Zubin S. answered • 11/12/20

Pure Math Ph.D Student Specializing in Logic

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".