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

More Related Content

What's hot

Similar to Computational logic Propositional Calculus proof system

Recently uploaded

Computational logic Propositional Calculus proof system