Infinite class versus finite class

The University of Wales, Trinity Saint David, London. 

Infinite class versus finite class

Preface

March 14 2019 edition, slightly edited for a version posted on academia.edu, September 10, 2023.

LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York. (Philosophy)

I earned my PhD at the Lampeter campus.

The review of the Langer text continues, and my learning symbolic logic continues:

Key symbols

≡df = Equivalence by definition
: = Equal (s)
ε = Epsilon and means is
⊃ = Is the same as
⊨ is Entails
˜ = Not
∃ = There exists
∃! = There exists
∴ = Therefore
. = Therefore
< = Is included
v = a logical inclusive disjunction (disjunction is the relationship between two distinct alternatives).
x = variable
. = Conjunction meaning And
0 = Null class
cls = Class
int = Interpretation

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Infinite class versus finite class

Philosopher Langer on page 214 further explains that in symbolic logic the uniqueness of 0 and 1 is guaranteed and therefore a more important equation as the system of the Law of Tautology can be shown. (214).

Tautology is repeating the same idea, not identically.

These propositions are called 'tautology' because they show that no matter how many times a term is mentioned in a sum or in a product (within symbolic logic, my add), a product is not changed by being multiplied or by something in it, nor a sum by having one of its summands added to it. (215).

Cited 

Britannica (online)

Britannica/Tautology

tautology, in logic, a statement so framed that it cannot be denied without inconsistency. Thus, “All humans are mammals” is held to assert with regard to anything whatsoever that either it is not a human or it is a mammal. But that universal “truth” follows not from any facts noted about real humans but only from the actual use of human and mammal and is thus purely a matter of definition. 

To simplify, Langer writes:

A class of dogs is simply a class of dogs. (215).

Adding of multiplying dogs in that class, does not change the fact it is only and simply a class of dogs.

Therefore, my examples demonstrate that addition or multiplication does not change a class of dogs.

z= Dogs

z x z = z

z + z = z 
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Langer explains that the propositions using tautology will use no exponents. (215). In other words, in multiplication, there will not be a smaller exponent number present, to the right of the base number. (215).

This is in the context of multiplication.

Therefore, z x z = z, and z2, z3 and related, etcetera is not used. (My example, based on Langer (1953)(1967: 215).

In a similar way with addition 2z cannot be arrived at with z + z = z.

With addition, 23, 34 etcetera is not arrived at. (My example, based on Langer (1953)(1967: 215).

Cited

...if 1 is added to anything, the sum is 1, and if anything is multiplied by 0 the product is 0. Langer (1953)(1967: 215).

My philosophical example based on reviewing the Langer text:

The infinite class (God) is simply infinite, nothing can be multiplied or added to that class.

The finite class is simply finite, nothing can be multiplied or added to that class.

This logic would counter philosophies and theologies reasoning the finite can become infinite.

i = infinite
f = finite

i ˜ ⊃ f

The infinite is not the same as the finite.

i ˜ ⊨ f

The infinite does not entail the finite.

And I add:

i . ˜ f

The infinite is therefore, not the finite.

i : ˜ f

The infinite equals not, the finite.

(∃!  i) + (∃! f) ˜ : 

There exists the infinite, plus there exists the finite, they are not equal. The class of dogs is simply a class of dogs. (215). Using the same logic, there is an infinite class and an finite class. They are separate.

Theologically and philosophically, this idea could be used to document God/The First Cause as within the infinite class and separate from creation, which would be within the finite class. 

Certainly, there are other, and in my opinion, more clear ways to explain this type of theology and philosophy, but I am attempting to stay true to Langer's symbolic logic. 

For example, there are different types of dogs, but for our case here, only one class of dogs. There are of course, different finite entities and types of finite entities (human being versus angelic, for example), but for our case, only one finite class, in contrast to one infinite class.

Note that in the incarnation, the infinite nature of God the Son, within the Trinity, does not mix with his finite human nature. This remains true as the resurrected Christ, Jesus Christ, has two natures, divine (infinite) and human (finite).

ASHBY, E G. (1986) 'Colossians' in The International Bible Commentary, Grand Rapids, Zondervan.

CRANFIELD, C.E.B. (1992) Romans: A Shorter Commentary, Grand Rapids, William B. Eerdmans Publishing Company. 

ERICKSON, MILLARD J. (1994) Christian Theology, Grand Rapids, Baker Book House.

HEBBLETHWAITE, BRIAN, 'Incarnation' in A New Dictionary of Christian Theology, London, SCM Press. 

HEWLETT, H.C. (1986) 'Philippians' in The International Bible Commentary, Grand Rapids, Zondervan.

LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York. (Philosophy)

MARTINS, RALPH P. (1987) Philippians, Grand Rapids, IVP.

MOUNCE, R.H. (1995) The New American Commentary: Romans, Nashville, Broadman & Holman Publishers.

REYMOND, R.L. (1996) 'Incarnation' in Evangelical Dictionary of Theology, Grand Rapids, Baker Books.

The Editors of Encyclopaedia Britannica/Brian Duignan (2023) Tautology, Britannica. https://www.britannica.com/topic/tautology

THEISSEN, HENRY, CLARENCE (1956) Introductory Lectures in Systematic Theology, Grand Rapids, Eerdmans. 

WRIGHT, N.T. (1989) Colossians and Philemon, Grand Rapids, IVP.