Basic Concepts in SD
1. Derivability.
Definition. A sentence P is DERIVABLE from a set
S IFF there is a derivation in which the primary
assumptions consist of the members of S and P
appears within the scope of only those assumptions.
To indicate that P is derivable from S we write
S├ P
See problems 5.4E #2, #7
To do these problems, assume the members of the set
to the left of the turnstile as separate assumptions and
derive the sentence to the right.
2. Validity of arguments.
Definition. An argument is VALID in SD IFF the
conclusion is derivable in SD from the set consisting
of the premises.
To represent an argument we write it in standard
form:
Premises
_____________
Conclusion
See problems 5.4E #3, #8 , #13, #17 (translations as
well)
To do these problems, assume the sentence(s) above
the horizontal line as assumptions and derive the
sentence below the line.
3. Theorem.
Definition: P is a theorem of SD iff there is a
derivation of P from the empty set of assumptions.
See problems 5.4E #4, #9, #14
To do these problems, either a) make an assumption
for ~I or ~E of the opposite of the theorem you want
to derive and try to reach a contradiction, or b) if the
theorem is a conditional, make an assumption of the
antecedent and derive the consequent (usually by
indirect proof), or c) if the theorem is a biconditional,
do two subderivations showing each side of the
biconditional.
4. Equivalence.
Definition. P and Q are equivalent in SD IFF there is
a derivation using P as an assumption and deriving Q
and a derivation using Q as an assumption and
deriving P.
See problems 5.4E #5, #10, #15
To do these problems, first assume the sentence on
the left and derive the sentence on the right. Then, in
a separate derivation, assume the sentence on the
right and derive the sentence on the left.
5. Inconsistency.
Definition. A set S of sentences is inconsistent in SD
IFF there is a derivation taking the members of S as
assumptions and leading to a contradiction.
See problems 5.4E # 6, #11, #16, #18 (translations as
well.)
To do these problems, assume the sentences in the set
as assumptions and try to derive a contradiction.
(Since the set is inconsistent, it contains a
contradiction. Therefore, anything ought to be
derivable from it. Ockham was right! Anything
follows from a contradiction. Thus any contradiction
you want should be derivable. Be brave! )