Contents
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5.1 The Derivation System SD
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5.2 Applying the Derivation Rules of SD
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5.3 Basic Concepts of SD
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5.4 Strategies for Constructing Derivations in SD
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5.5 The Derivation System SD+
Notes
SD validity
An argument of SL is valid in SD if and only if the
conclusion of the argument is derivable in SD 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. Because SD is sound (cf. soundness proof),
it is also the case that an argument of SL is valid
in SD if and only if the argument is truth-functionally
valid. Note that the first definition does not imply
this second one.
SD inconsistency.
A set $\Gamma$ of sentences of SL is inconsistent in SD if
and only if both a sentence $\scr{P}$ of SL and its negation $\sim\mathscr{P}$
are derivable in SD from $\Gamma$. Because SD is both sound
and complete (cf. soundness and completeness proofs),
it is also the case that a set $\Gamma$ of sentences of SL
is inconsistent in SD if and only if $\Gamma$ is
truth-functionally inconsistent. Note that the first
definition does not imply this second one.
SD consistency.
A set $\Gamma$ of sentences of SL is consistent in SD if
and only if it is not inconsistent in SD.
SD derivable.
A sentence $\scr{P}$ 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 $\mathscr{P}$ occurs in the
scope of only those assumptions. Because SD is
complete (cf. completeness proof), it is also the
case that a sentence $\mathscr{P}$ of SL is derivable in SD
from a set $\Gamma$ of sentences of SL if and only if $\mathscr{P}$
is truth-functionally entailed by $\Gamma$. Note that
the first definition does not imply this second one.
SD equivalence.
Sentences $\mathscr{P}$ and $\mathscr{Q}$ of SL are equivalent in SD if
and only if $\mathscr{Q}$ is derivable in SD from $\mathscr{P}$ and $\mathscr{P}$
is derivable in SD from $\mathscr{Q}$. Because SD is both
sound and complete (cf. soundness and completeness
proofs), it is also the case that sentences $\mathscr{P}$ and $\mathscr{Q}$
of SL are equivalent in SD if and only if $\mathscr{P}$ and $\mathscr{Q}$
are truth-functionally equivalent. Note that the
first definition does not imply this second one.
theorem in SD.
A sentence $\mathscr{P}$ of SL is a theorem in SD if and only
if $\mathscr{P}$ is derivable in SD from the empty set. Because
SD is sound (cf. soundness proof), it is also the
case that a sentence $\mathscr{P}$ of SL is a theorem in SD if
and only if $\mathscr{P}$ is truth-functionally true. Note that
the first definition does not imply this second one.
corresponding material biconditional
Alex Notes