Logic is included in truth

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Logic is included in truth

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

The review continues...

Me 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
---

From back in April, 2019 (please see archives for my previous work and reviews):

Importantly, philosopher Langer explains that there is no guarantee that there is truth in a logical system. (189). Logic does not necessarily promote a fact, rather 'it stands for the conceptual possibility of a system'. (189). Logic documents with the deduction of premises. It stands for 'the consistency of all propositions'. (189). It is standing for logical validity. (189), not factual certainty or truth. (189). This is standard from philosophy, logic, texts. Certainly not something Langer or I manufactured as original.

Langer demonstrates the following as logical:

Napoleon discovered America

Napoleon died before 1500 A.D. (189).

Conclusion

America was discovered before 1500 A.D. (189).

These two premises imply that America was discovered before 1500 and Langer opines that a third proposition that would be derived (a conclusion, my add) would also be logical and valid. (189). 

Indeed the first two premises are historically false. (189). They are still logically consistent, while the consequent is true that America was discovered before 1500 A.D. (189).

Also logical, but a true premise: n= Napoleon d= Discover a= America

n ˜ (d+a)

Napoleon did not discover America.

January 13 2020

Langer mentions that the text shows that for every proposition there is also an analogous one (221). If there is an entity that when multiplied with any term, leaves that term unchanged, then there is also one that can be added to that term without altering it (221).

Langer's theorem:

-(a + b) = -a x -b (221)

She notes that this theorem is the complement of a + b. (221).

The complement is the amount added to something to make it whole. Each entity needs to complete the other is a universe of discourse. (143).

Langer writes (paraphrased) that she is not explicitly explaining her argument here. (222). She states in regards to breaking down the theorems...

'But this is left to the brave and ambitious reader.' (223).

But in philosophical terms, for the sake of logic, her theorems represent the law of duality. (223). The law of duality between + and x. (223). The theorems which explain the relation between sums (+) and products (x) express this law of duality. (223). The relation between addition and multiplication.

Webster

Definition of algebraic sum : the aggregate of two or more numbers or quantities taken with regard to their signs (as + or −) according to the rules of addition in algebra the algebraic sum of −2, 8, and −1 is 5

Study.com

What Is a Product? When speaking mathematically, the term product means the answer to a multiplication problem. For example: 5 * 3 = 15 

15 is the product The term product first showed up in England in the 1400s and comes from the Latin word productum, which means 'to produce.'

Philosophical relevance?

Philosopher (and Mathematician) Langer opines that everything she noted about sums is also true of products, (224). I am a philosopher and not a mathematician, but seems to me, she is demonstrating the logic and consistency of symbolic logic within algebra, mathematics and philosophy.

Overall, I reason, symbolic logic has minimal practical use, even within most philosophy. But I appreciate that Langer demonstrates the consistency of logic, and as well that the logical is not necessarily true. But, in my embraced philosophy and theology, the truth is always logical. In other words, the truth always can be made sense of with reasonable premises and conclusions.

l = Logic
t = Truth

l ˜ = t

(Logic does not equal truth, strictly philosophically speaking)

l < t

(Logic is included in truth)

(l < t) ˜ ⊨ (l = t)

(Logic is included in truth, does not entail logic equals truth)

Research and study within four academic degrees and years of academic website writing has shown me that I have read and researched many logically presented premises and conclusions, meaning, many logical arguments, that are not likely true.

BLACKBURN, SIMON (1996) Oxford Dictionary of Philosophy, Oxford, Oxford University Press. 

CONWAY DAVID A. AND RONALD MUNSON (1997) The Elements of Reasoning, Wadsworth Publishing Company, New York. 

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

PIRIE, MADSEN (2006)(2015) How To Win Every Argument, Bloomsbury, London. 

Langer's theorem: 

quote 

-(a + b) = -a x -b (221) She notes that this theorem is the complement of a + b. (221). 

Further 

-a + (a + b) = 1 (221) -1 + (1 + 1 = 2) = 1 (My add) -a x ab = 0 -1 x a-b (1-1 = 0) = 0 (With assist from 221) 

There is a law of absorption: a absorbs any sum of itself and any term multiplied with any other product. (217). a x (a + b) means the common part of a and (a + b) which is just a. (217). a + (a x b) means the class of a or a x b is also just a. (217-218). Not just b. Same with c, d, e, f, g, h, i, etcetera.

Brought to you by the letter 'R'/Moderate left meets moderate right

I finally lead with a photo
of this textbook.

Brought to you by the letter 'R'

'R' quite convenient...

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

The review 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

∧ = Logical conjunction

# = Higher in pitch

Monday, May 18, 2020: White house versus black house 

Previously 

This book review of sorts, since 2016, has now advanced to Chapter X: Abstraction and Interpretation. Philosopher Langer explains that logic is the study of forms and these forms are derived within systems from common human experiences, reality and life. (240). This is done by abstraction. (240). 

She further explains that the science of logic is a continued progression from the concrete to the abstract. (240). That would be concrete ideas and things to abstract symbolic logic. 'From contents with certain forms to those forms without contents, from instances to kinds, from examples to concepts.' (240). Langer explains that the first step is to the replacement of individual elements by formalized elements of variable meaning. (240). These are formalized elements, as in symbols within symbolic logic. The meaning of these elements is 'presently fixed.' (240). Not to be interpreted by their original terms. (240). Langer states that the symbolic logic has them interpreted in 'an entirely new way.' (240). 

From Langer's explanation, what the symbolic logic provides is through quantifiers, are the old elements (which set meanings in contexts, my add), by new terms, which are general terms. (240). Symbolic logic provides a degree of formation from specific elements to quantified variables that are general terms. (240).

August 9, 2020

Properties of Relations is section 2 in Chapter X: Abstraction and Interpretation. Philosopher, Langer explains that with a general or abstract proposition, it is stated 'there is at least one relation, R having certain properties; and the form of the proposition to be expressive of those properties. Relations which have all their logical properties in common are of the same type, and are possible values of the same variable R.' (246).

Langer explains that the most fundamental characteristic of a relation is its degree. (246). Forming dyads, triads, tetrrads, etc.. (246).  Sets of 2, 3, 4, etc..my add. A symbol of R2 (246) is also in the form of a R b. (246). The symbolic logic symbols of 'a' and 'b' here are considered identical. (246). These are known as reflexive. (246). 

Taking one of the examples:

(a) . ˜ (a nt a) (247).

(A) therefore not (house 'a' is north of house 'a')

In other words, house 'a' is not north of itself.

A non-reflexive symbol possibly, but not necessarily, combines a term with itself. (257). 

Langer example:

(∃a) . a likes a (247)

(A exists) therefore 'a' likes 'a' 

(∃a) . ˜ (a likes a) (247)

(A exists) therefore 'a' does not like 'a'

Langer implies that a creature may or may not like itself. (247).

Practical philosophy

I have noted within the review series that it is significant academic work to make Langer's complex, technical, textbook, practical. In reality, propositions, premises and conclusions will be far more presented in written language as opposed to symbolic logic. Regardless of the format presentations should be logical and reasoned for accuracy. 

Perhaps the comparison of reflexive and non-reflexive terms demonstrates the requirement for the sake of reason, of considering possibilities, as in counter propositions that are in disagreement with propositions held to. This requirement was a definitive and definite aspect of my British MPhil/PhD works, even more so that my previous Canadian work. 

Considering contrasting statements and arguments does not necessarily lead to a change of view, but it may lead to a superior, wider, deeper understanding of subjects under review.

Moderate left meets moderate right

Zoom last night. The moderate left meets the moderate right.
British Columbia. Florida, Norway...

People will be ticked

The losing Toronto Maple Leafs, at this point tonight, might join the Edmonton Oilers, Winnipeg Jets, Pittsburgh Penguins and four other teams with a reasonable possibility of winning the first pick overall in the next entry draft. The winner will be announced Monday. The pick is expected to be Alexis Lafrenière, a possible all-star. If the Leafs lose tonight, the result will offend many Leaf/Oiler fans and non-fans, alike.

White house versus black house

Lampeter, Wales

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

The review 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
∧ = Logical conjunction
---

White house versus black house

This book review of sorts, since 2016, has now advanced to Chapter X: Abstraction and Interpretation. Philosopher Langer explains that logic is the study of forms and these forms are derived within systems from common human experiences, reality and life. (240). This is done by abstraction. (240).

She further explains that the science of logic is a continued progression from the concrete to the abstract. (240). That would be concrete ideas and things to abstract symbolic logic.

'From contents with certain forms to those forms without contents, from instances to kinds, from examples to concepts.' (240).

Langer explains that the first step is to the replacement of individual elements by formalized elements of variable meaning. (240). These are formalized elements, as in symbols within symbolic logic. The meaning of these elements is 'presently fixed.' (240). Not to be interpreted by their original terms. (240). Langer states that the symbolic logic has them interpreted in 'an entirely new way.' (240).

From Langer's explanation, what the symbolic logic provides is through quantifiers, are the old elements (which set meanings in contexts, my add), by new terms, which are general terms. (240). Symbolic logic provides a degree of formation from specific elements to quantified variables that are general terms. (240).

Encyclopaedia Brittanica

Cited

Quantification, in logic, the attachment of signs of quantity to the predicate or subject of a proposition.
---

Langer continues by explaining that it has been established in her text (and my reviews) that K =int as houses. (241). K (a, b, c, d...etc) is various types of houses. (241). At the same time, in her text (and my reviews) nt =interpretation as north of. (241).

It could be written that:

wh= White house

bh= Black house

wh ˜ ⊃ bh

The white house is not the same as the black house.

(wh) ˜ ⊃ (bh)

The white house is not the same as the black house.

(wh) . nt (bh)

The white house is therefore north of the black house.

(bh) ˜ nt (wh)

The black house is not north of the white house.

(bh) ˜ ⊨ (wh)

The black house does not entail the white house.

Practical philosophy

Ten Chapters and over four years into this textbook review:

Positive: Philosophically, the book assists the reader to better understand the technical differences between logic and truth, the logical and the true.

I now have a greater familiarity with the terms and symbols. I can decently read the equations in Langer's textbook, correctly. Potentially reading symbolic logic, more than the limited amount I read for my MPhil/Ph.D. work, in philosophical journals and books was a reason I bought the Langer textbook for review.

Negative: It is quite clear that most commonly for typical readers, academics, and most philosophers, written prose and standard language is generally a more clear, reasonable and proficient method for presenting concepts, premises and conclusions than is symbolic logic.

Langer states that the symbolic logic has them interpreted in 'an entirely new way.' (240).

Many times in everyday writing and in academia, explaining the concrete reasonably and in truth is more beneficial for most readers than creating an abstraction with its own internal rules that requires significant new learning from the reader. I reason that symbolic logic does have its merits at some technical points.

Laurel Bern photoshop via online websites. There is nothing
 politically intended by me as it just fits this section of the Langer text.

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 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

---

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 
---

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. 

Be like Bill of the human being class

Be like Bill of the human being class

Photo: Buenos-Aires-homes-1-of-1 gateway to south america 2019

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

This book review of sorts, since 2016, 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

∧ = Logical conjunction 

From the previous related entry

Sunday, August 09, 2020 Brought to you by the letter 'R'/Moderate left meets moderate right 

Properties of Relations is section 2 in Chapter X: Abstraction and Interpretation. 

Philosopher, Langer explains that with a general or abstract proposition, it is stated 'there is at least one relation, R having certain properties; and the form of the proposition to be expressive of those properties. Relations which have all their logical properties in common are of the same type, and are possible values of the same variable R.' (246). 

Langer explains that the most fundamental characteristic of a relation is its degree. (246). Forming dyads, triads, tetrrads, etc.. (246). Sets of 2, 3, 4, etc..my add. A symbol of R2 (246) is also in the form of a R b. (246). The symbolic logic symbols of 'a' and 'b' here are considered identical. (246). These are known as reflexive. (246). 

Taking one of the examples: 

(a) . ˜ (a nt a) (247). 

(A) therefore not (house 'a' is north of house 'a') 

In other words, house 'a' is not north of itself. A non-reflexive symbol possibly, but not necessarily, combines a term with itself. (257). 

Langer example:

(∃a) . a likes a (247).

(A exists) therefore 'a' likes 'a' 

(∃a) . ˜ (a likes a) (247) 

(A exists) therefore 'a' does not like 'a' 

Langer implies that a creature may or may not like itself. (247).

March 18 2021

Cited

'A transitive relation is such that if it relates two terms to a mean (average my add), it relates the extremes to each other. The significance of this trait lies in the fact that it allows us to pass, by the agency of a mean term, to more and more terms of which is thus related to every one of the foregoing elements. This creates a chain of related terms; in ordering a whole universe of elements, such a relation which transfers itself from couple to couple when new terms are added one at a time, is of inestimable value (too great to accurately calculate in value, my add). This is the type of relation by virtue of which we reason from two premises, united by a mean or "middle terms," to a conclusion''. (248).

I will not use Langer's now non-politically correct and offensive to many in 2021, language, that was used commonly in the 1950's and 1960's. But the following is based on Langer on page 248.

All Canadians are human beings

All human beings are mortals

-----------------------------------

Therefore all Canadians are mortals

---

British philosopher, Pirie documents that the standard three line argument requires that one term be repeated in the first two lines, and not be within the conclusion. (171). This is in the context of syllogistic reasoning. (171). Another British philosopher, Blackburn, explains that a syllogism (above) is the presentation of one proposition from two premises. (368). In other words, two premises (propositions) and then a conclusion.

Note that academic arguments do not have to be syllogistic to be logically valid and reasonable.

Cited link for reference

Thursday, July 13, 2017: Quaternio terminorum: The fallacy of four 

---

Therefore < = Is included

< Canadians, human beings, mortals (based on Langer 248). 

They are taken as three classes as transitive, but if there was no relation between the classes it would be intransitive. (248). Back on page 115, Langer writes that a class has members that have certain character. (115).

Related equations

Canadians=c

Human beings=h

Mortals=m

(∃c) < (∃h) ∴ (∃m)

Canadians exist, is included in human beings exist, therefore mortals exist

(∃c) ⊨ (∃h) = (∃!m)

Canadians exist, entails human beings exist, equals mortals exist

Practical philosophy

The use of a class (term) and related classes (terms) as transitive can assist in the development of valid, logical, reasonable, premises and conclusions (arguments).

BLACKBURN, SIMON (1996) Oxford Dictionary of Philosophy, Oxford, Oxford University Press. 

PIRIE, MADSEN (2006)(2015) How To Win Every Argument, Bloomsbury, London. 

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

From Facebook: Most of the time I reason it wise to be like Bill.

Value of Symbolic Logic for Science & Philosophy

Value of Symbolic Logic for Science & Philosophy

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

Preface

Unlike with my review of the Pirie text, the Langer review text never ended. But I will end this non-exhaustive review with this article, and of course continue to use the book as reference. My PhD was in philosophical theology and philosophy of religion, and my website work consists mainly of these academic disciplines along with biblical studies and philosophy. I am not a scientist or mathematician, but I have reviewed symbolic logic, which has mathematic symbols, for presenting propositions and premises.

Of course when I use science and mathematics, it needs to be accurate. This book review has strengthened my understanding of formal logic as a system, just as the Pirie text review has helped me to better understand informal logic. 

A formal fallacy occurs when a logical form is not used, and therefore is illogical in structure, and an informal fallacy occurs when there are errors in reasoning with a premise (s) and conclusion. In the similar way, formal logic is concerned with a logical form, to follow the rules of a logical system, to avoid being illogical. Informal logic is attempting to avoid fallacious reasoning with use of premise (s) and conclusion. 

Key symbols from Langer text

≡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 

∧ = Logical conjunction

# = Higher in pitch

---

The Value of Logic for Science and Philosophy

Langer opines that the development of logic, such as is used within symbolic logic, is not dependent on psychology or metaphysics. (332). In contrast, the author reasons that logic has greatly influenced the development of science (332-333), and at the same time has 'shifted many a philosophical point of view' (332-333). Using logic it is asked, what are the presuppositions of a view? (333). What are the premises of a view? (333). I agree that presenting logical, reasonable and true premises is crucial within credible academic work.

Langer explains that philosophy, unlike science, does not use sense experience to check errors all the time (333). My add, philosophy is not empirical, at least primarily. It is using reason. I would not go so far to state that the empirical does not at times influence reason, of course it does. Theology may be considered 'philosophy in regards to God', my Reformed, biblical, Christian theology holds to the post-mortem doctrine of the resurrection of Jesus Christ (the gospels/Acts/Revelation, as examples) and the future post-mortem resurrection (1-2 Thessalonians, Revelation 20-22, as examples) of regenerate (John 3, Titus 3, 1 Peter 1, as examples) believers based on the historical, empirically viewed resurrection of the God-man, Jesus Christ.

Cited

'Here let it only be said that general logic is to philosophy what mathematics is to science; the realm of its possibilities, and the measure of its reason.' (334).

Author summary of book

Langer writes that logistics is a specialized system of logic (334), with the purpose to show that the fundamental assumptions of mathematics are all purely logical notions (334), and therefore all mathematics may be deduced within a system of logic. (334).

A number is defined as a class of classes having a certain membership (335). That ''0" is the number in the class of empty classes (335). That "I" is the class of all classes with only identical members (335). 

Cited

'The process of forming a "member" is to define the numerosity of a given class without reference to the number, and then establish the class of all classes similar it. Two classes are similar if the members of one may be put into one to one correspondence with the members of the other. The concept "number", itself, denotes the class of all such classes of similar classes.' (335).

Generalized System of Classes 

Earlier in the Langer text, I reviewed the following: 

Monday, December 11, 2017: An empty class 

This review has progressed where we are now at the point in the textbook where philosopher, Langer explains that we have passed from a system of individuals and predicates, such as a class of white houses (wt) and a class of brick houses (bk). (171). 

This leads to a system of certain classes

< = Is included as in houses = white houses and brick houses. (171). 

Etcetera, including red houses (rd), green houses (gn), wood houses (wd). 

This means that in any universe whose elements are classes there is one class having the logical properties of 'the class of no houses'. (172). This is also known as an empty class, and this class is included in every class of the universe. (172). Langer explains that in each universe there is one 'greatest class' which is analogous to 'the class of all houses'. (172-173). This includes every class is the universe. (173). Langer means in this context, the universe of discourse for symbolic logic. 

Therefore, for any class, there is at least one class 0 included. Therefore, for any class, there is at least one class 1 included. 

(∃0) (a) : 0 < a 

There exists at least one class 0 that for any class a, 0 is included in a. (173). 

(∃1) (a) : 0 < a 

There exists at least one class 1 that for any class a, 1 is included in a. (173). 

0 represents there is a class of no houses in this universe of discourse. 

1 represents there is a class of houses in this universe of discourse. This specific system. (173). 

For any Universe of discourse, such as K (houses) whose elements are classes contains a 0 and a 1. (173). There are houses and non-houses. 

There are Christians and non-Christians, there are Canadians and non-Canadians, etcetera. 

(∃!) (cr) : 0 < cr 

There exists at least one class 0 that for any class cr (Christians), 0 is included in a. 

There is a class of no Christians, in this universe of discourse. 

(∃!) (cr) : 1< cr 

There exists at least one class 1 that for any class cr (Christians), 1 is included in a. 

There is a class of Christians, in this universe of discourse.

---

Boolean

Boolean is an aspect of algebra that is not powerful enough to support mathematics (335). But is used to present values instead of numbers, such as in symbolic logic. I reason symbolic logic also lacks the complexity of premise based, written argumentation. Similarly to symbolic logic, having developed and presented one sentence propositions for both MPhil and PhD questionnaires and surveys, these lack the context needed to develop deeper, sophisticated ideas. When answering these types of questionnaires, one is often left with filling in context and answering based on those deductions. The same could be stated for reviewing argumentation that is strictly using symbolic logic.

The calculus of elementary propositions

The calculus of elementary propositions is extended to general proposition by asserting that the function in an analyzed proposition is true. (336). Not with any specific argument (336), this has to do with format (my add). Because it is format, it has to do with the individual argument, presented this way (336). The calculus of elementary propositions is found to follow the pattern of the elementary calculus. (336).

Any individual, as in the quantifier (x) (336) is taken as primitive (336). Based on what Langer wrote, 

(x) : ax . ⊃ . bx

x equals ax therefore is the same therefore as bx

ax entails bx because the symbols that serve as functions are interchangeable. (336). Every function defines a class, 'namely the class of arguments which it is true.' (336). This class is its extension. (336).

Every function defines a class, namely the class of arguments for which it is true. (336). The class and its extensions. If a class is taken in extension, it can then be stated to be in classes. (336). Therefore, the calculus of classes may be derived from the calculus of general propositions. (336).

Relationship

Defining the relation between classes (336), the author explains that transitions from one sub-system to another have created some difficulties which have been met by developing the 'theory of logical types'. (337). This concept back to Properties of Relations is section 2 in Chapter X: Abstraction and Interpretation. 

With a general or abstract proposition, it is stated 'there is at least one relation, R having certain properties; and the form of the proposition to be expressive of those properties. Relations which have all their logical properties in common are of the same type, and are possible values of the same variable R.' (246). 

The most fundamental characteristic of a relation is its degree. (246). Forming dyads, triads, tetrrads, etc.. (246). Sets of 2, 3, 4, etc..my add. 

A symbol of R2 (246) is also in the form of a R b. (246). The symbolic logic symbols of 'a' and 'b' here are considered identical. (246). These are known as reflexive. (246). Taking one of the examples:

(a) . ˜ (a nt a) (247).

(A) therefore not (house 'a' is north of house 'a') 

In other words, house 'a' is not north of itself. A non-reflexive symbol possibly, but not necessarily, combines a term with itself. (257).

Langer example: 

(∃a) . a likes a (247). (A exists) 

therefore 'a' likes 'a' 

(∃a) . ˜ (a likes a) (247) (A exists) 

therefore 'a' does not like 'a' 

Langer implies that a creature may or may not like itself. (247). 

A transitive relation is such that if it relates two terms to a mean (average my add), it relates the extremes to each other. The significance of this trait lies in the fact that it allows us to pass, by the agency of a mean term, to more and more terms of which is thus related to every one of the foregoing elements. This creates a chain of related terms; in ordering a whole universe of elements, such a relation which transfers itself from couple to couple when new terms are added one at a time, is of inestimable value (too great to accurately calculate in value, my add). This is the type of relation by virtue of which we reason from two premises, united by a mean or "middle terms," to a conclusion''. (248). 

I will not use Langer's now non-politically correct and offensive to many in 2021, language, that was used commonly in the 1950's and 1960's. But the following is based on Langer on page 248. 

All Canadians are human beings 

All human beings are mortals 

----------------------------------- 

Therefore all Canadians are mortals 

--- 

Related equations 

Canadians=c 

Human beings=h

Mortals=m 

(∃c) < (∃h) ∴ (∃m) 

Canadians exist, is included in human beings exist, therefore mortals exist 

(∃c) ⊨ (∃h) = (∃!m) 

Canadians exist, entails human beings exist, equals mortals exist 

Practical philosophy 

The use of a class (term) and related classes (terms) as transitive can assist in the development of valid, logical, reasonable, premises and conclusions (arguments).

Langer finale

For the author, symbolic logic for science is a close relation to mathematics. (337). Logic is indispensable for philosophy because analysis of concepts is the only practical check for error. (338). I agree that propositions/statements always need to be checked for error. I agree that premises and conclusions need to be checked for errors. Symbolically presenting these premises may or may not add clarity to a situation, depending on the writer and as well, the reader. But admittedly, at times, I have found it useful to review premises individually before placing them within an argument in prose form, especially on website work. Such premises could theoretically be presented with symbolic logic and I have done so. Langer opines that symbolic logic 'offers a great deal of direct philosophical material'. (338).

ANDERSON, RAY S. (2001) The Shape of Practical Theology, Downers Grove, Illinois, InterVarsity Press.

BAVINCK, HERMAN (1918)(2006) Reformed Dogmatics Volume 2: God and Creation, John Bolt (gen.ed.), Translated by John Vriend, Baker Academic, Grand Rapids. 

BAVINCK, HERMAN (1918)(2006) Reformed Dogmatics Volume 3: Sin and Salvation in Christ, John Bolt (gen.ed.), Translated by John Vriend, Baker Academic, Grand Rapids. 

BLACKBURN, SIMON (1996) Oxford Dictionary of Philosophy, Oxford, Oxford University Press.

CALVIN, JOHN (1539)(1998) The Institutes of the Christian Religion, Book II, Translated by Henry Beveridge, Grand Rapids, The Christian Classic Ethereal Library, Wheaton College.

CALVIN, JOHN (1539)(1998) The Institutes of the Christian Religion, Book IV, Translated by Henry Beveridge, Grand Rapids, The Christian Classic Ethereal Library, Wheaton College. 

CALVIN, JOHN (1543)(1996) The Bondage and Liberation of the Will, Translated by G.I. Davies, Grand Rapids, Baker Book House.

DARROW, CLARENCE (1928)(1973) ‘The Myth of the Soul’ in The Forum, October, in Paul Edwards and Arthur Pap (eds), A Modern Introduction To Philosophy, New York, The Free Press.

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

ERICKSON, MILLARD (2003) What Does God Know and When Does He Know It?, Grand Rapids, Zondervan. 

FLEW, ANTONY, R.M. HARE, AND BASIL MITCHELL (1983) (1996) ‘The Debate on the Rationality of Religious Belief’, in L.P. Pojman (ed.), Philosophy, The Quest for Truth, New York, Wadsworth Publishing Company. 

FRANKE, JOHN R. (2005) The Character of Theology, Baker Academic, Grand Rapids.

GEBARA, IVONE (2002) Out of the Depths, Translated by Ann Patrick Ware, Minneapolis, Fortress Press.

KAVANAGH, AIDAN (1999) ‘Initiation, Christian’, in Alan Richardson and John Bowden (eds.), A New Dictionary of Christian Theology, Kent, SCM Press Ltd.

KIERKEGAARD, SOREN (1847-1848)(1955)(1966) On Authority and Revelation, Translated by Walter Lowrie, New York, Harper and Row, Publishers, Incorporated.

KIERKEGAARD, SOREN (1848-1849)(1961) Christian Discourses & The Lilies of the Field and The Birds of the Air & Three Discourses at The Communion on Fridays, Translated by Walter Lowrie, New York, Oxford University Press. 

KLEIN, WILLIAM W., CRAIG, C. BLOMBERG, AND ROBERT L. HUBBARD, JR. (1993) Introduction to Biblical Interpretation, London, Word Publishing. 

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

MARSHALL, ALFRED (1975)(1996) The Interlinear KJV-NIV, Grand Rapids, Zondervan.

MOLTMANN, JÜRGEN (1993) The Crucified God, Minneapolis, Fortress Press.

MOLTMANN, JÜRGEN (1999) ‘Perseverance’, in Alan Richardson and John Bowden (eds.), New Dictionary of Christian Theology, Kent, SCM Press Ltd. 

MOUNCE, ROBERT H. (1990) The Book of Revelation, Grand Rapids, William B. Eerdmans Publishing Company. 

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

MURRAY, JOHN (1937-1966)(1977) Collected Writings of John Murray, Vol. 2: Select Lectures in Systematic Theology, Edinburgh, The Banner of Truth Trust. 

PACKER, J.I. (1996) ‘Regeneration’ in Walter A. Elwell (ed.), Evangelical Dictionary of Theology, Grand Rapids, Baker Books. 

PHILLIPS, D.Z. (2005) The Problem of Evil and the Problem of God, Fortress Press, Minneapolis. 

PIRIE, MADSEN (2006)(2015) How To Win Every Argument, Bloomsbury, London.

SCHLEIERMACHER, FRIEDRICH (1799)(1961) On Religion, in Elie Kedourie, Nationalism, New York, Praeger University Series. 

SCHLEIERMACHER, FRIEDRICH (1821)(1928)(1976) The Christian Faith, Edited by H.R. Mackintosh and J.S. Stewart, Philadelphia, Fortress Press.

SCHRECK, ALAN (1984) Catholic and Christian, Ann Arbor, Michigan, Servant Books. 

SHEDD, WILLIAM G.T. (1874-1890)(1980) Dogmatic Theology, Volume 1, Nashville, Thomas Nelson Publishers. 

SHEDD, WILLIAM G.T. (1874-1890)(1980) Dogmatic Theology, Volume 2, Nashville, Thomas Nelson Publishers. 

THIESSEN, HENRY C. (1956) Introductory Lectures in Systematic Theology, Grand Rapids, Wm. B. Eerdmans Publishing Company. 

WEBER, OTTO (1955)(1981) Foundations of Dogmatics, Volumes 1 and 2, Translated and annotated by Darrell L. Guder, William B. Eerdmans Publishing Company. 

WHALE, J.S. (1958) Christian Doctrine, Glasgow, Fontana Books.

---

  

The clash of universes?

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

The continuation of text review:

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

Primitive concepts, terms and relations, within symbolic logic are not explained, but are simply 'taken for granted'. (167). These meanings are provided by interpretation only in the context provided. (167). The symbols for houses and related is an example as these symbolic interpretations. (167).

For clarity, philosopher, Langer writes that there is a new context assumed. (167). In this context the formal context has elements which are certain classes. (168).

Let us note that Langer adds another symbol: cls, which is the usual symbol for class.

From page 168:

K= int (interpreted) as class of houses
B = int (interpreted) as class of brick houses
W = int (interpreted) as class of white houses
-B = int (interpreted) as class of  not-brick houses
-W = int (interpreted) as class of not-white houses
B x W =int (interpreted) as class of white brick houses
---
0 = int (interpreted) as class of no houses
I = int (interpreted) as class of all houses

On page 170, Langer states that a very important point is that there is a difference between:

K = The universe of discourse (Is this context established by Langer)

&

I = The universe class

Langer warns against identifying the universe of discourse with the greatest class that is within it. (170). Langer explains that the error of equating the universe of discourse with the universe of class, was made by John Venn is his Symbolic Logic of 1881. Langer instead reasons that I does not equate with K, but rather I is an element within K. (170).

My equations

˜ (I ⊨ K)

The universe class does not entail the universe of discourse.

˜ (I ⊃ K)

The universe class is not the same as the universe of discourse.

In other words the universe of discourse contains the universe class. The universe class does not contain the universe of discourse.

The principles of logical proof

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

The review continues... Me 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
---

Previous entry

March 14, 2019

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 in 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).

Summary

Cited

'A calculus is any system wherein we may calculate from some given properties of our elements to others not explicitly stated.' (235).

Calculus is expressed in symbols in general terms and their relations in general it is in algebra. (236). The classes provided through general propositions is genuine algebra. (236).

The principles of logical proof...

Importantly, philosopher Langer explains that there is no guarantee that there is truth in a logical system. (189). Logic does not necessarily promote a fact, rather 'it stands for the conceptual possibility of a system'. (189). Logic documents with the deduction of premises. It stands for 'the consistency of all propositions'. (189). It is standing for logical validity. (189), not factual certainty or truth. (189). This is standard from philosophy, logic, texts. Certainly not something Langer or I manufactured as original.

In many cases when a person states that a premise or argument is logical, the person means that it is true. But a premise or argument can be logical and false. Therefore, it would be more accurate in many cases to claim that a premise or argument is true and or reasonable.

Stanford Encyclopedia of Philosophy

Cited

On standard views, logic has as one of its goals to characterize (and give us practical means to tell apart) a peculiar set of truths, the logical truths...

Langer demonstrates the following as logical:

Napoleon discovered America
Napoleon died before 1500 A.D. (189).
Conclusion

America was discovered before 1500 A.D. (189).

These two premises imply that America was discovered before 1500 and Langer opines that a third proposition that would be derived (a conclusion, my add) would also be logical and valid. (189). 

Indeed the first two premises are historically false. (189). They are still logically consistent, while the consequent is true that America was discovered before 1500 A.D. (189).

Also logical, but a true premise:

n= Napoleon
d= Discover
a= America

n ˜ (d+a)

Napoleon did not discover America.

Ambiguous Language

Poland: trekearth

Back to a review of the Langer text on Symbolic Logic, after a break since March as I was facilitating on a local church course and posting related articles.

The previous Langer post needs to be restated for context:

Chapter 2: The Essentials Of Logical Structure 

Langer provides further equations continued from the Chapter: pages 53-54

1. I played bridge with my three cousins

2. I played chess with my three cousins (53)

A=Speaker

B, C, D=Three cousins (54)

A br B, C, D (54)

If in chess each player was played separately

A ch B
A ch C
A ch D
A ch (B-C-D)* (54)

*The hyphen which could also be a + expresses an operation when the two terms are united as one. (54)

So this could be A ch (B+C+D)

I take it here the author means uniting B-C-D, as she explains this will be explained more later and must at this point be taken in faith. (54)

It is actually three terms, but I take the point and she means two or more.

---

'When a relation-symbol stands in a construct, the number of terms grouped with it reveals the degree of the relation. But when it is not actually used, but merely spoken of, it is sometimes convenient to have some way of denoting its degree. This may be done by adding a numerical subscript; for example, "kd2" means that "killing" is dyadic (a pair), "bt3" that "between" is triadic.' (55).

The examples of different degrees are provided:

ch2
br4 (55)

The author states that two beings named 'John' are not likely to be treated as the same in the language of discourse (56). It is made apparent in context that there is this John and that John. (56).

Symbolic logic provides a new medium of such expression. (57).

For example the following

John a

John b

Are a symbolic way of differentiating between two different persons named John using arbitrary symbols as Langer calls them. (58). Although the example is mine.

Langer writes natural language has a tendency to let one word have and embody many meanings and this leads to in philosophical terms fallacious argumentation and reasoning. (55).

In fact, twisted arguments can be created. (55).

A reason for the use of symbolic logic and reasoning as alternative within philosophy.

In a religious context, philosophy of religion crosses over with theology and there are at times theological arguments that are presented both in natural language and with symbolic logic, and so therefore learning both modes of argumentation is beneficial.

---

Continued

Ambiguous Language

Langer explains 'Our linguistic means of conveying relations are highly ambiguous. But the expression of relations is the chief purpose of language. If we were interested only in things and not in their arrangement and connection, we could express ourselves with our forefingers.' (56).

An interesting author example, and the idea of supposed human communication as ape-like creatures within the concept of Darwinian Evolution comes to mind. Assuming that at one point evolving 'humanity' perhaps did communicate by such methods.

This being the case, if indeed one would accept such views over explanations that include both reasonable scientific induction and deduction and a literal view and not mythological view of the historical religious history of Genesis and Scripture. This reasonable approach that can include both plain literal and figurative literal biblical interpretations based on what biblical language and context dictates.

But I digress.

The author explains that in the example, the two John's are not likely to be confused. This is because these relations can be explicitly known in discourse. (56).

However, Langer expresses the idea that terms and concepts not explained clearly through discourse need to be explained when obscurity in communication occurs. (57). In order to escape error 'another sort of discourse' is required. (57).

This being symbolic logic. (57).

A more precise symbolism can bring logic out of language. (57).

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

There exists white houses

Vancouver: This 6:30 am

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

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
---

Previously 

The Universe of Classes

We documented a system of houses (157), as example of K=interpreted as houses and a dyadic. (157) 'When a relation-symbol stands in a construct, the number of terms grouped with it reveals the degree of the relation. But when it is not actually used, but merely spoken of, it is sometimes convenient to have some way of denoting its degree. This may be done by adding a numerical subscript; for example, "kd2" means that "killing" is dyadic (a pair), "bt3" that "between" is triadic.' (55).

In such a dyadic system, all the elements have to expressible in terms of two elements. (157). There is a fixed element that relates to an element on the other end of the pole, so to speak. (157). Every class is therefore relative to some given element. (157). The defining form of the class must be in dyad (157) such as with Langer's example of K nt2. These are the houses north of a stated element. Langer explains that if the elements had begun with a triadic (group of three not two as in dyadic relation), such as using the term 'between', then two fixed elements would have been used to generate a class. (157). The class between a and b or the class of terms not between a and b.

Using dyadic: K=interpreted as houses nt=interpreted as north of... K= (a, b, c... =nt2 ) a, b, c... are houses north of x within this deductive system and universe of discourse.

September 25, 2017

Langer

'Ordinarily, however, we do not invoke relationship to a given term in order to form a class; we form such classes as 'white houses.' two-storeyed houses,' etc., without reference to any given term. What sort of formal context does this requires? What relation functions among the elements of our universe to generate a class of 'white houses'? (157).

'If the is no relation among the elements of the universe...how can we have any elementary structure of terms.' (157). As in the propositional forms in any given context, to define classes of elements. (157).

The answer Langer provides is:

Predication

For example, the term 'being white' (Reader again, please be aware of the context and time of this textbook being 1953 and 1967. The text is not playing philosophical or political games with any modern context.) If there is no second term, such as 'white house', it cannot be stated that x has any relation to any other term. (158). This relation of 'nomadic degree', (158), is called a predicate. (158).

To add some unfortunate confusion, Langer then states that within the philosophical community (of that day) there is 'considerable disagreement' on whether or not it is fair to call a predicate a nomadic degree. (158).

To be blunt, this is another example of how philosophy often offers shades of gray/grey...

Langer example:

wt= Is white

wt x=A definition of the class of things white, without relating what is white to any other term. (Based on 159).

Therefore the term wt needs to be connected with other terms.

(∃! x) = wt · x (Based on 159).

There exists x equals white equals and means x. But it connects to nothing else.

However:

K=int 'houses'

(∃! wt) . (K)

There exists white and houses.

There exists white houses.

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