Proving the implications of the truth functional notions

1
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

2
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.

3
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).

4
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.

5
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).

6
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.

7
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 cant know
  which P is so we cannot know the truth functional
  status (consistent or inconsistent) of P

8
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).

9
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.

10
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.

11
Proving implications

• Part 1
• If the corresponding material conditional of an
  argument
• P
• Q
• R
• ---
• 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.

12
Proving implications

• Part 2
• If the argument
• P
• Q
• R
• ---
• 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.

13
Proving implications

• Part 2
• If the argument
• P
• Q
• R
• ---
• 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.

14
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.

15
Truth functional properties andtruth 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.

16
Truth functional properties andtruth 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?

17
Truth functional properties andtruth functional
consistency

• Sentences P and Q are truth functionally
  equivalent IFF (P ? Q is truth functionally
  inconsistent.
• Why?

18
Truth functional properties andtruth 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)

19
Truth functional properties andtruth 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

20
Truth functional properties andtruth functional
consistency

• So, the argument
• A v B
• B
• -------
• A
• is truth functionally valid IFF
• A v B, B, A
• is truth functionally inconsistent.