Transcript Relational Predicate Logic

Chapter Ten
Relational Predicate Logic
1. Relational Predicates
We now broaden our coverage of predicate logic to include
relational predicates. This allows us to symbolize
sentences such as “Kareem is taller than Mugsy” as Tkm.
With relational predicates the order in which the letters occurs
is significant.
2. Symbolizations Containing
Overlapping Quantifiers
Symbolizations may contain quantifiers with overlapping
scopes.
When the overlapping quantifiers are of the same types, the
order in which they occur is not relevant to the meanings
of the sentences.
But when an existential and a universal quantifier are both
involved, order becomes crucial.
3. Expansions and Overlapping
Quantifiers
One way better to understand sentences which include
overlapping quantifiers is to become familiar with the
expansions of such multiply quantified sentences.
4. Places and Times
The symbolization of statements concerning places or times
are especially interesting and can cause trouble.
5. Symbolizing “Someone,” “Somewhere,”
“Sometime,” and So On
The word “someone” can be misleading, for it sometimes
functions as an existential quantifier and sometimes as a
universal quantifier.
The words “somewhere,” “something,” “sometime,” and so
on can also be misleading in this way.
6. Invalidity and Consistency in
Relational Predicate Logic
We demonstrate invalidity and consistency in relational
predicate logic using the same techniques we employed for
monadic predicate logic.
Invalidity and Consistency in Relational
Predicate Logic, continued
For invalidity we produce an interpretation that makes the
premises all true and the conclusion false.
For consistency we need only an interpretation that makes all
the sentences true.
Invalidity and Consistency in Relational
Predicate Logic, continued
As in monadic predicate logic we can provide a complete
interpretation, or we can use the more mechanical method
where we replace the quantified sentences with their
expansions.
7. Relational Predicate Logic
Proofs
The rules for predicate logic proofs outlined in
Chapter Nine were devised to handle relational
predicate logic as well.
However, relational predicate logic is more complex
as we can encounter lines with more than one
quantifier and more than one type of variable.
8. Strategy for Relational
Predicate Logic Proofs
• If a premise contains more than one quantifier, you may
have to use EI after you have used UI. But you should
usually remove the existential quantifier as soon as
possible.
• Sometimes it helps when removing variables to introduce
new variables.
9. Theorems and Inconsistency in
Predicate Logic
The conclusion of a valid deduction in which there are no
given premises is a theorem of logic.
Theorems are sometimes referred to as logical truths, or
truths of logic.
Theorems and Inconsistency in
Predicate Logic, continued
A logical contradiction, or a logical falsehood, is a single
statement that can be proved false without the aid of
contingent information.
10. Predicate Logic Metatheory
There are two ways a system of proof rules could be
deficient: If there are valid arguments that cannot be
proven, the rules would be incomplete; if there are invalid
arguments that can be proven, the rules would be unsound.
11. A Simpler Set of Quantifier
Rules
The quantifier rules UI, EI, UG, EG and QN,
together with the eighteen valid argument forms
plus CP and IP, form a complete set of rules for
quantifier logic.
A Simpler Set of Quantifier Rules,
continued
But there are simpler sets of quantifier rules.
One very simple set includes only two of the four
QN rules, together with Rule UI and Rule EI.
A Simpler Set of Quantifier Rules,
continued
Every inference permitted by the simpler rules is
also permitted by the standard rules, although the
reverse is not true.