Download Predicate Logic Terms and substitution

Predicate Logic
Terms and substitution
The Lecture
Logical laws for quantifiers
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Logical laws for quantifiers

Idea: If ∀xA is true, then A is true for
every value of x, but what does this
mean?
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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?
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Terms
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Terms


Constants and variables are called
terms.
x,y,z,c,d,0,1,…
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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.
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The value of a term
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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).
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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
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Changing variables
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Changing variables

For the logical laws of quantifiers we
have to look at ways in which variables
can be changed in a formula.
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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.
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Bound variables can be changed
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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)
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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.
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But one has to be careful!
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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)
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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
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The point
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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.
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When exactly can a bound variable be
changed to another?
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When exactly can a bound variable be
changed to another?

To understand this properly, we need a
new concept of freedom.
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The concept “free for”
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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.
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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).
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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).
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bound
Convention
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Convention

We agree that a constant is always free
for any variable in any formula.
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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.
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Substitution
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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.
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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)
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Basic fact about substitution
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Basic fact about substitution

Substitution lemma: If t is free for y
in A, then the following are equivalent:
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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 ⊨s(a/y) A, where a=tM⟨s⟩
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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 ⊨s(a/y) A, where a=tM⟨s⟩
Proof: Exercise.
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Changing bound variables
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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.
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Change of bound variables made easy

You can always change a bound
variable to a variable which does not
occur in the original formula.
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Valid formulas about quantifiers
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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
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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
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Recap
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Recap

We need substitution in formulating
logical laws concerning quantifiers.
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Recap

We need substitution in formulating
logical laws concerning quantifiers.

In order that substitution goes right we
need the concept of “free for”.
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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”.
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