Chapter 5 (Derivations)

Question 1

Derivability in SD

Answer 1

A sentence P of Sl is derivable in SD from a set (GAMMA) of sentences of SL if and only if there is a derivation in SD in which all the primary assumptions are members of GAMMA and P occurs in the scope of only those assumptions.

Question 2

Validity in SD

Answer 2

An argument of SL is valid in SD if and only if the conclusion of the argument is derivable from the set consisting of the premises. An argument of SL is invalid in SD if and only if it is not valid in SD.

Question 3

Theorem in SD

Answer 3

A sentence P of SL is a theorem in SD if and only if P is derivable in SD from the empty set.

Question 4

Equivalence in SD

Answer 4

Sentences P and Q of SL are equivalent in SD if and only if Q is derivable in SD from {P} and P is derivable in SD from {Q}.

Question 5

Inconsistency in SD

Answer 5

A set (GAMMA) of sentences of SL is inconsistent in SD if and only if both a sentence P of SL and its negation ~P are derivable in SD from GAMMA. A set (GAMMA) of sentences of SL is consistent in SD if and only if it is not inconsistent in SD.