Logic as a Tool Chapter 1: Understanding Propositional Logic 1.4 - PowerPoint PPT Presentation

Inductive definition (2) of the set of propositional formulae Thus, the set of propositional formulae FOR can also be defined as the least set of words in the alphabet of propositional logic such that: Goranko

Inductive definition (2) of the set of propositional formulae Thus, the set of propositional formulae FOR can also be defined as the least set of words in the alphabet of propositional logic such that: 1. Every Boolean constant is in FOR . Goranko

Inductive definition (2) of the set of propositional formulae Thus, the set of propositional formulae FOR can also be defined as the least set of words in the alphabet of propositional logic such that: 1. Every Boolean constant is in FOR . 2. Every propositional variable is in FOR . Goranko

Inductive definition (2) of the set of propositional formulae Thus, the set of propositional formulae FOR can also be defined as the least set of words in the alphabet of propositional logic such that: 1. Every Boolean constant is in FOR . 2. Every propositional variable is in FOR . 3. If A is in FOR then ¬ A is in FOR . Goranko

Inductive definition (2) of the set of propositional formulae Thus, the set of propositional formulae FOR can also be defined as the least set of words in the alphabet of propositional logic such that: 1. Every Boolean constant is in FOR . 2. Every propositional variable is in FOR . 3. If A is in FOR then ¬ A is in FOR . 4. If each of A and B is in FOR then each of ( A ∧ B ) , ( A ∨ B ) , ( A → B ) , and ( A ↔ B ) is in FOR . Goranko

Inductive definition (2) of the set of propositional formulae Thus, the set of propositional formulae FOR can also be defined as the least set of words in the alphabet of propositional logic such that: 1. Every Boolean constant is in FOR . 2. Every propositional variable is in FOR . 3. If A is in FOR then ¬ A is in FOR . 4. If each of A and B is in FOR then each of ( A ∧ B ) , ( A ∨ B ) , ( A → B ) , and ( A ↔ B ) is in FOR . This pattern of converting the inductive definition into explicit one is general and can be applied to every inductive definition. Goranko

Inductive definition (2) of the set of propositional formulae Thus, the set of propositional formulae FOR can also be defined as the least set of words in the alphabet of propositional logic such that: 1. Every Boolean constant is in FOR . 2. Every propositional variable is in FOR . 3. If A is in FOR then ¬ A is in FOR . 4. If each of A and B is in FOR then each of ( A ∧ B ) , ( A ∨ B ) , ( A → B ) , and ( A ↔ B ) is in FOR . This pattern of converting the inductive definition into explicit one is general and can be applied to every inductive definition. For correctness of the definition above, it must be proved that the set FOR defined above exists , that it is unique , and that it equals FOR ∗ . Goranko

Inductive definition (2) of the set of propositional formulae Thus, the set of propositional formulae FOR can also be defined as the least set of words in the alphabet of propositional logic such that: 1. Every Boolean constant is in FOR . 2. Every propositional variable is in FOR . 3. If A is in FOR then ¬ A is in FOR . 4. If each of A and B is in FOR then each of ( A ∧ B ) , ( A ∨ B ) , ( A → B ) , and ( A ↔ B ) is in FOR . This pattern of converting the inductive definition into explicit one is general and can be applied to every inductive definition. For correctness of the definition above, it must be proved that the set FOR defined above exists , that it is unique , and that it equals FOR ∗ . Equivalent inductive definition of FOR ∗ in a Backus-Naur normal form: A := p | ⊤ | ⊥ | ¬ A | ( A ∧ A ) | ( A ∨ A ) | ( A → A ) | ( A ↔ A ) where p ∈ PVAR . Goranko

Induction principles and proofs by induction With every inductive definition, a scheme for proofs by induction can be associated. Goranko

Induction principles and proofs by induction With every inductive definition, a scheme for proofs by induction can be associated. The construction of this scheme is uniform from the inductive definition, as illustrated below. Goranko

Induction principles and proofs by induction With every inductive definition, a scheme for proofs by induction can be associated. The construction of this scheme is uniform from the inductive definition, as illustrated below. Principle of induction on the words in an alphabet: Given an alphabet A , let P be a property of words in A such that: Goranko

Induction principles and proofs by induction With every inductive definition, a scheme for proofs by induction can be associated. The construction of this scheme is uniform from the inductive definition, as illustrated below. Principle of induction on the words in an alphabet: Given an alphabet A , let P be a property of words in A such that: 1. The empty string ǫ has the property P . Goranko

Induction principles and proofs by induction With every inductive definition, a scheme for proofs by induction can be associated. The construction of this scheme is uniform from the inductive definition, as illustrated below. Principle of induction on the words in an alphabet: Given an alphabet A , let P be a property of words in A such that: 1. The empty string ǫ has the property P . 2. If the word w in A has the property P and a ∈ A , then the word wa has the property P . Goranko

Induction principles and proofs by induction With every inductive definition, a scheme for proofs by induction can be associated. The construction of this scheme is uniform from the inductive definition, as illustrated below. Principle of induction on the words in an alphabet: Given an alphabet A , let P be a property of words in A such that: 1. The empty string ǫ has the property P . 2. If the word w in A has the property P and a ∈ A , then the word wa has the property P . Then, every word w in A has the property P . Goranko

Induction on natural numbers Goranko

Induction on natural numbers Principle of (mathematical) induction on natural numbers. Let P be a property of natural numbers such that: Goranko

Induction on natural numbers Principle of (mathematical) induction on natural numbers. Let P be a property of natural numbers such that: 1. 0 has the property P . Goranko

Induction on natural numbers Principle of (mathematical) induction on natural numbers. Let P be a property of natural numbers such that: 1. 0 has the property P . 2. For every natural number n , if n has the property P then Sn has the property P . Goranko

Induction on natural numbers Principle of (mathematical) induction on natural numbers. Let P be a property of natural numbers such that: 1. 0 has the property P . 2. For every natural number n , if n has the property P then Sn has the property P . Then, every natural number n has the property P . Goranko

Induction on natural numbers Principle of (mathematical) induction on natural numbers. Let P be a property of natural numbers such that: 1. 0 has the property P . 2. For every natural number n , if n has the property P then Sn has the property P . Then, every natural number n has the property P . Here is the same principle, stated in set-theoretic terms: Let P be a set of natural numbers such that: 1. 0 ∈ P . 2. For every natural number n , if n ∈ P then Sn ∈ P . Then, every natural number n is in P , i.e. P = N . Goranko

Structural induction on propositional formulae Goranko

Structural induction on propositional formulae The general pattern for defining a principle of induction for a set X defined by an inductive definition: replace throughout the definition “ is in the set X ” with “ satisfies the property P ”. Goranko

Structural induction on propositional formulae The general pattern for defining a principle of induction for a set X defined by an inductive definition: replace throughout the definition “ is in the set X ” with “ satisfies the property P ”. Applied to the inductive definition of propositional formulae, it produces the following principle of structural induction on propositional formulae. Goranko

Structural induction on propositional formulae The general pattern for defining a principle of induction for a set X defined by an inductive definition: replace throughout the definition “ is in the set X ” with “ satisfies the property P ”. Applied to the inductive definition of propositional formulae, it produces the following principle of structural induction on propositional formulae. Let P be a property of propositional formulae such that: Goranko

Structural induction on propositional formulae The general pattern for defining a principle of induction for a set X defined by an inductive definition: replace throughout the definition “ is in the set X ” with “ satisfies the property P ”. Applied to the inductive definition of propositional formulae, it produces the following principle of structural induction on propositional formulae. Let P be a property of propositional formulae such that: 1. Every Boolean constant satisfies the property P . Goranko

Structural induction on propositional formulae The general pattern for defining a principle of induction for a set X defined by an inductive definition: replace throughout the definition “ is in the set X ” with “ satisfies the property P ”. Applied to the inductive definition of propositional formulae, it produces the following principle of structural induction on propositional formulae. Let P be a property of propositional formulae such that: 1. Every Boolean constant satisfies the property P . 2. Every propositional variable satisfies the property P . Goranko

Structural induction on propositional formulae The general pattern for defining a principle of induction for a set X defined by an inductive definition: replace throughout the definition “ is in the set X ” with “ satisfies the property P ”. Applied to the inductive definition of propositional formulae, it produces the following principle of structural induction on propositional formulae. Let P be a property of propositional formulae such that: 1. Every Boolean constant satisfies the property P . 2. Every propositional variable satisfies the property P . 3. If A satisfies the property P then ¬ A satisfies the property P . Goranko

Structural induction on propositional formulae The general pattern for defining a principle of induction for a set X defined by an inductive definition: replace throughout the definition “ is in the set X ” with “ satisfies the property P ”. Applied to the inductive definition of propositional formulae, it produces the following principle of structural induction on propositional formulae. Let P be a property of propositional formulae such that: 1. Every Boolean constant satisfies the property P . 2. Every propositional variable satisfies the property P . 3. If A satisfies the property P then ¬ A satisfies the property P . 4. If each of A and B satisfies the property P then each of ( A ∧ B ) , ( A ∨ B ) , ( A → B ), and ( A ↔ B ) satisfies the property P . Goranko

Structural induction on propositional formulae The general pattern for defining a principle of induction for a set X defined by an inductive definition: replace throughout the definition “ is in the set X ” with “ satisfies the property P ”. Applied to the inductive definition of propositional formulae, it produces the following principle of structural induction on propositional formulae. Let P be a property of propositional formulae such that: 1. Every Boolean constant satisfies the property P . 2. Every propositional variable satisfies the property P . 3. If A satisfies the property P then ¬ A satisfies the property P . 4. If each of A and B satisfies the property P then each of ( A ∧ B ) , ( A ∨ B ) , ( A → B ), and ( A ↔ B ) satisfies the property P . Then every propositional formula satisfies the property P . Goranko

General framework for inductive definitions and principles The necessary ingredients for an inductive definition are: Goranko

General framework for inductive definitions and principles The necessary ingredients for an inductive definition are: • A universe U . In the examples, the universes are sets of words in a given alphabet. Goranko

General framework for inductive definitions and principles The necessary ingredients for an inductive definition are: • A universe U . In the examples, the universes are sets of words in a given alphabet. • A subset B ⊆ U of initial (basic) elements. In the examples, the sets of initial elements are respectively: { ǫ } ; { 0 } ; {⊤ , ⊥} ∪ PVAR . Goranko

General framework for inductive definitions and principles The necessary ingredients for an inductive definition are: • A universe U . In the examples, the universes are sets of words in a given alphabet. • A subset B ⊆ U of initial (basic) elements. In the examples, the sets of initial elements are respectively: { ǫ } ; { 0 } ; {⊤ , ⊥} ∪ PVAR . • A set F of operations (constructors) in U . Goranko

General framework for inductive definitions and principles The necessary ingredients for an inductive definition are: • A universe U . In the examples, the universes are sets of words in a given alphabet. • A subset B ⊆ U of initial (basic) elements. In the examples, the sets of initial elements are respectively: { ǫ } ; { 0 } ; {⊤ , ⊥} ∪ PVAR . • A set F of operations (constructors) in U . In the examples, these are: – the operation of appending a symbol to a word; Goranko

General framework for inductive definitions and principles The necessary ingredients for an inductive definition are: • A universe U . In the examples, the universes are sets of words in a given alphabet. • A subset B ⊆ U of initial (basic) elements. In the examples, the sets of initial elements are respectively: { ǫ } ; { 0 } ; {⊤ , ⊥} ∪ PVAR . • A set F of operations (constructors) in U . In the examples, these are: – the operation of appending a symbol to a word; – the operation of prefixing a natural number by A ; Goranko

General framework for inductive definitions and principles The necessary ingredients for an inductive definition are: • A universe U . In the examples, the universes are sets of words in a given alphabet. • A subset B ⊆ U of initial (basic) elements. In the examples, the sets of initial elements are respectively: { ǫ } ; { 0 } ; {⊤ , ⊥} ∪ PVAR . • A set F of operations (constructors) in U . In the examples, these are: – the operation of appending a symbol to a word; – the operation of prefixing a natural number by A ; – the propositional connectives regarded as operations on words. Goranko

The set inductively defined over B by applying the operations in F Goranko

The set inductively defined over B by applying the operations in F Intuitively, the set of elements of U inductively defined over B by applying the operations in F , hereby denoted by C ( B , F ), is the set defined by the following inductive definition: Goranko

The set inductively defined over B by applying the operations in F Intuitively, the set of elements of U inductively defined over B by applying the operations in F , hereby denoted by C ( B , F ), is the set defined by the following inductive definition: 1. Every element of B is in C ( B , F ). Goranko

The set inductively defined over B by applying the operations in F Intuitively, the set of elements of U inductively defined over B by applying the operations in F , hereby denoted by C ( B , F ), is the set defined by the following inductive definition: 1. Every element of B is in C ( B , F ). 2. For every operation f ∈ F , such that f : U n − → U , if every x 1 , . . . , x n is in C ( B , F ) then f ( x 1 , . . . , x n ) is in C ( B , F ). Goranko

The set inductively defined over B by applying the operations in F Intuitively, the set of elements of U inductively defined over B by applying the operations in F , hereby denoted by C ( B , F ), is the set defined by the following inductive definition: 1. Every element of B is in C ( B , F ). 2. For every operation f ∈ F , such that f : U n − → U , if every x 1 , . . . , x n is in C ( B , F ) then f ( x 1 , . . . , x n ) is in C ( B , F ). For a precise mathematical meaning and two equivalent characterisations of the set C ( B , F ), see Section 1.4 in the book. Goranko

Induction principle for the inductively defined set C ( B , F ) Proposition (Induction principle for C ( B , F )) Let P be a property of elements of U, such that: Goranko

Induction principle for the inductively defined set C ( B , F ) Proposition (Induction principle for C ( B , F )) Let P be a property of elements of U, such that: 1. Every element of B has the property P . Goranko

Induction principle for the inductively defined set C ( B , F ) Proposition (Induction principle for C ( B , F )) Let P be a property of elements of U, such that: 1. Every element of B has the property P . 2. For every operation f ∈ F , such that f : U n − → U, if every x 1 , . . . , x n has the property P then f ( x 1 , . . . , x n ) has the property P . Goranko

Induction principle for the inductively defined set C ( B , F ) Proposition (Induction principle for C ( B , F )) Let P be a property of elements of U, such that: 1. Every element of B has the property P . 2. For every operation f ∈ F , such that f : U n − → U, if every x 1 , . . . , x n has the property P then f ( x 1 , . . . , x n ) has the property P . Then every element of C ( B , F ) has the property P . Goranko

Induction principle for the inductively defined set C ( B , F ) Proposition (Induction principle for C ( B , F )) Let P be a property of elements of U, such that: 1. Every element of B has the property P . 2. For every operation f ∈ F , such that f : U n − → U, if every x 1 , . . . , x n has the property P then f ( x 1 , . . . , x n ) has the property P . Then every element of C ( B , F ) has the property P . For a proof, see Section 1.4 in the book. Goranko

Recursive definitions on inductively definable sets Given an inductively defined set C ( B , F ) in a universe U , how to define a function h on that set by using the inductive definition? Goranko

Recursive definitions on inductively definable sets Given an inductively defined set C ( B , F ) in a universe U , how to define a function h on that set by using the inductive definition? To idea: first define h on the set B , and then provide rules prescribing how the definition of that function propagates over all operations. Goranko

Recursive definitions on inductively definable sets Given an inductively defined set C ( B , F ) in a universe U , how to define a function h on that set by using the inductive definition? To idea: first define h on the set B , and then provide rules prescribing how the definition of that function propagates over all operations. Formally, in order to define by recursion a mapping h : C ( B , F ) → X where X is a fixed target set, we need: Goranko

Recursive definitions on inductively definable sets Given an inductively defined set C ( B , F ) in a universe U , how to define a function h on that set by using the inductive definition? To idea: first define h on the set B , and then provide rules prescribing how the definition of that function propagates over all operations. Formally, in order to define by recursion a mapping h : C ( B , F ) → X where X is a fixed target set, we need: 1. A mapping h 0 : B → X . Goranko

Recursive definitions on inductively definable sets Given an inductively defined set C ( B , F ) in a universe U , how to define a function h on that set by using the inductive definition? To idea: first define h on the set B , and then provide rules prescribing how the definition of that function propagates over all operations. Formally, in order to define by recursion a mapping h : C ( B , F ) → X where X is a fixed target set, we need: 1. A mapping h 0 : B → X . 2. For every n -ary operation f ∈ F a mapping F f : U n × X n → X . Goranko

Recursive definitions on inductively definable sets Given an inductively defined set C ( B , F ) in a universe U , how to define a function h on that set by using the inductive definition? To idea: first define h on the set B , and then provide rules prescribing how the definition of that function propagates over all operations. Formally, in order to define by recursion a mapping h : C ( B , F ) → X where X is a fixed target set, we need: 1. A mapping h 0 : B → X . 2. For every n -ary operation f ∈ F a mapping F f : U n × X n → X . Now the mapping h is defined as follows: Goranko

Recursive definitions on inductively definable sets Given an inductively defined set C ( B , F ) in a universe U , how to define a function h on that set by using the inductive definition? To idea: first define h on the set B , and then provide rules prescribing how the definition of that function propagates over all operations. Formally, in order to define by recursion a mapping h : C ( B , F ) → X where X is a fixed target set, we need: 1. A mapping h 0 : B → X . 2. For every n -ary operation f ∈ F a mapping F f : U n × X n → X . Now the mapping h is defined as follows: 1. If a ∈ B then h ( a ) := h 0 ( a ). Goranko

Recursive definitions on inductively definable sets Given an inductively defined set C ( B , F ) in a universe U , how to define a function h on that set by using the inductive definition? To idea: first define h on the set B , and then provide rules prescribing how the definition of that function propagates over all operations. Formally, in order to define by recursion a mapping h : C ( B , F ) → X where X is a fixed target set, we need: 1. A mapping h 0 : B → X . 2. For every n -ary operation f ∈ F a mapping F f : U n × X n → X . Now the mapping h is defined as follows: 1. If a ∈ B then h ( a ) := h 0 ( a ). 2. For every n -ary operation f ∈ F : h ( f ( a 1 , . . . , a n )) := F f ( a 1 , . . . , a n , h ( a 1 ) , . . . , h ( a n )) . Goranko

Recursive definitions on inductively definable sets Given an inductively defined set C ( B , F ) in a universe U , how to define a function h on that set by using the inductive definition? To idea: first define h on the set B , and then provide rules prescribing how the definition of that function propagates over all operations. Formally, in order to define by recursion a mapping h : C ( B , F ) → X where X is a fixed target set, we need: 1. A mapping h 0 : B → X . 2. For every n -ary operation f ∈ F a mapping F f : U n × X n → X . Now the mapping h is defined as follows: 1. If a ∈ B then h ( a ) := h 0 ( a ). 2. For every n -ary operation f ∈ F : h ( f ( a 1 , . . . , a n )) := F f ( a 1 , . . . , a n , h ( a 1 ) , . . . , h ( a n )) . Sometimes the mappings F f only take as arguments h ( a 1 ) , . . . , h ( a n ). Goranko

Example: primitive recursion on natural numbers Goranko

Example: primitive recursion on natural numbers As a particular case, functions on natural numbers can be defined by primitive recursion, using the inductive definition given earlier. Goranko

Example: primitive recursion on natural numbers As a particular case, functions on natural numbers can be defined by primitive recursion, using the inductive definition given earlier. 1. Basic scheme of primitive recursion: h (0) = a; h ( n + 1) = h ( Sn ) = F S ( n , h ( n )) Goranko

Example: primitive recursion on natural numbers As a particular case, functions on natural numbers can be defined by primitive recursion, using the inductive definition given earlier. 1. Basic scheme of primitive recursion: h (0) = a; h ( n + 1) = h ( Sn ) = F S ( n , h ( n )) For example, the scheme h (0) = 1 ; h ( Sn ) = ( n + 1) h ( n ) defines the factorial function : h ( n ) = n !. Goranko

Example: primitive recursion on natural numbers As a particular case, functions on natural numbers can be defined by primitive recursion, using the inductive definition given earlier. 1. Basic scheme of primitive recursion: h (0) = a; h ( n + 1) = h ( Sn ) = F S ( n , h ( n )) For example, the scheme h (0) = 1 ; h ( Sn ) = ( n + 1) h ( n ) defines the factorial function : h ( n ) = n !. 2. General scheme of primitive recursion with parameters: h ( m , 0) = F 0 ( m ) ; h ( m , Sn ) = F S ( m , n , h ( m , n )) . Goranko

Example: primitive recursion on natural numbers As a particular case, functions on natural numbers can be defined by primitive recursion, using the inductive definition given earlier. 1. Basic scheme of primitive recursion: h (0) = a; h ( n + 1) = h ( Sn ) = F S ( n , h ( n )) For example, the scheme h (0) = 1 ; h ( Sn ) = ( n + 1) h ( n ) defines the factorial function : h ( n ) = n !. 2. General scheme of primitive recursion with parameters: h ( m , 0) = F 0 ( m ) ; h ( m , Sn ) = F S ( m , n , h ( m , n )) . For example, the scheme h ( m , 0) = m; h ( m , n + 1) = h ( m , n ) + 1 defines the function addition : h ( m , n ) = m + n . Goranko

Example: truth valuations of propositional formulae Recall that the set FOR is built on a set of propositional variables PVAR and a truth assignment is a mapping s : PVAR → { T , F } . Goranko

Example: truth valuations of propositional formulae Recall that the set FOR is built on a set of propositional variables PVAR and a truth assignment is a mapping s : PVAR → { T , F } . Now, given any truth assignment s : PVAR → { T , F } we define a mapping α : FOR → { T , F } that extends it to a truth valuation – a function computing the truth values of all formulae in FOR , defined by recursion on the inductive definition of FOR as follows: Goranko

Example: truth valuations of propositional formulae Recall that the set FOR is built on a set of propositional variables PVAR and a truth assignment is a mapping s : PVAR → { T , F } . Now, given any truth assignment s : PVAR → { T , F } we define a mapping α : FOR → { T , F } that extends it to a truth valuation – a function computing the truth values of all formulae in FOR , defined by recursion on the inductive definition of FOR as follows: 1. α ( ⊤ ) = T , α ( ⊥ ) = F . Goranko

Example: truth valuations of propositional formulae Recall that the set FOR is built on a set of propositional variables PVAR and a truth assignment is a mapping s : PVAR → { T , F } . Now, given any truth assignment s : PVAR → { T , F } we define a mapping α : FOR → { T , F } that extends it to a truth valuation – a function computing the truth values of all formulae in FOR , defined by recursion on the inductive definition of FOR as follows: 1. α ( ⊤ ) = T , α ( ⊥ ) = F . 2. α ( p ) = s ( p ) for every propositional variable p . Goranko

Example: truth valuations of propositional formulae Recall that the set FOR is built on a set of propositional variables PVAR and a truth assignment is a mapping s : PVAR → { T , F } . Now, given any truth assignment s : PVAR → { T , F } we define a mapping α : FOR → { T , F } that extends it to a truth valuation – a function computing the truth values of all formulae in FOR , defined by recursion on the inductive definition of FOR as follows: 1. α ( ⊤ ) = T , α ( ⊥ ) = F . 2. α ( p ) = s ( p ) for every propositional variable p . 3. α ( ¬ A ) = F ¬ ( α ( A )), where F ¬ : { T , F } → { T , F } is defined as follows: F ¬ ( T ) = F , F ¬ ( F ) = T . Goranko

Example: truth valuations of propositional formulae Recall that the set FOR is built on a set of propositional variables PVAR and a truth assignment is a mapping s : PVAR → { T , F } . Now, given any truth assignment s : PVAR → { T , F } we define a mapping α : FOR → { T , F } that extends it to a truth valuation – a function computing the truth values of all formulae in FOR , defined by recursion on the inductive definition of FOR as follows: 1. α ( ⊤ ) = T , α ( ⊥ ) = F . 2. α ( p ) = s ( p ) for every propositional variable p . 3. α ( ¬ A ) = F ¬ ( α ( A )), where F ¬ : { T , F } → { T , F } is defined as follows: F ¬ ( T ) = F , F ¬ ( F ) = T . 4. α ( A ∧ B ) = F ∧ ( α ( A ) , α ( B )), where F ¬ : { T , F } 2 → { T , F } is defined as follows: F ∧ ( T , T ) = T and F ∧ ( T , F ) = F ∧ ( F , T ) = F ∧ ( F , F ) = F . (That is, F ∧ computes the truth table of ∧ .) Goranko

Truth valuations of propositional formulae, continued Goranko

Truth valuations of propositional formulae, continued 1. α ( A ∨ B ) = F ∨ ( α ( A ) , α ( B )), where F ∨ : { T , F } 2 → { T , F } is defined respectively as follows: F ∧ ( T , T ) = F ∧ ( T , F ) = F ∧ ( F , T ) = T and F ∧ ( F , F ) = F . Goranko

Truth valuations of propositional formulae, continued 1. α ( A ∨ B ) = F ∨ ( α ( A ) , α ( B )), where F ∨ : { T , F } 2 → { T , F } is defined respectively as follows: F ∧ ( T , T ) = F ∧ ( T , F ) = F ∧ ( F , T ) = T and F ∧ ( F , F ) = F . 2. α ( A → B ) = F → ( α ( A ) , α ( B )), where F → : { T , F } 2 → { T , F } is defined according to the truth table of → . Goranko

Truth valuations of propositional formulae, continued 1. α ( A ∨ B ) = F ∨ ( α ( A ) , α ( B )), where F ∨ : { T , F } 2 → { T , F } is defined respectively as follows: F ∧ ( T , T ) = F ∧ ( T , F ) = F ∧ ( F , T ) = T and F ∧ ( F , F ) = F . 2. α ( A → B ) = F → ( α ( A ) , α ( B )), where F → : { T , F } 2 → { T , F } is defined according to the truth table of → . 3. α ( A ↔ B ) = F ↔ ( α ( A ) , α ( B )), where F ↔ : { T , F } 2 → { T , F } is defined according to the truth table of ↔ . Goranko

Truth valuations of propositional formulae, continued 1. α ( A ∨ B ) = F ∨ ( α ( A ) , α ( B )), where F ∨ : { T , F } 2 → { T , F } is defined respectively as follows: F ∧ ( T , T ) = F ∧ ( T , F ) = F ∧ ( F , T ) = T and F ∧ ( F , F ) = F . 2. α ( A → B ) = F → ( α ( A ) , α ( B )), where F → : { T , F } 2 → { T , F } is defined according to the truth table of → . 3. α ( A ↔ B ) = F ↔ ( α ( A ) , α ( B )), where F ↔ : { T , F } 2 → { T , F } is defined according to the truth table of ↔ . The so defined mapping α is called the truth-valuation of the propositional formulae generated by the truth assignment s . Goranko

Truth valuations of propositional formulae, continued 1. α ( A ∨ B ) = F ∨ ( α ( A ) , α ( B )), where F ∨ : { T , F } 2 → { T , F } is defined respectively as follows: F ∧ ( T , T ) = F ∧ ( T , F ) = F ∧ ( F , T ) = T and F ∧ ( F , F ) = F . 2. α ( A → B ) = F → ( α ( A ) , α ( B )), where F → : { T , F } 2 → { T , F } is defined according to the truth table of → . 3. α ( A ↔ B ) = F ↔ ( α ( A ) , α ( B )), where F ↔ : { T , F } 2 → { T , F } is defined according to the truth table of ↔ . The so defined mapping α is called the truth-valuation of the propositional formulae generated by the truth assignment s . Using such recursive definitions, various other natural functions associated with propositional formulae can be defined likewise, such as length, number of occurrences of logical connectives, set of occurring propositional variables, etc. Goranko