Module: Sentential logic
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Democratic societies can’t force people. Therefore they have to control what they think.
- Noam Chomsky
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Please complete these exercises by filling in the blanks with the correct truth-values. You should find the first four sequents to be valid, and the next four invalid. There might be more than one correct answer even though the answer provided here allows only one of them. Sorry about this.
( | P | → | ( | Q | & | ( | R | → | S | ))) | P | ~ | S | ⊧ | ~ | ( | Q | & | R | ) |
( | P | ↔ | ( | R | → | ( | P | ∨ | ~ | Q | ))) | ~ | ( | R | → | ( | P | ∨ | Q | )) | ⊧ | Q |
( | ~ | ( | P | & | Q | ) | & | ~ | ( | ~ | P | ∨ | ~ | Q | )) | ⊧ | ~ | ( | ~ | ( | P | ∨ | Q | ) | ↔ | ( | ~ | P | & | ~ | Q | )) |
~ | (( | ~ | P | & | Q | ) | ∨ | ( | R | → | S | )) | ( | ~ | ( | ~ | P | ∨ | ~ | Q | ) | & | ~ | ( | R | → | S | )) | ⊧ | ( | ~ | (( | R | → | S | ) | & | ~ | P | ) | ↔ | ( | P | ∨ | ( | R | ↔ | S | ))) | |
((( | P | ∨ | Q | ) | → | R | ) | ↔ | (( | ~ | P | & | ~ | Q | ) | ↔ | R | )) | ⊧ | ( | P | ∨ | Q | ) |
( | P | → | ( | Q | & | ~ | R | )) | ( | ~ | P | ∨ | ~ | ( | ~ | Q | ∨ | ~ | S | )) | ⊧ | ( | S | → | ( | ~ | P | ∨ | T | )) |
( | ~ | R | → | ~ | Q | ) | (( | ~ | P | & | R | ) | & | ~ | Q | ) | ⊧ | ~ | ( | P | ↔ | ( | ~ | R | ∨ | Q | )) |
(( | P | → | Q | ) | → | ( | ~ | P | ∨ | Q | )) | ( | ~ | ( | P | ∨ | Q | ) | & | ( | ~ | P | & | ~ | Q | )) | ⊧ | ~ | ( | ~ | (( | P | → | Q | ) | & | ( | P | ∨ | ~ | Q | )) | ∨ | ( | P | ↔ | Q | )) |