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Logic and grammar differ:”Tom is handsome and intelligent” in grammar is a single proposition. But in logic are two: “Tom is beautiful”. “Tom is intelligent”. That become p and q. And even if the subject is the same logic matters little. p and q are independent.

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the elementary proposition

Logic and grammar differ:”Tom is handsome and intelligent” in grammar is a single proposition. But in logic are two: “Tom is beautiful”. “Tom is intelligent”. That become p and q.

And even if the subject is the same logic matters little. p and q are independent.

elementary proposition p is independent of q. The dependence is secured by ^, ~,v, that is, by the logic.

the elementary proposition
a perfect mirror
A perfect mirror?
  • Logic was a perfect mirror for Wittgenstein they thought they had finally solved the problems of philosophy.But in Vienna in 1928 was Brouwer. At that conference went even Wittgenstein. The statement that blows Wittgenstein. Wittgenstein was that mathematics was not a form of logic but was based on intuition. The other was that in mathematics p = ~~p; but ~~p is not equal to p.
true false provable absurd
true,false,provable,absurd
  • Another difference between the logic and mathematical logic was that we speak of true and false. In math provable or unprovable or absurd.
  • 2n = 2 is provable.
  • 20 = 2 is absurd, unprovable.
linear logic
Linear logic
  • This movement of the logic will then be expanded: there will be many logics, intuitionistic logic, which is inspired by Brouwer, fuzzy logic, linear logic, etc.., The sequent calculus
  • For example demonstrates linear logic intuitionistic logic. The proofs are sequent of this type.
  • A⊢B In B there can be no more than an expression A,C,B⊢B but not A⊢BvC
modus ponens modus tollens
Modus ponens,modus tollens
  • We show that the modus tollens is the opposite of ponens
  • A⊢B
  • moving from right to left or vice versa must change sign
  • ~B,A⊢0
  • ~B⊢ ~A
brouwer
Brouwer
  • A⊢A
  • A,~A⊢0
  • A⊢~~A
  • But
  • 0⊢A,~A no!
  • ~~A = A
after the meeting in brouwer
After the meeting in Brouwer
  • Publishes a problematic memory on color.As you know the elementary propositions are indipendent of each other.
  • But if we describe a color, such as red at the same time I describe all the other colors.All colors;just have the red,green and blu.For example a table is red and not blu and not green.
slide8
but...
  • the proposition becomes p^~q^~r or,to better indicate the color red,green blu r^~g^~b that is: a table is red and not green and not blu.But is not an elementary proposition.becomes a conjuction of three propositions.
as might be the truth functions
as might be the truth functions
  • also saying that the table is red and not green, to reduce the lines that,as you know,becomes 8 with three propositions, we get a complex statement and not a simple proposition :a table is red
specific description a description indefinite
specific description, a description indefinite
  • specific=lenght 25 m;indefinite= lenght20/30
  • specific becomes “p” indefinite becomes “q”.It is not possible that p is true and q is false that :~[p^~q] wich is equivalente a p implies q that is, in this case, a tautology
  • but p→q is not a tautology