Logic in Action Chapter 3: Syllogistic Reasoning - PowerPoint PPT Presentation

Logic in Action Chapter 3: Syllogistic Reasoning http://www.logicinaction.org/ ( http://www.logicinaction.org/ ) 1 / 13

Reasoning About Predicates Beyond Propositional Logic How would you decide whether the following inferences are valid or not using the Propositional Logic tools ? ( http://www.logicinaction.org/ ) 2 / 13

Reasoning About Predicates Beyond Propositional Logic How would you decide whether the following inferences are valid or not using the Propositional Logic tools ? All politicians are rich. No student is politician. ? No student is rich. ( http://www.logicinaction.org/ ) 2 / 13

Reasoning About Predicates Beyond Propositional Logic How would you decide whether the following inferences are valid or not using the Propositional Logic tools ? All politicians are rich. No student is politician. ? No student is rich. All politicians are rich. There is a student that is a politician. ? There is a student that is rich. ( http://www.logicinaction.org/ ) 2 / 13

Reasoning About Predicates Beyond Propositional Logic How would you decide whether the following inferences are valid or not using the Propositional Logic tools ? All politicians are rich. No student is politician. ? No student is rich. All politicians are rich. There is a student that is a politician. ? There is a student that is rich. How to deal with individuals and their properties? ( http://www.logicinaction.org/ ) 2 / 13

Reasoning About Predicates Syllogisms A syllogism is a inference with specific characteristics. ( http://www.logicinaction.org/ ) 3 / 13

Reasoning About Predicates Syllogisms A syllogism is a inference with specific characteristics. Only two premises. ( http://www.logicinaction.org/ ) 3 / 13

Reasoning About Predicates Syllogisms A syllogism is a inference with specific characteristics. Only two premises. The premises and conclusion can have only the following forms: ( http://www.logicinaction.org/ ) 3 / 13

Reasoning About Predicates Syllogisms A syllogism is a inference with specific characteristics. Only two premises. The premises and conclusion can have only the following forms: 1 “All A are B ” . ( http://www.logicinaction.org/ ) 3 / 13

Reasoning About Predicates Syllogisms A syllogism is a inference with specific characteristics. Only two premises. The premises and conclusion can have only the following forms: 1 “All A are B ” . 2 “Some A are B ” . ( http://www.logicinaction.org/ ) 3 / 13

Reasoning About Predicates Syllogisms A syllogism is a inference with specific characteristics. Only two premises. The premises and conclusion can have only the following forms: 1 “All A are B ” . 2 “Some A are B ” . 3 “All A are not B ” (i.e., “No A is B ” ). ( http://www.logicinaction.org/ ) 3 / 13

Reasoning About Predicates Syllogisms A syllogism is a inference with specific characteristics. Only two premises. The premises and conclusion can have only the following forms: 1 “All A are B ” . 2 “Some A are B ” . 3 “All A are not B ” (i.e., “No A is B ” ). 4 “Some A are not B ” (i.e., “Not all A are B ” ). ( http://www.logicinaction.org/ ) 3 / 13

Reasoning About Predicates Syllogisms A syllogism is a inference with specific characteristics. Only two premises. The premises and conclusion can have only the following forms: 1 “All A are B ” . 2 “Some A are B ” . 3 “All A are not B ” (i.e., “No A is B ” ). 4 “Some A are not B ” (i.e., “Not all A are B ” ). where A and B are predicates , that is, they represent collection of objects. ( http://www.logicinaction.org/ ) 3 / 13

Reasoning About Predicates Syllogisms A syllogism is a inference with specific characteristics. Only two premises. The premises and conclusion can have only the following forms: 1 “All A are B ” . 2 “Some A are B ” . 3 “All A are not B ” (i.e., “No A is B ” ). 4 “Some A are not B ” (i.e., “Not all A are B ” ). where A and B are predicates , that is, they represent collection of objects. The inference involves just three predicates. ( http://www.logicinaction.org/ ) 3 / 13

Reasoning About Predicates Examples Some mugs are beautiful. All fruit is nutritious. All mugs are useful. All fruit is tasty. Some useful things are beautiful. Some tasty things are nutritious. All schools are buildings. Some travellers are not caucasian. Some schools are tents. None of the tourists is a traveller. No buildings are tents. Some tourists are not caucasian. ( http://www.logicinaction.org/ ) 4 / 13

Reasoning About Predicates Square of opposition ( http://www.logicinaction.org/ ) 5 / 13

Reasoning About Predicates Square of opposition All A are B . ( http://www.logicinaction.org/ ) 5 / 13

Reasoning About Predicates Square of opposition All A are B . Some A are B . ( http://www.logicinaction.org/ ) 5 / 13

Reasoning About Predicates Square of opposition All A are B . Some A are B . All A are not B (No A is B ). ( http://www.logicinaction.org/ ) 5 / 13

Reasoning About Predicates Square of opposition All A are B . Some A are B . All A are not B (No A is B ). Some A are not B (Not all A are B ). ( http://www.logicinaction.org/ ) 5 / 13

Reasoning About Predicates Square of opposition All A are B . Some A are B . All A are not B (No A is B ). Some A are not B (Not all A are B ). All A are B All A are not B Some A are B Some A are not B ( http://www.logicinaction.org/ ) 5 / 13

Sets and Operations on Sets Sets A set is a collection of objects. ( http://www.logicinaction.org/ ) 6 / 13

Sets and Operations on Sets Sets A set is a collection of objects. If object a is in the set A , we write a ∈ A . ( http://www.logicinaction.org/ ) 6 / 13

Sets and Operations on Sets Sets A set is a collection of objects. If object a is in the set A , we write a ∈ A . A set can be defined by a property: { x | ϕ ( x ) } ( http://www.logicinaction.org/ ) 6 / 13

Sets and Operations on Sets Sets A set is a collection of objects. If object a is in the set A , we write a ∈ A . A set can be defined by a property: { x | ϕ ( x ) } Usually there is a domain U from where the objects are taken from. { x ∈ U | ϕ ( x ) } ( http://www.logicinaction.org/ ) 6 / 13

Sets and Operations on Sets Operations on sets: two predicates Domain : Humans P oliticians S tudents ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: two predicates A set : P oliticians P oliticians S tudents P ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: two predicates A set : S tudents P oliticians S tudents S ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: two predicates Complement : No S tudents P oliticians S tudents S ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: two predicates Complement : No P oliticians P oliticians S tudents P ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: two predicates Union : P oliticians or S tudents P oliticians S tudents P ∪ S ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: two predicates Union : S tudents or P oliticians P oliticians S tudents S ∪ P ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: two predicates Intersection : P oliticians and S tudents P oliticians S tudents P ∩ S ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: two predicates Intersection : S tudents and P oliticians P oliticians S tudents S ∩ P ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: two predicates Difference : P oliticians that are no S tudents P oliticians S tudents P \ S ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: two predicates Difference : S tudents that are no P oliticians P oliticians S tudents S \ P ( http://www.logicinaction.org/ ) 7 / 13

Sets and Operations on Sets Operations on sets: three predicates P S R All possible combinations are represented in this diagram. ( http://www.logicinaction.org/ ) 8 / 13

Sets and Operations on Sets Operations on sets: three predicates P S R P ( http://www.logicinaction.org/ ) 8 / 13

Sets and Operations on Sets Operations on sets: three predicates P S R S ( http://www.logicinaction.org/ ) 8 / 13

Sets and Operations on Sets Operations on sets: three predicates P S R R ( http://www.logicinaction.org/ ) 8 / 13

Sets and Operations on Sets Operations on sets: three predicates P S R P ∪ S ∪ R ( http://www.logicinaction.org/ ) 8 / 13

Sets and Operations on Sets Operations on sets: three predicates P S R S ∪ P ( http://www.logicinaction.org/ ) 8 / 13

Sets and Operations on Sets Operations on sets: three predicates P S R P ∩ R ( http://www.logicinaction.org/ ) 8 / 13

Sets and Operations on Sets Operations on sets: three predicates P S R P ∩ ( S ∪ R ) ( http://www.logicinaction.org/ ) 8 / 13

Sets and Operations on Sets Operations on sets: three predicates P S R R \ ( P ∪ S ) ( http://www.logicinaction.org/ ) 8 / 13