**Slides / Bullets**
- Tautologies
- A tautology is a sentence whose truth table contains nothing but T’s in the column under the main connective.

- Logical truth
- A sentence is logically true or logically necessary if it must be true (as a matter of logic); i.e. if it’s true in all possible circumstances.
- Any argument whose conclusion is logically true is valid.
- Even an argument with no premises at all.
- Conversely: any sentence that follows from the null set of premises is a logical truth.

- Tautology and logical truth
- All tautologies are logical truths.
- Not all logical truths are tautologies.
- SameRow(a, a)
- b = b
- ¬Between(a, b, b)
- ¬(Large(a) ∧ Small(a))

- TT-possibility
- A sentence is TT-possible if its truth table contains at least one T under the main connective.
- What is the relation between the following claims:
- P is TT-possible
- ¬P is TT-possible
- P is a tautology
- ¬P is a tautology

- Answer: P is TT-possible if and only if ¬P is not a tautology; ¬P is TT-possible if and only if P is not a tautology.

- Logical possibility and TT-possibility
- A sentence is logically possible if it might (as far as logic is concerned) be true; if it is true in some possible circumstance.
- P is logically possible if and only if ¬P is not logically necessary.

- All logically possible sentences are TT-possible, but not all TT-possible sentences are logically possible.

- For next week:
- Read: 4.1; optionally, 4.2-4.4
- Do the ‘You try it’ exercises.
- If you’ve never done so, you might also try getting the hang of the ‘Boolean search’ mechanism of some Internet search engine - see exercise ***.

- Do exercises 3.21 (48%), 4.2 (24%), 4.4-4.7 (7% each).