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Adequate set of connectives
Lila Kari
The University of Western Ontario
Adequate set of connectives
CS2209, Applied Logic for Computer Science
1 / 13
Adequate set of connectives
• A remarkable property of the standard set of connectives
(¬, ∧, ∨, →, ↔) is the fact that for every truth table
p
1
q
1
...
...
0
0
...
*
*
*
*
there is a formula (depending on the variables p, q, . . .
and using only the standard connectives) that has
exactly this truth table.
• For every truth table there is Boolean function
f (p, q, . . . ) with exactly this truth table.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
2 / 13
Post’s observation
• Any set of connectives with the capability to express all
truth tables is said to be adequate.
• Post (1921) observed that the standard connectives are
adequate.
Emil Post, 1897-1954
Adequate set of connectives
CS2209, Applied Logic for Computer Science
3 / 13
Proving adequacy of a connective set
We can show that a set S of connectives is adequate if we can
express all the standard connectives in terms of S.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
4 / 13
Proving adequacy of a connective set
We can show that a set S of connectives is adequate if we can
express all the standard connectives in terms of S.
• Formulas A → B and ¬A ∨ B are tautologically
equivalent.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
4 / 13
Proving adequacy of a connective set
We can show that a set S of connectives is adequate if we can
express all the standard connectives in terms of S.
• Formulas A → B and ¬A ∨ B are tautologically
equivalent.
• Then → is definable in terms of (or is reducible to or
can be expressed in terms of) ¬ and ∨.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
4 / 13
Proving adequacy of a connective set
We can show that a set S of connectives is adequate if we can
express all the standard connectives in terms of S.
• Formulas A → B and ¬A ∨ B are tautologically
equivalent.
• Then → is definable in terms of (or is reducible to or
can be expressed in terms of) ¬ and ∨.
• Similarly, ∨ is definable in terms of ¬ and → because
A ∨ B is tautologically equivalent to ¬A → B.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
4 / 13
Theorem. {¬, ∧, ∨} is an adequate set of connectives.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
5 / 13
Theorem. {¬, ∧, ∨} is an adequate set of connectives.
Proof.
For any formulas A, B,
¬A |=|¬A
A ∧ B |=|A ∧ B
A ∨ B |=|A ∨ B
A → B |=|¬A ∨ B
A ↔ B |=|(A → B) ∧ (B → A).
Adequate set of connectives
CS2209, Applied Logic for Computer Science
5 / 13
Theorem. {¬, ∧, ∨} is an adequate set of connectives.
Proof.
For any formulas A, B,
¬A |=|¬A
A ∧ B |=|A ∧ B
A ∨ B |=|A ∨ B
A → B |=|¬A ∨ B
A ↔ B |=|(A → B) ∧ (B → A).
All the five standard connectives can be expressed in terms of
¬, ∧, ∨, therefore {¬, ∧, ∨} is an adequate set of connectives.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
5 / 13
Theorem. {¬, ∧, ∨} is an adequate set of connectives.
Proof.
For any formulas A, B,
¬A |=|¬A
A ∧ B |=|A ∧ B
A ∨ B |=|A ∨ B
A → B |=|¬A ∨ B
A ↔ B |=|(A → B) ∧ (B → A).
All the five standard connectives can be expressed in terms of
¬, ∧, ∨, therefore {¬, ∧, ∨} is an adequate set of connectives.
Corollary {¬, ∧}, {¬, ∨}, and {¬, →} are adequate.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
5 / 13
One connective is enough
Schroder showed in 1880 that each of the standard connectives is
definable in terms of a single binary connective ↓, where the truth
table associated with ↓ is
p
1
1
0
0
Adequate set of connectives
q
1
0
1
0
p↓q
0
0
0
1
CS2209, Applied Logic for Computer Science
6 / 13
Proof of adequacy
We can express ↓ in terms of the standard connectives by
p ↓ q |=|¬p ∧ ¬q, and also the standard connectives in terms of ↓ by
Adequate set of connectives
CS2209, Applied Logic for Computer Science
7 / 13
Proof of adequacy
We can express ↓ in terms of the standard connectives by
p ↓ q |=|¬p ∧ ¬q, and also the standard connectives in terms of ↓ by
¬p |=| p ↓ p
Adequate set of connectives
CS2209, Applied Logic for Computer Science
7 / 13
Proof of adequacy
We can express ↓ in terms of the standard connectives by
p ↓ q |=|¬p ∧ ¬q, and also the standard connectives in terms of ↓ by
¬p |=| p ↓ p
p ∧ q |=| (p ↓ p) ↓ (q ↓ q)
Adequate set of connectives
CS2209, Applied Logic for Computer Science
7 / 13
Proof of adequacy
We can express ↓ in terms of the standard connectives by
p ↓ q |=|¬p ∧ ¬q, and also the standard connectives in terms of ↓ by
¬p |=| p ↓ p
p ∧ q |=| (p ↓ p) ↓ (q ↓ q)
p ∨ q |=| (p ↓ q) ↓ (p ↓ q)
Adequate set of connectives
CS2209, Applied Logic for Computer Science
7 / 13
Proof of adequacy
We can express ↓ in terms of the standard connectives by
p ↓ q |=|¬p ∧ ¬q, and also the standard connectives in terms of ↓ by
¬p
p∧q
p∨q
p→q
|=|
|=|
|=|
|=|
p↓p
(p ↓ p) ↓ (q ↓ q)
(p ↓ q) ↓ (p ↓ q)
((p ↓ p) ↓ q) ↓ ((p ↓ p) ↓ q)
Adequate set of connectives
CS2209, Applied Logic for Computer Science
7 / 13
Proof of adequacy
We can express ↓ in terms of the standard connectives by
p ↓ q |=|¬p ∧ ¬q, and also the standard connectives in terms of ↓ by
¬p
p∧q
p∨q
p→q
p↔q
|=|
|=|
|=|
|=|
|=|
p↓p
(p ↓ p) ↓ (q ↓ q)
(p ↓ q) ↓ (p ↓ q)
((p ↓ p) ↓ q) ↓ ((p ↓ p) ↓ q)
((p ↓ p) ↓ q) ↓ ((q ↓ p) ↓ p).
Thus it follows that a single connective ↓ is adequate.
Consequently, to test a given S for being adequate it suffices to test
if ↓ can be expressed by S.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
7 / 13
Sheffer stroke
In 1913 Sheffer showed that the Sheffer stroke | with associated truth
table
p
1
1
0
0
q
1
0
1
0
p|q
0
1
1
1
is also a single binary connective in terms of which the standard
connectives can be expressed.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
8 / 13
Proving inadequacy
How do we show that a given set S of connectives is not adequate?
Adequate set of connectives
CS2209, Applied Logic for Computer Science
9 / 13
Proving inadequacy
How do we show that a given set S of connectives is not adequate?
Show that some standard connective cannot be expressed by S.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
9 / 13
Proving inadequacy
How do we show that a given set S of connectives is not adequate?
Show that some standard connective cannot be expressed by S.
Example. The set S = {∧} is not adequate.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
9 / 13
Proving inadequacy
How do we show that a given set S of connectives is not adequate?
Show that some standard connective cannot be expressed by S.
Example. The set S = {∧} is not adequate.
Proof. To see this, note that a formula depending on only one
variable and which uses only the connective ∧ has the property that
its truth value for a value assignment that makes p = 0 is always 0.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
9 / 13
Proving inadequacy
How do we show that a given set S of connectives is not adequate?
Show that some standard connective cannot be expressed by S.
Example. The set S = {∧} is not adequate.
Proof. To see this, note that a formula depending on only one
variable and which uses only the connective ∧ has the property that
its truth value for a value assignment that makes p = 0 is always 0.
In order to define the negation ¬p in terms of ∧, there should exist a
formula f depending on the variable p and using only the connective
∧ such that ¬p |=|f .
Adequate set of connectives
CS2209, Applied Logic for Computer Science
9 / 13
Proving inadequacy
How do we show that a given set S of connectives is not adequate?
Show that some standard connective cannot be expressed by S.
Example. The set S = {∧} is not adequate.
Proof. To see this, note that a formula depending on only one
variable and which uses only the connective ∧ has the property that
its truth value for a value assignment that makes p = 0 is always 0.
In order to define the negation ¬p in terms of ∧, there should exist a
formula f depending on the variable p and using only the connective
∧ such that ¬p |=|f .
However, for a value assignment v such that v (¬p) = 1, we have
v (p) = 0 and therefore v (f ) = 0, which shows that ¬p and f cannot
be tautologically equivalent.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
9 / 13
A ternary connective
Let us use the symbol τ for the ternary connective whose truth table
is given by
p
1
1
1
1
0
0
0
0
Adequate set of connectives
q
1
1
0
0
1
1
0
0
r τ
1 1
0 1
1 0
0 0
1 1
0 0
1 1
0 0
CS2209, Applied Logic for Computer Science
10 / 13
A ternary connective
It is easy to see that for any value assignment v we have
v (τ (p, q, r )) = v (q) if v (p) = 1 and v (r ) if v (p) = 0.
Adequate set of connectives
CS2209, Applied Logic for Computer Science
11 / 13
A ternary connective
It is easy to see that for any value assignment v we have
v (τ (p, q, r )) = v (q) if v (p) = 1 and v (r ) if v (p) = 0.
This is the familiar if-then-else connective from computer science,
namely
if p then q else r
Adequate set of connectives
CS2209, Applied Logic for Computer Science
11 / 13
Comments
We can consider now propositional logic based not upon the five
common connectives, but upon any adequate set of connectives, for
instance {¬, ∧}.
Let Lp0 be a sublanguage of Lp obtained by deleting from Lp the
three connectives ∨, →, ↔, and let Form(Lp0 ) be the set of formulas
of Lp0 .
Theorem. Form(Lp0 ) = Form(Lp ).
Adequate set of connectives
CS2209, Applied Logic for Computer Science
12 / 13
Proof
Proof.
Obviously, Form(Lp0 ) ⊆ Form(Lp ).
Conversely, for A ∈ Form(Lp ) we define (by recursion) its translation
A0 into Lp0 as follows:
A0 = A for atomic variable A,
(¬A)0 = ¬A0 ,
(A ∧ B)0 = A0 ∧ B0 ,
(A ∨ B)0 = ¬(¬A0 ∧ ¬B0 ),
(A → B)0 = ¬(A0 ∧ ¬B0 ),
(A ↔ B)0 = (A → B)0 ∧ (B → A)0 =
= ¬(A0 ∧ ¬B0 ) ∧ ¬(¬A0 ∧ B0 )
Adequate set of connectives
CS2209, Applied Logic for Computer Science
13 / 13