a lattice of intuitionistic existential graphs systems€¦ · a lattice of intuitionistic...
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A lattice of Intuitionistic
Existential Graphs systems
Arnold Oostra
Joint work with a host of undergraduate students
Universidad del Tolima
René Magritte – La recherche de l’absolu
Outline
1. Brief introduction to Existential Graphs
2. A series of novel Existential Graphs systems
3. About equivalence proofs
Charles S. Peirce (1839 – 1914)
Gamma
Beta
Existential Graphs: My chef d’œvre
Alpha Classical Propositional Calculus
Classical First Order Logic
Classical Modal Logics
the second derivative is positive
the function has a minimum
M
S
Elements
Sheet of assertion
Letters
Cuts
Basic Connectives
A B
A
A B
A B
A B
A
A
B
B
A
Areas and Parity
A
B
A
A
C B
C D
E
• An area is a portion of the sheet of assertion
limited by cuts
• An area is even or odd if there is an even or odd
number of cuts around it
Transformation Rules
(In) Insertion
in odd areas
(Er) Erasure
in even areas
(It) Iteration
towards the inside
(De) Deiteration
from the outside
(Do) Double Cut
B B A A C
A A
B A B A A
A B B
B A A B
In
Er
It
De
Do
B C B
A A
Deduction A, A B
B
B A
A
De
A
B
Do A
B
Er
B
Do
In
It
Theorem A A
A
A A
Deduction A B, B C
A C Premises
B C
A B
Premises
B
A
C
B
Deduction A B, B C
A C
Premises
Iteration
B
A
C
B
C B
Deduction A B, B C
A C
Premises
Iteration
Deiteration
B
A
C
B
C
Deduction A B, B C
A C
Premises
Iteration
Deiteration
Double cut
B
A
C
B
C
Deduction A B, B C
A C
Premises
Iteration
Deiteration
Double cut
Erasure
A
C
Deduction A B, B C
A C
Premises
Iteration
Deiteration
Double cut
Erasure
A C
Deduction A B, B C
A C
is a student
is bright
Beta Graphs
it rains
Gamma Graphs…
René Magritte – La recherche de l’absolu
A Question
Are there Existential Graphs systems for
Intuitionistic Logic?
(Fernando Zalamea)
First idea: Just eliminate “double cut erasure”
Not even Modus Ponens turns out to be provable:
In Intuitionistic Logic the connectives are independent
Real problem: We need new signs for and
No easy solution
A
A B
A
B
B ?
1. Implicative Logic { , }
For implication take two cuts, one inside the other,
but joined at one point. We call this graph a scroll.
A B
Peirce himself ocassionally used this diagram: (!!!)
A B
With Andrea Y. Gómez
Transformation Rules
(In) Insertion
in odd areas
(Er) Erasure
in even areas
(It) Iteration
towards the inside
(De) Deiteration
from the outside
(Sc) Scrolling
B B A A C
A A
C
A B A A
AB
B
In
Er
It
De
Sc
B C B
A A B C
B
A A B
Deduction A, A B
B
B A
A
De
A
B
Sc A
B
Er
B
Theorem 1. The above system of graphs corresponds
to the { , } segment of Intuitionistic Proposi-
tional Calculus (aka Implicative Logic with Con-
junction, aka Positive Implicative Logic with
Conjunction)
2. Segment { , , }
Next introduce a constant graph and define:
is
But then
is
Insertion, erasure, iteration and deiteration with
the empty cut become provable
With these transformation rules, most properties of
(intuitionistic) negation also are provable
A A
Sc
Seminario Permanente Peirce…
Theorem 2. The above system of graphs corresponds
to the { , , } segment of Intuitionistic Proposi-
tional Calculus (without axiom x ).
3. Segment { , , }
For the full intuitionistic negation it suffices to add
the following rule
This is equivalent to an elegant form of Insertion in
odd:
With this adaptation of the original five rules, all pro-
perties of intuitionistic negation become provable.
Theorem 3. The above system of graphs corresponds
to the full { , , } segment of IPC.
A
In
A
Seminario…
Premise
Deduction A B
( A B )
B A
Premise
Deduction A B
( A B )
A B
Premise
Insertion
Deduction A B
( A B )
A B B
Premise
Insertion
Deduction A B
( A B )
A B B
Premise
Insertion
Iteration
Deduction A B
( A B )
A B B B
Premise
Insertion
Iteration
Deiteration
Deduction A B
( A B )
A B B
Premise
Insertion
Iteration
Deiteration
Erasure
Deduction A B
( A B )
A B
Premise
Insertion
Iteration
Deiteration
Erasure
Definition
Deduction A B
( A B )
A B
Premise
Insertion
Iteration
Deiteration
Erasure
Definition
Deduction A B
( A B )
A B
For disjunction take the following diagram:
A B
(“multiple scroll” )
We make some conventions:
is
is
and so on and on…
4. Segment { , , }
B A
A
A B
C A B C
B
C A B C
Seminario…
Transformation Rules
(In) Insertion
in odd areas
(Er) Erasure
in even areas
(It) Iteration
in the same cut
(De) Deiteration
from the same cut
(Sc) Scrolling
B B
A A
A A
A
A
B
In
Er
It
De
Sc
B
A B
B
A A B
C
B
B
B D
C D
C C A B
B
Theorem 4. The above system of graphs corresponds
to the { , , } segment of Intuitionistic Proposi-
tional Calculus.
Theorem 5. The system of graphs that combines
Theorems 2 and 4 corresponds to the { , , , }
segment of IPC (without axiom x ).
Theorem 6. The system of graphs that
combines Theorems 3 and 4
corresponds to the full IPC.
The main result
Seminario…
Slide No. 54
Note: Just adding the identity line, we get Beta-like
Existential Graphs for First Order Intuitionistic
Logic
Note: And allowing dotted cuts or loops, we get
Graphs systems for Intuitionistic Modal Logics
Glivenko’s Theorem
If CPC├ φ then IPC├ φ.
A double-star exercise:
A B B A
Intermediate logics
Theorem 7. Any finitely axiomatizable intermediate
(or, superintuitionistic) logic has as a graphic
system the one obtained by adding to the system of
Theorem 6 one additional “axiomatic” graph.
Examples:
LC
Classical
B B
A B A B
,
, , , ,
, , , , ,
IPC
CPC
Intermediate
Logics
Finitely
Axiomatizable
René Magritte – La recherche de l’absolu
Hints of proofs
Graphs
system
Alternate
system
(without
parity) Algebraic
system Seman-
tics
Alternate classical system
If A B and B C then A C
If A B then A C B C
If A B then
Edgar D. Rodríguez and Jorge E. Taboada --- Camilo Fuentes
Inspired in a paper by Y. Poveda -- without Dubuc :-)
A B A
A A A
A B A A B
A A
B A
A definition of Boolean algebra
Our finding:
(S, , ’, 1)
x y = y x
(x y) z = x (y z)
x x = x
x 1 = x
x y’ = x (x y)’
x x’ = 1’
x’’ = x
Caicedo’s improvement:
(S, , ’ )
x y = y x
(x y) z = x (y z)
x x = x
x y’ = x (x y)’
x x’ = y y’
x’’ = x
Edgar D. Rodríguez and Jorge E. Taboada
Hilbert semilattices
Definition:
(M, , , 1)
• (M, , 1) is a Hilbert a:
x (y x) = 1
(x(yz))((xy)(xz)) = 1
x 1 = 1
x y = y x = 1 implies x = y
• For the order induced,
x, y have infimum x y
• Compatibility
x y z iff x y z
Our finding:
(M, , , 1)
• (M, , 1) is a semilattice
with maximum
• Axioms for
(x y) z = (x y) (y z)
x (y z) = x ((x y) z)
x = 1 x
If x y then y z x z
If x y then z x z y
Mauricio Castillo
Heyting algebras
Our finding:
(H, , , , 0, 1)
• (H, , , 0, 1) is a bounded lattice
• Axioms for
(x y) z = (x y) (y z)
x (y z) = x ((x y) z)
x = 1 x
If x y then y z x z
If x y then z x z y
