Logic as a Tool Chapter 3: Understanding First-order Logic 3.3 - PowerPoint PPT Presentation

The sets of bound and free variables in a formula Definition (The set of free variables in a formula) Let L be any first-order language. The mapping FVAR : FOR ( L ) → P ( VAR ) ( where P ( VAR ) is the powerset of VAR ) is defined recursively on the structure of formulae in L as follows. Goranko

The sets of bound and free variables in a formula Definition (The set of free variables in a formula) Let L be any first-order language. The mapping FVAR : FOR ( L ) → P ( VAR ) ( where P ( VAR ) is the powerset of VAR ) is defined recursively on the structure of formulae in L as follows. 1. For every atomic formula A = p ( t 1 , ..., t n ), FVAR ( A ) := VAR ( t 1 ) ∪ . . . ∪ VAR ( t n ). Goranko

The sets of bound and free variables in a formula Definition (The set of free variables in a formula) Let L be any first-order language. The mapping FVAR : FOR ( L ) → P ( VAR ) ( where P ( VAR ) is the powerset of VAR ) is defined recursively on the structure of formulae in L as follows. 1. For every atomic formula A = p ( t 1 , ..., t n ), FVAR ( A ) := VAR ( t 1 ) ∪ . . . ∪ VAR ( t n ). 2. If A is a formula in L then FVAR ( ¬ A ) := FVAR ( A ). Goranko

The sets of bound and free variables in a formula Definition (The set of free variables in a formula) Let L be any first-order language. The mapping FVAR : FOR ( L ) → P ( VAR ) ( where P ( VAR ) is the powerset of VAR ) is defined recursively on the structure of formulae in L as follows. 1. For every atomic formula A = p ( t 1 , ..., t n ), FVAR ( A ) := VAR ( t 1 ) ∪ . . . ∪ VAR ( t n ). 2. If A is a formula in L then FVAR ( ¬ A ) := FVAR ( A ). 3. If A and B are formulae in L and • ∈ {∧ , ∨ , → , ↔} then FVAR ( A • B ) := FVAR ( A ) ∪ FVAR ( B ) Goranko

The sets of bound and free variables in a formula Definition (The set of free variables in a formula) Let L be any first-order language. The mapping FVAR : FOR ( L ) → P ( VAR ) ( where P ( VAR ) is the powerset of VAR ) is defined recursively on the structure of formulae in L as follows. 1. For every atomic formula A = p ( t 1 , ..., t n ), FVAR ( A ) := VAR ( t 1 ) ∪ . . . ∪ VAR ( t n ). 2. If A is a formula in L then FVAR ( ¬ A ) := FVAR ( A ). 3. If A and B are formulae in L and • ∈ {∧ , ∨ , → , ↔} then FVAR ( A • B ) := FVAR ( A ) ∪ FVAR ( B ) 4. If A is a formula in L and x is an individual variable, then FVAR ( ∀ xA ) = FVAR ( ∃ xA ) = FVAR ( A ) \ { x } . Goranko

The sets of bound and free variables in a formula Definition (The set of free variables in a formula) Let L be any first-order language. The mapping FVAR : FOR ( L ) → P ( VAR ) ( where P ( VAR ) is the powerset of VAR ) is defined recursively on the structure of formulae in L as follows. 1. For every atomic formula A = p ( t 1 , ..., t n ), FVAR ( A ) := VAR ( t 1 ) ∪ . . . ∪ VAR ( t n ). 2. If A is a formula in L then FVAR ( ¬ A ) := FVAR ( A ). 3. If A and B are formulae in L and • ∈ {∧ , ∨ , → , ↔} then FVAR ( A • B ) := FVAR ( A ) ∪ FVAR ( B ) 4. If A is a formula in L and x is an individual variable, then FVAR ( ∀ xA ) = FVAR ( ∃ xA ) = FVAR ( A ) \ { x } . Exercise: define similarly the set of bound variables BVAR ( A ) for every formula A . Goranko

Open and closed formulae Goranko

Open and closed formulae Theorem The truth of a formula A in a given structure S only depends on the assignment of values to the free variables occurring in A. Goranko

Open and closed formulae Theorem The truth of a formula A in a given structure S only depends on the assignment of values to the free variables occurring in A. That is, if v 1 , v 2 are variable assignments in S such that v 1 | FVAR ( A ) = v 2 | FVAR ( A ) , then S , v 1 | = A iff S , v 2 | = A . Goranko

Open and closed formulae Theorem The truth of a formula A in a given structure S only depends on the assignment of values to the free variables occurring in A. That is, if v 1 , v 2 are variable assignments in S such that v 1 | FVAR ( A ) = v 2 | FVAR ( A ) , then S , v 1 | = A iff S , v 2 | = A . Proof: by structural induction of formulae, using the recursive definition of FVAR ( A ). Exercise. Goranko

Open and closed formulae Theorem The truth of a formula A in a given structure S only depends on the assignment of values to the free variables occurring in A. That is, if v 1 , v 2 are variable assignments in S such that v 1 | FVAR ( A ) = v 2 | FVAR ( A ) , then S , v 1 | = A iff S , v 2 | = A . Proof: by structural induction of formulae, using the recursive definition of FVAR ( A ). Exercise. An open formula is a formula with no bound variables, i.e. not containing any occurrences of quantifiers. Goranko

Open and closed formulae Theorem The truth of a formula A in a given structure S only depends on the assignment of values to the free variables occurring in A. That is, if v 1 , v 2 are variable assignments in S such that v 1 | FVAR ( A ) = v 2 | FVAR ( A ) , then S , v 1 | = A iff S , v 2 | = A . Proof: by structural induction of formulae, using the recursive definition of FVAR ( A ). Exercise. An open formula is a formula with no bound variables, i.e. not containing any occurrences of quantifiers. A closed formula, or a sentence is a formula with no free variables. Goranko

Open and closed formulae Theorem The truth of a formula A in a given structure S only depends on the assignment of values to the free variables occurring in A. That is, if v 1 , v 2 are variable assignments in S such that v 1 | FVAR ( A ) = v 2 | FVAR ( A ) , then S , v 1 | = A iff S , v 2 | = A . Proof: by structural induction of formulae, using the recursive definition of FVAR ( A ). Exercise. An open formula is a formula with no bound variables, i.e. not containing any occurrences of quantifiers. A closed formula, or a sentence is a formula with no free variables. Thus, the truth of a sentence in a structure does not depend on the variable assignment. Goranko

Using bound and free variables in a formula Goranko

Using bound and free variables in a formula Free variables have their own values in a given formula (determined by a variable assignment) Goranko

Using bound and free variables in a formula Free variables have their own values in a given formula (determined by a variable assignment), while bound variables only play a dummy role and can be replaced (with care!) by one another. Goranko

Using bound and free variables in a formula Free variables have their own values in a given formula (determined by a variable assignment), while bound variables only play a dummy role and can be replaced (with care!) by one another. ∃ x ( 5 < x ∧ x 2 + x − 2 = 0 ) For instance, the sentence ∃ y ( 5 < y ∧ y 2 + y − 2 = 0 ), means exactly the same as in the sense that both formulae always have the same truth value. Goranko

Using bound and free variables in a formula Free variables have their own values in a given formula (determined by a variable assignment), while bound variables only play a dummy role and can be replaced (with care!) by one another. ∃ x ( 5 < x ∧ x 2 + x − 2 = 0 ) For instance, the sentence ∃ y ( 5 < y ∧ y 2 + y − 2 = 0 ), means exactly the same as in the sense that both formulae always have the same truth value. ∀ x ( 5 < x ∨ x 2 + x − 2 = 0 ) Likewise, means the same as ∀ y ( 5 < y ∨ y 2 + y − 2 = 0 ). Goranko

Using bound and free variables in a formula Free variables have their own values in a given formula (determined by a variable assignment), while bound variables only play a dummy role and can be replaced (with care!) by one another. ∃ x ( 5 < x ∧ x 2 + x − 2 = 0 ) For instance, the sentence ∃ y ( 5 < y ∧ y 2 + y − 2 = 0 ), means exactly the same as in the sense that both formulae always have the same truth value. ∀ x ( 5 < x ∨ x 2 + x − 2 = 0 ) Likewise, means the same as ∀ y ( 5 < y ∨ y 2 + y − 2 = 0 ). On the other hand, the meaning of 5 < x ∧ x 2 + x − 2 = 0 is essentially different from the meaning of 5 < y ∧ y 2 + y − 2 = 0 . Goranko

Using bound and free variables in a formula Free variables have their own values in a given formula (determined by a variable assignment), while bound variables only play a dummy role and can be replaced (with care!) by one another. ∃ x ( 5 < x ∧ x 2 + x − 2 = 0 ) For instance, the sentence ∃ y ( 5 < y ∧ y 2 + y − 2 = 0 ), means exactly the same as in the sense that both formulae always have the same truth value. ∀ x ( 5 < x ∨ x 2 + x − 2 = 0 ) Likewise, means the same as ∀ y ( 5 < y ∨ y 2 + y − 2 = 0 ). On the other hand, the meaning of 5 < x ∧ x 2 + x − 2 = 0 is essentially different from the meaning of 5 < y ∧ y 2 + y − 2 = 0 . Both formulae state the same, but about different individuals. Goranko

Reusing variables as free and bound in a formula Goranko

Reusing variables as free and bound in a formula The same variable can occur both free and bound in a formula: x > 5 → ∀ x ( 2 x > x ) . Goranko

Reusing variables as free and bound in a formula The same variable can occur both free and bound in a formula: x > 5 → ∀ x ( 2 x > x ) . However, the free occurrence of x is unrelated to the bound occurrences of x : x > 5 → ∀ x ( 2 x > x ) . Goranko

Reusing variables as free and bound in a formula The same variable can occur both free and bound in a formula: x > 5 → ∀ x ( 2 x > x ) . However, the free occurrence of x is unrelated to the bound occurrences of x : x > 5 → ∀ x ( 2 x > x ) . Thus, the formula above has the same meaning as x > 5 → ∀ y ( 2 y > y ) , Goranko

Reusing variables as free and bound in a formula The same variable can occur both free and bound in a formula: x > 5 → ∀ x ( 2 x > x ) . However, the free occurrence of x is unrelated to the bound occurrences of x : x > 5 → ∀ x ( 2 x > x ) . Thus, the formula above has the same meaning as x > 5 → ∀ y ( 2 y > y ) , but not the same meaning as y > 5 → ∀ x ( 2 x > x ) . Goranko

Binding a variable by different quantifiers in a formula Goranko

Binding a variable by different quantifiers in a formula Different occurrences of the same variable can be bound by different quantifiers: ∃ x ( x > 5 ) ∨ ∀ x ( 2 x > x ) . Goranko

Binding a variable by different quantifiers in a formula Different occurrences of the same variable can be bound by different quantifiers: ∃ x ( x > 5 ) ∨ ∀ x ( 2 x > x ) . Again, the occurrences of x , bound by the first quantifier, have nothing to do with those bound by the second one. Goranko

Binding a variable by different quantifiers in a formula Different occurrences of the same variable can be bound by different quantifiers: ∃ x ( x > 5 ) ∨ ∀ x ( 2 x > x ) . Again, the occurrences of x , bound by the first quantifier, have nothing to do with those bound by the second one. For instance, the two x ’s claimed to exist in the formula ∃ x ( x > 5 ) ∧ ∃ x ( x < 3 ) . need not (and, in fact, cannot) be the same. Goranko

Binding a variable by different quantifiers in a formula Different occurrences of the same variable can be bound by different quantifiers: ∃ x ( x > 5 ) ∨ ∀ x ( 2 x > x ) . Again, the occurrences of x , bound by the first quantifier, have nothing to do with those bound by the second one. For instance, the two x ’s claimed to exist in the formula ∃ x ( x > 5 ) ∧ ∃ x ( x < 3 ) . need not (and, in fact, cannot) be the same. Thus, the formula above has the same meaning as each of ∃ y ( y > 5 ) ∧ ∃ x ( x < 3 ) , Goranko

Binding a variable by different quantifiers in a formula Different occurrences of the same variable can be bound by different quantifiers: ∃ x ( x > 5 ) ∨ ∀ x ( 2 x > x ) . Again, the occurrences of x , bound by the first quantifier, have nothing to do with those bound by the second one. For instance, the two x ’s claimed to exist in the formula ∃ x ( x > 5 ) ∧ ∃ x ( x < 3 ) . need not (and, in fact, cannot) be the same. Thus, the formula above has the same meaning as each of ∃ y ( y > 5 ) ∧ ∃ x ( x < 3 ) , ∃ x ( x > 5 ) ∧ ∃ z ( z < 3 ) , Goranko

Binding a variable by different quantifiers in a formula Different occurrences of the same variable can be bound by different quantifiers: ∃ x ( x > 5 ) ∨ ∀ x ( 2 x > x ) . Again, the occurrences of x , bound by the first quantifier, have nothing to do with those bound by the second one. For instance, the two x ’s claimed to exist in the formula ∃ x ( x > 5 ) ∧ ∃ x ( x < 3 ) . need not (and, in fact, cannot) be the same. Thus, the formula above has the same meaning as each of ∃ y ( y > 5 ) ∧ ∃ x ( x < 3 ) , ∃ x ( x > 5 ) ∧ ∃ z ( z < 3 ) , ∃ y ( y > 5 ) ∧ ∃ z ( z < 3 ). Goranko

Nested bindings of a variable in a formula Goranko

Nested bindings of a variable in a formula Different bindings of the same variable can be nested, e.g.: ∀ x ( x > 5 → ∃ x ( x < 3 )) . Goranko

Nested bindings of a variable in a formula Different bindings of the same variable can be nested, e.g.: ∀ x ( x > 5 → ∃ x ( x < 3 )) . Again, the occurrences of x in the subformula ∃ x ( x < 3 ) are bound by ∃ and not related to the first two occurrences of x , bound by ∀ : ∀ x ( x > 5 → ∃ x ( x < 3 )) . Goranko

Nested bindings of a variable in a formula Different bindings of the same variable can be nested, e.g.: ∀ x ( x > 5 → ∃ x ( x < 3 )) . Again, the occurrences of x in the subformula ∃ x ( x < 3 ) are bound by ∃ and not related to the first two occurrences of x , bound by ∀ : ∀ x ( x > 5 → ∃ x ( x < 3 )) . Thus, the formula above has the same meaning as each of ∀ x ( x > 5 → ∃ y ( y < 3 )) , Goranko

Nested bindings of a variable in a formula Different bindings of the same variable can be nested, e.g.: ∀ x ( x > 5 → ∃ x ( x < 3 )) . Again, the occurrences of x in the subformula ∃ x ( x < 3 ) are bound by ∃ and not related to the first two occurrences of x , bound by ∀ : ∀ x ( x > 5 → ∃ x ( x < 3 )) . Thus, the formula above has the same meaning as each of ∀ x ( x > 5 → ∃ y ( y < 3 )) , ∀ z ( z > 5 → ∃ x ( x < 3 )) , Goranko

Nested bindings of a variable in a formula Different bindings of the same variable can be nested, e.g.: ∀ x ( x > 5 → ∃ x ( x < 3 )) . Again, the occurrences of x in the subformula ∃ x ( x < 3 ) are bound by ∃ and not related to the first two occurrences of x , bound by ∀ : ∀ x ( x > 5 → ∃ x ( x < 3 )) . Thus, the formula above has the same meaning as each of ∀ x ( x > 5 → ∃ y ( y < 3 )) , ∀ z ( z > 5 → ∃ x ( x < 3 )) , ∀ z ( z > 5 → ∃ y ( y < 3 )). Goranko

Renaming of a bound variable in a formula Goranko

Renaming of a bound variable in a formula Using the same variable for different purposes in a formula can be confusing, and is often unwanted, so we may want to eliminate it. Goranko

Renaming of a bound variable in a formula Using the same variable for different purposes in a formula can be confusing, and is often unwanted, so we may want to eliminate it. Renaming of the variable x in a formula A is the substitution of all occurrences of x bound by the same occurrence of a quantifier in A with another variable, not occurring in A . Goranko

Renaming of a bound variable in a formula Using the same variable for different purposes in a formula can be confusing, and is often unwanted, so we may want to eliminate it. Renaming of the variable x in a formula A is the substitution of all occurrences of x bound by the same occurrence of a quantifier in A with another variable, not occurring in A . E.g., a possible renaming of ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ x ( x < y )) Goranko

Renaming of a bound variable in a formula Using the same variable for different purposes in a formula can be confusing, and is often unwanted, so we may want to eliminate it. Renaming of the variable x in a formula A is the substitution of all occurrences of x bound by the same occurrence of a quantifier in A with another variable, not occurring in A . E.g., a possible renaming of ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ x ( x < y )) is the formula ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ z ( z < y )) Goranko

Renaming of a bound variable in a formula Using the same variable for different purposes in a formula can be confusing, and is often unwanted, so we may want to eliminate it. Renaming of the variable x in a formula A is the substitution of all occurrences of x bound by the same occurrence of a quantifier in A with another variable, not occurring in A . E.g., a possible renaming of ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ x ( x < y )) is the formula ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ z ( z < y )) However, neither of the following formulae is a correct renaming: Goranko

Renaming of a bound variable in a formula Using the same variable for different purposes in a formula can be confusing, and is often unwanted, so we may want to eliminate it. Renaming of the variable x in a formula A is the substitution of all occurrences of x bound by the same occurrence of a quantifier in A with another variable, not occurring in A . E.g., a possible renaming of ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ x ( x < y )) is the formula ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ z ( z < y )) However, neither of the following formulae is a correct renaming: ( z > 5 ) ∧ ∀ x (( x > 5 ) → ¬∃ x ( x < y )) , Goranko

Renaming of a bound variable in a formula Using the same variable for different purposes in a formula can be confusing, and is often unwanted, so we may want to eliminate it. Renaming of the variable x in a formula A is the substitution of all occurrences of x bound by the same occurrence of a quantifier in A with another variable, not occurring in A . E.g., a possible renaming of ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ x ( x < y )) is the formula ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ z ( z < y )) However, neither of the following formulae is a correct renaming: ( z > 5 ) ∧ ∀ x (( x > 5 ) → ¬∃ x ( x < y )) , ( x > 5 ) ∧ ∀ z (( z > 5 ) → ¬∃ z ( z < y )) , Goranko

Renaming of a bound variable in a formula Using the same variable for different purposes in a formula can be confusing, and is often unwanted, so we may want to eliminate it. Renaming of the variable x in a formula A is the substitution of all occurrences of x bound by the same occurrence of a quantifier in A with another variable, not occurring in A . E.g., a possible renaming of ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ x ( x < y )) is the formula ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ z ( z < y )) However, neither of the following formulae is a correct renaming: ( z > 5 ) ∧ ∀ x (( x > 5 ) → ¬∃ x ( x < y )) , ( x > 5 ) ∧ ∀ z (( z > 5 ) → ¬∃ z ( z < y )) , ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ y ( y < y )) . Goranko

Renaming of a bound variable in a formula Using the same variable for different purposes in a formula can be confusing, and is often unwanted, so we may want to eliminate it. Renaming of the variable x in a formula A is the substitution of all occurrences of x bound by the same occurrence of a quantifier in A with another variable, not occurring in A . E.g., a possible renaming of ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ x ( x < y )) is the formula ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ z ( z < y )) However, neither of the following formulae is a correct renaming: ( z > 5 ) ∧ ∀ x (( x > 5 ) → ¬∃ x ( x < y )) , ( x > 5 ) ∧ ∀ z (( z > 5 ) → ¬∃ z ( z < y )) , ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ y ( y < y )) . Claim: The result of renaming a variable in a formula A always has the same truth value as A Goranko

Renaming of a bound variable in a formula Using the same variable for different purposes in a formula can be confusing, and is often unwanted, so we may want to eliminate it. Renaming of the variable x in a formula A is the substitution of all occurrences of x bound by the same occurrence of a quantifier in A with another variable, not occurring in A . E.g., a possible renaming of ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ x ( x < y )) is the formula ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ z ( z < y )) However, neither of the following formulae is a correct renaming: ( z > 5 ) ∧ ∀ x (( x > 5 ) → ¬∃ x ( x < y )) , ( x > 5 ) ∧ ∀ z (( z > 5 ) → ¬∃ z ( z < y )) , ( x > 5 ) ∧ ∀ x ( x > 5 → ¬∃ y ( y < y )) . Claim: The result of renaming a variable in a formula A always has the same truth value as A , so it is logically equivalent to A . Goranko

Clean formulae Goranko

Clean formulae A formula A is clean if no variable occurs both free and bound in A and every two occurrences of quantifiers bind different variables. Goranko

Clean formulae A formula A is clean if no variable occurs both free and bound in A and every two occurrences of quantifiers bind different variables. Thus, ∃ x ( x > 5 ) ∧ ∃ y ( y < z ) is clean, Goranko

Clean formulae A formula A is clean if no variable occurs both free and bound in A and every two occurrences of quantifiers bind different variables. Thus, ∃ x ( x > 5 ) ∧ ∃ y ( y < z ) is clean, while ∃ x ( x > 5 ) ∧ ∃ y ( y < x ) Goranko

Clean formulae A formula A is clean if no variable occurs both free and bound in A and every two occurrences of quantifiers bind different variables. Thus, ∃ x ( x > 5 ) ∧ ∃ y ( y < z ) is clean, while ∃ x ( x > 5 ) ∧ ∃ y ( y < x ) and ∃ x ( x > 5 ) ∧ ∃ x ( y < x ) are not. Goranko

Clean formulae A formula A is clean if no variable occurs both free and bound in A and every two occurrences of quantifiers bind different variables. Thus, ∃ x ( x > 5 ) ∧ ∃ y ( y < z ) is clean, while ∃ x ( x > 5 ) ∧ ∃ y ( y < x ) and ∃ x ( x > 5 ) ∧ ∃ x ( y < x ) are not. Claim: Every formula can be transformed into a clean formula by means of several consecutive renamings of variables. Goranko

Clean formulae A formula A is clean if no variable occurs both free and bound in A and every two occurrences of quantifiers bind different variables. Thus, ∃ x ( x > 5 ) ∧ ∃ y ( y < z ) is clean, while ∃ x ( x > 5 ) ∧ ∃ y ( y < x ) and ∃ x ( x > 5 ) ∧ ∃ x ( y < x ) are not. Claim: Every formula can be transformed into a clean formula by means of several consecutive renamings of variables. E.g., ( x > 5 ) ∧ ∀ x (( x > 5 ) → ¬∃ x ( x < y )) Goranko

Clean formulae A formula A is clean if no variable occurs both free and bound in A and every two occurrences of quantifiers bind different variables. Thus, ∃ x ( x > 5 ) ∧ ∃ y ( y < z ) is clean, while ∃ x ( x > 5 ) ∧ ∃ y ( y < x ) and ∃ x ( x > 5 ) ∧ ∃ x ( y < x ) are not. Claim: Every formula can be transformed into a clean formula by means of several consecutive renamings of variables. E.g., ( x > 5 ) ∧ ∀ x (( x > 5 ) → ¬∃ x ( x < y )) can be transformed into a clean formula as follows: Goranko

Clean formulae A formula A is clean if no variable occurs both free and bound in A and every two occurrences of quantifiers bind different variables. Thus, ∃ x ( x > 5 ) ∧ ∃ y ( y < z ) is clean, while ∃ x ( x > 5 ) ∧ ∃ y ( y < x ) and ∃ x ( x > 5 ) ∧ ∃ x ( y < x ) are not. Claim: Every formula can be transformed into a clean formula by means of several consecutive renamings of variables. E.g., ( x > 5 ) ∧ ∀ x (( x > 5 ) → ¬∃ x ( x < y )) can be transformed into a clean formula as follows: ( x > 5 ) ∧ ∀ x 1 (( x 1 > 5 ) → ¬∃ x ( x < y )) , Goranko

Clean formulae A formula A is clean if no variable occurs both free and bound in A and every two occurrences of quantifiers bind different variables. Thus, ∃ x ( x > 5 ) ∧ ∃ y ( y < z ) is clean, while ∃ x ( x > 5 ) ∧ ∃ y ( y < x ) and ∃ x ( x > 5 ) ∧ ∃ x ( y < x ) are not. Claim: Every formula can be transformed into a clean formula by means of several consecutive renamings of variables. E.g., ( x > 5 ) ∧ ∀ x (( x > 5 ) → ¬∃ x ( x < y )) can be transformed into a clean formula as follows: ( x > 5 ) ∧ ∀ x 1 (( x 1 > 5 ) → ¬∃ x ( x < y )) , ( x > 5 ) ∧ ∀ x 1 (( x 1 > 5 ) → ¬∃ x 2 ( x 2 < y )) . Goranko

Substitution of a term for a variable in a formula Goranko

Substitution of a term for a variable in a formula Unform substitution of a term t for a variable x in a formula A means that all free occurrences x in A are simultaneously replaced by t . The result of the substitution is denoted A [ t / x ]. Goranko

Substitution of a term for a variable in a formula Unform substitution of a term t for a variable x in a formula A means that all free occurrences x in A are simultaneously replaced by t . The result of the substitution is denoted A [ t / x ]. Example: given the formula A = ∀ x ( P ( x , y ) → ( ¬ Q ( y ) ∨ ∃ yP ( x , y ))) Goranko

Substitution of a term for a variable in a formula Unform substitution of a term t for a variable x in a formula A means that all free occurrences x in A are simultaneously replaced by t . The result of the substitution is denoted A [ t / x ]. Example: given the formula A = ∀ x ( P ( x , y ) → ( ¬ Q ( y ) ∨ ∃ yP ( x , y ))) we have A [ f ( y , z ) / y ] = ∀ x ( P ( x , f ( y , z )) → ( ¬ Q ( f ( y , z )) ∨ ∃ yP ( x , y ))) , Goranko

Substitution of a term for a variable in a formula Unform substitution of a term t for a variable x in a formula A means that all free occurrences x in A are simultaneously replaced by t . The result of the substitution is denoted A [ t / x ]. Example: given the formula A = ∀ x ( P ( x , y ) → ( ¬ Q ( y ) ∨ ∃ yP ( x , y ))) we have A [ f ( y , z ) / y ] = ∀ x ( P ( x , f ( y , z )) → ( ¬ Q ( f ( y , z )) ∨ ∃ yP ( x , y ))) , while A [ f ( y , z ) / x ] = A because x does not occur free in A . Goranko

Substitution of a term for a variable in a formula Unform substitution of a term t for a variable x in a formula A means that all free occurrences x in A are simultaneously replaced by t . The result of the substitution is denoted A [ t / x ]. Example: given the formula A = ∀ x ( P ( x , y ) → ( ¬ Q ( y ) ∨ ∃ yP ( x , y ))) we have A [ f ( y , z ) / y ] = ∀ x ( P ( x , f ( y , z )) → ( ¬ Q ( f ( y , z )) ∨ ∃ yP ( x , y ))) , while A [ f ( y , z ) / x ] = A because x does not occur free in A . Intuitively, A [ t / x ] is supposed to say about the individual denoted by t the same as what A says about the individual denoted by x . Goranko

Substitution of a term for a variable in a formula Unform substitution of a term t for a variable x in a formula A means that all free occurrences x in A are simultaneously replaced by t . The result of the substitution is denoted A [ t / x ]. Example: given the formula A = ∀ x ( P ( x , y ) → ( ¬ Q ( y ) ∨ ∃ yP ( x , y ))) we have A [ f ( y , z ) / y ] = ∀ x ( P ( x , f ( y , z )) → ( ¬ Q ( f ( y , z )) ∨ ∃ yP ( x , y ))) , while A [ f ( y , z ) / x ] = A because x does not occur free in A . Intuitively, A [ t / x ] is supposed to say about the individual denoted by t the same as what A says about the individual denoted by x . Question: is that always the case? Goranko

Substitution of a term for a variable in a formula Unform substitution of a term t for a variable x in a formula A means that all free occurrences x in A are simultaneously replaced by t . The result of the substitution is denoted A [ t / x ]. Example: given the formula A = ∀ x ( P ( x , y ) → ( ¬ Q ( y ) ∨ ∃ yP ( x , y ))) we have A [ f ( y , z ) / y ] = ∀ x ( P ( x , f ( y , z )) → ( ¬ Q ( f ( y , z )) ∨ ∃ yP ( x , y ))) , while A [ f ( y , z ) / x ] = A because x does not occur free in A . Intuitively, A [ t / x ] is supposed to say about the individual denoted by t the same as what A says about the individual denoted by x . Question: is that always the case? Is a substitution of a term for a formula always ’safe’? Goranko

Capture of a variable in substitution Goranko

Capture of a variable in substitution The formula A = ∃ y ( x < y ) is true in N for any value of x . Goranko

Capture of a variable in substitution The formula A = ∃ y ( x < y ) is true in N for any value of x . However, A [( y + 1 ) / x ] = ∃ y ( y + 1 < y ), which is false in N . Goranko

Capture of a variable in substitution The formula A = ∃ y ( x < y ) is true in N for any value of x . However, A [( y + 1 ) / x ] = ∃ y ( y + 1 < y ), which is false in N . Therefore, the formula A [( y + 1 ) / x ] does not say about the term y + 1 the same as what A says about x . Goranko

Capture of a variable in substitution The formula A = ∃ y ( x < y ) is true in N for any value of x . However, A [( y + 1 ) / x ] = ∃ y ( y + 1 < y ), which is false in N . Therefore, the formula A [( y + 1 ) / x ] does not say about the term y + 1 the same as what A says about x . What went wrong? Goranko

Capture of a variable in substitution The formula A = ∃ y ( x < y ) is true in N for any value of x . However, A [( y + 1 ) / x ] = ∃ y ( y + 1 < y ), which is false in N . Therefore, the formula A [( y + 1 ) / x ] does not say about the term y + 1 the same as what A says about x . What went wrong? The occurrence of y in the term y + 1 got captured by the quantifier ∃ y , because we mixed the free and the bound uses of y . Goranko