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:

  1. The yellow columns represent the truth values for the set's three sentences in each of the eight possible assignments of truth values.
  2. 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.
  3. In the first row, i.e., in the "T,T,T" truth value assignment, each of the sentences in the set is true.
  4. 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.
  5. 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.
  6. Assuming the given interpretation of 'A','B', and 'L', row four represents a possibility in which neither Agnes nor Bob will attend law school.
  7. Assuming the given interpretation of 'A','B', and 'L', row four represents a possibility in which each member of our set is true.
  8. On no row do you find that all three members of the set are true.