truth, deduction, computation lecture a
DESCRIPTION
My logic lectures at SCU Intuitionistic LogicTRANSCRIPT
Truth, Deduction, ComputationLecture AIntuitionistic logic
Vlad PatryshevSCU2013
Clean Up the Field
● Nullary functions● Nullary predicates● ⊥ and ⊤ (remember, we have a monoid)● or maybe more? (it depends)
P v ¬P? Really?
● Home(claire) v ¬Home(claire)● How about incomplete logical system?● Or something like probability?● Or properties that change with time● Live or dead?
○ clinical death○ Terry Schiavo
Getting rid of Pv¬P
How do we define P→Q if not via ¬PvQ?
→ Intro P
Q
P→Q
Getting rid of Pv¬P
An even better definition
P∧Q
R
P
Q→R⇔
Digression: Currying
P∧Q ⊢ R ⇔ P ⊢ Q→R
f(x:X, y:Y):Z ⇔ fc(x:X):(Y=>Z)f(x,y) = fc(x)(y)
Digression: Currying - now in JavaFunction2<X,Y,Z> f = new Function2<X,Y,Z>() { public Z apply(X x, Y y) { Z z = /*bla-bla-bla with x and y*/ … return z; }}
Function<X,Function<Y,Z>> fc = curry<X,Y,Z>(f)/* where */
public static curry<X,Y,Z>(final Function2<X,Y,Z> f) { return new Function<X,Function<Y,Z>>() { public Function<Y,Z> apply(final X x) { return new Function<Y,Z>() { public Z apply(Y y) { return f(x,y) } } } }}
Now define ¬
Definition:¬P is the same as P→⊥
1. P∧¬P ⊢ ⊥ …right?
2. ¬¬P ⊢ P …right? can we prove it? how?
3. P ⊢ ¬¬P …right?
4. ¬(P∧Q) ⊢ ¬P∨¬Q …right?
5. ¬P∨¬Q ⊢ ¬(P∧Q) …right?
6. ¬(P∨Q) ⊢ ¬P∧¬Q …right?
7. ¬P∧¬Q ⊢ ¬(P∨Q) …right?
⊢⊨¬∨∧→↔⊤⊥
Okay, an example. True, False, ?
P Q P∧Q
⊤ ⊤ ⊤
⊤ ? ?
⊤ ⊥ ⊥
? ⊤ ?
? ? ?
? ⊥ ⊥
⊥ ⊤ ⊥
⊥ ? ⊥
⊥ ⊥ ⊥
P Q P∨Q
⊤ ⊤ ⊤
⊤ ? ⊤
⊤ ⊥ ⊤
? ⊤ ⊤
? ? ?
? ⊥ ?
⊥ ⊤ ⊤
⊥ ? ?
⊥ ⊥ ⊥
Okay, an example. ⊤rue, False, ?
P Q P∧Q
⊤ ⊤ ⊤
⊤ ? ?
⊤ ⊥ ⊥
? ⊤ ?
? ? ?
? ⊥ ⊥
⊥ ⊤ ⊥
⊥ ? ⊥
⊥ ⊥ ⊥
Q R Q→R
⊤ ⊤ ⊤
⊤ ? ?
⊤ ⊥ ⊥
? ⊤ ⊤
? ? ⊤
? ⊥ ⊥
⊥ ⊤ ⊤
⊥ ? ⊤
⊥ ⊥ ⊤
Using
P^Q ⊢ RP ⊢ Q→R
P ¬P
⊤ ⊥
? ⊥
⊥ ⊤
Can we have 7.30 in our logic?
We had 3-valued; can we have 4? 5?
⊤, ‘sure!’, ‘maybe’, ‘on weekends’, ‘probably not’, ⊥
Can we build a logic based on it?Yes we can… remember, we need a monoid! (And more)
We can have Boolean, if n is 2k
⊤, ‘on weekends’, ‘if it rains’, ⊥ - and all combinations
¬P∨P - check!
Like parallel universes
We can have Infinity
[0,1]; ⊥=0, ⊤=1
x ∨ y = max(x,y)
x ∧ y = min(x,y)
Monoids, distributive law.
Define →x ∧ y < z ⇔ x < (y → z)
Negation?
We can have Infinity
[0,1]; ⊥=0, ⊤=1
x ∨ y = max(x,y)
x ∧ y = min(x,y)
Monoids, distributive law.
Define →x ∧ y < z ⇔ x < (y → z)
Negation?
What We Need From Logical Values
DefinitionPartial Order - a transitive reflexive antisymmetric binary relation
Examples● ⊤⊥● ⊤⊥?● N, [0, 1]● tree
But we need more
Definitionmin (a.k.a. ∧), max (a.k.a. v) functions in Partial Order
min(a,b) (if exists) - such that x < min(a,b) ⇔ x < a ∧ x < b
max(a,b) (if exists) - such that x > max(a,b) ⇔ x > a ∧ x > b
Not all partial orders have it:
Still not enough
Definitionmin, max functions in Partial Ordermin(a,b) (if exists) - such that x < min(a,b) ⇔ x < a ∧ x < b
max(a,b) (if exists) - such that x > max(a,b) ⇔ x > a ∧ x > b
Closer… meet Lattice
DefinitionLattice is a partial order where min and max are defined for all elements
We need ⊤ and ⊥
This one is no good (and it also does not have min/max):
Bounded Lattice
DefinitionBounded Lattice is a lattice with top ⊤ and bottom ⊥
Still not good enough...
what do we miss?
P∧Q ⊢ R ⇔ P ⊢ Q → R
Now we have Heyting Algebra
That’s it for today