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3.1. SYNTAX
3.1
Syntax
3.1.1
Formulas
Just like any natural language, a formal language is based on an alphabet from
which its words and sentences can be formed.
Definition 3.1 An alphabet Σ consists of a finite or countably infinite set.
The elements of this set are called symbols. Without loss of generality we
assume that Λ !∈ Σ .
Examples for alphabets are
Σ = {1, 2}, Σ = {pi | i ∈ N}, Σ = {pi | i ∈ N} ∪ {¬, ∧, ∨, →, ←, (, )}
or the alphabet of the German language. In some applications alphabets are
restricted to consist of finite and non-empty sets. We will see later that we
need countably infinite sets. If an alphabet is empty, then one usually cannot
base many applications on it. However, from the viewpoint of logic there is no
need to require non-empty alphabets.
We can form words over alphabets. Informally speaking, words are just finite
sequences of elements from an alphabet. They are formally defined as follows:
Definition 3.2 Let Σ be an alphabet. The set of words (or strings) over Σ
is denoted by Σ∗ and is defined as the smallest set satisfying the following
conditions:
1. Λ ∈ Σ∗ .
2. If w ∈ Σ∗ and a ∈ Σ , then aw ∈ Σ∗ .
Λ is called empty word or empty string.
For Σ = {1, 2} we obtain the set
Σ∗ = {Λ, 1Λ, 2Λ, 11Λ, 12Λ, 21Λ, 22Λ, 111Λ, . . . }.
Because Λ denotes the empty word, the symbol Λ is often omitted when
writing non-empty words and we obain
Σ∗ = {Λ, 1, 2, 11, 12, 21, 22, 111, . . . }.
Strictly speaking, we have to show that a smallest set satisfying the two conditions in Definition 3.2 exists. Thereby sets are compared with respect to
the subset-relation. A set satisfying the two conditions in Definition 3.2 is the
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smallest set, if there is no proper subset satisfying the the conditions. Obviously, there are sets satisfying the conditions. The intersection of two sets
satisfying the conditions satisfies the conditions again. Hence, the smallest set
is the intersection of all sets satisfying the two conditions in Definition 3.2.
Let us consider the set Σ = {1, 2} as example again. Σ∗ as specified above
meets both conditions in Definition 3.2. This holds also for the set
Σ∗∗ = {Λ, 1, 2, !, 11, 12, 1!, 21, 22, 2!, 111, . . . }.
Here, an additional word ! and all words necessary to satisfy the second
condition in Definition 3.2 were added to Σ∗ . We find Σ∗ ⊂ Σ∗∗ . Hence, Σ∗∗
is not the smallest set satisfying the conditions in Definition 3.2.
Now, let us turn to the alphabet of propositional logic. So-called propositional
variables denote simple statements like for example “the cow is violet” or “the
rooster crows on the dung”, where the internal structure of such a statement is
not to be analyzed further at the moment. Simple propositions can be combined
to more complex propositions using connectives. Such connectives correspond
to, for instance, negation, conjunction, disjunction, implication, or equivalence
of one or more propositions. Some examples are:
• “the cow is not violet” (negation),
• “the cow is violet and the rooster crows on the dung” (disjunction),
• “the cow is violet or the rooster crows on the dung” (disjunction),
• “if the cow is violet then the rooster crows on the dung” (implication),
• “the cow is violet if and only if the rooster crows on the dung” (equivalence).
Connectives are denoted by ◦/n , where ◦ is the connective itself, and n is a
natural number indicating the arity of the connective. Besides that, there are
the parantheses as special characters.
Definition 3.3 An alphabet of propositional logic consists of
• a (countably) infinite set R = {p1 , p2 , . . .} of propositional variables,
• the set {¬/1, ∧/2, ∨/2, → /2, ↔ /2} of connectives, and
• the special characters “(” and “)”.
For simplicity, propositional variables are often denoted by p, q, . . . instead of
p1 , p2 , . . .. Likewise, indices are sometimes omitted. Propositional letters are
often called nullary relation symbols in the literature. Different alphabets of
3.1. SYNTAX
69
propositional logic differ in R . By abuse of notation, alphabets of propositional
logic are often denoted by R .
We can now build words over the alphabet according to Definition ??. A
language over an alphabet Σ does usually not contain all words occurring in
Σ∗ . For example, in English the words “sleep” or “run” are allowed, whereas
the woods “leesp” or “urn” are not (although they are sometimes used in
cartoons or advertisements). A language over an alphabet Σ is typically a
subset of Σ∗ . This subset is formed according to certain rules. In English, the
words are explicitely mentioned in a dictionary. In addition, there are rules
specifying when an ending like “ing” can be added to a word like “sleep” to
obtain the new wword “sleeping”. Similar rules exist specifying sentences. The
same holds for the language of propositional logic, where the well-formed words
are usually called formulas.
Definition 3.4 A propositional atomic formula, briefly called atom, is a propositional variable.
Definition 3.5 The set of propositional formulas is the smallest set L(R) of
strings over an alphabet of propositional logic with propositional variables R
satisfying the following properties:
1. If F is an atomic formula, then F ∈ L(R).
2. If F ∈ L(R) , then ¬F ∈ L(R).
3. If ◦ is a binary connective and F, G ∈ L(R) , then (F ◦ G) ∈ L(R).
Examples for formulas are p1 , ¬p2 , and ¬((p1 → p2 ) ∨ p1 ). On the other
hand, p1 p2 , p1 ∧ p2 , and (¬p1 ) are no formulas, although they are strings
over the alphabet R of propositional logic (see also Exercise 3.1).
We should again show that there is a smallest set satisfying the three conditions
mentioned in Definition 3.5. Obviously, there are sets satisfying all the conditions. An example is the set of all strings over the alphabet of propositional
logic. Obviously, the intersection of two sets satsfying these conditions satisfies
the conditions as well. The smallest set is the intersection of all sets satisfying
the conditions. As an example consider the following set of formulas:
{p1 , p2 , . . . , !, ¬p1 , ¬p2 , . . . , ¬!, (p1 ∧ p1 ), (p1 ∧ p2 ), . . . , (p1 ∧ !), . . .}.
This set satisfies the three conditions in Definition 3.5, but it is not minimal
because it contains the strings ! , ¬! , (p1 ∧ !) etc.
In the following, A (possibly indexed) denotes an atom and F , G and H
(possibly indexed) denote propositional formulas, if not stated otherwise.
Definitions 3.3, 3.4 and 3.5 may look a bit strange on first sight. On second
sight, however, the reader should realize that he or she is used to similarly defined notions. For example, consider arithmetic expressions. They are defined
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over an alphabet which may contain the rational numbers Q , the operators
+/2, −/2, ÷/2 and ×/2 as well as the special charaters “(“ and “)”. Over
such an alphabet we can define the arithmetic expressions as the smallest set
satisfying the following conditions:
1. If T is a rational number, then T is an arithmetic expression.
2. If ◦/2 is an operator and T1 , T2 are arithmetic expressions, then (T1 ◦
T2 ) is also an arithmetic expression.
3
Hence, we obtain the arithmetic expressions like 19
3 , (3 + 5) or ( 2 ÷ (3 × 17)) .
Comparing this definition with Definitions 3.3, 3.4 and 3.5 we find that rational
numbers correspond to propositional variables or atoms, operators correspond
to connectives and arithmetic expressions to formulas. One should observe that
even in arithmetics one can find unary operators like the factorial or the square.
Another example is the set of natural numbers. It is based on an alphabet
consisting of the symbol 0 , the unary operator s/1 and the special charaters
“(“ and “)”. The set of natural numbers is the smallest set satisfying the
following conditions:
1. 0 is a natural number.
2. If N is a natural number, so is s(N ) .
We thus obtain natural numbers like 0, s(0), s(s(0)) etc.
As a final example we recall the set of lists as defined in Prolog (see Section 2.4.1. They were defined according to the principle discussed in this section. For example, the strings [ ] , .(1, [ ]) and .([ ], [ ]) are strings.
3.1.2
Induction and Recursion
Let us return to the set of propositional logic. This is a set of structured objects over the alphabet of propositional logic. To prove properties1 of formulas
or to define functions over formulas, we need techniques known as structural
induction and structural recursion, respectively.
Theorem 3.6 Principle of structural induction: Every propositional formula
has a property E , if the following conditions are met:
1. Induction base: Every atom has property E.
1
We will see later that properties are usually predicate logic formulas with a freely occurring variable.
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71
2. Induction steps:
If F has property E , then ¬F has property E as well.
If ◦/2 is a connective, and F and G have property E , then (F ◦ G)
has property E as well.
Proof Let E be the set of all propositional formulas that have property E.
According to the induction base and the induction steps, E satisfies the three
properties of Definition 3.5. Because the set L(R) of propositional formulas
is the smallest set satisfying these conditions, L(R) ⊆ E holds. Because
E ⊆ L(R) , we have E = L(R).
"
Let E , for example, denote the property “the number of opening parentheses occurring in a propositional formula is identical to the number of closing
parentheses occurring in that formula.” To show that all formulas have this
property, it suffices to prove the induction base and the induction steps. An
application of Theorem 3.6 then yields the desired result. The induction base
follows immediately from the fact that there are no parentheses at all in atoms.
In the first induction step, we assume that E holds for the formula F . Because parenthesis occurs in ¬F if and only if they occur in F , E holds for
¬F as well. In the second induction step, we assume that E holds for F
and G. Let i be the number of opening and closing parentheses in F and j
the number of opening and closing parentheses in G. Then there are exactly
i + j + 1 opening and closing parentheses in (F ◦ G), which proves that E
holds for (F ◦ G) as well.
We have proven the principle of structural induction for propositional formulas. It can easily be applied to other structured objects that are or that can be
defined recursively. Assume we want to prove a property over lists of terms,
where lists are defined as in Section 2.4.1. The corresponding induction principle reads as follows: Every list has a property E if the following conditions
are met:
1. Induction base: The empty list [ ] has property E .
2. Induction step: If K is a term and the list R has property E , then .(K, R)
has property E as well.
Using this principle we can show that each list contains the symbol “ [ ] ”.
In a similar way we can define the principle of structural induction for natural
numbers in successor representation. Each natural number has a property E
if the following conditions are met:
1. Induction base: 0 has property E.
2. Induction step: If the natural number N has property E, then s(N ) has
property E as well .
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Using this principle we can show that each natural number is smaller than or
equal to its square.
As a final example I want to discuss the principle of structural induction for
words over an alphabet Σ. Each word occurring in Σ∗ has property E, if the
following conditions are met:
1. Induction base: Λ has property E.
2. Induction step: If w ∈ Σ∗ has property E and a ∈ Σ, then aw has
property E as well .
Using this principle we can, for example, show for Σ = {1, 2} that for all
w ∈ Σ∗ we find: the symbol 3 does not occur in w.
Let us now turn to the question of how to define functions over sets of structered
objects. For example, how can we compute the number of occurrences of binary
connectives in a formula? How can we compute the number of occurrences of
atoms in a formula? The following principle provides a possible solution to
these and similar questions.
Theorem 3.7 Principle of structural recursion: Let M be an arbitrary set.
Furthermore, assume that the following functions are given:
fooR : R → M,
foo¬ : M → M,
foo◦ : M × M → M for ◦ ∈ {∧, ∨, →, ↔}.
Then, there is exactly one function foo : L(R) → M satisfying the following
conditions:
1. Recursion base: foo(A) = fooR (A) for all A ∈ R .
2. Recursion steps:
foo(¬G) = foo¬ (foo(G)) for all G ∈ L(R) .
foo((G1 ◦ G2 )) = foo◦ (foo(G1 ), foo(G2 )) for all G1 , G2 ∈ L(R) .
In many applications the set M and the functions foo R , foo ¬ and foo ◦ are
not explicitely defined. Often the principle is introduced only informally: There
is exactly one function foo defined over L(R) satisfying the following conditions:
1. Recursion base: The value of foo for atoms is defined explicitly.
2. Recursion steps:
The value of foo for ¬F is defined depending only on the value of foo
for F .
The value of foo for (F ◦ G) is defined depending only on the values od
foo for F and G .
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3.1. SYNTAX
The interested reader is invited to prove this theorem in Exercise 3.2. It is
similar to the proof of Theorem 3.6. Theorem 3.7 can be applied to answer the
above mentioned questions. Let

if F is an atom,
 0
foo 1 (F )
if F is of the form ¬G,
foo 1 (F ) =

foo 1 (G1 ) + foo 1 (G2 ) + 1 if F is of the form (G1 ◦ G2 ).
foo 1 /1 computes the number of occurrences of binary connectives in a formula.
For example, we find foo 1 (p1 ) = 0, foo 1 (¬p1 ) = 0 and foo 1 (¬((p1 → p2 ) ∨
p1 )) = 2.
Consider now:

if F is an atom,
 1
foo 2 (G)
if F is of the form ¬G,
foo 2 (F ) =

foo 2 (G1 ) + foo 2 (G2 ) if F is of the form (G1 ◦ G2 ).
foo 2 /1 computes the number of occurrences of atoms in a formula. For example, we find foo 2 (p1 ) = 1, foo 2 (¬p1 ) = 1 and foo 2 (¬(p1 ∨ p1 )) = 2 . The last
example illustrates, that foo 2 computes in deed occurrences of atoms.
Another example is:

if F is an atom,
 F
¬foo 3 (G)
if F is of the form ¬G,
foo 3 (F ) =

foo 3 (G1 ) ◦ foo 3 (G2 ) if F is of the form (G1 ◦ G2 ).
foo 3 elimininates all paranthesis occurring in F :
foo 3 ((¬(p1 ∧ p2 ) → p3 )) = ¬p1 ∧ p2 → p3 .
As final example we consider:

∅
if F is an atom,



{1π | π ∈ foo 4 (G)} if F is of the form ¬G,
foo 4 (F ) = {Λ} ∪
{1π1 | π1 ∈ foo 4 (G1 )} ∪ {2π2 | π2 ∈ foo 4 (G2 )}



if F is of the form (G1 ◦ G2 ).
foo 4 (F ) is often called the set of positions or occurrences in F . For example,
we obtain:
foo 4 (p1 )
foo 4 (¬p1 )
foo 4 (¬¬p1 )
foo 4 ((p1 ∨ p2 ))
foo 4 (((p1 ∨ ¬p2 ) → (p3 ∧ p4 )))
=
=
=
=
=
{Λ},
{Λ, 1},
{Λ, 1, 11},
{Λ, 1, 2},
{Λ, 1, 11, 12, 121, 2, 21, 22}
Positions in a formula can nicely be illustrated by considering a tree representation of terms. Tree representations of Prolog-terms were already discussed
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p1 • Λ
¬ • Λ
¬ • Λ
p1 • 1
¬ • 1
∨ •Λ
p1 • 1
p2 • 2
p1 • 11
Figure 3.1: The tree representation of the formulas p1 , ¬p1 , ¬¬p1 and (p1 ∨
p2 ) together with the associated positions.
in Figure 2.5. In a similar wat we can represent formulas as trees. In Figures 3.1 und 3.2 several formulas are depicted as trees, and for each node the
corresponding position is shown.
We have presented the principle of structural recursion for propositional formulas. It can easily be applied to other structured objects. To illustrate this, we
consider again the lists of terms as defined in Section 2.4.1. The corresponding
recursion principle reads as follows: There is exactly one function foo defined
over the set of lists of terms satisfying the following conditions:
1. Recursion base: The value of foo for the empty list [ ] is defined explicitly.
2. Recursion step: The value of foo for .(K, R) is defined depending only
on the value of foo for R .
We can now use this definition to define a function length/1 over lists of terms:
%
0
if X is the empty list [ ],
length(X) =
length(R) + 1 if X is of the form .(K, R).
The interested reader is invited to compare this mathematically precise definition to the program for length/2 specified in Section 2.4.4 (see Exercise 3.3).
The principle of structural recursion can also be defined for natural numbers.
There is exactly one function foo defined over the set of natural numbers in
successor representation satisfying the following conditions:
1. Recursion base: The value of foo for 0 is defined explicitly.
2. Recursion step: The value of foo for s(N ) is defined depending only on
the value of foo for N .
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3.1. SYNTAX
→ •Λ
∨ • 1
∨ • 11
∧ • 2
¬ • 12
p3 • 21
p4 • 22
p2 • 121
Figure 3.2: The tree representation of the formula ((p1 ∨ ¬p2 ) → (p3 ∧ p4 ))
together with the associated positions.
Using this principle we can, for example, define a function +2/1 over natural
numbers, which adds 2 to its argument:
%
s(s(0))
if N = 0,
+2(N ) =
s(+2(N # )) if N is of the form s(N # ).
Thus, we find that +2(0) = s(s(0)) and +2(s(s(0))) = s(s(s(s(0)))). Another
example is the function !/1 , which is usually written in postfix notation:
%
s(0)
if N = 0,
N! =
N × N # ! if N is of the form s(N # ).
It is quite easy to implement the functions +2/1 and !/2 in Prolog.
Finally, we consider again strings over an alphabet Σ. There is exactly one
function foo defined over Σ∗ satisfying the following conditions:
1. Recursion base: The value of foo for Λ is defined explicitly.
2. Recursion step: The value of foo for aw ∈ Σ∗ is defined depending only
on the value of foo for w.
Using this principle we can, for example, define the length of a string as follows:
%
0
if w = Λ,
length(w) =
1 + length(w# ) if w is of the form aw#.
The interested reader is again asked to implement the function length/1 in
Prolog (see Exercise ??).