Proving the implications of the truth functional notions  How to prove claims that are the implications of the truth functional notions  Remember that P, Q, R, and S are meta varibles that range over individual sentences of SL  If a claim is of the form “if and only if” you must prove two if/then claims.  If a claim is not of the form “if and only if” (but only show that “if/then” you need to identify what follows ‘if’ and show that if it holds what follows ‘then’ does as well

Proving implications  Suppose that P and Q are truth-functionally indeterminate sentences. Does it follow that P & Q is truth functionally indeterminate?  If P and Q are truth functionally indeterminate then there is at least one TVA on which P is true and at least one TVA on which Q is true, and there is at least one TVA on which P is false and at least one TVA on which Q is false.  So P & Q can be neither truth-functionally true, nor truth-functionally false. Hence it will be truth- functionally indeterminate.

Proving implications  Suppose that P is a truth-functionally true sentence and Q is truth-functionally indeterminate.  Based on this information, can you determine if P  Q is truth functionally true, truth functionally false, or truth functionally indeterminate? If so, which is it?  Yes. It is truth functionally indeterminate. There will be at least one TVA on which it is true (when Q is true) and at least one TVA on which it is false (when Q is false).

Proving implications  Suppose that two sentences, P and Q, are truth functionally equivalent. Show that it follows that the sentences P and P & Q are truth functionally equivalent.  If P and Q are truth functionally equivalent, there is no TVA on which they have different truth values.  So on every TVA on which P and Q are true, P & Q will be true; and on every TVA on which P is false, P & Q will be false as well.  So P and P & Q are truth functionally equivalent.

Proving implications  Suppose that two sentences, P and Q, are truth functionally equivalent. Show that it follows that ~P v Q is truth functionally true.  If P and Q are truth functionally equivalent, there is no TVA on which P and Q have different truth values. Thus on any TVA on which Q is false, ~P is true because P is false, and on any TVA on which ~P is false, Q is true because P is true. So there is no TVA on which ~P v Q is false (so the sentence is truth functionally true). So there is no TVA on which ~P v Q is false (so the sentence is truth functionally true).

Proving implications  Prove that {P} is truth functionally inconsistent if and only if ~P is truth functionally true.  If {P} is truth functionally inconsistent, there is no TVA on which its member (P) is true. Hence P (the only member of the set) is truth functionally false and its negation, ~P, is truth functionally true.  If ~P is truth functionally true, P is truth functionally false. Hence {P} is truth functionally inconsistent because there is no TVA on which its member is true.

Proving implications  If {P} is truth functionally consistent must {~P} be truth functionally consistent as well?  If {P} is truth functionally consistent, there is at least one truth value assignment on which its member, P, is true.  This only tells us that P is not truth functionally false. If P is truth-functionally true, ~P is truth functionally false and {~P} is not truth functionally consistent. If P is truth functionally indeterminate then {~P} would be truth functionally consistent. But we can’t know which P is so we cannot know the truth functional status (consistent or inconsistent) of {~P}

Proving implications  Prove that if P  Q is truth functionally true, then {P, ~Q} is truth functionally inconsistent.  If P  Q is truth functionally true, then there is no TVA on which P and Q have different truth values.  So either both are truth functionally true sentences, both are truth functionally false sentences, or P and Q are truth functionally equivalent (on any TVA, they have the same truth values).

Proving implications  If both are truth functionally true, then ~Q is truth functionally false and the set {P, ~Q} is truth functionally inconsistent as there is no TVA on which ~Q will be true and thus none on which all the members of the set are true.  If both are truth functionally false, then P is truth functionally false and the set {P, ~Q} is truth functionally inconsistent because there is no TVA on which P is true and thus none on which all the members of the set are true.

Proving implications  If P and Q are truth functionally equivalent, then when P is true, ~Q is false; and when ~Q is true, P is false. So there is no TVA on which both P and ~Q are true and, so, the set {P, ~Q} is truth functionally inconsistent.

Proving implications Part 1: If the corresponding material conditional of an argument PQR---S which is [ (P & Q) & R]  S] is truth functionally true, there is no TVA on which (P & Q) & R is true and S is false. Thus there is no TVA on which the premises of the argument are all true and the conclusion S is false and so the argument is truth functionally valid.

Proving implications Part 2: If the argument PQR---S is truth functionally valid, then there is no TVA on which P, Q, and R are all true and S is false. Thus there is no TVA on which the material conditional that has the conjunction (P & Q) & R as its antecedent and S as its consequent is false, and hence the material conditional is truth functionally true.

Proving implications Part 2: If the argument PQR---S is truth functionally valid, then there is no TVA on which P, Q, and R are all true and S is false. Thus there is no TVA on which the material conditional that has the conjunction (P & Q) & R as its antecedent and S as its consequent is false, and hence the material conditional is truth functionally true.

Proving implications  Show that P ╞ Q and Q ╞ P IFF P and Q are truth functionally equivalent.  If P ╞ Q and Q ╞ P, then there is no TVA on which P is true and Q is false and no TVA on which Q is true and P is false. So there is no TVA on which P and Q have different truth values. So P and Q are truth functionally equivalent.  If P and Q are truth functionally equivalent, then there is no TVA on which P and Q have different truth values. So on any TVA on which P is true, Q is true and so P ╞ Q, and on any TVA on which Q is true, Q ╞ P.

Truth functional properties and truth functional consistency  It turns out that the truth functional concepts of truth functional truth, truth functional falsehood, truth functional indeterminacy, truth functional validity, and truth functional entailment can each be defined in an additional way: in terms of truth functional consistency.  And this affords an additional way of explicating each of them.

Truth functional properties and truth functional consistency  A sentence P is truth functionally false IFF {P} is truth functionally inconsistent.  Why?  A sentence P is truth functionally true IFF {P} is truth functionally consistent.  Why?  A sentence P is truth functionally indeterminate IFF both {~P} and {P} are truth functionally consistent.  Why?

Truth functional properties and truth functional consistency  Sentences P and Q are truth functionally equivalent IFF {~(P  Q} is truth functionally inconsistent.  Why?

Truth functional properties and truth functional consistency  Where  is a set of sentences of SL and P is any sentence of SL, we may form a set that contains all the members of  and P, represented by  ⋃ {P} (the union of  and the unit set of P)

Truth functional properties and truth functional consistency  An argument is truth functionally valid IFF the set containing as its only members the premises of the argument and the negation of the conclusion is truth functionally inconsistent.  So, the argument A v B ~B A

Truth functional properties and truth functional consistency  So, the argument A v B ~B A is truth functionally valid IFF {A v B, ~B, ~A} is truth functionally inconsistent.