Existential Graphs: Beta Introduction to Logic

Alpha Review: Symbolization ‘P’ ‘not P’ ‘P and Q’ ‘P or Q’ ‘if P then Q’ F EG PP PP P  Q P  Q P  Q P Q QP P QP

Alpha Review: Inference Rules Double Cut (De)Iteration Erasure Insertion        12k  1   12k+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: a Placing this on the SA asserts the existence of the object we named: 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 a relationship R between two or more objects, write the predicate symbol between the objects: R To express that an object does not have a certain property P, use the cut: P

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: Cube 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

The Boolean Square of Opposition  x P(x)   x  P(x)  x  P(x)  x P(x)  x  P(x)   x P(x) : Contradictories PP P P ‘Everything is P’‘Nothing is P’ ‘Something is P’‘Something is not P’

The Aristotelean Square of Opposition  x (P(x)  Q(x))   x (P(x)  Q(x))  x (P(x)  Q(x))  x (P(x)  Q(x)) : Contradictories P P ‘Every P is Q’‘No P is Q’ ‘Some P is Q’‘Some P is not Q’  x (P(x)  Q(x))   x (P(x)  Q(x)) Q P Q P Q P Q

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 Q P P Q IN It may be useful, however, to define Insertion in such a way that the above two steps can be taken at once: P Q P IN

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 PP Q E E Again, it may be useful to define Erasure in such a way that the above two steps can be taken at once: P Q P E

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 P Q DC

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 y 1 … y n can be copied into any level and attached to objects z 1 … z n as long as the copy exists at a nested level with regard to the original and to each of the objects z 1 … z n and there exists a series of objects x 1 … x n such that each y i and z i are nested objects with regard to x i. Example: P P IT P P

Example  x Cube(x)  x (Cube(x)  Small(x))   x Small(x) C C S C C S C C S C S C S S IT IN DE DCE