tautologies in predicate logic

tautologies in predicate logic

Let FO

(

Σ

)

be a

first order language

over

signature

Σ

. Recall that the axioms for FO

(

Σ

)

are (

universal

)

generalizations

of wff’s belonging to one of the following six schemas:

1.

A

→

(

B

→

A

)

2.

(

A

→

(

B

→

C

)

)

→

(

(

A

→

B

)

→

(

A

→

C

)

)

3.

¬

⁢

¬

⁢

A

→

A

4.

∀

x

(

A

→

B

)

→

(

∀

x

A

→

∀

x

B

)

, where

x

∈

V

5.

A

→

∀

x

⁢

A

, where

x

∈

V

is not free in

A

6.

∀

x

⁢

A

→

A

⁢

[

a

/

x

]

, where

x

∈

V

,

a

∈

V

⁢

(

Σ

)

, and

a

is

free for

x

in

A

where

V

is the set of

variables

and

V

⁢

(

Σ

)

is the set of variables and

constants

, with

modus ponens

as its

rule of inference

: from

A

and

A

→

B

we may infer

B

.

The first three

axiom schemas

and the modus ponens tell us that predicate logic is an

extension

of the

propositional logic

. On the other hand, we can also view predicate logic as a part of propositional logic if we treat all quantified

formulas

as atoms. This can be done precisely as follows:

Call a wff of FO

(

Σ

)

quasi-atom

if it is either atomic, or of the form

∀

x

⁢

A

, where

A

is a wff of FO

(

Σ

)

. Let

Γ

be the set of all quasi-atoms of FO

(

Σ

)

.

Proposition 1

.

Every wff of FO

(

Σ

)

can be uniquely built up from

Γ

using only

logical connectives

→

and

¬

.

Proof.

Induction

on the complexity of wff. For any wff

A

of FO

(

Σ

)

, it has one of the following forms:

B

→

C

,

¬

⁢

B

, or

∀

x

⁢

B

, where

B

,

C

are wff’s. If

A

were

B

→

C

or

¬

⁢

B

, by induction, since

B

and

C

were in

Γ

,

A

is in

Γ

as a result. If

A

were

∀

x

⁢

B

, then

A

is quasi-atomic, and therefore in

Γ

by the

definition

of

Γ

.

Unique readability

follows from the unique readability of wff’s of propositional logic.
∎

Since no

quantifiers

are involved in the built-up process, the

language

based on

Γ

as the set of atoms can be considered as the language of propositional logic. Call this logic PL

(

Γ

)

. So the atoms of PL

(

Γ

)

are precisely the quasi-atoms of FO

(

Σ

)

.

Definition

. We call a wff of FO

(

Σ

)

a

tautology

if it is a tautology of PL

(

Γ

)

(via

truth tables

).

Proposition 2

.

In FO

(

Σ

)

, every tautology is a

theorem

, but not

conversely

.

Proof.

Suppose wff

A

is a tautology in FO

(

Σ

)

. Then it is a tautology in PL

(

Γ

)

by definition, and therefore a theorem in PL

(

Γ

)

since propositional logic is

complete

with respect to

truth-value semantics

. If

A

1

,

\ldot

,

A

n

is a

deduction

of

A

(with

A

n

=

A

), then each

A

i

is either an axiom, or is obtained by an application of modus ponens. If

A

i

is an axiom (of PL

(

Γ

)

), it is an

instance

of one of the first three axiom schemas of FO

(

Σ

)

above, which means that

A

i

is an axiom of FO

(

Σ

)

. Furthermore, since modus ponens is a rule of inference for FO

(

Σ

)

,

A

1

,

…

,

A

n

is a deduction of

A

in FO

(

Σ

)

, which means that

A

is a theorem of FO

(

Σ

)

.

On the other hand, any wff that is an instance of one of the last two axiom schemas of FO

(

Σ

)

is a theorem that is not a tautology. For example,

∀

x

(

x

=

y

)

→

(

z

=

y

)

is an instance of the fourth axiom schema, which takes the form

C

→

D

, where

C

is a quasi-atom, and

D

an atom, both of which are atoms of PL

(

Γ

)

. Any truth-valuation that takes

C

to

1

and

D

to

0

, takes

C

→

D

to

0

. Therefore,

∀

x

(

x

=

y

)

→

(

z

=

y

)

is not a tautology.
∎

Title

tautologies in predicate logic

Canonical

name

TautologiesInPredicateLogic

Date of creation

2013-03-22 19:32:25

Last modified on

2013-03-22 19:32:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B10

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