Introduction to Symbolic Logic
June 30, 2005
 Introduction to Predicate Derivations
 We now have a syntax and a semantics for PL

We can symbolize English sentences

We can show using interpretations semantic properties of PL sentences
and arguments

Semantics can tell us if a PL argument is valid based on the possible
meaning of each premise and conclusion

A PL argument is quantificationally valid iff it is not possible to
construct an interpretation in which the premises are true and the
conclusion is false

Remember that there is no decision procedure for determining the
quantificational validity of an argument; to show that an argument
is valid, we could end up having to examine every possible
interpretation

If we could devise a syntactic way for showing validity of PL
arguments, it might be more systematic

In SL, this is where SD came in (after truth tables)

We then briefly showed that an argument is truth-functionally
valid iff it is valid in SD

We hope to have a similar result for PL, but first we need to design
new syntactic rules that allow for the new elements of PL
 What is new in PL?

Predicates with individual constants – a predicate that has all of its
holes filled with constants behaves just like a sentence of SL
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
Quantifiers and Variables – the quantifiers are two new logical
operators

Recall that each connective in SD is associated with two syntactic
rules: introduction and elimination

We need to be able to work with the quantifiers the same way, so
essentially we need 4 new rules to create a new derivation system
for PL, PD
 Using the SD rules on PL sentences

Ignoring the new aspects of PL for the moment, PL sentences can be
constructed in the exact same way as SL sentences

This means that the same rules of SD can be applied in a PD proof in
the same way

Remember that the SD rules only work with their corresponding
sentence types!

It is the same in PD (ex. E still requires a conditional and its
antecedent), but now the PL sentences that are being connected can
be complex PL sentences themselves
 Example
 Here we have a negation as our conclusion. Or goal analysis
strategy tells us that we should use ~I.
 If we ever want to prove a quantified sentence or use one that appears in
the proof, we need introduction and elimination rules for each kind of
quantifier
 Universal Elimination
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
Whenever we eliminate a quantifier we need to fill the hole left by the
eliminated variable with an individual constant

Remember that a PL expression that contains free variables is not a
sentence, so, if you remove the quantifier, then you must remove
all occurrences of the variable that was bound by the quantifier

When we replace variables with constants, this is called variable
substitution

P(a/x) means that x is replaced with a wherever it appears in P

Def. If P is a sentence of PL of the form (x)Q or (x)Q, and a is an
individual constant, then Q(a/x) is a substitution instance of P. The
constant a is the instantiating constant.

A substitution instance is formed by dropping the initial quantifier
and replacing all remaining occurrences of the variable that the
quantifier contains with some constant
 Examples
 (y)Hay is a substitution instance of (x)(y)Hxy
 Hab is not a substitution instance of (x)(y)Hxy because
only one substitution can be performed at once and it can
only be performed on the initial quantifier

Universal Elimination gives us a way to substitute any individual
constant for the variable bound by the universal

This makes sense given what we intend  to mean, but it is
important to keep in mind that the E is a syntactic rule so
knowledge of the semantics of  is not needed to use it.

E can be applied to any sentence whose type is universal

Note that it cannot be applied to a universal within a sentence if it
is not the main logical operator
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
There are no restrictions on what constants can be substituted for a 

The constant can already be in the proof or it could be completely
new

The main thing to remember is to replace every occurrence of the
variable with the constant so that you don’t end up with an open
sentence

Example
 Existential Introduction

I allows us to introduce an existential sentence into our proof, either
as a conclusion or a means to an end

An existential sentence can always be created from a substitution
instance

That is, if a sentence appears with an individual constant, we can
infer that that means there is at least one thing in the UD that has
the predicate’s property or relation

I does not require that every occurrence of an individual constant be
existentially generalized

Remember that ‘Rmm’ can be a substitution instance of 3 different
sentences: (x)Rxx, (x)Rxm, and (x)Rmx

Any one of these three sentences could be derived from ‘Rmm’
using I

Example 1
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
Example 2

(x)(y)Cxy |-- (x)(y)Cxy
 Universal Introduction

So far the two new derivation rules have been very simple because of
the intended meaning for the quantifiers

If we know that a sentence is a universal, we know that we can pull
any constant out of the UD bag and do a substitution (E)

If we know that a sentence with a constant is true (i.e. it’s in the
proof), then we know that there is at least one time when
something from the UD bag has that property (I)

Universal Introduction allows us to prove or introduce a  sentence,
thereby saying that the sentence holds for any constant in the UD

This is a very large claim!

We can’t possibly show that a sentence is true for every constant,
but, if we can show that an arbitrarily chosen constant holds, then we
can derive the universal
 A constant is arbitrary iff the constant doesn’t occur in any
sentence on which the validity of the inference depends
 The constant cannot appear in the inferred sentence (the one
we get when we use I) or any undischarged assumption (P
or PA)

To use I, we must construct a subderivation that uses an arbitrary
constant

Example 1:
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 How did we get that?!
 The structure of this proof was determined by goal analysis
for when the goal is a . We will talk about this much more
tomorrow, but for now, notice that the immediate
subformula of the conclusion is a conditional. We know that
if we could get a substitution of that conditional using an
arbitrary constant, then we could derive the  we want
 Note that ‘a’ does not appear in either of the premises and
that the PA is discharged on line 8. This makes ‘a’ arbitrary
so line 9 is eligible for I

Provided that:
 a does not occur in an undischarged assumption
 a does not occur in (x)P
 Existential Elimination

The final new rule of PD allows us to make use of an  when it appears
in the proof

Remember that  only says that there is at least one thing in the UD
that can be substituted for the variable, but we don’t know what that
thing is

E works a little bit like E (groan!)
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
It basically says: assume for the moment that the thing in the UD
that the  is talking about is named (some arbitrary constant), then
prove your desired conclusion
 Provided that:
 a does not occur in an undischarged assumption
 a does not occur in (x)P
 a does not occur in Q

Example

Note that the desired conclusion never changes when you use this
rule as a strategy ( just like E)

We will see that this is another rule that takes effect before we
begin goal analysis. That is, whenever an  appears as a premise or
PA, we have to switch to E so that we can make use of it

Big Example

Cf &Bfl, (x)(Cx & Bxf), (x)(y)(z)[(Bxy & Byz)  Bxz] |-(z)[Cz & (Bzf & Bzl)]
 Homework 7 – due Monday
 All starred problems from section 10.1

There are two starred problems after each subsection introduces a new
rule
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