Question

Consider the following statement:

“All mathematicians must be good logicians and all good logicians must justify their claims.”

a) Write this statement symbolically as a conjunction of two conditional statements. Use three components (p, q, and r) and explicitly state what these are in your work. Remember: a component in logic is a simple declarative sentence that is either true or false.

b) Write the negation of this statement

Answer 1

Okay, so suppose we have the form: "All p must be q". The question is does this mean \(p\implies q\) or \(q \implies p\)? You have to test it out a bit.

Answer 2

So I'd start out with something like... "All dogs must be animals". Now does this mean "If you're a dog, then you're an animal." or does it mean "If you're an animal, you must be a dog"?

Answer 3

@princesspixie Does this help?

Answer 4

a little but what does a dis junction mean?

Answer 5

Disjuntion isn't mentioned, but I'd assume it's two propositions combined with 'or'.

Answer 6

oh i meant to say conjunction sorry!

Answer 7

conjunction is when you take two propositions and combine them with and or or.

Answer 8

but the sentence already has the word and in it so thats why im confused..

Answer 9

Which just means you use and \( \wedge\) for it.

Answer 10

??

Answer 11

can you give me an example..

Answer 12

this would be the first answer ? :
“All mathematicians must be good logicians ^ good logicians must justify their claims.”

Answer 13

No, because you need to convert them into conditionals.

Answer 14

“if you are a mathematicians then you must be a good logician and all good logicians must justify their claims.” ??

Answer 15

Yeah, now I'd say that \(p\) means 'person is mathematician' \(q\) means 'person is good logician' and \(r\) means 'justifies their claims'

Answer 16

So we end up with \[
(p \implies q)\wedge(q \implies r)
\]

Answer 17

ok, is that part of the answer?

Answer 18

That is part a.

Answer 19

ok, great thanks! how would i negate it now? just throw a not in front of the original sentence?

Answer 20

Yeah, throw in a \( \lnot \) .

Answer 21

and you may have to simplify

Answer 22

\( \neg ( ( p \implies q ) \wedge ( q \implies r ) ) \)

\( \neg ( p \implies q ) \vee \neg ( q \implies r ) \)

\( (p \wedge \neg q) \vee (q \wedge \neg r) \)

Answer 23

which means, the statement is true,

whenever a 'mathematician is not a logician'

or

whenever a 'logician is not a justfier'

Answer 24

which forms a logical negation of the original statement

Answer 25

ok thank you :)

Answer 26

np.. yw :)

Answer 27

so should the negation be :
a mathematician is not a good logicians and logicians do not justify claims ?

Answer 28

needs to be an or in the middle

Answer 29

so would it be : a mathematician is not a good logician or logicians would justify their claims?

Answer 30

@wio