Contents
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8.1 Informal Semantics for PL
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8.2 Quantificational Truth, Falsehood, and Indeterminacy
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8.3 Quantificational Equivalence and Consistency
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8.4 Quantificational Entailment and Validity
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8.5 Truth-Functional Expansions
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8.6 Semantics for Predicate Logic with Identity
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8.7 Formal Semantics of PL and PLI
Notes
PL interpretation
An interpretation consists in the specification of
a UD (which can never be an empty set) and the
interpretation of each sentence letter, predicate,
and individual constant in the language PL.
PL entailment
A set $\Gamma$ of sentences of PL quantificationally
entails a sentence $\mathscr{P}$ of PL if and only if
there is no interpretation on which every member of
$\Gamma$ is true and $\mathscr{P}$ is false.
PL consistency
A set of sentences of PL is quantificationally
consistent if and only if there is at least one
interpretation on which all the members of the set
are true.
PL inconsistency
A set of sentences of PL is quantificationally
inconsistent if and only if the set is not
quantificationally consistent.
PL validity
An argument of PL is quantificationally valid if
and only if there is no interpretation on which
every premise is true and the conclusion is false.
PL invalidity
An argument of PL is quantificationally invalid if
and only if the argument is not quantificationally
valid.
PL equivalence
Sentences $\mathscr{P}$ and $\mathscr{Q}$ are
quantificationally equivalent if and only if there
is no interpretation on which $\mathscr{P}$ and
$\mathscr{Q}$ have different truth-values.
quantificationally true
A sentence $\mathscr{P}$ of PL is quantificationally
true if and only if $\mathscr{P}$ is true on every
interpretation.
quantificationally false
A sentence $\mathscr{P}$ of PL is quantificationally
false if and only if $\mathscr{P}$ is false on every
interpretation.
PL indeterminate
A sentence $\mathscr{P}$ of PL is quantificationally
indeterminate if and only if $\mathscr{P}$ is
neither quantificationally true nor
quantificationally false.
Alex Notes