Propositional logic versus first-order (predicate) logic

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Giorgi Japaridze Logic
First Order Logic
Episode 5

• Propositional logic versus first-order
  (predicate) logic
• The universe of discourse
• Constants, variables, terms and valuations
• Predicates as generalized propositions
• Boolean operations as operations on predicates
• Substitution of variables
• Quantifiers
• The language of first-order logic
• Interpretation, truth and validity
• The undecidability of the validity problem
• A Gentzen-style deductive system
• Soundness and Gödels completeness
• Peano arithmetic and Gödels incompleteness

0
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Propositional logic versus predicate logic
5.1
The language of propositional logic is very
poor and does not allow us to talk about many
things that we would like to be able to talk
about. That is because propositional logic fails
to look inside propositions and see any
further structure in them. For example,
propositional logic would not see any connection
between Bob likes Jane and There is someone
who likes Jane, even though one statement
logically implies the other.

This limitation of expressive power is
overcome in predicate logic, which is also
called first-order logic. It is based not just on
propositions, but on predicates (relations).
Propositions are simple special cases of
predicates. Hence, propositional logic is just a
simple fragment of the more expressive predicate
logic.
In a sense, the expressive power of
first-order logic is universal it allows us to
talk about virtually anything.
Note In this episode, first-order logic
will be presented in a way which may seem quite
different from the treatments that you have
probably seen elsewhere. Yet, our approach is
equivalent to the more traditional ones.
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The universe of discourse
5.2
Relations are always considered in the
context of some set. For example, when we
mention lt, we may say that we mean it as a binary
relation on the set N of natural numbers. This
formally means that lt is a subset of N?N. Such a
context-setting set (in this example N) is said
to be the universe of discourse.
When applying first-order logic, we always
have some universe of discourse in mind. For
example, if first-order logic is used for
building a formal arithmetic, the universe of
discourse would be N. And if logic is used for a
biological classification system, the universe of
discourse would contain (the names of) all plants
and animals.
In our treatment, we assume that the
universe of discourse is always N. There is no
(much) loss of generality in doing so. After all,
plants, people, chemical elements, rational
numbers --- all objects that have or can have
names --- can be encoded as natural numbers.
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Constants, variables, terms and valuations
5.3
We identify the elements of our universe of
discourse with their decimal representations,
and call the elements of 0,1,2,...,17,...
constants. The letters a, b, c, d will be
typically used as metavariables for constants.
Next, we fix another countably infinite set
of expressions and call its elements variables.
The letters x,y,z will be typically used as
metavariables for variables.
A term means either a variable or a constant.
The letter t will be typically used as a
metavariable for terms.
A valuation is any function that assigns a
constant to each variable. The letter e will be
typically used as a metavariable for valuations.
We extend the domain of each valuation e to
all terms by stipulating that, for any constant
c, e(c)c.
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Predicates revisited
5.4
From now on, by a predicate we will always
mean a function p that assigns a value
ep??,? (true or false) to each valuation
e. Note that we write ep instead of p(e).
When ep? , we say that predicate p is
true at e. And when ep?, we say that p
is false at e.
For example, the predicate x is even, or
Even(x), is defined by
eEven(x)
? if e(x) is even ? otherwise.
And the predicate x is greater than y, or xgty,
is defined by
exgty
? if e(x)gte(y) ? otherwise.
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Constant predicates propositions as special
cases of predicates
5.5
We say that a predicate p is constant if its
value does not depend on valuation. That is, p is
constant iff, for any two valuations e and e, we
have epep.
xgty xgtx xgt0 x?0 225
no
yes
Examples. Are the following predicates constant?
no
yes
yes
The last example above illustrates that
propositions are nothing but constant
predicates. In general, propositional logic is
nothing but first-order logic restricted to
constant predicates.
We say that a predicate p depends on a
variable x iff there are two valuations e and e
such that (a) e and e agree on all variables
except x, and (b) ep?ep. Constant
predicates (propositions) thus do not depend on
any variables.
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Boolean operations as operations on predicates
5.6
In Episode 4, Boolean operations were
defined as operations on propositions, i.e.
functions of the type propositionsn?proposition
s (n0, n1 or n2). They easily extend to
operations on predicates, i.e. functions of the
type predicatesn ?predicates, by the
following definition For every valuation e and
all predicates p and q e?p ?(ep), i.e.,
?p is true at e iff p is false at e ep?q
(ep) ?(eq), i.e., p?q is true at e iff so
are both p and q ep?q (ep)?(eq), i.e.,
p?q is true at e iff so is either p or q or
both ep?q (ep)?(eq), i.e., p?q is true
at e iff either p is false at e, or

q is true at e, or both.
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Substitution of variables
5.7
We often fix a tuple x1,...,xn of pairwise
distinct variables for a given predicate p, and
write p (when first mentioning it) as
p(x1,...,xn). Note by doing so, we do not
necessarily mean that p depends on all of the
variables x1,...,xn, or that p does not depend
on any other variables.

• When p(x1,...,xn) is as above and t1,...,tn
  are any terms, p(t1,...,tn) is
• written to mean the predicate such that, for any
  valuation e, we have
• ep(t1,...,tn)ep(x1,...,xn), where e is the
  valuation satisfying the
• following two conditions
• e(x1)e(t1), ..., e(xn)e(tn)
• e agrees with e on all other variables.

Example. Let both p(x,y) and q(x) mean x is
a multiple of y. Then
p(15,3) p(x,3) p(y,y) p(y,z)
q(7) q(z) q(y)
15 is a multiple of 3 ?
7 is a multiple of y
x is a multiple of 3
z is a multiple of y
y is a multiple of y ?
y is a multiple of y
y is a multiple of z
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Quantifiers
5.8

• Quantifiers in classical logic are functions
  of the type
• predicates?variables ?
  predicates.
• There are two quantifiers
• universal quantifier ?, with ?xp read as for
  all x, p
• existential quantifier ?, with ?xp read as
  there is x such that p.

They can be defined as big conjunction
and big disjunction ?xp(x)
p(0) ? p(1) ? p(2) ? p(3) ? ...
?xp(x) p(0) ? p(1) ? p(2) ? p(3) ?
...
More formally, for any variable x, predicate
p(x) and valuation e, we have
e?xp(x) ? iff, for every constant c,
ep(c)? e?xp(x) ? iff there is
a constant c such that ep(c)?.
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Examples
5.9
Let e be the valuation which assigns 5 to x and
assigns 0 to all other variables. Which of the
following predicates are true at e and which are
false?
yltx zlty ?z(zltx) ?z(zlty) ?x(xltx) ?z(zy ? 0ltz)
true
?x?y(xlty) ?y?x(xlty) ?y?x(x?y) ?x?y(x?y) 234
23x
true
false
false
true
false
false
true
false
false
true
true
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The language of classical first-order logic
5.10
In addition to the components that the
language of propositional logic has, the language
of first-order logic contains constants,
variables, quantifiers and predicate letters,
for which we use p,q,r,s as metavariables. With
each predicate letter is associated a natural
number called its arity. When the arity of p is
n, we say that p is n-ary.
An atom of this language is p(t1,...,tn),
where p is an n-ary letter and t1,...,tn are any
terms. When the arity of p is 0, we write p
instead of p( ). The atoms of propositional
logic remain atoms of first-order logic, as we
understand them as 0-ary letters. This includes ?
and ?, which are now treated as 0-ary logical
predicate letters and hence logical atoms.

• Formulas are defined inductively by
• Atoms are formulas
• If F is a formula, so is ?(F)
• If E and F are formulas, so are (E)?(F),
  (E)?(F), (E)?(F)
• If F is a formula and x is a variable, ?x(F) and
  ?x(F) are formulas.

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Free and bound terms normal formulas
5.11
An occurrence of a term t in a formula F is
said to be bound iff it is in the scope of ?t or
?t. Otherwise the occurrence is free. For
example, in formula ?y(p(x,y)? ?xp(x,y)), the
first occurrence of x is free while the other
occurrences of x, as well as all occurrences of
y, are bound.
A formula is said to be normal iff no
variable has both free and bound occurrences in
it. From now on, we will implicitly assume
that all formulas that we deal with are normal.
That is, from now on, we agree that the word
formula means normal formula.
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Interpretations
5.12
An interpretation for first-order logic is a
function that assigns some predicate
p(x1,...,xn) (with the fixed attached tuple
x1,...,xn of pairwise distinct variables) to
each n-ary nonlogical predicate letter p.
Such an interpretation is said to be admissible
for a formula F (or F-admissible) if, for any
n-ary predicate letter p of F, the
predicate p(x1,...,xn) assigned to p does not
depend on any variables that are not among
x1,...,xn but occur in F. In the sequel, we
always implicitly assume that the interpretations
we consider are admissible for the formulas that
we are talking about. Note In the
literature, interpretations are more commonly
called models or structures.
An interpretation extends to a function
formulas?predicates by stipulating that
(p(t1,...,tn))p(t1,...,tn) ??
?? (?F)?(F) (E?F) E?F
(E?F) E?F (E?F) E?F (?xF)?x(F)
(?xF)?x(F).
Usually we prefer to write F(t1,...,tn)
instead of (F(t1,...,tn)).
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Examples
5.13
Let p be a 3-ary predicate letter, and be
an interpretation that assigns to it the
predicate p(x,y,z) which is true at a given
valuation e iff e(x)e(y)e(z). What are
the meanings of the following formulas (into what
predicates do they turn) under this
interpretation?
p(x,y,z) ---
xyz
p(z,4,y) ---
z4y
p(x,3,5) ---
x35
i.e., x8
p(x,x,x) ---
xxx
i.e., x0
x?y
?zp(x,y,z) ---
?zp(z,x,z) ---
x0
?x?y?z(p(x,y,z)?p(x,z,y)) ---
?
?z1?z2?z3(p(z1,y,y)?p(z2,z1,z1)?p(z3,z2,z2)?p(x,z3
,z3)) ---
x16y
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Validity
5.14
A formula F of first-order logic is said to
be valid iff, for every interpretation and
every valuation e, we have eF?.
Are the following formulas valid?
p(x)
No
?x?yq(x,y)??y?xq(x,y)
Yes
p(x)??p(x)
?x?yq(x,y)??y?xq(x,y)
Yes
No
?x(p(x)??p(x))
?x?y(p(x)??p(y))
Yes
Yes
?x?y(p(x)??p(y))
?xp(x)??x?p(x)
No
Yes
Theorem 5.1. The problem of telling whether a
given formula of first-order logic is valid is
recursively enumerable but not decidable.
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A Gentzen-style deductive system
5.15
As in system G2 from Episode 4, we
understand sequents as finite sets of (now first
order) formulas. Furthermore, as in Episode 4,
we only consider formulas without ?, ?, ? and
without ? applied to nonatomic formulas. ??xF
should be understood as ?x?F, and ??xF as ?x?F.
Below are the rules of system G3. In those
rules, G is any set of formulas, E and F are any
formulas, x is any variable, H(x) is any
formula, t is any term with no bound occurrence
in H(x) or G, H(t) is the result of replacing in
H(x) all free occurrences of x by t, y is any
variable which does not occur in H(x) and G, and
H(y) is the result of replacing in H(x) all free
occurrences of x by y. Remember also that we
require all formulas to be normal (Slide 5.11).
For safety, here we also require that sequents,
seen as formulas (i.e. disjunctions of their
elements) be normal.
Axiom ?-Introduction
?-Introduction
G, E, F
G, E G, F
no premises
?
?
A
G, E?F
G, E?F
G,?E,E
?-Introduction
?-Introduction
G, ?xH(x), H(t)
G, H(y)
?
?
G, ?xH(x)
G, ?xH(x)
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Examples
5.16
A

?y?q(z1,y), ?xq(x,z2), ?q(z1,z2), q(z1,z2)
?
?y?q(z1,y), ?xq(x,z2), ?q(z1,z2)
?
A G3-proof of ?x?yq(x,y)??y?xq(x,y).
?y?q(z1,y), ?xq(x,z2)
?
?y?q(z1,y), ?y?xq(x,y)
?
?x?y?q(x,y), ?y?xq(x,y)
?
?x?y?q(x,y) ? ?y?xq(x,y)
A
?x?y(p(x)??p(y)), p(0), ?p(z), p(z), ?p(u)
? ?
?x?y(p(x)??p(y)), p(0)??p(z), p(z)??p(u)
?
A G3-proof of ?x?y(p(x)??p(y)).
?x?y(p(x)??p(y)), p(0)??p(z), ?y(p(z)??p(y))
?
?x?y(p(x)??p(y)), p(0)??p(z)
?
?x?y(p(x)??p(y)), ?y(p(0)??p(y))
?
?x?y(p(x)??p(y))
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The soundness and completeness of G3
5.17
Theorem 5.2. For any formula F of first-order
logic, we have Soundness If F is provable
in G3, then F is valid. Completeness If F
is valid, then F is provable in G3.
The soundness part of this theorem is
relatively easy to prove just as for G2, it can
be done by verifying that all rules preserve
validity. The completeness part is harder.
It was first proven in 1930 by Kurt Gödel. For
that reason, and for the reason of completeness
being the more important part, Theorem 5.2 (or
the same theorem for any other equivalent
deductive system) is called Gödels completeness
theorem.
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Peano arithmetic
5.18
Language , , ??, , 0 (a means a1)
Underlying logic an extension of G3 which
understands as the equality predicate.
Axioms 1. ?x?y (xy ? xy) 2. ?x (x ?
0) 3. ?x (x0 x) 4. ?x?y xy (xy) 5.
?x (x?0 0) 6. ?x?y x?y (x?y)x 7. Q(0)
? ?x (Q(x) ? Q(x)) ? ?xQ(x)
Gödels Incompleteness Theorem These axioms are
not sufficient to prove every true arithmetical
sentence. Neither would be sufficient any bigger
set of axioms.
Axiom 7 is a scheme, for every formula Q If Q
contains additional variables z1,...,zn, then the
whole thing should be prefixed with ?z1 ...
?zn This axiom is called the induction scheme