truth-value semantics for classical propositional logic

truth-value semantics for classical propositional logic

In classical

propositional logic

, an

interpretation

of a

well-formed formula

(wff)

p

is an assignment of truth (=1) or falsity (=0) to

p

. Any interpreted wff is called a

proposition

.

An

interpretation

of all wffs over the

variable

set

V

is then a

Boolean function

on

V

¯

. However, one needs to be careful, for we do not want both

p

and

¬

⁢

p

be interpreted as true simultaneously (at least not in classical propositional logic)! The proper way to find an interpretation on the wffs is to start from the atoms.

Call any Boolean-valued function

ν

on

V

a

valuation

on

V

. We want to extend

ν

to a Boolean-valued function

ν

¯

on

V

¯

of all wffs. The way this is done is

similar

to the construction of wffs; we build a

sequence

of

functions

, starting from

ν

on

V

0

, next

ν

1

on

V

1

, and so on… Finally, we take the union of all these “approximations” to arrive at

ν

¯

. So how do we go from

ν

to

ν

1

? We need to interpret

¬

⁢

p

and

p

∨

q

from the valuations of atoms

p

and

q

. In other words, we must also interpret

logical connectives

too.

Before doing this, we define a

truth function

for each of the logical connectives:

•

for

¬

, define

f

:

{

0

,

1

}

→

{

0

,

1

}

given by

f

⁢

(

x

)

=

1

-

x

.

•

for

∨

, define

g

:

{

0

,

1

}

2

→

{

0

,

1

}

given by

g

⁢

(

x

,

y

)

=

max

⁡

(

x

,

y

)

.

As we are trying to

interpret

¬

(not) and

∨

(or), the choices for

f

and

g

are clear. The values

0

,

1

are interpreted as the usual integers (so they can be subtracted and ordered, etc…). Hence

f

and

g

make sense.

Next, recall that

V

i

are sets of wffs built up from wffs in

V

i

-

1

(see

construction of well-formed formulas

for more detail). We define a function

ν

i

:

V

i

→

{

0

,

1

}

for each

i

, as follows:

•

ν

0

:=

ν

•

suppose

ν

i

has been defined, we define

ν

i

+

1

:

V

i

+

1

→

{

0

,

1

}

given by

ν

i

+

1

⁢

(

p

)

:=

{

ν

i

⁢

(

p

)

if

⁢

p

∈

V

i

,

f

⁢

(

ν

i

⁢

(

q

)

)

if

⁢

p

=

¬

⁢

q

⁢

for some

⁢

q

∈

V

i

,

g

⁢

(

ν

i

⁢

(

q

)

,

ν

i

⁢

(

r

)

)

if

⁢

p

=

q

∨

r

⁢

for some

⁢

q

,

r

∈

V

i

.

Finally, take

ν

¯

to be the union of all the approximations:

ν

¯

:=

⋃

i

=

0

∞

ν

i

.

Then, by

unique readability

of wffs,

ν

¯

is an interpretation on

V

¯

.

Remark

. If

⟂

is included in the

language

of the logic (as the symbol for falsity), we also require that

ν

i

⁢

(

⟂

)

=

ν

¯

⁢

(

⟂

)

=

0

.

Remark

V

¯

can be viewed as an

inductive set

over

V

with respect to the

¬

and

∨

, viewed as

operations

on

V

¯

. Furthermore,

V

¯

is

freely generated

by

V

, since each

V

i

+

1

can be partitioned into sets

V

i

,

{

(

p

∨

q

)

∣

p

,

q

∈

V

i

}

, and

{

(

¬

⁢

p

)

∣

p

∈

V

i

}

, and each

partition

is non-empty. As a result, any valuation

ν

on

V

uniquely extends to a valuation

ν

¯

on

V

¯

.

Definitions

. Let

p

,

q

be wffs in

V

¯

.

•

p

is

true

or

satisfiable

for some valuation

ν

if

ν

¯

⁢

(

p

)

=

1

(otherwise, it is

false

for

ν

).

•

p

is true for every valuation

ν

, then

p

is said to be

valid

(or

tautologous

). If

p

is false for every

ν

, it is

invalid

. If

p

is valid, we write

⊧

p

.

•

p

implies

q

for a valuation

ν

if

ν

¯

⁢

(

p

)

=

1

implies

ν

¯

⁢

(

q

)

=

1

.

p

semantically implies

if

p

implies

q

for every valuation

ν

, and is denoted by

p

⊧

q

.

•

p

is

equivalent

to

q

for

ν

if

ν

¯

⁢

(

p

)

=

ν

¯

⁢

(

q

)

. They are

semantically equivalent

if they are equivalent for every

ν

, and written

p

≡

q

.

Semantical equivalence is an

equivalence relation

on

V

¯

.

The above can be easily generalized to sets of wffs. Let

T

be a set of propositions.

•

T

is true or satisfiable for

ν

if

ν

¯

⁢

(

T

)

=

{

1

}

(otherwise, it is false for

ν

).

•

T

is valid if it is true for every

ν

; it is

invalid

if it is false for every

ν

. If

T

is valid, we write

⊧

T

.

•

T

implies

p

for

ν

if

ν

¯

⁢

(

T

)

=

{

1

}

implies

ν

¯

⁢

(

p

)

=

1

.

T

semantically implies

p

if

T

implies

p

for every

ν

, and is denoted by

T

⊧

p

.

•

T

1

implies

T

2

for

ν

if, for every

p

∈

T

2

,

T

1

implies

p

for

ν

.

T

1

semantically implies

T

2

if

T

1

implies

T

2

for every

ν

, and is denoted by

T

1

⊧

T

2

.

•

T

1

is equivalent to

T

2

for

ν

if for some valuation

ν

,

T

1

implies

T

2

for

ν

and

T

2

implies

T

1

for

ν

.

T

1

and

T

2

are semantically equivalent if

T

1

⊧

T

2

and

T

2

⊧

T

1

, written

T

1

≡

T

2

.

Clearly,

⊧

p

iff

∅

⊧

p

, and

T

⊧

p

iff

T

⊧

{

p

}

.

Title

truth-value semantics for classical propositional logic

Canonical name

TruthvalueSemanticsForClassicalPropositionalLogic

Date of creation

2013-03-22 18:51:20

Last modified on

2013-03-22 18:51:20

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

20

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B05

Synonym

entail

Defines

truth-value semantics

Defines

valuation

Defines

interpretation

Defines

valid

Defines

invalid

Defines

satisfiable

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