Transcript Existential Graphs: Beta

Existential Graphs: Beta
Introduction to Logic
Alpha Review: Symbolization
F
EG
‘P’
P
P
‘not P’
P
P
‘P and Q’
PQ
P
Q
‘P or Q’
PQ
P
Q
‘if P then Q’
PQ
P
Q
Alpha Review: Inference Rules
Double Cut

(De)Iteration
 
Erasure

Insertion




    



2k

1
2k+1 1





2k


1
2k+1 1
Beta: Symbolization
• The Beta part of Existential Graphs corresponds to
predicate logic.
• Just as predicate logic is an extension of
propositional logic, Beta is an extension of Alpha.
• Beta Graphs correspond to statements from
Predicate Logic.
• To define Beta Graphs, we need to define how we
symbolize objects, individuals constants, identity,
predicates, and quantifiers.
Objects
In Beta, objects are symbolized using a dot:
Putting a dot on the SA asserts
the existence of an object:
Notice that F and other traditional logic systems
don’t have the ability to express that something exists
in a simple fashion: ‘x’ is not a statement in F.
Moreover, F and other traditional logic systems take
this very claim as a given, i.e. F makes the Assumption
of Existential Import. EG does not do this.
Individual Constants
We can use individual constants
to refer to specific objects. To do this,
simply write the name next to the object:
Placing this on the SA asserts
the existence of the object we
named:
a
a
Again, F and other traditional logic systems
have no means of expressing this in a
straightforward way.
Identity
To express that two objects are identical,
place the two dots representing those
two objects next to each other:
This can be done for any number of dots,
resulting in a line:
A line on the SA asserts that there are a whole bunch
of objects, all identical to each other. As such, it makes
a claim that is logically equivalent to the claim that is
made by putting a single dot on the SA. Therefore, we
can interchange between lines and dots.
Non-Identity
To express that two objects are not identical,
we use the cut to negate the claim that the two
objects are identical. This is best seen using lines
instead of dots:
To claim the existence of three distinct objects:
Predicates
To express that an object has a certain
property P, simply write the predicate
symbol next to the object:
P
To express that an object does not have
a certain property P, use the cut:
P
To express a relationship R between two
or more objects, write the predicate
symbol between the objects:
R
Quantifiers
Obviously, the dots or lines will serve
as existential quantifiers. That is, a
claim like x Cube(x) will be
symbolized in Beta as:
To express a claim like x Cube(x),
we use the Quantifier Negation or
Quantifier DeMorgan Equivalences.
So, since x Cube(x)  x Cube(x),
we express x Cube(x) in Beta as:
Cube
Cube
The Boolean Square of
Opposition
‘Everything is P’
‘Nothing is P’
x P(x)

x P(x)
P
P
x P(x)

x P(x)
x P(x)
P
P
x P(x)
‘Something is P’
‘Something is not P’
: Contradictories
The Aristotelean Square of
Opposition
‘Every P is Q’
‘No P is Q’
P
x (P(x)Q(x))

x (P(x)Q(x))
Q
P
Q
x (P(x)Q(x))

x (P(x)Q(x))
P
x (P(x)Q(x))
P
Q
P
‘Some P is Q’
x (P(x)Q(x))
Q
‘Some P is not Q’
: Contradictories
Beta Inference Rules
• Beta inherits the four inference rules from Alpha,
but adds a few things to those rules to deal with
objects and predicates.
• Beta does not introduce any new rules.
• In defining the rules, the notions of double cut,
level, and nested level remain the same, while
graphs are understood to be Beta Graphs.
• We do, however, have to define a new notion,
namely that of a nested object.
Nested Objects
• An object y is nested with regard to object x
if and only if there is a line going from x to
y that does not go outside any cut. Example:
a
b
In this graph, b is nested with
regard to a.
a
b
In this graph, b exists at a nested level
with regard to a, but b is not a nested
object with regard to a.
Beta Inference Rules: Insertion
The only addition to the rule of Insertion is that on
an odd level, two lines may be connected. Example:
P
IN
P
Q
IN
P
Q
It may be useful, however, to define Insertion in such
a way that the above two steps can be taken at once:
P
IN
P
Q
Beta Inference Rules: Erasure
The only addition to the rule of Erasure is that on
an even level, two lines may be disconnected. Example:
P
Q
E
P
Q
E
P
Again, it may be useful to define Erasure in such
a way that the above two steps can be taken at once:
P
Q
E
P
Beta Inference Rules: Double Cut
The only addition to the rule of Double Cut is that any
lines that pass all the way through a double cut can
be ignored. Example:
P
Q
DC
P
Q
Beta Inference Rules:
(De)Iteration
The only addition to the rule of Iteration is that any
n-ary predicate R (including the empty predicate)
relating objects y1 … yn can be copied into any level
and attached to objects z1 … zn as long as the copy
exists at a nested level with regard to the original and to
each of the objects z1 … zn and there exists a series of
objects x1 … xn such that each yi and zi are nested
objects with regard to xi. Example:
P
IT
P
IT
P
P
Example
C
C
C
IN
IT
C
x Cube(x)
x (Cube(x)  Small(x))
 x Small(x)
C
S
C
S
S
DE
C
C
E
S
DC
S
S