For this statement, "Either music is not dropped from the curriculum or the students will become cultural philistines; furthermore,

Question

For this statement,

"Either music is not dropped from the curriculum or the students will become cultural philistines; furthermore, the students will not become cultural philistines if and only if music is dropped from the curriculum" [use M, C] 

According to its truth table, which property best characterizes it?  

a.It is bad.

b.It is contingent

c.It is self-contradictory.

d. It is a tautology.

Answer 1

Denote the main statement by M.

Define an atomic sentence to be any sentence which does not contain any words or phrases which may be interpreted as "or", "and", "not" or "if and only if" . For example the word 'furthermore' may be interpreted as "and".

Additionally, consider the following shorthand where A and B are arbitrary atomic sentences:

"or" is given by the symbol "∨" where A∨B is true only when least one of A or B is true.

"and" is given by the symbol "∧" where A∧B is true only when both A and B are true and false otherwise.

"not" is given by the symbol "¬" where ¬A is true only when A is false.

"if and only if" given by the symbol "↔" where A↔B is true only when A and B have the same truth value.

"if then" given by the symbol "→" where A→B is true if either both A and B are true or if A is false.

That is, we may build a set S recursively first by taking all atomic sentences which we wish to consider so S would satisfy the following properties:

i) if C is an atomic sentence then C belongs to S.

ii) if C and D belong to S then C∨D belongs to S.

iii) if C and D belong to S then C∧D belongs to S.

iv) if C belong to S then ¬C belongs to S.

v) if C and D belong to S then C↔D belongs to S.

vi) if C and D belong to S then C→D belongs to S.

Thus we say that this set S is closed under logical conjunction i.e that it is closed under the logical conjunction by {∨, ∧, ¬, ↔, →}.

Next define a truth valuation by a function E: S ---> {T, F} where for any sentence Φ in S we have that E(Φ) = T exactly when Φ is true in the truth table sense and similarly E(Φ) = F exactly when Φ is false as one would determine via truth tables. Now assume C and D belong to S. Then E(C∨D) = T if at least one of the following hold i) E(C) = T ii) E(D) = T. Similarly E will return T or F exactly as expected according to the definitions of the logical conjuncts {∨, ∧, ¬, ↔, →} above.

Now let's dissect M according to all the machinery which we have established here.

Let A be the sentence "music is dropped from the curriculum".

Let B be the sentence "students will become cultural philistines."

Then M is the sentence: ((¬A)∨B)∧((¬B)↔A).

Let C be the sentence: ((¬A)∨B).

Let D be the sentence: ((¬B)↔A).

Notice E(C) = T when E(A) = F or E(B) = T.

E(D) = T when E(A) =/= E(B)

E(M) = T when both E(C) = T and E(D) = T i.e when E(A) = F and E(B) = T. E(M) is false otherwise.

Thus M is best characterized by the description that "it is contingent".