Symbolic Logic

I put this on my website a long time ago, maybe around 1996, as an HTML page. This is it moved to my blog.

The structure of this explanation is lifted from ‘An Introduction to Symbolic Logic’ by Susanne K. Langer, without her permission.

1.

Content:

The things or material in a system.

Form:

The way in which the contents are related in a system.

Abstraction:

Separating Form from Content, sometimes by discovering analogies.

Interpretation:

Finding possible Content for Forms.

2.

Degree:

Number of Elements used by a Relation. e.g. Dyadic: ‘is north of’, Triadic: ‘is
between’.

Proposition:

Asserts that the Elements are related by the Relation. e.g. ‘Edinburgh’ nt ‘Swindon’.

Symbols:

=int means ‘equals by interpretation’. e.g ‘nt2’ =int ‘is north of’.~ means ‘is not
true’. e.g. ~’Swindon nt Edinburgh’.

3.

Context:

Consists of elements and relations.

e.g. K(‘Brighton’, ‘Swindon’, ‘Edinburgh’) nt2 =int ‘cities’2 =int ‘is north of’

Universe Of Discourse:

All Elements in the context. e.g. K(a,b,c,…) K=int ‘cities’

Constituent Relations:

The relations used in the context. e.g. nt2 =int ‘is north of’

Elementary Propositions:

The statements that may be made by relating Elements.

e.g. ‘Edinburgh’ nt ‘Swindon’.

Truth Value:

Whether the Elementary Proposition is true or false. e.g. ‘Swindon’ nt ‘Edinburgh’ is false.

Symbols:

⋅ means Conjunction (‘and’)
∨ means Disjunction (‘or’)
⊃ means Implication (‘implies that’)

Logical Relations:

When the truth of one Elementary Proposition is dependent upon the truth of others they
are Logically Related.

e.g. (‘Edinburgh’ nt ‘Swindon’) ⋅ (‘Swindon’ nt ‘Brighton’) ⊃ (‘Edinburgh’ nt ‘Brighton’)

System of Elements:

Context with Elementary Propositions connected by Logical Relations

4.

Variables:

As in algebra. e.g. x means ‘Edinburgh’ or ‘Swindon’ or ‘Brighton’ etc.

Allows us to summarise Logical Relations of the same form.

e.g. (a nt b) ⋅ (b nt c)  (a nt c)

Quantifiers:

The Logical Relation may hold for All or Some of the variables.

Universal Quantifier:
(a) means ‘for all a’. e.g. (a) : (a nt b)  ~(b nt a)

Particular Quantifier:

(∃a) means ‘for at least one a’. e.g. (∃a): (a nt ‘Swindon’)

Propositional Form:

Elementary Proposition or Logical Relation using Variables, whose Truth Value would depend upon the actual Elements substituted. e.g ‘a nt b’.

General Proposition:

Propositional Form with Quantifiers.

5.

Class:

∈ means ‘is a member of’. e.g. ‘Murray’ ∈ B, where B =int ‘Class of Humans’.

General Propositions (see above) concern Classes of Elements.

Defining Form:

Defines the Class in terms of Propositional Forms. The Class contains all elements for which the Propositional Form is True.

Intension:

Meaning of a concept. e.g. the class ‘town’.

Extension:

The elements to which the concept applies. e.g. the class of ‘towns’.

N.B. Classes with unrelated Intensions may share some Elements in Extension.

e.g. ‘Towns north of Swindon’ and ‘Towns with Universities’.

Inclusion:

(x): (x ∈ A)  (x ∈ B) means Class A is included in Class B, by stating that any Element in A is therefore in B.

Unit Class (I):

Has one member, meaning that if two Elements are both in A then they must both be the
same Element.

(∃x) (y): (x ∈ A) ⋅ [ (y ∈ A) ⊃ (x=y) ]

The Null Class (o):

Has no Elements. There is a single Null Class, because two null classes could not be distinguished.

The Universe Class:

Contains all Elements. There is a single Universe Class.

Mutual Inclusion:

Classes have same Elements so each Class is included in the other.

(x): [ (x ∈ A) ⊃ (x Î B) ] ⋅ [ (x ∈ B) ⊃ (x ∈ A) ]

Class symbols:

< means Inclusion. e.g. A < B means (x): (x ∈ A) ⊃ (x ∈ B)
X means Conjunction (and). e.g. A X B means the Elements which are in A and in B. X is often omitted e.g. AB
+ means Disjunction (or). e.g. A + B means the Elements which are in A or in B.
– means Complement. e.g. -A means the Elements not in A.
= means Mutual Inclusion e.g. A = B means (A < B) . (B < A)

N.B. A<A, A<I, 0<A

Mutual Exclusion:

Classes have no Elements in common.

A X B = o

Dichotomy:

The fact that I = A + -A.

Exclusion:

A < -B means A is excluded from B.

7.

Predicate:

e.g. A, -A, A X B, A + B

Predicative Propositions:

Propositions about Predicates

System of Classes:

K(a,b,c…) <, similar to System of Elements but with Classes instead of Elements and < as the constituent relation.

Dots instead of brackets:

e.g. :. a : b . cd : e instead of ( a . ( b . ( c . d) ) . e)

You’ll get the hang of it.

8.

Calculus of Classes:

Describes the System of Classes for all Classes, just as the Calculus of Numbers
describes the System of Numbers for all Numbers.

Shows how to deduce some Propositions from others.

Postulates.

Basic Propositions of the system. e.g. (a, b) . a + b = b + a.

There are ten Postulates of the Calculus of Classes, analogous to the uses of Venn Diagrams.

Axioms:

Self-evident Postulates that are assumed because they can not be deduced.

9.

Boolean Algebra of Classes

Generalised Calculus of Classes, just as Algebra of Numbers is generalised Calculus of
Numbers.

Laws of Duality

Conjunction can be defined in terms of Disjunction and vice-verse.

Primitive Propositions

The propositions used to prove theorems in Boolean Algebra. They may use either
Conjunction or Disjunction. They show the following:

Operational Assumptions:

Existence of complements, sums, products.

Existential Assumptions:

Existence of Universe Class, Null Class, more than one Class.

Laws of Combination:

Tautology e.g. ‘a + a = a’
Commutation e.g. ‘a + b = b + a’
Association e.g. ‘(a + b) + c = a + (b + c)’
Distribution e.g. ‘a + (b X c) = (a + b) X (a + c)’
Absorption e.g. ‘a + ab = a’

Laws of the Unique Elements:

Universe Class e.g. ‘a + 1 = 1’
Null Class e.g. ‘a + 0 = 0’

Laws of Negation:

Complementation e.g. ‘a + -a = 1’
Contraposition ‘a = -b . ⊃ . b = -a’
Double Negation e.g. ‘a = -(-a)’
Expansion e.g. ‘ab + a-b = a’
Duality e.g. ‘-(a + b) = -a X -b’

10.

System in Abstracto.

K R, where K is the universe of something and R is some way of relating these things.

Properties of Relations:

Reflexiveness e.g. ‘(a). a R a’
Symmetry e.g. ‘(a, b). a R b .  . b R a’
Transitivity e.g. ‘(a, b, c): a R b . b R c .  . a R c’

11.

Propositional Calculus.

Uses a Universe of Propositions which are either True (1) or False (O)

p means ‘p is true’ or ‘p=1’, leading to ‘p=(p=1)’.

12.

Calculus of Elementary Propositions.

Used by Principia Mathematica by Russel & Whitehead. Improves on above flawed notation.

† means ‘it is asserted that’ e.g. ‘†: p V q .  . q V p;

13.

Function and Argument:

A Proposition consists of a Function and Arguments.

e.g. ϕx instead of p, where f is the function and x is the argument.

We may quantify the argument instead of the whole proposition to show that functions which are not identical are formally equivalent, allowing us to express ‘(x): mortal(x) =
will_die(x)’

Logistics:

Seeks to create a logical foundation for mathematics.

Recommended Reading:

An Introduction to Symbolic Logic: Susanne K. Langer

Godel, Escher, Bach : An Eternal Golden Braid, Douglas R. Hofstadter

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