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The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q).
These examples, one from mathematics and one from natural language, illustrate the concept of vacuous truths: "For any integer x, if x > 5 then x > 3." [11] – This statement is true non-vacuously (since some integers are indeed greater than 5), but some of its implications are only vacuously true: for example, when x is the integer 2, the statement implies the vacuous truth that "if 2 > 5 ...
Interesting number paradox: The first number that can be considered "dull" rather than "interesting" becomes interesting because of that fact. Potato paradox : If potatoes consisting of 99% water dry until they are 98% water, they lose 50% of their weight.
For instance, counterfactual conditionals would all be vacuously true on such an account, when in fact some are false. [8] In the mid-20th century, a number of researchers including H. P. Grice and Frank Jackson proposed that pragmatic principles could explain the discrepancies between natural language conditionals and the material conditional.
The existence of the free Burnside group and its uniqueness up to an isomorphism are established by standard techniques of group theory. Thus if G is any finitely generated group of exponent n, then G is a homomorphic image of B(m, n), where m is the number of generators of G. The Burnside problem for groups with bounded exponent can now be ...
In predicate logic, a predicate P over some domain is called decidable if for every x in the domain, either P(x) holds, or the negation of P(x) holds. This is not trivially true constructively. Markov's principle then states: For a decidable predicate P over the natural numbers, if P cannot be false for all natural numbers n, then it is true ...
In particular, if the Erdős–Straus conjecture itself (the case =) is false, then the number of counterexamples grows only sublinearly. Even more strongly, for any fixed k {\displaystyle k} , only a sublinear number of values of n {\displaystyle n} need more than two terms in their Egyptian fraction expansions. [ 17 ]
Since 10 is greater than 9, the pigeonhole principle says that at least one hole has more than one pigeon. (The top left hole has 2 pigeons.) In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. [1]