Download propositional logic

PROPOSITIONAL LOGIC
Jorma K. Mattila
LUT, Department of Mathematics and Physics
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Introduction
1.1
Logical Inference
The result of an inference is deduction or argument. A deduction or an argument
comes from the fact that from some premises it follows a certain conclusion. So,
inferences are drawing conclusions from sets of premises, or showing that the
conclusion follows from the premises:
Premises
=⇒
Conclusion
An argument can be either valid i.e. (logically) correct tai invalid. The following argument seems to be valid:
Socrates is a human.
All humans are mortal.
Socrates is mortal.
premise
premise
conclusion
(1.1)
The premises and the conclusion in the argument (1.1) are (or, were) true.
Thus the argument is (logically) correct.
The following argument may evidently not be valid.
Socrates is a human.
Some humans are mortal.
Socrates is not mortal.
premise
premise
conclusion
In (1.2) the premises are true but the conclusion false.
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(1.2)
To the validity of a argument the following necessary condition is given:
If the premises are true then the conclusion is true.
(1.3)
This is a very important property. according to this condition, a valid argument does not lead "outside the truth", whenever the premises are true. We say,
that a valid argument is truth preserving.
The condition (1.3) is not sufficient condition for a valid argument, as the
following example shows.
Socrates is human.
All humans are mortal.
Socrates is a philosopher.
premise
premise
conclusion
(1.4)
The premises are true and the conclusion is true in (1.4) but the argument is
incorrect. The condition (1.3) is only the demand for a argument that must at
least hold, if the argument would be valid.
An argument may be correct also in the following cases.
True premise
False premise
=⇒
False conclusion
False premises
=⇒
True conclusion
False premises
=⇒
False conclusion
Concerning the validity of an argument, there is no meaning whether its
premises are true or not, as the following example shows.
Socrates is a finn.
All the finns are immortal.
Socrates is immortal.
premise
premise
conclusion
(1.5)
Both the premises and the conclusion are false in (1.5).
Let us have still one argument:
Peter is a raven.
All ravens are black.
Peter is black.
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premise
premise
conclusion
(1.6)
Consider the arguments (1.1), (1.5) ja (1.6). They clearly have the same logical form i.e. the same logical construction, that can preliminarily be presented
in the following way:
a is A.
Every A is B.
Thus: a is b.
(a has the property A)
(All ones having the property A
has the property B)
(a has the propertyB)
(1.7)
We have the syntactic criterion for the validity of an argument:
The validity of an argument does not particularly depend
on the actual contents of the sentences in the argument
as on their logical form.
(1.8)
The syntactic criterion (1.8) is in connection with the semantical criterion
(1.3). We will consider the question, what the logical form means and how the
validity of an argument depends on it.
The king of arguments considered above are called deductions. In a deduction
the conclusion follows necessarily from the premises.
An argument where the conclusion does not necessarily or certainly follow
from the premises, but for example with a great probability, is an inductive argument.
Practical, or every-day argunebts, are usually not presented as complete arguments with all the premises needed. The following examples illustrate this way
of argumentation.
All humans are mortal.
So, Socrates is mortal.
or: ”As a human, Socrates is mortal.”
(1.9)
Peter is a raven.
(1.10)
So, Peter is black.
or: ”Peter is black, because he is a raven.”
The problem of the validity of a practical argument can be solved, if the
missing premises can be found.
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1.2
On Formal Theories
A formal language consists of a set of primitive symbols, called alphabets, and
formulas or sentences formed by special formation rules.
By a syntax of a formal language is meant the research of the language in its
own scope – without depending on meanings of the expressions or the use of the
language. "Grammar", specification of alphabets, formation rules for formulas,
and proof theory (relationships between formulas) of a formal language belongs
to the syntax of the language.
In the proof theory of a formal language some formulas are chosen as the
axioms, from which theorems can be deduced by the inference rules. Theorems
are called provable; their proofs are finite sequences of formulas. A formal language equipped with proof theory is called calculus.
Formal systems are theories, where the following conditions hold:
1. The shape rules are given; they tell what symbols are allowed in the theory.
2. The formation rules are given; they tell how correct expressions are formed
from the allowed symbols in the scope of the theory.
3. The axioms are given; theorems of the theory can be deduced from the
axioms.
4. The inference rules are given; they tell how new expressions can be deduced
from other expressions of the theory.
A formal language is completely determined, if the conditions (1) – (4) hold
in the scope of the theory. These conditions define the principle of calculus.
David Hilbert is (or was) the central developer for proof theory. He tried to
axiomatize a "universal" axiomatic theory for mathematic proofs.
The four main problems of proof theory are
1. consistency of the system: to show that there are no contradictions in the
system;
2. independency of the axioms: to show that the axioms are independent;
3. completeness of the system: to find out whether all the true expressions of
the system can be deduced from the axioms;
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4. the problem of solvability: is there any finite mechanical method (algorithm) by which all the true expressions of the system can be proved to be
true. If such a method exists, it is called method of solvability.
The problems mentioned above can be solved for example for propositional
logic. According to predicate logic, independence, completeness and consistency can be proved to hold in it, but not solvability (Church). According to arithmetics, no method of solvability exists. Arithmetics is also incomplete (Gödel).
To prove consistency is also sometimes difficult in different branches of mathematics.
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2.1
Fundamentals of Propositional Logic
Object language and metalanguage
A formal language can be either tools for use or an object of research. When a
language is an object of research, it is called an object language. On the other
hand, in this kind of research a language is also needed as tools. This language is
called metalanguage. Metalanguage is used when we present results concerning
the object language. These results are called metatheorems. In this way we have
difference between theorems (of the object language) and the research results of
the study on the object language.
2.2
Basic Concepts and Definitions
There are several synonyms to the word propositional logic, as sentence logic,
sentence calculus, and logic of zero order. We use the symbol L as the label of
the propositional logic considered here.
Alphabet and formation of formulas
We consider here the basic concepts of the syntax of L. Propositional logic as a
formal language can be defined in many different ways depending on the choice
of alphabet.
Definition 2.1. The alphabet of L consists of the following symbols, called the
primitives of L:
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(i) a numerable set od propositional variables {pi | i ∈ N},
(ii) implication symbol → ,
(iii) negation symbol ¬ ,
(iv) parentheses (, ) .
An arbitrary finite sequence of symbols of L is an expression of L. For
example, the sequence p1 → ¬p2 is an expression of L.
Well-formed formulas (write wff) of L are defined inductively as follows.
Definition 2.2. The well-formed formulas of L consist of
(i) pn for all n ∈ N ;
(ii) if A and B are wff’s then (A → B) is a wff ;
(iii) if A is a wff then (¬A) is a wff .
(iv) An expression is a wff only, if it is a wff according to the rules (i) − (iii).
The conditions (i) − (iii) of Def. 2.2 are the formation rules of wff’s of L.
The condition (i) defines atoms or atomic formulas of L. The conditions (ii)
and (iii) expresses how connected formulas can be deduced from atoms.
Example 2.1. The expression p3 → ((¬p2 ) → p1 )) is a wff of L. Because
p1 , p2 and p3 are wff’s by (i) and ¬p2 is a wff by (iii) of Def. (2.2) then the
expression ((¬p2 ) → p1 )) is a wff by (ii) of Def. (2.2), so the original expression
is a wff of L by (ii) Def. (2.2).
The set of wff’s of the language L is uniquely determined, as the following
metatheorem shows.
Metatheorem 2.1. There exists a unique set W of expressions of L, such that
(1) pn ∈ W for all n ∈ N,
(2) if A ∈ W, then (¬A) ∈ W,
(3) if A ∈ W, then (A → B) ∈ W,
0
0
(4) if W satisfies the conditions (1)-(3), then W is included in the set W .
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Proof. follows from the definitions Def. 2.1 and Def. 2.2 by mathematical induction.
When we want to prove that all the wff’s A of L have a property ϕ, we can
do it as follows:
(1) Show that ϕ(pn ) holds for all n ∈ N,
(2) Show that ϕ(¬A), if ϕ(A),
(3) Show that ϕ(A → B), if ϕ(A) and ϕ(B).
This kind of method is called induction on the length of a formula.
In the definition (given in metalanguage) (Def. 2.2) primitives pn , ¬, →,
( and ) of L are used autonomously. The symbols A and B in this definition are
not symbols of L but those of its metalanguage, so-called metavariables. They
have wff’s of L as their values.
When we indicate in metalanguage to wff’s of L, we obey the following
agreement:
(i) The most outer pair of parentheses can be omitted.
(ii) The pair of parentheses in a wff of negation form can be omitted – also in
the case where it exists as a part of a wff.
Operating order: Negation operates first, then implication.
Abbreviations: We adopt the following abbreviations:
df
A ∨ B ⇐⇒ ¬A → B,
df.
A ∧ B ⇐⇒ ¬(¬¬A → ¬B),
df
A ↔ B ⇐⇒ ¬(¬¬(A → B) → ¬(B → A)),
df
A|B ⇐⇒ ¬A ∨ ¬B,
df
A ↓ B ⇐⇒ ¬A ∧ ¬B.
df.
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(Obs. ⇐⇒ and ⇐⇒ are symbols of our metalanguage!) The abbreviations (2.1),
(2.2), (2.3), (2.4) and (2.5) represent the connectives disjunction, conjunction ,
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equivalency , Sheffer’s stroke , and Peirce’s arrow, respectively. In natural language these connectives have the interpretations ’or’, ’and’, ’if and only if ’,
’not-or-not’, and ’not-and-not’, respectively.
About further connections between connectives, we can express the conjunction (2.2) by means of negation and implication in the form
A ∧ B ⇐⇒ ¬(A → ¬B)
(2.6)
by (2.1). We can express the equivalency (2.3) by means of implication and
conjunction in the form
A ↔ B ⇐⇒ (A → B) ∧ (B → A)
(2.7)
by (2.1) and (2.2). Furthermore, we can express the connectives negation, conjunction, disjunction, and implication by using Sheffer’s stroke and Peirce’s
arrow as follows:
¬A
A∧B
A∨B
A→B
¬A
A∧B
A∨B
A→B
⇐⇒
⇐⇒
⇐⇒
⇐⇒
⇐⇒
⇐⇒
⇐⇒
⇐⇒
A|A ,
(A|B)|(A|B) ,
(A|A)|(B|B) ,
A|(B|B) ,
(A ↓ A) ,
(A ↓ A) ↓ (B ↓ B) ,
(A ↓ B) ↓ (A ↓ B) ,
((A ↓ A) ↓ B) ↓ ((A ↓ A) ↓ B).
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
In addition to these, we show how we can express disjunction, implication,
and equivalence by means of negation and conjunction:
A ∨ B ⇐⇒ ¬(¬A ∧ ¬B) ,
A → B ⇐⇒ ¬(A ∧ ¬B) ,
A → B ⇐⇒ ¬(A ∧ ¬B) ∧ ¬(B ∧ ¬A).
(2.16)
(2.17)
(2.18)
The interpretation of disjunction is so-called inclusive or. It means that the
formula A ∨ B can be true if at least one of the formulas A and B is true, especially, it is true when both of A and B are true, too. We have also the other kind
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of disjunction, named exclusive or A Y B, which is true only if either A is true
or B is true, but not both of them. We can define the exclusive disjunction by
means of negation, conjunction, and disjunction as follows:
df
A Y B ⇐⇒ (A ∧ ¬B) ∨ (¬A ∧ B).
(2.19)
Remark 2.1. By the alphabets we chose, the symbols ∨, ∧, ↔, |, ↓, and Y belong
to the metalanguage. By these symbols we refer to wff’s of L according to the
abbreviation formulas. Including these ’new’ connectives, the operating order
is as follows: as the strongest connective, negation operates first, then disjunction and conjunction being as strong, when their mutual operating order must be
shown by parentheses, and, at last, implication and equivalence being as strong
together when parentheses are needed to show their mutual operating order.
According to Sheffer’s stroke and Peirce’s arrow, they usually are used alone,
and so the operating order must always be shown by parentheses.
Remark 2.2. It is possible to choose either the connectives negation and disjunction, negation and conjunction, Sheffer’s stroke or Peirce’s arrow as primitives.
We call an expression being a wff itself, which is a uniform part of a wff A,
a part-wff (or part-formula) of A. Thus every connected formula can write in
some of the following forms:
(¬A) , (A ∨ B) , (A ∧ B) , (A → B) , and (A ↔ B)
where A and B are part-formulas of a given wff. The connective existing in this
kind of forms operating at last, i.e. corresponding to the outer parentheses, is
called main connective.
2.3
Semantics of Propositional Logic
A language is always a language about something. It considers, presents, illustrates or means something. The semantics, or model theory, of a language is a
theory about the meaning of the language, about what the language presents.
A theory of the semantics of a formal language often uses, as also the syntactical proof theory, mathematical concepts and methods, that often are more
exacting that syntactical proof theory.
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The central semantical concepts are truth and falsity. If a sentence describes
a state of affairs that holds, then the sentence is true. If a sentence describes a
state of affairs that does not hold, then the sentence is false. We give now the
main features of the semantics of L.
The truth values true and false can be denote in the following ways:
def.
(1) true = T
def.
(2) true = 1
def.
false = F ,
def.
false = 0.
Truth values of wff’s of L can be evaluated in the following way using valuation:
Definition 2.3. Each wff pn , (n ∈ N), of L is associated with the truth value T
or F by a valuation of L.
According to this definition, we say that a valuation gives a truth value distribution to the propositional variables of a given wff.
Definition 2.4. Valuation is extended for connected wff’s of L to be a mapping
from the set of wff’s of L onto the set {F, T } as follows:
1. If a wff A is of the form B → C then A := T , if B := F or C := T ,
otherwise A := F .
2. If A is of the form ¬B then A := T , if B := F , otherwise A := F .
3. If A is of the form B ∧ C then A := T , if both B := T and C := T ,
otherwise A := F .
4. If A is of the form B ∨ C then A := T , if at least one of the conditions
B := T and C := T holds, otherwise A := F .
5. If A is of the form B ↔ C then A := T if and only if B and C have the
same truth value, otherwise A := F .
Example 2.2. Let P := T and Q := F . Then ¬P := F . What is the truth value
of the wff P ⇒ (P ⇒ ¬Q) in this situation?
¬Q := T , P ⇒ ¬Q := T by Def. (2.4, thus P ⇒ (P ⇒ ¬Q) := T .
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The following metatheorem shows that any valuation gives to every wff of L
either the truth value true or false, and no valuation gives the both truth values to
a wff.
Metatheorem 2.2. Every wff A of L have uniquely either the value T or F in
any valuation.
Proof. Let A be an arbitrary wff of L. Suppose that a truth value distribution to
the propositional variables of A is given by some valuation. If A is an atom then
it already have a unique truth value. If A is a connected wff then we can evaluate
its truth value by Def. 2.4. The definition 2.4 guarantees the unique evaluation
of A. Because A was chosen arbitrarily, the result holds for any wff of L.
The idea of Metatheorem 2.2 can be considered also in the following way. By
means of Def. 2.4, every wff A defines a function
fA : {F, T }n −→ {F, T } ,
n∈N
(2.20)
where n is the number or propositional variables (or atoms) existing in A. The
function fA is called the truth function determined by A. Generally, a truth
function is a mapping from a set of truth distributions onto the set of truth values.
Thus for any truth functional expression it holds that a truth function associates
to every truth distribution uniquely a certain truth value. The domain of the
function (2.20) expresses the idea of a truth distribution. A truth distribution of
a given wff A is an n-tuple or truth values belonging to the product set {F, T }n
where n ∈ N is the number of the atoms of A. The number of truth distributions
of A is 2n .
Example 2.3. Consider the expression P ∧ ¬(Q → R) purely from the truth
functional view. It is a three-valued function f (P, Q, R), i.e. f is a function
f : {F, T }3 −→ {F, T }.
Elements of the set {F, T }3 are ordered triples of truth values. All its elements
form the truth distributions of the expression. The number of them is 23 = 8.
For example, let (P, Q, R) := (T, F, T ). Thus the wff P ∧ ¬(Q → R) has the
truth value F , i.e. f (T, F, T ) = F .
Remark 2.3. Truth value distributions are functions from the set of wff’s onto
the set of truth values, i.e. valuations. We can illustrate this fact using truth
tables.
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Satisfiability and Validity
Besides the concepts ’truth’ and ’falsity’, satisfiability and validity are essential
concepts in formal semantics.
Definition 2.5. A non-empty set M of propositional variables (propositional
letters) of L is a model of L.
In fact, a model is a collection of states of affairs, that can be given as a set of
those wff’s that describe these states of affairs. These states of affairs hold in the
model. Thus a wff is true iff (read: if and only if) the state of affairs described
by the wff holds. When we express this more exactly, we use correspondence
theory of truth: A wff A is true in a model M iff the state of affairs expressed by
A holds in M.
Definition 2.6 (Definability). A wff A is definable if there exsists a model M,
such that A := T in M, i.e. A ∈ M. Then we write M |= A.
Definition 2.7 (Validity). A wff A is valid, iff A ∈ M holds for all models M,
i.e. M |= A holds for all M. Then we write |= A.
Example 2.4. Let M = {P, Q}. Then M |= P ∧ Q, because M |= P and
M |= Q. Also M |= P ∨ R, because M |= P , and M |= ¬(P → ¬Q), because
M 2 ¬P holds, when M 2 P → ¬Q holds, too, from which it follows that
M |= ¬(P → ¬Q) holds.
A wff P is satisfiable because it is true in some model, for example in the
model of previous example. In the same way, a wff ¬P is satisfiable because it
is true in such models where P is false. The wff P is not valid because there
exists models where P is true, and on the other hand, there exists models where
P is false. Thus the wff P and its negation ¬P can be simultaneously satisfiable,
but the same model cannot satisfy them. E.g. a wff ¬(P → ¬Q) is satisfiable
because M |= ¬(P → ¬Q) holds.
A wff P → P is valid because it is true in any model. This can be seen for
example by truth tables.
The following metatheorems are clear and easy to prove.
Metatheorem 2.3. (a) A wff Q follows logically from a wff P , iff P → Q is
valid. (b) Wff’s P and Q are logically equivalent, iff P ↔ Q is valid.
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Metatheorem 2.4. A wff A is valid iff ¬A is not satisfiable.
Metatheorem 2.5. A wff A is satisfiable iff ¬A is not valid.
Definability and validity can be defined also for sets of wff’s. Then we are
speaking about simultaneous definability.
Definition 2.8. Let ∆ be a set of wff’s of L and ν the valuation of a model M.
The valuation ν satisfies (simultaneously) the set ∆, if ν satisfies the every wff of
∆. The set of wff’s ∆, that is satisfied simultaneously, is satisfiable or consistent.
Example 2.5. Let ∆ = {P → Q, ¬Q, P → P } and M = {R}. Then it is not
so, that
M |= P and M |= Q,
when the wff’s of ∆ have the following truth values: P → Q := T , ¬Q := T
and P → P := T i.e. M |= P → Q, M |= ¬Q and M |= P → P .
Logical (Semantical) Consequence
In semantics one can investigate under what conditions a wff A follows from
some set of wff’s ∆. We say that ∆ imolies a wff A if A is true always, when
the every wff of ∆ is true. We define this exactly.
Definition 2.9 (Semantical Consequence). Let ∆ be a set of wff’s of L. A wff A
follows logically from the set of wff’s ∆ (∆ implies semantically A) if A is true
in all these models, where the every wff of ∆ is true. Then we write ∆ |= A.
Example 2.6. Let ∆ = {P, Q → ¬P } and M = {P, R}. Then M |= ¬Q and
M |= Q → ¬P . It can be easily see that M is the only model, that satisfies
simultaneously ∆. Because it satisfies a wff ¬Q, too, then ∆ |= ¬Q.
Example 2.7. Let ∆ = {P → ¬P, ¬(P → ¬P )}. No model does not satisfy
∆. ∆ is (semantically) contradictory (semantically inconsistent). Then it holds
for every valuation, that if it simultaneously satisfies ∆, it satisfies a wff Q, too.
Thus ∆ |= Q, or inconsistent set of wff’s imply semantically what ever wff.
Metatheorem 2.6. ∆ |= A, iff |= A. (|= A means that A is valid.)
Metatheorem 2.7. If A ∈ ∆ then ∆ |= A.
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Metatheorem 2.8. If ∆ |= A then ∆ ∪ Ω |= A.
Metatheorem 2.9. If ∆ |= A and Ω ∪ {A} |= B then ∆ ∪ Ω |= B.
Metatheorem 2.10. If ∆ |= A → B then ∆ ∪ {A} |= B.
Metatheorem 2.11. If ∆ ∪ {A} |= B then ∆ |= A → B.
We give corresponding proof-theoretical results in the subsection of Proof
Theory.
By Def. 2.9, an argument is valid if the conclusion is true in all those models
where the premises are true. An argument is not valid if there exists a model,
such that the premises are true but the conclusion false.
Example 2.8. Show that the following argument is not valid.
If a cuckoo is calling,
the spring is far advanced.
The spring is far advanced.
Thus, a cuckoo is calling.
Let us formalize the argument as follows: K ⇔ ’a cuckoo is calling’, P ⇔
’the spring is far advanced’. We have the formal inference
1. K → P pr.
2. P
pr.
3. K
cl.
(= conclusion)
We have to find such a model where the sentences 1. and 2. are true but the
sentence 3. false. We try the model M = {P }, when P := T and K := F .
Then K → P := T , P := T , but K := F i.e. just so, that the premises are true
and the conclusion false.
Truth Tables
To find out the logical nature of wff’s of L, the truth table method is often used.
We give here the main things about the truth table method due to E. Post. First
we form the truth tables for negation, implication, conjunction, disjunction and
equivalence. We call these tables basic truth tables.
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P ¬P P Q P → Q P ∧ Q P ∨ Q P ↔ Q
T F T T
T
T
T
T
F T T F
F
F
T
F
F T
T
F
T
F
F E
T
F
F
T
The basic truth tables for connectives are constructed by finding all the possible valuations for the connectives. The number of these truth distributions is
2k , where k is the number of propositional variables of a wff. All the truth tables
of L can be constructed by means of the basic truth tables.
Example 2.9. We construct the truth table of the wff (P → ¬Q) ∨ (¬Q ∧ Q).
P
T
T
F
F
Q ¬Q ¬P P → ¬Q ¬Q ∧ Q (P → ¬Q) ∨ (¬Q ∧ Q)
T F F
F
F
F
F T
F
T
F
T
T F
T
T
F
T
F T
T
T
F
T
When we determine truth values, we proceed in suc a way that first we write
under the wff’s P and Q all the truth value distributions for P and Q to the
left and form truth values of the connectives for corresponding distributions by
taking them from the basic truth tables.
Definition 2.10 (Tautology). A wff is a tautology if its truth table consists of
only the truth values T , i.e. the wff is true for all truth value distributions.
From Def. 2.10 it follows
Metatheorem 2.12. A wff of L is a tautology iff it is valid.
Example 2.10. We solve by means of truth table, whether the wff ¬(P ∧ ¬Q) →
(¬P ∨ Q) is a tautology.
P
T
T
F
F
Q ¬P ¬Q P ∧ ¬Q ¬(P ∧ ¬Q) ¬P ∨ Q ¬(P ∧ ¬Q) → (¬P ∨ Q)
T F F
F
T
T
T
F F
T
T
F
F
T
T T
F
F
T
T
T
F T
T
F
T
T
T
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Because the truth table of the wff (on the right side) consists of only the tryth
value true, the wff is a tautology and thus valid, too.
By truth tables we can find out, whether
(i) a wff, or a set of wff’s, is satisfiable, refutable or inconsistent,
(ii) a wff is valid, and
(iii) an argument is valid.
Example 2.11. We find out by truth tables , whether the wff (P → Q)∧(P ∧¬Q)
is satisfiable.
P
T
T
F
F
Q ¬Q (P → Q) P ∧ ¬Q (P → Q) ∧ (P ∧ ¬Q)
T F
T
F
F
F T
F
T
F
T F
T
F
F
F T
T
F
F
The truth table of the wff consists only of the truth value false, so it is not satisfiable but it is inconsistent.
Example 2.12. We have to find out whether the wff’s ¬(P ∨ ¬Q) and ¬P ∧ Q
are logically equivalent. This is done by testing with truth tables, whether the
condition (b) of Metatheorem 2.3 holds, i.e. whether the equivalency
|= ¬(P ∨ ¬Q) ↔ (¬P ∧ Q).
is valid. This can be seen by forming the truth table of the wff.
As we have seen before, a logical inference can be presented by enumerating
the premises and the conclusion. It can be presented also in the implication
form, where the front part of the implication consists of the conjunction of all
the premises, and the conclusion part is the conclusion of the inference. Thus the
inference P1 , P2 , . . . , Pn |= Q can be given in the form
P1 ∧ P2 ∧ . . . ∧ Pn → Q.
When we consider the validity of an inference, the implication form is used.
16
Example 2.13. We consider the validity of the following inference:
1. P → ¬Q pr.
2. ¬Q ∧ P pr.
3. ¬P ∨ Q cl.
(= conclusion)
The implication form of the inference is (P → ¬Q) ∧ (¬Q ∧ P ) → (¬P ∨ Q).
If this wff is valid then the original inference is valid, too. The wff is valid if it
is a tautology. By forming the truth table of this wff we see that this wff is not a
tautolofy, and hence not valid. Thus the original inference is not valid.
Disproof Method
Disproof method is analogous to the method of indirect proof basing on reductio
ad absurdum, and it can be applied well for checking validity in practice.
The method works as follows: Try to show the wff under consideration to be
refutable. If we find it impossible then we have shown that the wff is valid.
Example 2.14. Show that the wff (A → B ∨ C) → (A → (¬C → B)) is valid.
(1) If the wff would be refutable then it is necessary, that A → (B ∨
C) on tosi ja A → (¬C → B) is false.
(2) In order to A → (¬C → B) to be false, it is necessary, that A is
true and ¬C → B false.
(3) In order to ¬C → B to be false, it is necessary, that ¬C is true
and B false.
(4) In order to ¬C to be true, it is necessary, that C would be false.
Thus, necessarily A := T , B := F ja C := F . But in this case the wff A →
(B ∨ C) is false, which is against to the demand of (1). Thus the wff cannot
be refuted, by means of which it is true in all truth value distributions and thus
valid.
Metatheorem 2.13. If |= P and |= P → Q then |= Q.
Proof. If Q would not be a tautology, i.e. |= Q does not hold, then Q would
have the value F in some truth distribution. Because P → Q is a tautology
by supposition, then P has the value F in this truth value distribution. Thus P
would not be a autology, which is against the supposition.
17
Remark 2.4. The previous metatheorem can be interpreted in such a way that
modus ponens preserves tautology.
Problem of Solvability
The problem of solvability of L can be settled, because it has a method of solvability, i.e. L has methods by means of which one can investigate every wff
of L mechanically with finite number of steps, whether it is valid or not. E.g.
truth tables serve one solvability method. Other methods are semantical trees,
disproof method and other applications of reductio ad absurdum.
2.4
Proof-theory for L
The axiomatization of L is performed by specifying a set of axioms and inference rules. The following axiomatization is basing on the axiom system due
to Łukasiewicz,although in such a way that, according to von Neumann’s idea,
axioms are presented as axiom schemes. This system is useful especially when
one wants to prove theorems in L.
The approach, where the method of study logic consists of axiomatization and
proofs, is called proof theory. We call axiom schemes shortly axioms.
Definition 2.11 (Axioms). If P , Q, and R are any wff’s of L then the following
wff’s are axioms:
(A1)
(A2)
(A3)
P → (Q → P );
(P → (Q → R)) → ((P → Q) → (P → R));
(¬Q → ¬P ) → (P → Q).
Remark 2.5. The number of axioms is infinite, because P , Q, and R can be any
wff’s of L. Axioms (A1) − (A3) are so-called axiom schemes, as we already
noticed above.
Definition 2.12 (Inference rule). The inference rule of the axiom system is Modus
(Ponendo) Ponens, labelled M P , that says: from P and P → Q, infer Q.
The rule (MP) is often presented as an inference diagram:
18
P
P →Q
Q
(M P )
The line in the diagram is so-called inference line. The premises are above
and the conclusion beneath of it.
Definition 2.13. Let a set ∆ be wff’s of the language L (premises). A finite
sequence of wff’s S1 , . . . , Sn is a ∆-inference of Sn iff for all i, 1 ≤ i ≤ n at
least one of the following conditions holds:
(i) Si is an axiom,
(ii) Si ∈ ∆,
(iii) There exist integers j, k < i, such that Sk is Sj → Si . (Then Si is inferred
from the previous wff’s Sj and Sj → Si by M P ).
The sequence S1 , . . . , Sn is the deduction of Sn from the set ∆. Then we write
∆ ` Sn and say that Sn is deducible from the set ∆. Especially, if ∆ is finite, for
example, ∆ = {A1 , . . . , Ak }, we write
A1 , . . . , A n ` S n .
There is a clear analogy between semantical implication and the concept of
deducibility:
∆ ` A : A is deducible from the set ∆,
∆ |= A : A is a semantical consequence of the set ∆.
Definition 2.14. If a wff P is deducible from an empty set of wff’s ∆ then P is a
theorem of L, denote ` P . A finite sequence of wff’s of L S1 , . . . , Sn (Sn = P )
is the proof of P in L if it fulfills the conditions (i) and (iii) of Def. 2.13. Then
we say that P is provable in L.
Metatheorem 2.14. ` P → P .
Proof.
1. (P → ((P → P ) → P )) → ((P → (P → P )) → (P → P ))
2. P → ((P → P ) → P )
A2
A1
19
3. (P → (P → P )) → (P → P )
M P, 1, 2
4. P → (P → P )
A1
5. P → P
M P, 4, 3
Metatheorem 2.15. ` ¬P → (P → Q).
Proof.
1. (¬Q → ¬P ) → (P → Q)
A3
2. ((¬Q → ¬P ) → (P → Q)) → (¬P → ((¬Q → ¬P ) → (P → Q))) A1
3. ¬P → ((¬Q → ¬P ) → (P → Q))
M P, 1, 2
4. (¬P → ((¬Q → ¬P ) → (P → Q))) → ((¬P → (¬Q → ¬P )) →
(¬P → (P → Q)))
A2
5. (¬P → (¬Q → ¬P )) → (¬P → (P → Q))
6. ¬P → (¬Q → ¬P )
M P, 3, 4
A1
7. ¬P → (P → Q)
M P, 6, 5
Metatheorem 2.16. ¬P, P ` Q.
Proof.
1. ¬P
pr.
2. ¬P → (¬Q → ¬P )
A1
3. ¬Q → ¬P
MP,1,2
4. (¬Q → ¬P ) → (P → Q)
A3
5. P → Q
MP,3,4
6. P
pr.
7. Q
MP,6,5
Metatheorem 2.17. ¬¬P ` P .
20
Proof.
1.
2.
3.
4.
5.
6.
¬¬P
¬¬P → (¬P → ¬¬¬P )
¬P → ¬¬¬P
(¬P → ¬¬¬P ) → (¬¬P → P )
¬¬P → P
P
pr.
M etat.2.15
M P, 1, 2
A3
M P, 3, 4
M P, 1, 5
In Metatheorem 2.17 Metatheorem 2.15 has been exploited. This is allowable
because the proof of Metatheorem 2.15 can be written on the row 2.
We want to execute the deduction ∆ ` P . If P ` Q has already been deduced,
where P ∈ ∆ then in the ∆-inference we can straightly write Q as in the case
of axiom or premise. Writing instead of Q the ∆-inference of Q, we get the
∆-inference of P originally according to Def. 2.13.
Metatheorem 2.18. ` A iff ∅ ` A.
Proof. (1◦ ) Let ` A. Then there exists a proof of A B1 , . . . , Bn , such that no
premises are including in it, by Def. 2.14. Such a sequence is a deduction of A
from the empty set by the same definition. Thus, such a deduction exists. Thus
∅ ` A holds.
(2◦ ) Let ∅ ` A, i.e. there exists a deduction B1 , . . . , Bn of A from the set ∅.
Because ∅ is empty, the deduction does not have any premises, and thus it is a
proof by Def. 2.14. Hence ` A.
Metatheorem 2.19. ∆ ` A iff for some finite subset P of ∆ holds P ` A.
Proof. (1◦ ) Let ∆ ` A. Then A has a sequence B1 , . . . , Bn where Bi ∈ ∆ , 1 ≤
i ≤ n. Since this sequence is finite (Def. (2.13)), it can include only finite
number of wff’s of ∆. Let ∆◦ = ∆ ∩ {B1 , . . . , Bn }. Then the elements of ∆◦
form the deduction of A from ∆◦ , because B1 , . . . , Bn ` A i.e. ∆◦ ` A. ∆◦
is finite, because it consists of the elements B1 , . . . , Bn , whose number is finite,
because they form the deduction of A from ∆. Thus, if ∆ ` A then there exists
a finite subset P of ∆ satisfying the condition P ` A.
21
(2◦ ) Let P be finite and P ` A. In addition, let ∆ be such that it contains
P. Then there exists a deduction B1 , . . . , Bn of A from P. Because the every
element of P is an element of ∆, too, then B1 . . . Bn is also a deduction of A
from ∆ by Def. (2.13), i.e. ∆ ` A.
Consider wff’s (2.3) and (2.4). The former can be created from the latter by
first transferring P and then ¬P to the right side of the symbol ` as follows:
¬P, P ` Q
¬P ` P → Q
` ¬P → (P → Q)
The Deduction theorem tells, that in this way it can be always do. The proof of
the wff (2.4) is much easier than that of the wff (2.3). The deduction theorem
on very useful, because the proof of a theorem can be replaced with a deduction
starting from premises.
Metatheorem 2.20 (Deduction Theorem). If ∆, P ` Q then ∆ ` P → Q.
Proof. We pass the proof. However, we mention that in the proof of the deduction theorem only the axioms A1 and A2 and the inference rule M P are deeded.
Thus the deduction theorem holds, even though the third axiom would be changed.
Metatheorem 2.21 (The inverse of Deduction Theorem). If ∆ ` P → Q then
∆, P ` Q.
Proof.
1.
2.
3.
P
..
.
∆
..
.
P →Q
Q
pr.
∆-deduction of (P → Q)
M P, 1, 2
In addition, we give the following metatheorems.
22
Metatheorem 2.22. If A ∈ ∆ then ∆ ` A.
Metatheorem 2.23. If ∆ ` A then ∆ ∪ P ` A.
Metatheorem 2.24. If ∆ ` A ja P, A ` B then ∆ ∪ P ` B.
Proof. Proofs as exercises.
Metatheorem 2.25.
(a)
(b)
(c)
(d)
(e)
(f )
(g)
(h)
` P → P,
` ¬P → (P → Q),
` ¬¬P → P,
` P → ¬¬P,
` (P → Q) → ((Q → R) → (P → R)),
` (P → Q) → (¬Q → ¬P ),
` Q → (¬R → ¬(Q → R)),
` (R → P ) → ((¬R → P ) → P ).
Proof. The cases (a) and (b) are proved already in the metatheorems 2.2 and 2.3.
The case (c) can be derived from the metatheorem 2.5 by DT . The proof of (d)
is as follows:
1.
2.
3.
¬¬¬P → ¬P
(¬¬¬P → ¬P ) → (P → ¬¬P )
P → ¬¬P
Metat. 2.25(c)
A3
M P, 1, 2
(e) It suffices to show that P → Q, Q → R, P ` R by DT .
1.
2.
3.
4.
5.
P
P →Q
Q
Q→R
R
pr.
pr.
M P, 1, 2
pr.
M P, 3, 4
(f ) It suffices to show that P → Q ` ¬Q → ¬P by DT .
23
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
¬¬P → P
(¬¬P → P ) → ((P → Q) → (¬¬P → Q))
(P → Q) → (¬¬P → Q)
P →Q
¬¬P → Q
(¬¬P → Q) → ((Q → ¬¬Q) → (¬¬P → ¬¬Q))
(Q → ¬¬Q) → (¬¬P → ¬¬Q
Q → ¬¬Q
¬¬P → ¬¬Q
(¬¬P → ¬¬Q) → (¬Q → ¬P )
¬Q → ¬P
Metat. 2.25(c)
Metat. 2.25(e)
M P, 1, 2
pr.
M P, 3, 4
Metat. 2.25(e)
M P, 5, 6
Metat. 2.25(d)
M P, 8, 7
A3
M P, 9, 10
(g) It suffices to show that Q ` ¬R → ¬(Q → R) by DT .
1.
2.
3.
4.
5.
6.
Q
Q→R
R
(Q → R) → R
(Q → R) → R) → (¬R → ¬(Q → R))
¬R → ¬(Q → R)
pr.
pr.
M P, 1, 2
DT, 2, 3
Metat. 2.25(f )
M P, 4, 5
An additional supposition Q → R was made in the step 2 of the proof, and its
elimination was done in the step 4 by DT . The wff’s bounded by the additional
supposition, is written to the right from the vertical base level.
(h) Show that R → P ` (¬R → P ) → P .
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
R→P
¬P
(R → P ) → (¬P → ¬R)
¬P → ¬R
¬R
¬R → (¬P → ¬(¬R → P ))
¬P → ¬(¬R → P )
¬(¬R → P )
¬P → ¬(¬R → P )
(¬P → ¬(¬R → P )) → ((¬R → P ) → P )
(¬R → P ) → P
24
pr.
pr.
Metat. 2.25(f )
M P, 1, 3
M P, 2, 4
Metat. 2.25(g)
M P, 5, 6
M P, 2, 7
DT, 2, 8
A3
M P, 9, 10
We mentioned induction according to the length of a wff above. We explained,
how the fact can be shown that all the wff’s of L have a certain property. We consider here a special case, where we want to prove that all the theorems of L have
a certain property ϕ. In this case, it suffices to show that
(1)
(2)
ϕ(A) if A is an axiom,
if A is deduced from the wff’s B and C by M P , and if ϕ(B)
and ϕ(C) then also ϕ(A).
Natural Deduction system
Consider a certain set of wff’s, which forms a natural deduction system called
Suppes-Genzen inference rule system. Naturally, modus ponens belongs to this
system. We give these wff’s in the form of inference schenes without proofs.
This natural inference system is flexible because all the most usual connectives
exist in the inference rules.
1. Modus (Ponendo) Ponens (M P )
P →Q
P
2. Modus (Tollendo) Tollens (T T )
P →Q
¬Q
¬P
Q
3. Modus (Tollendo) Ponens (T P )
P ∨Q
¬P
4. Commutative Law (KV )
P ∧Q
Q∧P
Q
5. Commutative Law (DV )
P ∨Q
Q∨P
25
Introducing and Eliminating Rules:
6. DN I
P
7. DN E
¬¬P
¬¬P
P
8. CI
P
Q
9. CE
P ∧Q
P
P ∧Q
10. DI
P
11. DE
P ∨P
P ∨Q
Q∨P
P
12. EI
P →Q
Q→P
13. EE
P ↔Q
P →Q
Q→P
P ↔Q
DeMorgan’s Laws:
14. DL1
¬P ∧ ¬Q
15. DL2
¬(P ∨ Q)
16. DL3
¬P ∨ ¬Q
17. DL4
¬(P ∧ Q)
¬(P ∨ Q)
¬P ∧ ¬Q
¬(P ∧ Q)
¬P ∨ ¬Q
18. Hypothetical Syllogism (HS) 19. Disjunctive Syllogism (DS)
P →Q
P ∨Q
Q→R
P →R
Q→S
P →R
R∨S
20. Deduction Theorem (DT )
[P ]
Q
21. Rule of Indirect Proof (RAA)
[¬Q]
P ∧ ¬P
P →Q
Q
20. Deduction Theorem (DT ): If a wff Q can be deduced from P and from a
set of premises, the wff P ⇒ Q can be deduced only from the premises.
26
21. Rule of Indirect Proof (ES): If from premises and from a wff ¬Q can
be deduced a logical contradiction, then the wff Q can be deduced only from the
premises.
Sone Hints for Finding Deductions
1. If it is difficult to start constructing a deduction, or it sticks after some steps,
try the indirect proof: Take the negation of the wff to be deduced as a new
premise and try to deduce a contradiction, i.e. a wff of the form P ∧ ¬P . If
this succeeds then the original wff to be deduced is concluded by (RAA).
2. If the wff to be deduced is a(multiple) implication then take the first (the
condition) part(s) of the implication(s) as a new premise and try to deduce
the conclusion part(s). The implication to be deduced can be got applying
(DT ). (If the implication is multiple, apply (DT ) in a suitable order as
many times as the implication has arrows.)
Consistency
We consider some essential concepts of formal theories and combine then into
L.
Non-contradictoriness of wff’s and sets of wff’s can be considered by means
of the concept of deducibility. Examples about contradictory or inconsistent sets
of wff’s are e.g. ∆1 = {P, ¬P } and ∆2 = {P → Q, P, ¬Q}. Contradiction
can be deduced from these sets.
A set of wff’s is contradictory, or inconsistent, if and only if ∆ ` A and
∆ ` ¬A. Contradiction can be deduced from a set of wff’s, if and only if all the
wff’s can be deduced from it. Hence a set of wff’s ∆ is consistent if there is a
wff such that it is not deducible from ∆.
Definition 2.15. A set ∆ of wff’s ofL is consistent in L if it includes a wff A,
such that ∆ ` A does not include in it. A set ∆ of wff’s ofL is inconsistent if it
is not consistent.
Metatheorem 2.26. The set ∅ of wff’s of L is inconsistent iff there exists a wff A
of L such that ∆ ` A and ∆ ` ¬A.
27
Proof. (i) If ∅ is inconsistent then there are no wff’s that cannot be deduced from
it by Def. (2.15). Then there exists a wff A such that ∆ ` A and ∆ ` ¬A.
(ii) Let A be a wff of L, such that ∆ ` A and ∆ ` ¬A. Then A, ¬A ` B by
Metatheorem (2.4) where B is any wff of L.
Because ∆ ` A and ∆ ` ¬A and A, ¬A ` B, then ∆ ` B by Metatheorem
(2.12). Then ∆ is inconsistent by Def. (2.15).
We give some results concerning consistency.
Metatheorem 2.27. A set ∆ of L is consistent iff every finite subset of ∆ is
consistent.
Metatheorem 2.28. ∆ ` A iff ∆ ∪ {¬A} is inconsistent.
Metatheorem 2.28) is the base of the inference rule RAA of the natural deduction system we considered above .
Metatheorem 2.29. If ∆ is consistent then for all wff’s A of L it holds that either
∆ ∪ {A} or ∆ ∪ {¬A} is consistent.
Definition 2.16. A wff A of L is inconsistent if the set {A} is inconsistent.
Metatheorem 2.30. A wff A of L is inconsistent iff ` ¬A.
Consistency, completeness and independency of L
It can be discussed about consistency in many different meaning. We give here
four definitions for consistency.
Definition 2.17. 1. A formal language is absolutely consistent if its every wff
is not a theorem.
2. A formal language is consistent with respect to interpretation (sound), jos
every theorem of the language is a tautology.
3. A formal language is canonically consistent if P is a certain theorem, ¬P
is not theorem.
4. A formal language is consistent with respect to negation if there is not such
a wff Q in the language, such that ` Q and ` ¬Q.
28
Seuraus 2.1. If a formal language is consistent with respect to negation then it
is canonically consistent.
Metatheorem 2.31. The axiomatization of L considered above is consistent in
all meanings (1)-(4) of Def. 2.17.
Proof. 1. Show first that every theorem is a tautology, i.e. the consistency with
respect to interpretation of L. Every axiom A1, A2, and A3 is a tautology
(it can easily be checked for example by truth tables). The inference rule
M P preserves tautology property, i.e. if |= P and |= P → Q then |= Q.
We show this using indirect proof. If Q were not a tautology then it would
be false in some truth value distribution. Because P → Q is a tautology
then P would be false in this truth value distribution. This means that P
would not be a tautology, which is against the supposition. Thus, because
M P preserves tautology property then every theorem is a tautology.
2. Every wff is not a tautology, for example, a propositional variable p is not
a tautology. It cannot be a theorem, because otherwise it should be a tautology by Def. 2.17 (2).
3. Suppose, that Def. 2.17 (4) would not hold, i.e. there would exist a wff P ,
such that ` P and ` ¬P . ` ¬P → (P → Q) by Metatheorem 2.15, where
Q is any wff. Thus, in terms of our supposition, we have a deduction
1.
2.
3.
4.
5.
¬P → (P → Q)
P
¬P
P →Q
Q
lause 2.25(b)
ol.
ol.
M P, 3, 1
M P, 2, 4
Hence ` Q, if every wff would be a theorem. This is against Def. 2.17 (1).
4. The case (3) of Def. 2.17 follows from the case (4).
Completeness has also many meanings. We give here three definitions for
completeness.
Definition 2.18.
29
1. A formal language is absolutely complete, if every wff P of the language is
either a theorem, or adding it to the set of theorems causes, that every wff
of the language is a theorem.
2. A formal language is complete with respect to interpretation, if every tautology of the language is a theorem.
3. A formal language is complete with respect to negation, if for any wff of
the language it holds, that it is a theorem or its negation is a theorem.
Metatheorem 2.32. The axiomatization of L presented above is not absolutely
complete.
Proof. Consider the wff p of L. It is not a theorem, because otherwise it would
be a tautology by the proof of Metatheorem 2.31. Add p to the set of axioms. If L
were absolutely complete then, for example, the wff ¬p could be deduced from
p, i.e. p ` ¬p. Thus ` p → ¬p by Deduction theorem. Because every theorem
is a tautology by Metatheorem 2.31, then p → ¬p is tautology. However, this is
impossible because if p is true then p → ¬p is false.
L is complete with respect to interpretation.
Metatheorem 2.33 (completeness with respect to interpretation of L). Every
tautology of L is a theorem of L, i.e. if |= P then ` P .
Proof. The metatheorem has a converse theorem.
Metatheorem 2.34 (Soundness). Every theorem of L is a tautology, i.e. if ` P
then |= P .
From the metatheorems 2.33 and 2.34 the main result of propositional logic
` P ⇔|= P.
(2.21)
can be concluded.
Metatheorem 2.35 (Extended Completeness Theorem). ∆ ` P ↔ ∆ |= P ,
where ∆ is a set of wff’s.
30
Some often existing problems
1. Show that the wff P is valid (invalid).
The problem can always be solved by truth table method. It can also to
be solved by deducing P (¬P ) and then using completeness result. The
problem can be solved with disproof method, too.
2. Show that ∆ |= P .
If ∆ is finite then the problem can be solved by truth table method. Other
methods described above can be applied, too.
3. Show that ∆ ` P .
The problem can be solved deductively or by using semantical methods and
the applying the completeness of L.
4. Show that the set of wff’s Σ is inconsistent (consistent).
The problem can be solved either by deducing a wff of the form P ∧ ¬P
from the set of wff’s or by showing that no truth value distribution makes
all the wff’s of Σ true, and then applying the extended completeness result.
If Σ is finite then the logical nature of Σ can be determined by truth table
method by examining the conjunction of the wff’s of Σ.
Σ is showed to be consistent by introducing a truth value distribution, which
satisfies all the wff’s of Σ and applying extended completeness theorem. A
set of wff’s cannot be shown to be consistent by deduction.
5. Show that the set of wff’s Σ is satisfiable (refutable).
Σ is shown to be satisfiable by giving such a truth value distribution that
satisfies all the wff’s of Σ. Σ is shown to be refutable by giving a such truth
value distribution that makes at least one wff of Σ false.
Example 2.15. Show the set of wff’s
Σ = {¬(C ∨ D), B → C, C → D, ¬B}
to be consistent.
1. ¬B is true iff B is false.
31
2. B → C is true if B is false iff either C is false or true. Choose C to be
false, and examine, what follows from it.
3. C → D is true if C is false iff either D is false or true. Choose D to be
false.
4. Then also ¬(C ∨ D) is true.
Thus for example the model M = {A} is a model of Σ, and thus Σ is consistent by the extended completeness theorem. A truth value distribution, where
A is true and other propositional variables of L false, corresponds the model
M = {A}.
2.5
Method of Resolution
We illustrate here the resolution method in terms of an example.
Resolution method is used in applying formal logic to logic programming.
Logic programming is involved in automatic theorem proving.
The base of logic programming can be briefly introduced as follows:
• identifying
• search
• back-propagation
"Common denominator": ARTIFICIAL INTELLIGENCE
The procedure of resolution method is simple and its grade of "mechanizing"
is so big, that the method can be carried out effectively by computers.
Example 2.16. Observation: Salaries do not rise
Consider the economical political situation for finding the possible reason to
this. We find following things:
• If salaries or prices will rise, the inflation comes.
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• If the inflation comes, then the government must control it, or people will
suffer.
• If people will suffer, then ministers fall into disfavour.
• The government does not control inflation, and ministers do not fall into
disfavour.
Is it possible to conclude from this that SALARIES DO NOT RISE?
First we consider the problem by means of classical logic. After that we clear
up whether there may be found a more mechanical way to solve this case, and
also other similar cases.
We formalize the sentences of our observation:
P := salaries will rise
H := prices will rise
F := the inflation comes
L := government must control the inflation
K := people will suffer
M := ministers fall into disfavour
Our observation in a formal form:


1. P ∨ H → T



2. F → L ∨ K
Premises :

3. K → M



4. ¬L ∧ ¬M
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Deduction :


















5.
¬M
KE, 4
6.
7.
8.
9.
10.
¬L
¬K
¬L ∧ ¬K
¬(L ∨ K)
¬F
KE, 4
TT, 3,5
KT, 6,7
DM, 8
TT, 2,9








11. ¬(P ∨ H) TT, 1, 10





12. ¬P ∧ ¬H DM, 11




13. ¬P
KE, 12
So, the goal sentence ¬P follows logically from the premises.
Programmable inconveniences:
• A uniform presentation of the formulas is missing.
• The number of inference rules needed is big.
• Presentation form of formulas is heterogeneous.
Solution of the problem:
• Write the sentences of our observation in the so-called disjunctive form,
when the whole description can be presented in th conjunctive normal form.
• Each sentence of this kind can be presented as so-called Horn clauses or in
so-called Kowalski-form:
formula
K ∨ ¬L1 ∨ . . . ∨ ¬Ln
K (⇔ true → K)
¬L1 ∨ . . . ∨ ¬Ln
Horn clause
K1 , ¬L1 , . . . , Ln
K
¬L1 , . . . , ¬Ln
Kowalski-form
K ← L1 ∨ . . . ∨ Ln program sentence
K←
fact
← L1 , . . . , L n
goal sentence
• Exploit the formal principle of indirect proof.
We have the so-called resolution method as the result. In accordance with the
principle of indirect proof, add the negation of the original goal sentence ti the
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set of premises and examine whether the resulting set of sentences (or formulas)
is logically contradictory. If this happens, the method creates an empty formula
(=contradiction, logically false formula), write .
Logic program of the example case:
In Horn clauses:
P = {{¬P, F }, {¬H, F }, {¬F, L, K}, {¬K, M }, {¬L}, {¬M }, {P }},
In Kowalski-form:
P = {F ← P ; F ← H; L, K ← F ; M ← K; ← L; ← M ; P ←}.
The logic program P creates the following resolution refutation:
← L, M
↓
← L, K
↓
←F
↓
←P
↓
M ←K
.
L, K ← F
.
F ←P
.
P ←
.
The result is empty formula , and thus ¬P is a logical consequence of the
premises of the situation.
References
[1] Stanley N. Burris, Logic for Mathematics and Computer Science, Prentice
Hall, 1998.
[2] Winfried.K. Grassmann, Jean-Paul Tremblay Logic and Discrete Mathematics. A Computer Science Perspective, Addison-Wesley, Pearson Education Limited, 2002.
35
[3] R. Johnsonbaugh, Discrete Mathematics, Prentice Hall, 4. painos, 2001.
[4] Veikko Rantala, Ari Virtanen, Logiikkaa. Teoriaa ja sovelluksia, Matemaattisten tieteiden laitos, Tampereen yliopisto, B 43, 1995
36