Transcript Validity and Inconsistency in Predicate Logic, continued

Chapter Eight
Predicate Logic Semantics
1. Interpretations in Predicate
Logic
• An argument is valid in predicate logic iff there is no
valuation on which the premises are true and the
conclusion false.
• Sentences are consistent just in case there is some
valuation on which they are all true
• Two sentences are logically equivalent iff there is no
valuation ion which they differ in truth-value.
Interpretations in Predicate Logic,
continued
But while validity, consistency, and logical equivalence in
predicate logic are the same as they are in sentential logic,
more is involved in specifying a valuation in predicate
logic.
Interpretations in Predicate Logic,
continued
In predicate logic, although we have individual variables and
quantification using such variables, we have no predicate
variables, only individual variables.
Interpretations in Predicate Logic,
continued
We introduce predicate constants into a specific context, but
must treat them as though they have other interpretations.
To speak of different interpretations of a predicate letter is
to treat it as though it were a variable for which we are
substituting another predicate: We assign it to different
domains in different contexts.
Interpretations in Predicate Logic,
continued
At this point we shall treat predicates intensionally, relying on
commonsense meanings of them.
2. Proving Invalidity
In sentential logic we could prove that an argument was
invalid by producing an interpretation on which all the
premises were true and the conclusion false.
In predicate logic too we produce counterexamples to show
that arguments are invalid.
3. Using Expansions to Prove
Invalidity
An easy way to prove invalidity is to use an interpretation
with a very small domain.
Using Expansions to Prove Invalidity,
continued
Technique:
• Construct the expansion of the premises and conclusion for
a small domain.
• Use the shortcut technique from sentential logic, ascribing
truth values consistently.
• Attempt to find an interpretation on which the premises are
true and the conclusion false.
Using Expansions to Prove Invalidity,
continued
While many invalid arguments can be shown to be
invalid using a two-individual domain, not all can.
Some may require expansions of three or more
individuals.
4. Consistency in Predicate Logic
Just as in sentential logic, in predicate logic the method for
proving consistency of the premise of an argument is
basically part of the method used for proving invalidity.
Consistency in Predicate Logic,
continued
In predicate logic, to show that an argument is invalid we
produce an interpretation in which the premises are true
and the conclusion is false.
To show that the premises are consistent, we need only to
show that there is an interpretation on which they are all
true.
Consistency in Predicate Logic,
continued
Note that whenever one proves an argument invalid, one also
proves that the premises are consistent.
5. Validity and Inconsistency in
Predicate Logic
An argument is valid just in case among all of its many
interpretations there is not one where the premises are true
and the conclusion false.
A set of sentences is inconsistent iff there is not a single
interpretation in which all the sentences are true.
Validity and Inconsistency in
Predicate Logic, continued
To demonstrate validity or inconsistency in predicate
logic we must either use a proof or a truth tree.
Key Terms
• Consistent
• Intensional interpretation of a predicate
• Interpretation
• Logically equivalent
• Valid