometimes "iff".These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas ...
Term logic treated All, Some and No in the 4th century BC, in an account also touching on the alethic modalities. In 1827, George Bentham published his Outline of a New System of Logic: With a Critical Examination of Dr. Whately's Elements of Logic, describing the principle of the quantifier, but the book was not widely circulated. [12]
In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition to another proposition "not ", written , , ′ [1] or ¯. [ citation needed ] It is interpreted intuitively as being true when P {\displaystyle P} is false, and false when P {\displaystyle P} is true.
Formulas and are logically equivalent if and only if the statement of their material equivalence is a tautology. [ 2 ] The material equivalence of p {\displaystyle p} and q {\displaystyle q} (often written as p ↔ q {\displaystyle p\leftrightarrow q} ) is itself another statement in the same object language as p {\displaystyle p} and q ...