Moral objections to it being true that a falsehood implies any absurdity
someone wants can maybe be reduced by realizing that if the hypothesis
is false, then the whole conditional is moot anyhow (i.e., conclusion
isn’t happening, at least not because of that hypothesis) and so
it might as well be true
But it’s a little deeper than that, there are important places later
in logic where having such an implication be true rather than false is
important
Examples
Which of the following are legal mathematical statements?
From reading: statement = declarative and either true or false
Roses are red
Ambiguous re “all” roses vs “some” roses so
truth or falsehood can’t be determined — not a statement
But with either of those so-called “quantifiers” it would
be a valid statement
x represents a real number
Not a statement, can’t determine truth value without knowing x
x = 3 and x < 10
Not statement, 2 comparisons with undefined x
If x = 3, then x < 10
True statement
This sentence is false
Not statement — paradox, neither true nor false
This sentence is true
Not statement — sentence is consistent with both true and false
Which of the following are true? How do you know?
From reading: check truth or falsehood by looking for proof, incl prior
knowledge, or check examples for plausibility or falsehood via
counterexample, guesses/intuition
Every odd number greater than 1 is prime
False, many counterexamples: 9, 25, etc.
Every prime number greater than 2 is odd
True, even numbers divisible by 2 and so can’t be prime
If p(x) is a polynomial with n roots (values of x at which p(x) = 0),
then p′(x) has n-1 roots
False as literally phrased, e.g., counterexample could be p(x) = 2
But small changes, e.g., n > 1 and n distinct
roots, could make it a true statement
If Math 239 has English 142 as a prerequisite, then Prof. Baldwin is an aardvark
True, property of conditionals
Every even integer greater than or equal to 4 is the sum of 2 prime numbers
Examples suggest this is true
This is part of Goldbach’s conjecture, which has famously resisted
proof for around 250 years
The set of rational numbers is closed under exponentiation