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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

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  1. wio
    • 3 years ago
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    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
    • 3 years ago
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    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
    • 3 years ago
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    @princesspixie Does this help?

  4. princesspixie
    • 3 years ago
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    a little but what does a dis junction mean?

  5. wio
    • 3 years ago
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    Disjuntion isn't mentioned, but I'd assume it's two propositions combined with 'or'.

  6. princesspixie
    • 3 years ago
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    oh i meant to say conjunction sorry!

  7. wio
    • 3 years ago
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    conjunction is when you take two propositions and combine them with and or or.

  8. princesspixie
    • 3 years ago
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    but the sentence already has the word and in it so thats why im confused..

  9. wio
    • 3 years ago
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    Which just means you use and \( \wedge\) for it.

  10. princesspixie
    • 3 years ago
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    ??

  11. princesspixie
    • 3 years ago
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    can you give me an example..

  12. princesspixie
    • 3 years ago
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    this would be the first answer ? : “All mathematicians must be good logicians ^ good logicians must justify their claims.”

  13. wio
    • 3 years ago
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    No, because you need to convert them into conditionals.

  14. princesspixie
    • 3 years ago
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    “if you are a mathematicians then you must be a good logician and all good logicians must justify their claims.” ??

  15. wio
    • 3 years ago
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    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
    • 3 years ago
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    So we end up with \[ (p \implies q)\wedge(q \implies r) \]

  17. princesspixie
    • 3 years ago
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    ok, is that part of the answer?

  18. wio
    • 3 years ago
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    That is part a.

  19. princesspixie
    • 3 years ago
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    ok, great thanks! how would i negate it now? just throw a not in front of the original sentence?

  20. wio
    • 3 years ago
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    Yeah, throw in a \( \lnot \) .

  21. ganeshie8
    • 3 years ago
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    and you may have to simplify

  22. ganeshie8
    • 3 years ago
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    \( \neg ( ( p \implies q ) \wedge ( q \implies r ) ) \) \( \neg ( p \implies q ) \vee \neg ( q \implies r ) \) \( (p \wedge \neg q) \vee (q \wedge \neg r) \)

  23. ganeshie8
    • 3 years ago
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    which means, the statement is true, whenever a 'mathematician is not a logician' or whenever a 'logician is not a justfier'

  24. ganeshie8
    • 3 years ago
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    which forms a logical negation of the original statement

  25. princesspixie
    • 3 years ago
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    ok thank you :)

  26. ganeshie8
    • 3 years ago
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    np.. yw :)

  27. princesspixie
    • 3 years ago
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    so should the negation be : a mathematician is not a good logicians and logicians do not justify claims ?

  28. wio
    • 3 years ago
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    needs to be an or in the middle

  29. princesspixie
    • 3 years ago
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    so would it be : a mathematician is not a good logician or logicians would justify their claims?

  30. princesspixie
    • 3 years ago
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    @wio

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