Transcript EI - Cengage

Chapter Eleven
Rationale Behind the Precise
Formulation of the Four
Quantifier Rules
1. Cases Involving the Five
Major Restrictions
Restriction 1 on EI:
We must acknowledge that knowing that somebody is happy
does not justify our asserting that a particular named
individual is happy. Thus, when EI is used to drop an
existential quantifier, the variables thus freed cannot
validly be replaced by individual constants.
Cases Involving the Five Major
Restrictions, continued
Restriction 2 on EI:
We cannot derive the claim that there is some object that is
(e.g.) both red and square from the fact that something is
red and something is square. So, a variable introduced free
into a proof by EI must not occur free previously in the
proof.
Cases Involving the Five Major
Restrictions, continued
Restriction 1 on UG:
That a particular item had a certain property does not prove
the universalizability of this property. So, we cannot use
UG on a constant.
Cases Involving the Five Major
Restrictions, continued
Restriction 2 on UG:
That there are certain objects does not by itself justify the
conclusion that everything is an object of that type. So, we
must forbid the use of UG on a variable introduced free
into a proof by EI.
Cases Involving the Five Major
Restrictions, continued
We cannot use UG on a variable free in a line obtained by EI
whether that variable became free by using EI or not.
This restriction is nonintuitive!
Cases Involving the Five Major
Restrictions, continued
Restriction 3 on UG:
This restriction rules out the use of UG within the scope of an
assumed premise on a variable free in that assumed
premise.
The point of this is to make sure that the variable bound in a
UG step names an arbitrary individual.
2. One-to-One Correspondence Matters
We might naively characterize an application of EI or UI as a
process in which a quantifier is dropped and all the
variables thus freed are replaced by a particular variable.
But there are two cases where this one-to-one correspondence
cannot be required if our logic is to be complete.
One-to-One Correspondence Matters,
continued
We cannot require a one-to-one correspondence between x
and y variables in the application of UI; all we can require
is that for each occurrence of the variable freed by the UI
step, there corresponds a variable bound by the quantifier
on which we performed UI.
One-to-One Correspondence Matters,
continued
We cannot require one-to-one correspondence
between x and y variables in the application of
EG.
This is handled by the last clause in restriction 1 on
EG.
One-to-One Correspondence Matters,
continued
In using EG or UG, the replacements for the occurrences of
only one variable in the original formula are to be bound in
the resulting formula by the newly introduced quantifier.
This is eliminated by the last clause in restriction 4 on UG.
One-to-One Correspondence Matters,
continued
If one occurrence of some variable x is freed by UI and
replaced by a free variable, then all x variables freed by
this application of UI must be replaced by free y variables.
One-to-One Correspondence Matters,
continued
In the use of UG, if a free x in the original formula is replaced
by a y that becomes bound in the resulting formula, then all
free occurrences of x in the original formula must be
replaced by bound y variables in the resulting formula.
This is taken care of by restriction 4 on UG.
3. Accidentally Bound Variables and
Miscellaneous Cases
When a quantifier is dropped by UI or EI, all the variables
thus freed must be uniformly replaced by free variables
(or, in the case of UI, by free variables or constants).
The rule UI and the third restriction on rule EI take care of
this.
Accidentally Bound Variables and
Miscellaneous Cases, continued
In using UG or EG, the variables to be quantified by
the newly introduced quantifier must not be bound
by some other quantifier.
(This is prevented by restrictions 4 and 1)
4. Predicate Logic Proofs with
Flagged Constants
There is an alternative system of predicate logic
proof rules which is both sound and complete.
Predicate Logic Proofs with Flagged
Constants, continued
In this alternative system, the rule QN is the same;
there two rules, UI and EI, for taking off
quantifiers, and UG and EG are used for putting
them back on (although these rules are stated
differently).
Predicate Logic Proofs with Flagged
Constants, continued
The typical sequence is still the same: use UI or EI,
use sentential logic, then use UG or EG.
Predicate Logic Proofs with Flagged
Constants, continued
The difference is that this alternative system uses
flagged constants instead of freed variables in the
intermediate steps of the proof.
Let us call this the flagging system.
Predicate Logic Proofs with Flagged
Constants, continued
When we flag a constant we “raise a red flag” to note
that there is something special about it.
Predicate Logic Proofs with Flagged
Constants, continued
Flagged constants are subject to three restrictions:
• They may not appear in the conclusion of the
proof.
• They must be new to the proof.
• Any constant introduced within a subproof can
only be used within that subproof.
Predicate Logic Proofs with Flagged
Constants, continued
The rule UG is where the flagging method differs
most from the standard system. In the flagging
system UG involves as subproof. However, the
first step consists not of an assumption but of a
flagging step.