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T3.5: 5 of 9
Logical Consistency and Inconsistency
Now, let's think about the notion of consistency and consider
the following set of sentences of SL:
{~(AvB), L>B, LvA}
Notice the "set bracket" symbols: '{' and '}'. These are not
a part of SL but are our way in English to set off a collection of sentences.
The collection in question has three members which might well symbolize
these three English sentences:
Neither Agnes nor Bob will attend law school. If he gets
a loan, Bob will attend law school. Either Bob will get a loan or Agnes
will attend law school.
Now, doesn't this group of sentences seem a bit funny? We can see why
it is problematic with a truth table. Spend a few moments considering
the following table:
| |
A |
B |
L |
|
~ |
(A |
v |
B) |
, |
L |
> |
B |
, |
L |
v |
A |
| row one |
T |
T |
T |
|
F |
T |
T |
T |
|
T |
T |
T |
|
T |
T |
T |
| row two |
T |
T |
F |
|
F |
T |
T |
T |
|
F |
T |
T |
|
F |
T |
T |
| row three |
T |
F |
T |
|
F |
T |
T |
F |
|
T |
F |
F |
|
T |
T |
T |
| row four |
T |
F |
F |
|
F |
T |
T |
F |
|
F |
T |
F |
|
F |
T |
T |
| row five |
F |
T |
T |
|
F |
F |
T |
T |
|
T |
T |
T |
|
T |
T |
F |
| row six |
F |
T |
F |
|
F |
F |
T |
T |
|
F |
T |
T |
|
F |
F |
F |
| row seven |
F |
F |
T |
|
T |
F |
F |
F |
|
T |
F |
F |
|
T |
T |
F |
| row eight |
F |
F |
F |
|
T |
F |
F |
F |
|
F |
T |
F |
|
F |
F |
F |
The above table says quite a bit about the set of sentences in question:
the possible truth values for the three sentences are specified. After
carefully considering it, click on each of the
correct statements below:
- The
yellow columns represent the truth values for the set's three sentences
in each of the eight possible assignments of truth values.
- The
yellow columns represent the truth values of the atomic components
of the set of sentences in each of the eight possible assignments of
truth values.
- In
the first row, i.e., in the "T,T,T" truth value assignment,
each of the sentences in the set is true.
- In
the first row, i.e., in the "T,T,T" truth value assignment,
each of the atomic components is true, but only two members of the set
are true.
- Assuming
the given interpretation of 'A','B', and 'L', row two represents a possibility
in which both Agnes and Bob will attend law school but in which Bob
does not get a loan.
- Assuming
the given interpretation of 'A','B', and 'L', row four represents a
possibility in which neither Agnes nor Bob will attend law school.
- Assuming
the given interpretation of 'A','B', and 'L', row four represents a
possibility in which each member of our set is true.
- On
no row do you find that all three members of the set are true.
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