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Formal Logic Inquiry
Posted: Wed Jun 13, 2018 7:08 pm
by draperde
Why does the conditional statement:
"If it is not the case that the park contains both laurels and oaks, then it contains firs and spruces."
translate in shorthand to:
"If ~L or ~O ---> F and S"
rather than:
"If ~L and ~O ---> F and S"?
Any help w/ this is appreciated.
Re: Formal Logic Inquiry
Posted: Fri Jun 15, 2018 5:06 pm
by attofishpi
draperde wrote: ↑Wed Jun 13, 2018 7:08 pm
Why does the conditional statement:
"If it is not the case that the park contains both laurels and oaks, then it contains firs and spruces."
translate in shorthand to:
"If ~L or ~O ---> F and S"
rather than:
"If ~L and ~O ---> F and S"?
Any help w/ this is appreciated.
Ive never done formal logic - but from a programming background the NOT still requires the AND of both L and O so...
I think Laurel sat on his laurel upon the oak and eventually was petrified that the log he saw and the log ic were unified into a con that i can confer was evergreen and thus any remaining laurels or oaks eventuated in the park containing firs and spruces.
ps..If the OR is required, I think its a trick question pertaining to logic in logs.
Re: Formal Logic Inquiry
Posted: Mon Jun 18, 2018 7:02 pm
by Averroes
draperde wrote: ↑Wed Jun 13, 2018 7:08 pm
Why does the conditional statement:
"If it is not the case that the park contains both laurels and oaks, then it contains firs and spruces."
translate in shorthand to:
"If ~L or ~O ---> F and S"
rather than:
"If ~L and ~O ---> F and S"?
Any help w/ this is appreciated.
In the conditional "If it is not the case that the park contains both laurels and oaks, then it contains firs and spruces," the antecedent is "it is not the case that the park contains both laurels and oaks" and the consequent is "it [the park] contains firs and spruces."
You are having problems only with the antecedent. The antecedent is correctly symbolised as ~(L and O), and not as (~L and ~O). There is a difference here. ~(L and O) is true and (~L and ~O) is false in either of the following cases, i.e.if the park contains
1. Laurels but not oaks,
2. Oaks but not laurels.
Now ~(L and O) is equivalent to (~L or ~O) by De Morgan theorem. This can be easily proved.
Code: Select all
1. (1) ~(L and O) premise
2. (2) ~ (~L or ~O) assumption
3. (3) L assumption
4. (4) O assumption
3, 4 (5) L and O 3, 4 &intro
1, 3, 4 (6) contradiction 1, 5 ~E
1, 4 (7) ~L 3, 6 ~intro
1, 4 (8) ~L or ~O 7 or intro
1, 2, 4 (9) contradiction 2, 8 ~E
1, 2 (10) ~O 4, 9 ~intro
1, 2 (11) ~L or ~O 10 or-intro
1, 2 (12) contradiction 2, 11 ~elim
1 (13) ~~ (~L or ~O) 2, 12 ~intro
1 (14) (~L or ~O) 13 Double negation□
You can also use boolean algebra to prove the deMorgan. Resources on the latter is readily available on the net.