Possible Worlds: An introduction to Logic and Its Philosophy

1. possibly true and possibly false; contingent
2. possibly true and possibly false; contingent
3. possibly false; necessarily false
4. possibly true; necessarily true
5. possibly true and possibly false; contingent
6. possibly false; necessarily false
7. possibly true and possibly false; contingent
8. possibly true and possibly false; contingent
9. possibly true and possibly false; contingent
10. possibly true and possibly false; contingent

1. Let's take as an example the set {J, K, L}, and we'll suppose that it is inconsistent. Now suppose that we add the proposition M, where M is the proposition that the preceding proposition is false. The new set, might then read: "{J, K, M, L}". Is this latter set now consistent? Absolutely not. For if M is true, then K is false; i.e. it is impossible for both M and K (whatever proposition K happens to be) to be true. Indeed adding M to a set of propositions will not only fail to create a new set that is consistent, it will create a new set that is self-inconsistent whatever the original set had been!
   
   
2. There is nothing 'special' about the sixth proposition. It no more 'induces' the inconsistency than does any one of the other propositions in the set. If you were to remove any one proposition – either the first or the second or the third ... or the sixth – the new set, consisting of any five of the original propositions, will be self-consistent.
   
   
3. Here are some possible answers (among an infinity of correct answers):
   
   
   • {All Chevrolets have six-cylinder engines, Tom's one car is a Chevrolet, Tom's car has an eight-cylinder engine.}
     
     
   • {Alicia loves Bradley, Bradley loves Alicia, It is false that both Bradley and Alicia love one another.}
     
     
   • {John borrowed $80 from Barbara and signed a note saying that he would repay the loan, John has given Barbara exactly $20, John is debt-free.}
   
   
4. (Again, this answer is just one among an infinite number of correct answers.)
   
   
   • {John has no sisters, John's only sister is his employer, John's only sister has no employees}
   
   
5. The definition is too restrictive: it excludes (i.e. fails to identify correctly) certain necessarily false sets of propositions, viz. those that are self-inconsistent but not all of whose members are themselves necessarily false. Examples are:
   
   
   • {Aziz is taller than Harry, Harry is taller than Aziz}
   • {All squares have four sides, Some square has five sides}
   • {Giuseppe drinks Coca-Cola, Giuseppe drinks Pepsi Cola, Giuseppe drinks only one brand of cola.}