Value of Symbolic Logic for Science & 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.
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
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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:
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.
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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
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Therefore all Canadians are mortals
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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).
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A version of this article to be placed on academia.edu on 20250101
