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: