Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
PROPOSITIONAL LOGIC Jorma K. Mattila LUT, Department of Mathematics and Physics 1 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. 1 (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. 2 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. 3 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; 4 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. 2 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: 5 (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 . 6 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 , 7 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 8 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. 9 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 . 10 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. 11 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. 12 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. 13 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. 14 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 15 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. 32 • 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 33 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 34 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