Classical propositional logic (quick review)

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Classical propositional logic(quick review)
Episode 4

• What logic is or should be
• Propositions
• Boolean operations
• The language of classical propositional logic
• Interpretation and truth
• Validity (tautologicity)
• Truth tables
• The NP-completeness of the nontautologicity
  problem
• Gentzen-style axiomatizations (sequent calculus
  systems)

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What logic is or should be
4.1
Below are a few theses about what your lecturer
believes logic is or should be
1. Logic is the most basic, general-purpose,
universal-utility tool for reasoning and acting
rationally.
2. All intellectual activities (sciences,
engineering, politics, everyday life) are or
should be based on logic, while logic, in turn,
is self-sufficient and needs none of those.
3. Logic makes no ontological commitments, and is
focused on form rather than content. It is
exactly this full abstraction from content that
makes logic general-purpose and
universal-utility.
4. All other sciences can be seen as applications
of logic to some particular domains. For this
very reason, such sciences are special- rater
than general-purpose tools, applicable only to
limited parts of the world.
5. Logic is the same to (all other) intellectual
activities as programming languages are to (all
other) application programs.
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Propositions
4.2
There are only two propositions ? (true) and ?
(false).
Classical logic sees no difference between two
true or two false propositions, so we have
224 snow is white ?
and
225 snow is black ?.
If you find this strange or confusing, remember
that, from the point of view of set theory, we
similarly have x xgt(x1) x x
is an elephant that can fly ?
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Boolean operations
4.3
Symbol Name Translation
Type
? Negation Not
?, ? ? ?, ? ?
Conjunction And ?, ???,
? ? ?, ? ? Disjunction Or
?, ???, ? ? ?, ? ?
Implication If ... then ?,
???, ? ? ?, ?
Note that ??p p p?q ? p?q ?(p?q)
?p??q ?(p?q) ?p??q
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The language of classical propositional logic
4.4
The language of classical propositional logic is
built from 1. Two logical atoms ? and ?. 2.
Infinitely many nonlogical atoms p, q, r, s, p1,
p2, p3, ... 3. Propositional connectives ?, ?,
?, ?. 4. Parentheses (, ).
Formulas are defined inductively by 1. Atoms
(logical or nonlogical) are formulas. 2. If F is
a formula, then so is ?(F). 3. If E and F are
formulas, then so are (E)?(F), (E)?(F), (E)?(F).
For readability, we will be typically omitting
certain parentheses, whenever this does not
create ambiguity. ? has the highest precedence,
so that, say, ?p?q should be understood as
(?p)?q rather than as ?(p?q). And ? has the
lowest precedence, so that, say, p?q?r should
be understood as (p?q)?r rather than as
p?(q?r).
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Interpretation and truth
4.5
An interpretation is a function that assigns a
particular proposition p (? or ?) to each
nonlogical atom p. Synonyms of interpretation
are model and truth assignment. Every
interpretation extends from nonlogical atoms to
all formulas by stipulating that

? ? ? ? (?F) ?(F) (E?F)
(E)?(F) (E?F) (E)?(F) (E?F) (E)?(F)
We say that formula F is true under
interpretation , or simply that F is true, if
F ?. Otherwise we say that it is false.
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Validity truth tables NP-completeness of the
nonvalidity problem
4.6
We say that a formula F is valid (or is a
tautology) if F is true under every
interpretation.
Examples of tautologies
? ? ? ?p?p p?(q?p) ??p.
How about this one? (p?q)?r ?
(p?q)?(p?r) There is a brute force method,
called the truth table method, that allows us to
get answers to such questions. It consists of
going through all possible interpretations to
see if all of them make the formula true.

What is the trouble with the truth table method?
Inefficiency. The truth table for a formula with
a couple of hundred atoms would have more rows
than the number of particles in the Universe!
Deductive methods (the following slides) are
more practical, even though, generally, we have
Theorem 4.1. The problem of telling whether a
formula is tautological is coNP-complete,
meaning that no truly efficient algorithms are
known for it.
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Deductive systems
4.7
There are various deductive systems for
classical propositional logic. They can be
divided into two major classes Hilbert-style and
Gentzen-style. Hilbert-style systems are
axiom-based while Gentzen-style systems are
rule-based. Gentzen-style systems have a number
of advantages, including existence of
straightforward proof search algorithms. In
this course we will deal only with Gentzen-style
systems. Another name for such systems is
sequent calculi. From now on we will
consider the version of the language of classical
propositional logic where formulas are not
allowed to contain logical atoms or implication.
This does not mean diminishing the expressive
power of the language, as ? can be considered an
abbreviation of p??p, ? considered an
abbreviation of p??p, and E?F an abbreviation of
?E?F. Furthermore, officially ? will only be
allowed on atoms. When applied to non-atomic
formulas, ? is considered an abbreviation defined
by
??E E
?(E?F) ?E??F
?(E?F) ?E??F
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A sequent calculus system for classical
propositional logic
4.8
A sequent is any finite sequence of formulas
(in the sense of Slide 4.7). Here are the rules
of one of several equivalent sequent calculus
systems for classical propositional logic. Let
us call it G1. In the following rules, E,F
stand for any formulas, and G,H stand for any
sequents. Above the horizontal line is (are) the
premise(s), and below the line is the
conclusion. A proof in such a system is a tree of
sequents, where each node is the conclusion of
some rule, and the children of the node are the
premises of that rule. Such a tree is a proof of
the sequent at its root. A formula F is
considered provable if the sequent F is provable
(of course, provable means has a proof).
Identity Exchange
Weakening Contraction
G,E,F,H
G
G,E,E
no premises
E
I
W
C
G,F,E,H
G,E
?E,E
G,E
?-Introduction
?-Introduction
G, E, F
G, E H, F
?
?
G, E?F
G, H, E?F
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An example of a proof in sequent calculus
4.9
Example. Find a proof of (p?q)?(r ?s) ?
(p?r)?(q?s).
First, we rewrite the formula as
(?p??q)?(?r?? s)?(p?r)?(q?s).
Next, we construct a proof of it in the bottom-up
fashion as follows
(?p??q)?(?r?? s), (p?r)?(q?s)
?
(?p??q)?(?r?? s)?(p?r)?(q?s)
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Soundness and completeness
4.10
Theorem 4.2. For any formula F, we have
1. (Soundness) If F is provable in G1, then F is
a tautology. 2. (Completeness) If F is a
tautology, then F is provable in G1.
Soundness
Completeness
Good formulas
Provable formulas
Provable formulas
Good formulas
In our present case, good tautological.
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A sequent calculus system without structural rules
4.11

• The Identity rule of G1 can be called an axiom,
  as it takes no premises.
• The rules ?-Introduction and ?-Introduction are
  called logical rules as they explain
• the logical meanings of connectives.
• The remaining rules of Exchange, Weakening and
  Contraction are called structural
• rules, as they only restructure sequents and
  otherwise are not related to any logical
• connectives.

Below comes a simpler system --- let us call it
G2 --- that has no structural rules at all. A
significant change, however, is that in G2
sequents are seen as sets rather than sequences
of formulas. G2 only has the following three
rules, where E,F are any formulas and G is any
sequent, i.e. any set of formulas.
Axiom ?-Introduction
?-Introduction
G, E, F
G, E G, F
no premises
?
?
A
G, E?F
G, E?F
G,?E,E
In view of Theorems 4.2 and 4.4, system G2 is
equivalent, in its deductive power, to our old
friend G1.
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Another example
4.12
Example. Find a proof of (?p??q)?(?r??
s)?(p?r)?(q?s) in G2.
We use a standard proof search method, which
can be applied to any formula. The method
is to start with the target formula, and keep
applying, in the bottom-up sense, the two logical
rules (in whatever order you like) as long as
there are conjunctions or disjunctions in the
sequent

A
?p, ?q, p, r
?p, ?q, q, s
A
A
?r, ?s, p, r
?r, ?s, q, s
A
?
?
?
?
?p, ?q, p?r
?p, ?q, q?s
?r, ?s, p?r
?r, ?s, q?s
?
?
?p, ?q, (p?r)?(q?s)
?r, ?s, (p?r)?(q?s)
?
?
?p??q, (p?r)?(q?s)
?r??s, (p?r)?(q?s)
?
(?p??q)?(?r?? s), (p?r)?(q?s)
?
(?p??q)?(?r?? s)?(p?r)?(q?s)
You will end up with a tree where all leaves
are sequents containing only atoms and negated
atoms. Such a tree is a G2-proof if (and only if)
every leaf contains some atom together with the
negation of the same atom.
One can show (see next slide) that the
target formula is provable iff this method
generates a proof tree.
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Soundness and completeness proof
4.13
We extend the concepts of truth and validity
(tautologicity) from formulas to sequents by
understanding each sequent F1,...,Fn as the
formula F1?...?Fn.

Lemma 4.3. (a) Every sequent which is (the
conclusion of) an axiom of G2 is valid. (b)
The two logical rules of G2 preserve validity in
both top-down and bottom-up directions. That is,
the premise(s) is (are) valid iff so is the
conclusion.
Proof Obvious. ?
Theorem 4.4. (Soundness and completeness of G2)
A sequent is provable in G2 iff it is valid
(tautological).
Proof. The Soundness of G2 is an immediate
consequence of Lemma 4.3 Axioms are valid and
the rules preserve validity, so every sequent in
a proof tree will be valid. As for
completeness, assume a given formula is valid.
Apply to it the method of Slide 4.12. In view of
Lemma 4.3, all sequents in the tree generated by
this method will be valid. This includes the
leaves of the tree. But leaves only contain
sequents where every formula is an atom or a
negated atom. Obviously such a sequent is valid
iff it is an axiom. Thus, the tree will be a
proof tree. ?