substitutability

substitutability

In any

logical system

, the way to obtain (well-formed)

formulas

from existing ones is by attaching

logical connectives

to existing ones. For example, in classical

propositional logic

, if

φ

and

ψ

are formulas, the following

φ

∨

ψ

,

¬

⁢

φ

,

ψ

∧

φ

are formed by attaching logical connectives

∨

,

¬

, and

∧

appropriately to

φ

and

ψ

.

Another convenient device is substitution:

replace all

occurrences

of some symbol

x

in a formula

φ

by an expression

ψ

.

We denote the resulting expression by

φ

⁢

[

ψ

/

x

]

.

For example, in classical propositional logic, if

φ

is

p

∧

(

¬

⁢

r

∨

p

)

, then

φ

⁢

[

(

p

∨

q

)

/

p

]

is

(

p

∨

q

)

∧

(

¬

⁢

r

∨

(

p

∨

q

)

)

On the other hand, the expressions

•

φ

⁢

[

¬

/

p

]

, which is

¬

∧

(

¬

⁢

r

∨

¬

)

, and

•

φ

[

q

/

∧

]

, which is

p

⁢

q

⁢

(

¬

⁢

r

∨

q

)

,

are not formulas. Thus, one must be careful when performing substitutions on formulas lest the resulting expressions are ill-formed. In other words, conditions must be placed on

x

and

ψ

in

φ

⁢

[

ψ

/

x

]

in order that

φ

⁢

[

ψ

/

x

]

is a (well-formed) formula. These conditions are called the

substitutability

conditions

. In this entry, we will concentrate on substitutability conditions on predicate logic. Details on substitions in propositional logic can be found in here (

http://planetmath.org/SubstitutionsInLogic

).

Substitution in First-Order Logic

Substitution works pretty much the same way for

first-order logic

as in propositional logic. However, the substitutability conditions are more subtle. Take a look at the following example:

∃

x

(

x

=

1

)

If we replace

x

by

0

, we end up with

∃

0

(

0

=

1

)

,

which is non-sensical (not a wff). This is because

x

occurs in the formula as a

bound variable

. (one reason why we distinguish the variables occurring in

first order

formulas into two types: free and bound).

We now formalize the notion of substitution in first-order logic. There are two parts: substitution for terms, and substitution for formulas.

Definition

. For any term

t

, any symbol

x

, and any expression

s

, define

t

⁢

[

s

/

x

]

inductively, as follows:

1.

if

t

is an individual variable or a

constant symbol

, then

t

⁢

[

s

/

x

]

is

s

if

t

is

x

, and

t

⁢

[

s

/

x

]

is

t

otherwise;

2.

if

t

is

f

⁢

(

t

1

,

…

,

t

n

)

, where

f

is an

n

-ary function symbol, and each

t

i

is a term, then

t

⁢

[

s

/

x

]

is

f

⁢

(

t

1

⁢

[

s

/

x

]

,

…

,

t

n

⁢

[

s

/

x

]

)

.

For example, if

t

is

x

+

y

, then

t

⁢

[

(

x

-

y

)

/

x

]

is

(

x

-

y

)

+

y

.

It is easy to see that

s

is a term and

x

an individual variable, then

t

⁢

[

s

/

x

]

is a term. In

addition

, by

induction

, one can easily show that if the formula

s

1

=

s

2

is true, so is the formula

t

⁢

[

s

1

/

x

]

=

t

⁢

[

s

2

/

x

]

.

Next, we define substitution for formulas. In light of the last example at the beginning of this

section

, we need to be a little careful.

Definition

. Let

φ

be a formula,

x

a symbol, and

s

an expression. The expression

φ

⁢

[

s

/

x

]

is again define inductively:

1.

if

φ

is

t

1

=

t

2

, then

φ

⁢

[

s

/

x

]

is

t

1

⁢

[

s

/

x

]

=

t

2

⁢

[

s

/

x

]

;

2.

if

φ

is

R

⁢

(

t

1

,

…

,

t

n

)

, then

φ

⁢

[

s

/

x

]

is

R

⁢

(

t

1

⁢

[

s

/

x

]

,

…

,

t

n

⁢

[

s

/

x

]

)

;

3.

if

φ

is

¬

⁢

ψ

, then

φ

⁢

[

s

/

x

]

is

¬

⁡

(

ψ

⁢

[

s

/

x

]

)

;

4.

if

φ

is

ψ

∨

σ

, then

φ

⁢

[

s

/

x

]

is

ψ

⁢

[

s

/

x

]

∨

σ

⁢

[

s

/

x

]

;

5.

if

φ

is

∃

y

⁢

ψ

, then

φ

⁢

[

s

/

x

]

is

∃

y

⁢

(

ψ

⁢

[

s

/

x

]

)

if

x

≠

y

, and

φ

⁢

[

s

/

x

]

is

φ

otherwise.

Again, substitutions involving logical connectives

→

,

∧

, and the

universal quantifier

∀

can be derived from the rules given above.

For example, if

φ

is

∃

x

(

x

=

y

∨

y

=

z

)

, then

φ

⁢

[

t

/

y

]

is

∃

x

(

x

=

t

∨

t

=

z

)

, whereas

φ

⁢

[

t

/

x

]

is just

φ

.

Given that

φ

is a formula, it is easy to see that if

x

is an individual variable, and

s

is a term, then

φ

⁢

[

s

/

x

]

is a formula.

In addition, it is easy to see that

sentences

are not affected by substitutions: if

φ

is a sentence, then

φ

⁢

[

s

/

x

]

is just

φ

. In other words, sentences can not be changed into formulas with

free variables

.

Conversely

, can a formula with free variables be changed into a sentence by substitution? Certainly. For example, if

φ

is

∃

x

(

y

<

x

)

,

then

φ

⁢

[

x

/

y

]

is

∃

x

(

x

<

x

)

.

Although syntactically correct, this is undesirable in many situations, particularly when we are interested in the

interpretations

of these formulas. In the example above, we have changed

∃

x

(

y

<

x

)

, which many very well be true in many interpretations, into

∃

x

(

x

<

x

)

, something with a fixed meaning (and always false if

<

is interpreted as the usual less than

relation

).

The problem with the situation described in the last paragraph arises because a free variable in

t

becomes bound in

φ

⁢

[

t

/

x

]

. To eliminate this undesirable situation, we define the notion of “free for”:

Definition

. Let

x

be an individual variable,

t

a term, and

φ

a formula. We define the relation

t

is free for, or substitutable for

x

in

φ

, inductively, as follows:

1.

φ

is an

atomic formula

;

2.

φ

is

¬

⁢

ψ

, and

t

is free for

x

in

ψ

;

3.

φ

is

ψ

∨

σ

, and

t

is free for

x

in

ψ

and in

σ

;

4.

φ

is

∃

y

⁢

ψ

, and either

–

x

∉

FV

⁡

(

φ

)

(

x

does not occur free in

φ

), or

–

y

does not occur in

t

, and

t

is free for

x

in

ψ

.

In words,

t

is free for

x

in

φ

iff whenever

z

is a variable in

t

, no

literal subformula

of

φ

of the form

∃

z

⁢

ψ

contains an occurrence of

x

which is free in

φ

.

For example,

f

⁢

(

x

,

y

)

is free for

x

in the following formulas:

P

⁢

(

x

,

y

)

,

P

⁢

(

x

)

∨

¬

⁢

Q

⁢

(

z

)

,

¬

⁢

∃

x

⁢

¬

⁢

R

⁢

(

x

,

y

)

,

and

¬

⁢

S

⁢

(

y

)

∨

∃

y

⁢

T

⁢

(

y

,

z

)

,

but not in the following formulas:

∃

y

⁢

P

⁢

(

x

,

y

)

,

Q

⁢

(

x

)

∨

∃

z

⁢

∃

y

⁢

R

⁢

(

x

,

y

)

,

and

S

⁢

(

y

)

→

∀

y

⁢

(

T

⁢

(

y

,

y

)

∧

¬

⁢

Q

⁢

(

x

)

)

.

Given any formula

φ

, we again write

φ

⁢

(

x

)

to mean that variable

x

occurs in

φ

. A

substitution instance

of

φ

⁢

(

x

)

is just

φ

⁢

[

t

/

x

]

, or

φ

⁢

(

t

)

for short. Furthermore, if

t

is free for

x

in

φ

, then

φ

⁢

(

t

)

is called a

free substitution instance

of

φ

⁢

(

x

)

.

It is easy, by induction, to show that if terms

t

1

and

t

2

are free for

x

in

φ

, and that the formula

t

1

=

t

2

is true, the substitution instances

φ

⁢

(

t

1

)

and

φ

⁢

(

t

2

)

are

logically equivalent

, as intended.

Title

substitutability

Canonical name

Substitutability

Date of creation

2013-03-22 19:12:15

Last modified on

2013-03-22 19:12:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

24

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B10

Classification

msc 03B05

Synonym

substitutable for

Related topic

Subformula

Related topic

FreeAndBoundVariables

Defines

free for

Defines

free substitution instance

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