Question 261272
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Let *[tex \Large p] represent "The koi are swimming in the pond"


Let *[tex \Large q] represent "The birds are chirping"


Then "If the koi are swimming in the pond, then the birds are chirping" can be represented by either:


If *[tex \Large p] then *[tex \Large q]


or


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ p\ \rightarrow\ q]


"The koi are not swimming in the pond" is the negation of statement *[tex \Large p], or *[tex \Large \~p]


But Material Implication says that


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ p\ \rightarrow\ q]


is equivalent to


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \~p\ \small\vee\LARGE\ q]


But since we are given *[tex \Large \~p], *[tex \Large \~p\ \small\vee\LARGE\ q] is true regardless of the value of *[tex \Large q].  Therefore knowing *[tex \Large \~p] implies nothing regarding the value of *[tex \Large q] and the argument is invalid.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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