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Lecture 6Topics
Formal deduction in propositional Logic
Thursday, February 7, 13
Announcements
• Exam: February 20
• Will cover Book Chapters 1.1-1.8 plus lectures given up to February 18
• TA
• Junia Valente ([email protected])
• Office Hours: Tu 10-11:30am and Thu 4-5pm, ECSS 4.701
Thursday, February 7, 13
Formal Proofs in Propositional Calculus• We have seen multiple “inference rules”
• Modus tollens, Hypothetical syllogism, Disjunctive syllogism, etc.
• A list of all inference rules would be long and confusing
• “True Logic” developing a deductive system for the language of propositions
Thursday, February 7, 13
Axiomatic Propositional
Calculus• Small set of rules of inference and axioms
from where all “theorems” can be derived
• Use Jan Lukasiewicz Axioms and only Modus Ponens
Thursday, February 7, 13
Axioms
A1 �� (� � �)A2 (�� (� � �))� ((�� �)� (�� �))A3 (¬�� ¬�)� (� � �)
Thursday, February 7, 13
Theorems
• Definition: A theorem is any inference obtained from a set of premises: the axioms and the rule of inference Modus Ponens
• Notation:
��
� � �
� � if � is empty
Thursday, February 7, 13
Class ExamplesThm1 : � �� �
Thm2 : �� �, � � � � �� �
R1 : � � � � �
R2 : �� (� � �),�� � � �� �
R3 : ¬�� ¬� � � � �
Thm3 : ¬� � �� �
Thm4 : ¬¬� � �
Thm5 : �,¬� � �
Thursday, February 7, 13
Example
Theorem 2 �� �, � � � � �� �
Proof 1.� � � P
2.(� � �)� (�� (� � �)) A13.�� (� � �) MP1, 24. (�� (� � �))� ((�� �)� (�� �)) A25.(�� �)� (�� �) MP3, 46.�� � P
7.�� � MP5, 6
Thursday, February 7, 13
Rules are directly derived from the Axioms (Rules can also be called Theorems if
you prefer) Rule 2 �� (� � �),�� � � �� �
Proof 1.�� (� � �) P
2.(�� (� � �))� ((�� �)� (�� �)) A23.(�� �)� (�� �) MP1, 24.�� � P
5.�� � MP3, 4
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Previous Theorems can be used in Proofs
Theorem 4 ¬¬� � �
Proof 1.¬¬� P
2.¬�� ¬¬¬� Thm3, 13.¬¬�� � R3, 24.� MP1, 3
Thursday, February 7, 13
Example
Theorem 5 �,¬� � �
Proof 1.¬� P
2.�� � Thm3, 13.� P
4.� MP2, 3
Thursday, February 7, 13
