## princesspixie 3 years ago 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

1. wio

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.

2. wio

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"?

3. wio

@princesspixie Does this help?

4. princesspixie

a little but what does a dis junction mean?

5. wio

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

6. princesspixie

oh i meant to say conjunction sorry!

7. wio

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

8. princesspixie

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

9. wio

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

10. princesspixie

??

11. princesspixie

can you give me an example..

12. princesspixie

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

13. wio

No, because you need to convert them into conditionals.

14. princesspixie

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

15. wio

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

16. wio

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

17. princesspixie

ok, is that part of the answer?

18. wio

That is part a.

19. princesspixie

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

20. wio

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

21. ganeshie8

and you may have to simplify

22. ganeshie8

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

23. ganeshie8

which means, the statement is true, whenever a 'mathematician is not a logician' or whenever a 'logician is not a justfier'

24. ganeshie8

which forms a logical negation of the original statement

25. princesspixie

ok thank you :)

26. ganeshie8

np.. yw :)

27. princesspixie

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

28. wio

needs to be an or in the middle

29. princesspixie

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

30. princesspixie

@wio