Samenvatting Skills for AI – Logic
College 1
Declarative sentence (proposition) = a statement that is true or false.
You can abstract an argument in the letters p, q, r , s etc. :
Abstraction: If p and not q, then r. Not r. p. Therefore, q.
With as logical formalization: (((p∧¬q)→r)∧ (¬r∧p)) → q
‘not’ is a unary
connective. All the other
ones are binary. Priority:
With a parse tree you can reconstruct a formula and see if its true or false.
With a truth table you can express functional behavior for each connective.
Formulas φ and ψ are semantically equivalent, notation φ ≡ ψ, if they have identical
columns in their truth tables.
, Conjunction() and disjunction() and the exclusive or() are associative. This means that if
they are alone in the formula, you don’t have to use brackets.
Tautology = always true. In a truth table there is a T/1 on every line. Example: p ∨ ¬p
Contradiction = always false. In a truth table there is a F/0 on every line. Example: p ∧ ¬p
Contingent = sometime true and sometimes false. True when the formula is neither a
tautology, nor a contradiction. Example: p ∧ ¬q
College 2
Bij deze formule
moet je dus bewijzen dat wanneer p is true, not r is true (r is false) en de formule is true: dan
moet q true zijn.
Semantic entailment
Dus bij P entails q:
Wanneer alle keren dat
p waar is, q ook waar is,
klopt het: valid.
Wanneer je een
counterexample vind (p
waar q niet waar) dan niet meer:
invalid
Counterexample
Deduction theorem
Slide 14???
Dan hebben bijde griekse letters exact
dezelfde truthtable.
Metalogic:
Reasoning at a higher abstraction level to answer concrete logical questions.
College 1
Declarative sentence (proposition) = a statement that is true or false.
You can abstract an argument in the letters p, q, r , s etc. :
Abstraction: If p and not q, then r. Not r. p. Therefore, q.
With as logical formalization: (((p∧¬q)→r)∧ (¬r∧p)) → q
‘not’ is a unary
connective. All the other
ones are binary. Priority:
With a parse tree you can reconstruct a formula and see if its true or false.
With a truth table you can express functional behavior for each connective.
Formulas φ and ψ are semantically equivalent, notation φ ≡ ψ, if they have identical
columns in their truth tables.
, Conjunction() and disjunction() and the exclusive or() are associative. This means that if
they are alone in the formula, you don’t have to use brackets.
Tautology = always true. In a truth table there is a T/1 on every line. Example: p ∨ ¬p
Contradiction = always false. In a truth table there is a F/0 on every line. Example: p ∧ ¬p
Contingent = sometime true and sometimes false. True when the formula is neither a
tautology, nor a contradiction. Example: p ∧ ¬q
College 2
Bij deze formule
moet je dus bewijzen dat wanneer p is true, not r is true (r is false) en de formule is true: dan
moet q true zijn.
Semantic entailment
Dus bij P entails q:
Wanneer alle keren dat
p waar is, q ook waar is,
klopt het: valid.
Wanneer je een
counterexample vind (p
waar q niet waar) dan niet meer:
invalid
Counterexample
Deduction theorem
Slide 14???
Dan hebben bijde griekse letters exact
dezelfde truthtable.
Metalogic:
Reasoning at a higher abstraction level to answer concrete logical questions.
