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Quiz 1 on Complex Conditional Translations to Lawgic

_oshun1__oshun1_ Alum Member
in Logical Reasoning 3652 karma
This is lesson 6/16 in the Advanced Logic Section. On DeMorgans law. Question 1. Translate the sentence into logic.

"Unless the leather is soaked and tanned for 3 days, the resulting boot will be smelly."
The right answer is
/S or /T ---> BS
contrap: /BS ---> S & T


Why does it turn from an “and” to an “or” statement if you choose /S & /T as the sufficient condition?
I thought that for “unless” conditionals, you just pick one statement and negate it and that’s the sufficient.

I’m choosing S & T as the sufficient, and BS as the necessary.

so
/S & /T —> BS
If the leather isn’t soaked and tanned, the boot will be smelly.

so the contrapositive would be…
/BS –> S or T
If the boot isn’t smelly, the leather was soaked or tanned.

I see that I’m wrong but I don’t understand why.

Is it just..whenever an & statement is negated, it turns into an or statement? and whenever an or statement is negated it turns into an & statement?

I’m super confused bc now I’m looking at the contrapositive of #2 on that quiz and it’s /S & /H —> E or D
If "and" statements turn into "or" statements when they’re negated then why isn’t that the case here >.<

P.S. I do know the splitting rules it's just hard to type onto here so I left it without splits.
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Comments

  • partyondudespartyondudes Alum Member
    edited August 2016 424 karma
    Hey @"surfy surf", from what I can tell, instead of negating (S and T) as /(S and T) at the start, you improperly jumped right to (/S and /T).

    Let's break the sentence down a bit:
    first condition: (S and T) are a package deal because they represent an entire condition (sufficient or necessary). If it helps, think of "(S and T)" as "X" instead
    second condition: BS
    logical indicator: unless (grp 3, negate sufficient)
    thus, the sentence reads as "unless X, BS", which translates to /X->BS or /(S and T)->BS
    applying De Morgan's law to "/(S and T)", we get "/S or /T".
    Thus, the translation simplifies to (/S or /T)->BS
    and the contrapositive is /BS->/(/S or /T) and further /BS->(S and T)

    Regarding question #2, the translation of the original sentence is (E or D)->(S or H). Taking a more simplistic approach, we could say "(E or D)" is represented by "A" and "(S or H)" is represented by "B". Thus, A->B. Clearly, the contrapositive of A->B is /B->/A and thus /(S or H)->/(E or D).
    Applying De Morgan's law, we get (/S and /H)->(/E and /D).

    So remember that when you negate the sufficient or necessary condition, you are first negating the entire condition and then applying De Morgan's law. De Morgan's law is triggered by the negation of the entire condition containing "and" or "or".

    Hope that helps! :) If not, you might try reviewing the De Morgan's law lesson again.
  • _oshun1__oshun1_ Alum Member
    3652 karma
    Thank you!!
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