Meta theory and review

1
Meta theory and review

• Assume (as it is proven in Chapter 6 of our text)
  that
• P ? Q if and only if P Q
• The proof demonstrates that the system SD is
  sound or truth preserving that you cannot, for
  example, derive a conclusion from a set of
  premises in SD unless the set of premises truth
  functionally entails the conclusion, and so forth.

2
Meta theory and review

• And because
• P ? Q if and only if P Q
• Sentences P and Q are equivalent in SD IFF they
  are truth-functionally equivalent
• And a set of sentences ? of SL is inconsistent in
  SD iff ? is truth-functionally inconsistent
• And a sentence P is a theorem of SD IFF P is
  truth functionally true.

3
Meta theory and review

• Explain why if
• P ? Q if and only if P Q
• We would not want the following rule in SD
• P v Q
• P
• Q

4
Meta theory and review

• Explain how and why the following rule in SD is
  truth preserving
• P v Q
• P
• R
• Q
• R
• R vE

5
Meta theory and review

• Explain why if a set P, Q is inconsistent in
  SD, any argument that has the set ? as its
  premises will be valid in SD.
• a. What do we know if we know the set is
  inconsistent in SD?
• P
• Q
• R
• R

6
Meta theory and review

• a. What do we know if we know the set is
  inconsistent in SD?
• P
• Q
• R
• R
• S A
• R R
• R R
• S E

7
Review

• Strategies for constructing derivations in SD
• Check to see if the final sentence to be derived,
  or any sentence you need to get, is contained in
  the primary assumption or assumptions (if there
  are any) or former derived sentences in a way
  that an elimination rule will work.
• If so, identify the elimination rule you might
  use to get the sentence you need by identifying
  the main connective of the sentence in which the
  sentence you want to derive appears.

8
Review

• If you need P, see if P appears
• As part of a conjunction P Q or Q P so that
  you can use E to get it.
• As the consequent of a material conditional Q ? P
  so that if you can get the antecedent Q you can
  get P by ? E.
• As one disjunct of a disjunction P v Q so that
  you might use vE to get it.
• As one half of a material biconditional P ? Q so
  that if you can get the other half you can get P
  by ?E.
• As something you might get by assuming P and
  showing that a contradiction follows so as to get
  P by E.

9
Review

• If the final sentence to be derived (or any
  sentence you need to get there) is not contained
  in the primary assumption or assumptions (if
  there are any) or former derived sentences in a
  way that an elimination rule will work.
• If so, assume you need to build the sentence and
  identify the introduction rule you need to use by
  identifying the main connective of the sentence
  you want to derive.

10
Review

• If a sentence you want to derive (either the
  final line of the derivation or one you need to
  get to that final line) needs to be built and is
  of the form P Q, identify the two conjuncts you
  need and add them to the derivation and then look
  to see how to get them.
• If that sentence is of the form P v Q, look to
  see which disjunct you might be able to get,
  write that in, and then work to get it and then P
  v Q by v I.

11
Review

• If it is of the form P ? Q, you need to construct
  a subderivation with P as the assumption and Q as
  its last line. Sketch that out and see how to get
  Q under the assumption of P. If you do derive Q,
  move out to the scope line to the left with P ?
  Q.
• If it is of the form P ? Q, sketch in 2
  subderivations one with P as the assumption and
  Q as what you derive, and vice versa. Work to
  finish each subderivation to move out one scope
  line to the left with P ? Q.

12
Review

• If it is of the form P , and if and only if there
  are negations available (possible sentences Q and
  Q) and there is no other way you can see to
  build P, try assuming P and try using E to
  derive P.
• If it is of the form P, and if and only if there
  are negations available (possible sentences Q and
  Q) and there is no other way you can see to
  build P, try assuming P and try using I to
  derive P.

13
Show that (A ? B) (C ? A), C v A ? B

• 1 (A ? B) (C ? A)
• 2 C v A
• B

• A
• A

14
Show that A v A and A A are equivalent in SD

• A v A A
• A A

• A A A
• A v A