Logical notation and Logical Equivalence Flashcards
What does ∴ mean?
Therefore
How else could we write this:
P1 : ¬(A → B)
Conc : (A → B) → A
¬(A → B) ∴ (A → B) → A
What is a tautology? Give an example.
A sentence that is true on all rows.
E.g. A ∨ ¬A or 2 + 2 = 4
What is a contradiction? Give an example.
A sentence that is false on all rows.
A & ¬A or 2 + 2 = 7
What is a contingent sentence?
A sentence that is neither a tautology nor a contradiction: There’s some row where it is T and some where it is F
When is a collection of sentences logically consistent?
If there’s some row of the truth table where they’re all T
E.g. the set {A, A & B, A ∨ B}
When is a collection of sentences logically inconsistent?
Give an example
If there’s no row of the truth table where they’re all T
E.g. the set {A → ¬B, A & B}
When are two sentences logically equivalent?
If they have the same truth value on every row
E.g. ¬A ∨ B and A → B
What is logical equivalence a kind of?
‘Equality’ relation for propositions
What kind of equality is logical equivalence?
A coarse kind of equality
Why is logical equivalence ‘coarse’
It lumps together lots of sentences without making fine distinctions
What is the only thing logical equivalence cares about?
Values in truth tables, not the meaning of the sentence
Give an example of logical equivalence with different meanings
Two tautologies with logical equivalence
What does logical equivalence useful for?
To check whether we’ve defined something correctly
What would be the logical operator for exclusive or?
Xor
