Predicate Logic Terms and substitution The Lecture Logical laws for quantifiers Last Jouko Väänänen: Predicate logic viewed Logical laws for quantifiers Idea: If ∀xA is true, then A is true for every value of x, but what does this mean? Last Jouko Väänänen: Predicate logic viewed Logical laws for quantifiers Idea: If ∀xA is true, then A is true for every value of x, but what does this mean? Another idea: If A is true for some value of x then ∃xA is true, but what does this mean? Last Jouko Väänänen: Predicate logic viewed Terms Last Jouko Väänänen: Predicate logic viewed Terms Constants and variables are called terms. x,y,z,c,d,0,1,… Last Jouko Väänänen: Predicate logic viewed Terms Constants and variables are called terms. x,y,z,c,d,0,1,… Characteristic of terms is that they have a value if we fix a model and an assignment. Last Jouko Väänänen: Predicate logic viewed The value of a term Last Jouko Väänänen: Predicate logic viewed The value of a term The value tM⟨s⟩ of a term t in a model M under the assignment s is defined as follows: If t is a constant c, then tM⟨s⟩ is cM. If t is a variable x, then tM⟨s⟩ is s(x). Last Jouko Väänänen: Predicate logic viewed The value of a term The value tM⟨s⟩ of a term t in a model M under the assignment s is defined as follows: If t is a constant c, then tM⟨s⟩ is cM. If t is a variable x, then tM⟨s⟩ is s(x). If we had function symbols like +, - and · there would be more terms: x+y, x·y, (x+y)·(x-y), (x·x)·x, etc polynomials Last Jouko Väänänen: Predicate logic viewed Changing variables Last Jouko Väänänen: Predicate logic viewed Changing variables For the logical laws of quantifiers we have to look at ways in which variables can be changed in a formula. Last Jouko Väänänen: Predicate logic viewed Changing variables For the logical laws of quantifiers we have to look at ways in which variables can be changed in a formula. There are some simple rules that govern change of variables. Last Jouko Väänänen: Predicate logic viewed Bound variables can be changed Last Jouko Väänänen: Predicate logic viewed Bound variables can be changed If you change x to z in ∀xR0(x,y), the meaning does not change: The following are equivalent: 1) M⊨s∀xR0(x,y) 2) M⊨s∀zR0(z,y) Last Jouko Väänänen: Predicate logic viewed Bound variables can be changed If you change x to z in ∀xR0(x,y), the meaning does not change: The following are equivalent: 1) M⊨s∀xR0(x,y) 2) M⊨s∀zR0(z,y) This is like Both are a1+a2+a3+a4+a5. Last Jouko Väänänen: Predicate logic viewed But one has to be careful! Last Jouko Väänänen: Predicate logic viewed But one has to be careful! If you change x to y in ∀xR0(x,y), the meaning does change: The following are not equivalent in general: 1) M⊨s∀xR0(x,y) 2) M⊨s∀yR0(y,y) Last Jouko Väänänen: Predicate logic viewed But one has to be careful! If you change x to y in ∀xR0(x,y), the meaning does change: The following are not equivalent in general: 1) M⊨s∀xR0(x,y) 2) M⊨s∀yR0(y,y) This is like Last Jouko Väänänen: Predicate logic viewed The point Last Jouko Väänänen: Predicate logic viewed The point When a bound variable is changed to another, no free occurrence should become bound. Easy case: If a bound variable is changed to a completely new variable, no free occurrence becomes bound. Last Jouko Väänänen: Predicate logic viewed When exactly can a bound variable be changed to another? Last Jouko Väänänen: Predicate logic viewed When exactly can a bound variable be changed to another? To understand this properly, we need a new concept of freedom. Last Jouko Väänänen: Predicate logic viewed The concept “free for” Last Jouko Väänänen: Predicate logic viewed The concept “free for” A variable x is free for another variable y in a formula A, if no free occurrence of y in A becomes a bound occurrence of x if x is substituted for y in A. Last Jouko Väänänen: Predicate logic viewed The concept “free for” A variable x is free for another variable y in a formula A, if no free occurrence of y in A becomes a bound occurrence of x if x is substituted for y in A. free x is free for y in ∀zR0(z,y): ∀zR0(z,x). Last Jouko Väänänen: Predicate logic viewed The concept “free for” A variable x is free for another variable y in a formula A, if no free occurrence of y in A becomes a bound occurrence of x if x is substituted for y in A. free x is free for y in ∀zR0(z,y): ∀zR0(z,x). x is not free for y in ∀xR0(x,y): ∀xR0(x,x). bound Last Jouko Väänänen: Predicate logic viewed Convention Last Jouko Väänänen: Predicate logic viewed Convention We agree that a constant is always free for any variable in any formula. Last Jouko Väänänen: Predicate logic viewed Convention We agree that a constant is always free for any variable in any formula. Reason: If a constant is substituted for a variable, it cannot give rise to new occurrences of bound variables, because a constant is not a variable at all. Last Jouko Väänänen: Predicate logic viewed Substitution Last Jouko Väänänen: Predicate logic viewed Substitution A(t/y) is the formula obtained from A by substituting the term t for y in every free occurrence of y in A. We never use this notation unless we know that t is free for y in A. Last Jouko Väänänen: Predicate logic viewed Substitution A(t/y) is the formula obtained from A by substituting the term t for y in every free occurrence of y in A. We never use this notation unless we know that t is free for y in A. A A(x/y) P0(y) P0(x) ∃z(zEy) ∃z(zEx) ∃z(R0(z,y)→∀xR1(x,z)) ∃z(R0(z,x)→∀xR1(x,z)) ∃z(R0(z,y)→∀xR1(x,y)) (not allowed) Last Jouko Väänänen: Predicate logic viewed Basic fact about substitution Last Jouko Väänänen: Predicate logic viewed Basic fact about substitution Substitution lemma: If t is free for y in A, then the following are equivalent: Last Jouko Väänänen: Predicate logic viewed Basic fact about substitution Substitution lemma: If t is free for y in A, then the following are equivalent: M ⊨s A(t/y) M M ⊨s(a/y) A, where a=t ⟨s⟩ Last Jouko Väänänen: Predicate logic viewed Basic fact about substitution Substitution lemma: If t is free for y in A, then the following are equivalent: M ⊨s A(t/y) M M ⊨s(a/y) A, where a=t ⟨s⟩ Proof: Exercise. Last Jouko Väänänen: Predicate logic viewed Changing bound variables Last Jouko Väänänen: Predicate logic viewed Changing bound variables We can change y to x in ∀yA, getting ∀xA(x/y) if x does not occur free but is free for y in A. The formulas ∀yA and ∀xA(x/y) are logically equivalent. Similarly we can change y to x in ∃yA, getting ∀xA(x/y), if x does not occur free but is free for y in A. The formulas ∃yA and ∃xA(x/y) are logically equivalent. Last Jouko Väänänen: Predicate logic viewed Change of bound variables made easy You can always change a bound variable to a variable which does not occur in the original formula. Last Jouko Väänänen: Predicate logic viewed Valid formulas about quantifiers Last Jouko Väänänen: Predicate logic viewed Valid formulas about quantifiers ∀yA→A(t/y), where t is free for y in A A(t/y)→∃yA, where t is free for y in A Last Jouko Väänänen: Predicate logic viewed Valid formulas about quantifiers ∀yA→A(t/y), where t is free for y in A A(t/y)→∃yA, where t is free for y in A Neither is valid if t is not free for y in A Last Jouko Väänänen: Predicate logic viewed Recap Last Jouko Väänänen: Predicate logic viewed Recap We need substitution in formulating logical laws concerning quantifiers. Last Jouko Väänänen: Predicate logic viewed Recap We need substitution in formulating logical laws concerning quantifiers. In order that substitution goes right we need the concept of “free for”. Last Jouko Väänänen: Predicate logic viewed Recap We need substitution in formulating logical laws concerning quantifiers. In order that substitution goes right we need the concept of “free for”. Also, in order that change of variables goes right we need the concept of “free for”. Last Jouko Väänänen: Predicate logic viewed.