Computational logic Propositional Calculus proof system

SRM INSTITUTE OF SCIENCE AND TECHNOLOGY
RAMAPURAM CAMPUS, CHENNAI-600 089
COMPUTATIONAL LOGIC
Dr.J.Faritha Banu
SRM IST- Ramapuram

Topics Covered in this Presentation are
 Propositional Calculus Introduction
 Terminologies
 Natural Deduction proof system
 Inference Rules
 Example Problems
 Sub Formula – Sub Proposition
 Soundness of Propositional Logic
 Completeness of Propositional Logic
 Gentzen sequent calculus
 Axiomatic System for PC

PROPOSITIONAL CALCULUS
 A proof starts with some well known facts and proceeds towards the result using
the allowed inferences.
 The accepted facts (proved theorems) are called axioms.
 The allowed inferences(Valid consequences) are called the rules of inference,
which link one or more propositions to another.
 A proof is then a sketch of how we reach the final proposition from the axioms by
means of the inference rules.
 Propositional Calculus (PC): Branch of symbolic logic that deals with
propositions and the relations between them, without examination of their
content.

INFORMAL METHOD
 Axiomatic systems are inconvenient as they insist on formality.
 Informal methods can be developed from the axiomatic systems.
INFORMAL SYSTEM – PROOF
NATURAL DEDUCTION
 Inference Rules: These are used in the Propositional natural deduction for
constructing proof. These rules are categorized to be either for introduction or for
elimination.
 Propositional Natural Deduction system (PND), This system has only one axiom,
which is ⊤; PND has the following inference rules (read ‘ i ’ for introduction and ‘e
’ for elimination

NATURAL DEDUCTION - Inference Rules
Natural Deduction of Propositional Logic: Rules of Conjunction, Disjunction, Implication, Negation
Conjunction (Introduction) Conjunction (Elimination) or Simplification
p p ∧ q p ∧ q
q p q
p ∧ q
Modus Ponens Modus Tollens
p→q p→q
p ¬q
q ¬ p

Disjunction(Introduction)
p q
p ∨ q p ∨ q
Disjunction(Elimination)
p ∨ q
p
(where p → r)
r
q ( where q → r)
r
r
Can be written as (p…..r) (q…r) , p v q ( ….. Means conditional)
r

NATURAL DEDUCTION - - Inference Rules
Top –True (Introduction) Top (Elimination)
Bottom -False (Introduction) Bottom -False (Elimination)

Negation (Introduction) Negation (Elimination)
Double Negation (Introduction) Double Negation
(Elimination)

NATURAL DEDUCTION- - Inference Rules
Conditional (Implication ) → (Introduction) → (Elimination)
Biconditional (Equivalence) ↔ (Introduction) ↔ (Elimination)

Proof 1 :
 Premises ( p ∧ q ) ∧ ( r ∧ ( s ∧ t ))
Conclusion s
Deduction:
1. ( p ∧ q ) ∧ ( r ∧ ( s ∧ t )) Premise
2. ( r ∧ ( s ∧ t )) Conjunction Elimination (1)
3. s ∧ t Conjunction Elimination (2)
4. s Conjunction Elimination (2)

Proof 2 :
 Premises ( p ∧ ( q ∨ r )) ∧ ( r ∧ ( s ↔ t ))
 Conclusion p ∧ r
Deduction:
1. ( p ∧ ( q ∨ r )) ∧ ( r ∧ ( s ↔ t )) Premise
2. p ∧ ( q ∨ r ) Conjunction Elimination(∧e)
(1)
3. p Conjunction Elimination(∧e)
(2)
4. r ∧ ( s ↔ t ) Conjunction Elimination (1)
5. r Conjunction Elimination (4)

Proof 3 : Premises ( p ∧ q ) → r
p ∧ s
q
 Conclusion r ∨ t
Deduction:
1. ( p ∧ q ) → r Premise
2. p ∧ s Premise
3. q Premise
4. p Conjunction Elimination (2)
5. s Conjunction Elimination (2)
6. p ∧ q Conjunction Introduction (4,3)
7. r Modus Ponens (1,6)
8. r ∨ t Disjunction (7)

Proof 4 : Premises p → ( q ∧ r )
¬( q ∧ r)
Conclusion ¬ p ∧ ¬ ( q ∧ r)
Deduction:
1. p → ( q ∧ r ) Premise
2. ¬( q ∧ r) Premise
3. ¬ p Modus Tollens (1)
4. ¬ p ∧ ¬ ( q ∧ r) Conjunction Introduction (3,2)

Proof 5 : Premises p → q
p → r
s → ¬( q ∨ r)
Conclusion ¬ s ∧ p
Deduction:
1. p → q Premise
2. p → r Premise
3. s → ¬( q ∨ r) Premise
4. p Conjuction elimination 2
5. q Modus ponens(4,1)
6. q ∨ r Disjunction introduction (5)
7. ¬ s Modus Tollens (3,6)
8. ¬ s ∧ p Conjuction introduction (4,7)

Proof 6: A proof for ⊢ p→(q→ p) is as follows:

Sub Formula – Sub Proposition:
 A sub-proposition of w, is a proposition corresponding to any subtree of the
parse tree of w.
Immediate sub-proposition of a proposition w is any proposition
corresponding to a subtree of the parse tree Tw of w whose depth is one less
than Tw.
The set of appropriate nonlogical constants is called the signature of the
propositional language.

 The function sub(P ) giving the sub formulas of a formula P is defined by
structural induction as follows:
sub(pi) = {pi}, for all atomic formulas pi
 sub(P ∧ Q) = {(P ∧ Q)} ∪ sub(P ) ∪ sub(Q)
sub(P ∨ Q) = {(P ∨ Q)} ∪ sub(P ) ∪ sub(Q)
sub(P → Q) = {(P → Q)} ∪ sub(P ) ∪ sub(Q)
sub(P ↔ Q) = {(P ↔ Q)} ∪ sub(P ) ∪ sub(Q)
sub(¬P ) = {¬P } ∪ sub(P ).
The set sub(P ) is called the set of sub formulas of P .

 Compute sub(P ) for P = ((p1 ∧ ¬p2) ∨ ¬p3).
sub(P ) = {P } ∪ sub(p1 ∧ ¬p2) ∪ sub(¬p3)
= {P } ∪ {(p1 ∧ ¬p2)} ∪ sub(p1) ∪ sub(¬p2) ∪ sub(¬p3)
= {P, (p1 ∧ ¬p2)} ∪ {p1} ∪ {¬p2} ∪ sub(p2) ∪ {¬p3} ∪ sub(p3)
= {P, (p1 ∧ ¬p2), p1, ¬p2, p2, ¬p3, p3}
Note : From the Parse tree write propositions for all the
subtree of the parse tree of w. A tree is subtree to itself,
so w also.
∨
∧ ¬
p1 ¬ p3
p2
Parse Tree

Soundness of Propositional Logic:
soundness property: if and only if every formula that can be proved in the system
is logically valid (Tautology) with respect to the semantics of the system.
Theorem :
Let Σ be a set of propositions, and let w be a proposition.
(1) If Σ ⊢ w in PC, then Σ ⊨ w.
(2) If Σ is satisfiable, then Σ is PC-consistent.
Ex: Modus ponens Proof
p (p=1)
p → q (p → q =1)
Q q=1

Completeness of Propositional Logic:
Completeness property: any formula that is true under all valuations is a theorem.
Theorem :
Let Σ be an infinite set of propositions, and let w be a proposition.
(1) Σ ⊨ w iff Σ has a finite subset Γ such that Γ ⊨ w.
(2) Σ is unsatisfiable iff Σ has a finite unsatisfiable subset.
(3) Σ is satisfiable iff each nonempty finite subset of Σ is satisfiable
Note:
 Soundness means that you cannot prove anything that's wrong
 Completeness means that you can prove anything that's right.

Gentzen sequent calculus
 Sequent calculus: Instead of constructing proofs, will prove that certain consequences
are provable. In a sequent calculus, one starts from a given sequent (consequence) and
goes on applying sequent rules to get newer sequents.
 Gentzen Sequent calculus: identifies some of the sequents as correct or self-evident
and tries to reduce everything to the self-evident ones, which terminate a proof.
 A sequent is of the form Σ ⊢ Γ, where Σ and Γ are sets of propositions. The empty sequent
‘⊢’ represents a consequence which never holds; and the universal sequent ⊤ represents
a valid consequence, which is used to terminate a proof.
 This system is also known as GPC, Gentzen’s Propositional Calculus.
 ⊤ p, r ⊢ q, s, t p, r ⊢ q p ⊢ q p ⊢ ⊢ q ⊢

 Let Σ, Γ, Δ, Ω be generic sets of propositions, and let x, y be generic
propositions. The inference rules or the sequent rules of GPC, along with their
mnemonics, are as follows:

 A derivation (GPC-derivation) is a tree whose root is a sequent, and it is
generated by applications of sequent rules on the leaves recursively.
 The new sequents are added as children of the original (leaf) node.
 Rules that have a single denominator are called stacking rules, and the ones
with two denominators are called branching rules.
 Sequents arising out of an application of a stacking rule are written one after
another from top to bottom, while those arising out of branching rules are written
with the help of slanted lines
 A proof of a sequent (GPC-proof) is a derivation with the sequent at its root and ⊤ at
all its leaves
 A set of propositions Σ is inconsistent in GPC iff Σ ⊢ ⊥.

GPC-proof for p → (q → p)
 ⊢ p → (q → p) ⊢ → (then x ⊢ y)
p ⊢ (q→p) ⊢ → (then x ⊢ y)
p , q ⊢ p
T

GPC-proof for ⊢ p→(¬q→¬(p→q))
⊢ p→ (¬q→¬(p→q)) ⊢ → (then x ⊢ y)
p ⊢ ¬q→¬(p→q) ⊢ → (then x ⊢ y)
 p , ¬q ⊢ ¬(p→q) ⊢ ¬ ( then x ⊢
 p, ¬q , (p→q) ⊢ ¬ ⊢ (then ⊢ x)
 p, (p→q) ⊢ q
→ ⊢ then ⊢ x y ⊢
 p ⊢ p, q p, q ⊢ q
 T T

GPC-proof for ⊢ (p→(q→r))→((p→q)→(p→r))
⊢ (p→(q→r))→((p→q)→(p→r)) ⊢ → (then x ⊢ y)
p →(q→r) ⊢ ((p→q)→(p→r) ⊢ → (then x ⊢ y)
 p→(q→r), p→q ⊢ p→r ⊢ → (then x ⊢ y)
 p→(q→r), p→q, p ⊢ r → ⊢ then ⊢ x y ⊢
 p→(q→r), p ⊢ p, r p→(q→r), q,p ⊢ r → ⊢ then ⊢ x y ⊢
 T q,p ⊢ p,r q→r,q,p ⊢ r
T
q,p ⊢ q, r r,q,p ⊢ r
T T

Show that the following sequent are provable or not using GPC
p, p→q ! ⊢ q
p, p→q ⊢ q → ⊢ then ⊢ x y ⊢
p ⊢p, q p, q ⊢ q
T T

AXIOMATIC SYSTEM PC
We choose the subset {¬,→} we ignore other connectives for simplicity.
We choose the subset {¬,→} we ignore other connectives for simplicity.
we use capital letters A,B,C, . . . as generic symbols for propositions.
The axiom schemes of PC are:
(A1) A→(B→A)
(A2) (A→(B→C))→((A→B)→(A→C))
(A3) (¬A→¬B)→((¬A→B)→A)
In addition to the axioms, PC has a inference rule - Modus Ponens:
{A, A→B} ⊨ B.

Proof using axiomatic system
EX: Show that ⊢ q→(p→ p).
1. p→ p Theorem
2. (p→ p)→(q→(p→ p)) A1
3. q→(p→ p) MP 1, 2
EX: Show that ⊢ (¬q→q)→q.
1. ¬q→¬q Theorem
2. (¬q→¬q)→((¬q→q)→q) A3
3. (¬q→q)→q 1, 2,MP

Proof using axiomatic system
⊨ PC r→(p→(q→ p)).
1. (p→(q→ p)) → r→(p→(q→ p)). A1 , A := p→(q→ p), B := r
2. p→(q→ p) A1, A := p, B := q
3. r→(p→(q→ p)) MP 1, 2,

References
1. Arindama Singh," Logics for Computer Science", PHI Learning Private
Ltd,2nd Edition, 2018
2. Wasilewska & Anita, "Logics for computer science: classical and non-
classical", Springer, 2018
3. Huth M and Ryan M, Logic in Computer Science: Modeling and Reasoning
about systems‖, Cambridge University Press, 2005
4. Dana Richards & Henry Hamburger, "Logic And Language Models For
Computer Science", Third Edition, World Scientific Publishing Co. Pte.
Ltd,2018

Computational logic Propositional Calculus proof system

Computational logic Propositional Calculus proof system

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