# Sorority World Sorority World Sorority World Sorority World

- Slides: 30

Sorority World

Sorority World

Sorority World

Sorority World

Sorority World

Sorority World

Logical Sentences

Logical Sentences

Logical Sentences Dana likes Cody. Abby does not like Dana does not like Abby. Bess likes Cody or Dana. Abby likes everyone that Bess likes. Cody likes everyone who likes her. Nobody likes herself.

Sorority World

Sorority World

Sorority World

Logical Entailment

Possible Worlds

Logical Entailment A set of premises logically entails a conclusion if and only if every world that satisfies the premises satisfies the conclusion. Premises: Conclusions: Dana likes Cody. Abby does not like Dana does not like Abby. somebody. Bess likes Cody or Dana. somebody. Abby likes everyone that Bess likes. Cody likes everyone who likes her. Nobody likes herself. Bess likes Cody. Bess does not like Everybody likes Everybody is liked by

Checking All Possible Worlds

Proofs

Proofs Abby likes everyone that Bess likes. Abby does not like Dana. Therefore, Bess does not like Dana. Bess likes Cody or Dana. Bess does not like Dana Therefore, Bess likes Cody.

Rules of Inference A rule of inference is a reasoning pattern consisting of some premises and some conclusions. If we believe the premises, a rule of inference tell us that we should also believe the conclusions.

Sample Rule of Inference All of Abby's friends are Bess's friends. All of Bess's friends are Cody's friends. Therefore, all of Abby's friends are Cody's friends.

Sample Rule of Inference All Accords are Hondas. All Hondas are Japanese. Therefore, all Accords are Japanese.

Sample Rule of of Inference All borogoves are slithy toves. All slithy toves are mimsy. Therefore, all borogoves are mimsy.

Sound Rule of Inference All x are y. All y are z. Therefore, x are z. Which patterns are correct? How many rules do we need?

Bad Rule of Inference All x are y. Some y are z. Therefore, some x are z.

Using Unsound Rule of Inference All Toyotas are Japanese cars. Some Japanese cars are made in America. Therefore, some Toyotas are made in America.

Using Unsound Rule of Inference All Toyotas are cars. Some cars are Porsches. Therefore, some Toyotas are Porsches.

Using Unsound Rule of Inference All of Abby's friends are Bess's friends. Some of Bess's friends are Abby's enemies. Therefore, some of Abby's friends are Abby's enemies. Example: Abby likes Cody, and Bess likes Cody. Bess likes Dana, but Abby does not like Dana.

Provability and Entailment A conclusion is provable for a set of premises if and only if every there is a finite sequence of sentences in which every element is either a premise or the result of applying a sound rule of inference to earlier members in the sequence. A set of premises logically entails a conclusion if and only if every world that satisfies the premises satisfies the conclusion.

Soundness and Completeness As we shall see, for well-behaved logics, logical entailment and provability are identical - a set of premises logically entails a conclusion if and only if the conclusion is provable from the premises. This is a very big deal.

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