Here's the question you clicked on:
princesspixie
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
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.
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"?
@princesspixie Does this help?
a little but what does a dis junction mean?
Disjuntion isn't mentioned, but I'd assume it's two propositions combined with 'or'.
oh i meant to say conjunction sorry!
conjunction is when you take two propositions and combine them with and or or.
but the sentence already has the word and in it so thats why im confused..
Which just means you use and \( \wedge\) for it.
can you give me an example..
this would be the first answer ? : “All mathematicians must be good logicians ^ good logicians must justify their claims.”
No, because you need to convert them into conditionals.
“if you are a mathematicians then you must be a good logician and all good logicians must justify their claims.” ??
Yeah, now I'd say that \(p\) means 'person is mathematician' \(q\) means 'person is good logician' and \(r\) means 'justifies their claims'
So we end up with \[ (p \implies q)\wedge(q \implies r) \]
ok, is that part of the answer?
ok, great thanks! how would i negate it now? just throw a not in front of the original sentence?
Yeah, throw in a \( \lnot \) .
and you may have to simplify
\( \neg ( ( p \implies q ) \wedge ( q \implies r ) ) \) \( \neg ( p \implies q ) \vee \neg ( q \implies r ) \) \( (p \wedge \neg q) \vee (q \wedge \neg r) \)
which means, the statement is true, whenever a 'mathematician is not a logician' or whenever a 'logician is not a justfier'
which forms a logical negation of the original statement
so should the negation be : a mathematician is not a good logicians and logicians do not justify claims ?
needs to be an or in the middle
so would it be : a mathematician is not a good logician or logicians would justify their claims?