Logic Class # 5

I have struggled with how to present this particular set of information. (Note 1) There is some drudgery involved, but the outcome is beautiful and worth the trip. So I finally have decided to present the outcome first, at least rough idea of where this is heading.

The objective here is to find a set of argument forms that are consistently valid. We can show that there are 256 different forms for Categorical Syllogisms, and we can show further that only 15 of those forms are valid arguments (!) That leaves 241 non-valid forms of Categorical Syllogisms. Is there any wonder that there are so many false arguments?

The fun part is that the 15 valid forms of Categorical Syllogisms have been given names. For example, one need not say to a logician any more than “Barbara”, and the logician will immediately know the exact form. I think that is a hoot, and it reminds me of the old joke about the comedian’s convention, where one comedian yells out, “number 47”, and the whole audience laughs…

In order to get to that point we have some things to understand first. Here is a map of where we are headed in this lesson:

1. Standard Form for Categorical Syllogisms.
2. The Concept of “Distribution”.
3. Affirmation / Negation.
4. Contraries and Contradictories.
5. Rules and Fallacies.
6. Moods and Figures.
7. A List of the Valid Categorical Syllogisms (of Standard Form)

We will need to define some things as we go along.

1. Standard Form for Categorical Syllogisms

The Structure of Categorical Propositions has a standard form. The form consists of this:

[subject] [some form of the verb “to be”] [ predicate]

Example:

[The king of France] [ is ] [bald].

This proposition is in a valid form, but is false: France has no king.

2. The Concept of Distribution

Distribution is defined as how much of the population of a set is covered by the proposition. If an element of the proposition refers to an entire population then it is said to be distributed (or “universal”); otherwise it is not distributed (also called “particular”).

Examples:

The modifier, “some”, does not cover the entire population and is thus not distributed.

The modifiers, “all” and “none”, refer to the entire population of the set, and are called distributed.

[All of P] [ is ] [some of Q]: Here P is distributed, and Q is not distributed.

3. Affirmation and Negation

Now, recalling the types of propositions, we can further refine them:

A. All Q is P;

This means that all of the Q category is a subset of the P category.
Q is distributed; P is not distributed.
Q is universal: P is particular.
The statement is affirmative.

E. No Q is P;

Q is completely outside and independent of the p category.
Q is distributed; P is distributed.
Both Q and P are universal.
The statement is negative

I. Some Q is P;

Part of the Q category is inside the P category.
Q is not distributed; P is not distributed.
Q and P are both particular.
The statement is affirmative.

O. Some Q is not P;

Part of the Q category is outside of the P category.
Q is not distributed; P is distributed.
Q is particular; P is universal.
The statement is negative.

4. Contraries and Contradictories

The relationship between the four types, A, E, I, O, is a square which demonstrates the contrary relationships, the contradictory relationships, and the sub-relationships:



[From Oxford Dictionary of Philosophy]


Here is a table for displaying the same information:
For example, when A is true, its contrary, E is false. When A is true, its contradictory, O is false.





5. Rules and Fallacies

There are some rules for Categorical Syllogisms, and there are associated fallacy names for their false usage:

Rule 1:
Don’t use four terms;
Associated Fallacy: four term fallacy.

Rule 2:
Distribute the middle term for one or more of the premises:
Associated Fallacy: undistributed middle fallacy.

Rule 3:
A distributed term in the conclusion must also be distributed in the premises.
Associated Fallacy: undistributed middle fallacy.

Rule 4:
Do not use two negative premises.
Associated Fallacy: exclusive premise fallacy.

Rule 5:
If one of the premises is negative, then the conclusion is negative as well.
Associated Fallacy: false affirmative from a single negative premise fallacy.

Rule 6.
No particular conclusion is derived from two universal premises.
Associated Fallacy: existential fallacy.

6. Moods and Figures

Here we come to an interesting part.

The structure of syllogism is called the “mood”. For example, the mood of a syllogism might be AOE, meaning that premise 1 is of the form A; premise 2 is of the form O; the conclusion is of the form E.

And the syllogism also has a “figure”, which describes the shape with respect to the middle term, M. (S= subject of the conclusion, aka “minor term”; P=predicate of the conclusion, aka “major term”):

Figure 1:

P1: M – P
P2: S – M
C: S – P
(The middle term is the subject in P1; the predicate in P2).

Figure 2:

P1: P – M
P2: S – M
C: S – P

Figure 3:

P1: M – P
P2: M – S
C: S – P

Figure 4:

P1: P – M
P2: M – S
C: S – P

Now we are finally able to fully define some arguments merely by referring to some codes. For example, AAA-1 means that the argument is formed of three “all Q are P “ types of statements, in the form of Figure 1.

So now we can list the 15 valid arguments and even categorize them with respect to their figures:

7. List of Valid Syllogisms of Standard Form, Categorized by Figure:

Figure 1 Valid Syllogisms:

1. AAA-1 Barbara
2. EAE-1 Celarent
3. AII-1 Darii
4. EIO-1 Ferio

Figure 2 Valid Syllogisms

5. AEE-2 Camestres
6. EAE-2 Cesare
7. AOO-2 Baroko
8. EIO-2 Festino

Figure 3 Valid Syllogisms

9. AII-3 Datisi
10. IAI-3 Disamis
11. EIO-3 Ferison
12. OAO-3 Bokardo

Figure 4 Valid Syllogisms

13. AEE-4 Camenes
14. IAI-4 Dimaris
15. EIO-4 Fresison

Of these, only EIO is valid in all Figures, 1 through 4.

The validity of these arguments has been determined by examining all 256 arguments for violation of the Rules which are listed above. Feel free to repeat that exercise if you wish. (Note 2)

In the next lesson we will investigate the translation of language-based arguments into the coded syllogistic form, so that we can more easily determine the validity of such everyday language arguments.

Notes:
(1) Much of this material was developed with the help of Copi & Cohen, Introduction to Logic, 2005, as well as other sources by Tarski, Frege, Kelley, Tomassi, etc.
(2) I did not do that. Some things are faith-based.

Logic Lesson 4: Types of Deductive Arguments

There will be some Greek words here, but don’t panic – you don’t have to know them by heart to understand logic. Much of logic was developed by Aristotle, and the Greek names have been kept. But the concepts are simple. So don’t let the Greek bother you.

There are some terms that are used in logic which you already use yourself:

And;
Or;
If … then;
If and only if.

These are connectives and are a part of deductive arguments.

Deductive arguments are based on propositions, and these propositions are demonstrations of the relationships of two (or more) categories. The categories we will use here are category Q, and category P. First we will list the relationships in words, and then in Venn diagrams.

Categorical Propositions

All Q is P; This means that all of the Q category is a subset of the P category.



No Q is P; Q is completely outside and independent of the p category.



Some Q is P; Part of the Q category is inside the P category.



Some Q is not P; Part of the Q category is outside of the P category.



These categories have been labeled A, E, I, O for analysis which we will attack later. The labels are shown here:

A. All Q is P; This means that all of the Q category is a subset of the P category.

E. No Q is P; Q is completely outside and independent of the p category.

I. Some Q is P; Part of the Q category is inside the P category.

O. Some Q is not P; Part of the Q category is outside of the P category.



Deductive Reasoning

Deduction works from a known valid category to try to determine the validity of a second category. An example we have already used is this:

Premise 1: If P is valid, then Q is valid; (This is a proposition, not an argument).

Premise 2: P is valid; (This is an assertion of truth or "what is").

Conclusion: Therefore, Q is valid. (This is a conclusion).

The argument contains three elements: Two premises consisting of a proposition and an assertion; and the conclusion.

The proposition, we can see, is the A proposition but stated a little differently: It says that the state of category Q is demonstrated by the state of P, or, the validity of Q is a subset of the validity of P. The Venn diagram again shows how this maps out, logically and categorically:


From the diagram it is easy to see that if P is valid, then Q must be valid also, because Q is part of P.

If the premises of the deductive argument are valid, then the conclusion cannot be other than valid: the conclusion is absolutely a function driven only by the premises. Also, one of the premises, the assertion, is claimed to be factual or True. It is the Truth of this premise that guarantees the Truth of the proposition – if the argument follows the proper format. In this case it is P which is asserted to be True.

This format is also written as follows:

If P, then Q
P;
Q.

Argument Failures

An argument fails if it doesn’t follow a proper format. An argument fails if the proposition is not really the case (i.e., the categories aren’t really related in the fashion shown in the proposition). And the argument fails if the assertion is not actually true or valid. Remember that it requires an overarching truth (or law, as in a law of nature, or an undeniably valid axiom) in order to validate the proposition.

Deductive Argument Cases

These cases are also known as Basic Rules of Inference, because the conclusion is inferred [note 1]from the premises. To change things up a little, the letters are reversed; don't get hung up on a letter always meaning the same thing - letters are just symbols, randomly attached to categories. Make your own Venn diagrams for these on scratch paper; it's really quite simple.

1. Modus Ponens

If Q, then P;
Q;
P.

Recognize this? We’ve been using it in the examples.

2. Modus Tollens

If Q, then P;
~ Q; (the tilde, ~, just means “not”, or a negation; here we have: ~Q = not Q)
~ P.

3. Hypothetical Syllogism

If Q, then P, and If P, then S;
Q;
S

This is also written in a more familiar form:

If Q, then P;
If P, then S;
If Q, then S.

So the validity of Q guarantees the validity of S.

4. Disjunctive Syllogism

Either Q Or P is true;
~P;
Q.

5. Constructive dilemma

[If Q, then P] And [If S, then T];
Q Or S;
P Or T.

6. Absorption

If Q then P;
Therefore, If Q, then (Q And P).

7. Simplification

P And Q;
P.

8. Conjunction

P;
Q;
P And Q.

9. Addition

P;
P Or Q.

As always, ask any question, make any comment.

Notes
Note 1: In logic the term inference has a more concrete meaning than it might have in general usage. In logic, inference means that the "conclusion definitely follows from...", assuming that the argument structure is correct and the premises are true and meaningful. This is slightly different from common usage, where inference can also mean to extrapolate meaning improperly from premises.

Logic Class Lesson 3: Propositional Arguments

Propositions are the “material of our reasoning”, according to Copi and Cohen. A proposition asserts that some condition is or isn’t the case. It is a statement, a declarative sentence, an assertion. Simple propositions point to a single fact: Q is P. This asserts that Q is a subset of P. For example, the statement, “purple is a color”, asserts that it is the case (it is true) that the set of “colors” contains “purple”.

This is merely an assertion of truth, and whether it is actually true remains to be verified independently; propositions (assertions) may be challenged.

In logic, arguments are not disputes nor are they debates. An argument is a statement or series of statements that attempts to establish that something is so.

A proposition is not an argument by itself. Arguments contain a group of propositions, where one proposition is asserted as a conclusion based on another proposition which is presumed valid, or several propositions presumed to be valid, called premises. Here is a sample of an argument which states that because [X] and [Y] are true, this is adequate support for the contention that the conclusion, [Z] is true:

Premise 1: It is the case that [X]. (This is a proposition or statement).
Premise 2: It is the case that [Y]. (This is a proposition or statement).

Conclusion: Because [X] and [Y], then [Z].

Arguments are classified as either inductive or deductive (note 1).

Inductive Arguments
Inductive arguments are based on premises that tend toward a conclusion because the premises observe a common characteristic that belongs to that conclusion. A famous example of induction is the Swan argument:

Because every swan I have observed is white, then it is probable that all swans are white.

Or,

Because instances [1] through [n] of Q are P, then it is probable that all instances of Q are P.

This is an inference drawn from individual members, 1 – n, of a set Q, that the character of the entire set is represented by the characteristics of this subset of individual members, P (where P includes members 1 – n), with some probability of accuracy. From the Swan argument it can be seen that an inductive argument never reaches a probability of 1.0, until 100% of the members of the set have been observed. White swans were always observed – until black swans were discovered in Australia.

Inductive arguments usually go from specific examples toward general cases (or laws). This has been used in natural science. For example, frogs all have certain features in common, salamanders have other features in common. Those features which are observed to be in common to frogs define the set [frogs], which logically speaking is a genus, with bullfrog being a species contained within the genus. Taking frogs and salamanders together, [amphibians] would be the genus and [frog] would be the species. These logical designations of genus and species differ from biological use of the terms, and are intended to define a superset / subset relationship.


Deductive Arguments
When an argument’s conclusion cannot be false because its premises are undeniably true and are necessary and sufficient to support the truth of the conclusion, the argument is deductive. Another way to say this is that a deductive argument produces a conclusion that is impossible to be false, based on the truth of the premises. An example of a deductive argument:

If [P] is true, then [Q] is true;
[P] is true;
Therefore, [Q] is true.

If [all the marbles in the jar are green], then [this marble in the jar is green];
All the marbles in the jar are green;
Therefore, this marble in the jar is green.

Note that the set is defined as “true” in the second premise. The argument proceeds from a general case which is declared valid to a subset of the general case which cannot be non-valid… or can it?

If the swan situation is declared deductively, it would look like this:

If [all swans are white] then [the next swan will be white];
All swans are white;
The next swan will be white.

It becomes obvious that the statement of the general case can be false, despite the valid format of the deductive argument. In this case the general case is false because the general case was based on an inductive conclusion. Induction does not produce absolute validity (until 100% of specific members are tested), so the second premise – all swans are white – is not a true general case. So deduction also has limitations: it is only as valid as its general case is valid. This is a serious consideration, because how are general cases determined? Very frequently through induction.

Perhaps there are cases where the general case of an inductive argument cannot be argued against, its validity being so obvious that it is unassailable. That would be the case for the swan argument, if and only if 100% of the swans that exist had been observed. At that point a valid deductive argument can be made that “any swan is either black or white”.

It is also possible to “define” categories to be valid.

A deductive statement with two premises, where the premises are defined categories used by biologists to classify animals:

If [frogs are amphibians] And [bullfrogs are frogs] then [bullfrogs are amphibians];
[frogs are amphibians] And [bullfrogs are frogs];
therefore, bullfrogs are amphibians.

The term amphibian is a class, not an animal per se; this class has been defined to include frogs, and frogs defined to include bullfrogs (but not toads). So the argument is hardly assailable because it is a serial definition. It is true by definition.

Here are some interesting points. The terms validity and invalidity, true and false, cannot apply to inductive conclusions, because the inductive process never produces certainty, nor can it produce truth. Those terms, validity and truth, can refer to the deductive conclusion though, because deduction should produce unquestionable truth, if the process being used is valid and the general case is true. In logic the term “truth” is restricted to mean “that which corresponds to a verifiable reality”, or “that which actually is the case”. And validity refers to whether the process itself is valid, not the conclusion.

Soundness
”When an argument is valid, AND all of its premises are true, it is called ‘sound’” (note 2). In science, the soundness of a proposed natural law is tested by deducing consequences of that law, and performing experiments designed to produce those consequences by creating an instance of the law in action. Deduction is about consequences, those eventualities that necessarily arise from the more general state or law. Even everyday consequences can be deduced, such as that heavy snowstorms impede traffic; as a consequence it will take more time to reach a destination.

In terms of consequence the simple deductive argument can be written as follows:

If [a law, P, is true], then [a necessary consequence, Q, of that law is true];
[ law, P, is true];
Therefore, [consequence Q is necessarily true].

This is simple deductive argumentation. There are more types of deductive arguments and these will be covered in the next lesson.

All questions and comments are welcome.

Note:
(1) There are other classes of argumentation too, such as abduction, which will be covered later.
(2) Copi and Cohen, Introduction to Logic, 12th Ed., pg 19.

Logic Class Lesson 2

Language is the currency with which we transact ideas. We use it for much more than that of course, but in terms of logic and analysis it is the idea that is of our concern. Because language consists of symbols – a sound or scratches on a surface – which contain meaning, then it is necessary to have a consistent meanings for each of those symbols. What you mean by a set of sounds must correspond with what it means to me when I hear those sounds. So we define those symbols to establish their meanings.

We go out of our way to catalog definitions into huge documents which everyone can reference. We take care to include definitions of new words, and to refer to older definitions as obsolete. Definitions are one of the most basic properties of language.

Yet when we come to analyzing argumentation, it is not uncommon to come to the realization that the words being used do not have the same meaning to both participants in the argument. If this is the case, then language is failing, and the meanings are being obscured.

Because definitions and the meanings of our concepts are crucially important to communicating our arguments accurately, we need to take some time with the concept of definitions before we get to arguments.

Purpose of Definitions
Definitions for a certain argument might need to be more precise than the word synonyms found in dictionaries. So we might need to create a word model of the concept that has the precision needed. According to Kelley:

a) A definition can clarify boundaries of a concept. (In what way is a cat not a dog? And also not a skunk?)

b) A definition can show relationships to other concepts (In what way is a cat like a dog but not like an alligator?)

c) A definition provides a summary statement about the referents of the concept. (If a concept is a container for all information about a certain class of things, a definition can summarize that information in a specific, condensed essence).

Types of Definitions
There are several types of definitions:

1. Stipulative; a new concept or symbol gets a definition by its creator.

2. Lexical; specifies a previously existing use of a term.

3. Precising; Used to remove ambiguity, to add precision to a term.

4. Theoretical; a comprehensive, perhaps scientific definition.

5. Persuasive; defined in a manner to “resolve a dispute by influencing attitudes or stirring emotions”; as in politically emotive language.

In logic, precising and theoretical types will likely be most common.

Definition Method
A term has a class or set of like concepts to which it belongs; this is called extension. It is possible that this set is too large to define completely.

A term also has a class or set of like concepts which belong totally to it; this is called intension. Intensional definitions include those that are accepted by public usage in everyday language, and this is called Conventional Intension, which is the commonly used set for creating definitions.

A term can be classified by the use of “Genus and Differentiation (Species)” to locate the concept within a specific class for similarities and subclass for differences. In categorizing a term (or concept), it can generally be placed within a class of similar items, or a genus. Within that class items can be again categorized into subclasses, a process also referred to as “differentiation into species”. Hence the terminology, Genus and Differentiation.

Examples of Genus and Difference definitions:

Term: father
Genus: parent
Difference / species: man (note 1)

Term: florin
Genus: coin
Difference / species: Italian (note 2)

Term: table
Genus: dining
Difference / species: round

Rules for Definitions
There are rules for Defining things. Here are two sets of similar rules for forming definitions:

Rules for Definitions From Kelley (note 4):

1. A definition should include a genus and a differentia [species].

2. A definition should not be too broad or narrow.

3. A definition should state the essential attributes of the concept’s referents. [i.e. go to the fundamentals of the concept].

4. A definition should not be circular.

5. A definition should not use negative terms unnecessarily.

6. A definition should not use vague, obscure or metaphorical language.

From Copi & Cohen (note 5):

1. A definition should state the essential attributes of the species.

2. A definition must not be circular.

3. A definition must be neither too broad nor too narrow.

4. Ambiguous, obscure, or figurative language must not be used in a definition.

5. A definition should not be negative where it can be affirmative.
(Copi’s rule set presumes Genus and Difference methodology).

The first three rules in Kelley’s set form a construction list for making definitions. The last three are quality check items to make sure the definition is sound. This organization of the rules (Kelley’s) seems the best organized to me, so let’s go through it in slightly more detail using examples.

A definition should include a genus and a differentia [species]. Placing the concept within a frame of reference relative to similar concepts is useful to understanding its meaning. So finding a proper genus which reflects that is necessary. The additional separation into a subclass makes the concept distinct from the others in the genus. Here’s an example of some selections for “table”:

a) Term: table

Genus: furniture
Difference / species: flat surface

Too broad; desks also have flat surfaces.

b) Term: table

Genus: furniture
Difference / species: end of couch

Too narrow; there are other table types.

c) Term: table

Genus: furniture
Difference / species: not a desk

Oops, unnecessary negative, trying to set an exclusive boundary for one other furniture species or type.

d) Term: table

Genus: furniture
Difference: bench

Synonym: circular reference: table = bench = table.

e) Term: table

Genus: furniture
Difference / species: Horizontal flat surface set on legs.

This definition contains proper referents (items pertinent to the concept and pertinent to differentiating the concept from other species).

Exercise:
Using Genus and Difference, define “cat”, being certain to differentiate it from “dog” with certainty without saying “not a dog”, and to similarly also exclude skunks, raccoons, ferrets, rabbits, etc. Also do not use circular synonyms, such as “feline”. Obviously: no search engines, dictionaries, reference books, etc. Do your own work. If you have a good definition and care to share it, please feel free to do so.

As always, all questions and comments are welcome.

Notes:
1) Copi and Cohen, Logic, p115.
2) Kelley, The Art of reasoning, p 37.
4) Kelley, p 43.
5) Copi and Cohen, p 115 –117.

Logic Class #1; Introduction

Welcome to Logic Class. There are some things I'd like you to know before we get started. This class will cover college level material, but it will be in a conversational language, at least as straightforward as I can manage to make it. Logic is not a difficult subject. Also, it might not be quite as complete and rigorous (no tests), but it will be more complete in some regards, which will become more apparent as we go along. Any material that is not normally included in textbooks will be annotated.

Logic is a natural function and is something we do frequently. We all already are logicians; but some have more skill than others. What this class will help with is to put a foundation and structure to what we already do. If I fail at this, please, please challenge me to do it better. The material will be derived primarily from college texts and from philosophers as appropriate; these will be credited. I will add my own comments as we go.

Don't worry about big, two-dollar words. They actually describe simple things; mostly they are just names. Besides, a dollar doesn't go that far these days.

I think that there are four main categories for this class to pursue:

1) Put a structure forward for analyzing arguments, propositions and assertions for values such as validity, correctness and truth;

2) Give examples for clarity;

3) Develop robust participation through discussion in the comments;

4) Use these principles for analyzing complete truth statements.

I hope that you will ask questions and make comments, and that everyone who is interested will engage in the conversation.

What is logic? The Definition.
There are a great many books that claim to be logical or to teach “critical thinking”, many of which are actually selling an ideology which is then supported with rationalization. There are many ideologies that claim logic as their focus; but these can be saturated with errors. But there are principles for logical thinking that we can access.

Logic has a specific history and is a "science" which is based upon specific principles. Logic is taught in college courses, and there are college texts available.

Here are some definitions from several college texts on Logic:

”To study logic is to study argument. Argument is the stuff of logic…" "The central problem which worries the logician is just this: how, in general, can we tell good arguments from bad ones?”
"Logic"; Paul Tomassi, Routledge, 2004; p2.

”The core of logic has always been the study of inference.”
"The Art of Reasoning"; David Kelley, W.W.Norton & C0, 1988; p2.

”Logic is the study of the methods and principles used to distinguish good (correct) from bad (incorrect) reasoning.”
"Introduction to Logic", 5th Ed.; Irving M. Copi, McMillan & Co, 1978, p3.

"The distinction between correct and incorrect reasoning is the central problem with which logic deals.
"Introduction to Logic", 5th Ed.; Irving M. Copi, McMillan & Co, 1978, p5.

”Every argument confronted raises this question: Does the conclusion reached follow from the premises used or assumed? There are objective criteria with which that question can be answered; in the study of logic we seek to discover and apply those criteria.”... ”But where judgments that must be relied upon are to be made, their most solid foundation will be correct reasoning. With the methods of and techniques of logic we can distinguish reliably between sound and faulty reasoning.”
"Introduction to Logic", 12th Ed; Irving M. Copi & Carl Cohen, p4, 5; p3.

Logic is not an unsupported, fluke invention of man, nor is it merely a pragmatic program for organizing mental objects. I have come across people who have stated just those opinions of logic. One person claimed that logic could be bent to prove anything whatsoever. That's not logic, it is illogic.

Logic is a reflection of the order that is visible in the immutable laws of the universe, and their effect on our ability comprehend. If there were no universal order there could be no order to thinking about it. It is the ordered functioning of the universe that allows ordered thinking.

Certain things about the orderly nature of the universe can be observed; there are rules for existence that never are violated, at least in the non-quantum, macro universe which is the one in which we live and think. These observations of characteristics of the universe can reveal that even though the most basic rules are not provable, they are indisputable within our limits of observation and they are useful in describing order.

These useful observations regarding order in the universe cover two separate arenas: the basic properties of existence, and the basic properties of validity and truth. They can then be developed into “principles” which guide our thinking in certain categories.

When these principles are violated, an incorrect view of the subject at hand is likely. So following these principles is necessary in order to achieve valid thinking.

Informal vs. Formal Logic
Formal logic assigns symbols to premises and then manipulates the symbols using a mathematics of logic. This allows the form of logic to be analyzed without any confusing interference from the meaning contained in the words of the premises. It allows for increased complexity to be reduced in order to be more easily managed without error.

Informal logic addresses premises directly, with a minimum of mathematical manipulation or symbols. We will use informal logic in this series of classes, so that much of the time the meaning is preserved and visible during the analysis.

Pragmatic Logic and Propositional Logic
I have defined an additional split in logic, Pragmatic Logic and Philosophical Logic. Pragmatic Logic is a procedure for producing a conclusion. The conclusion is based on accurate use of prior premises (assertions) which support the conclusion. This procedure is intended to produce conclusions that are necessarily valid, given that the premises are used in the prescribed fashion. Testing for this type of logic includes looking for informal fallacies.

However, Pragmatic Logic looks only at the process, and ignores the possibility that one or more of the premises might be wrong. If the process is correct, Pragmatic Logic is happy. Basic Propositional Logic, which exams statements that propose a conclusion, fall under Pragmatic Logic.

These terms and more will be defined as we proceed: proposition, argument, premise, valid, correct, truth.

Philosophical Logic and The First Principles
Philosophical Logic goes beyond Pragmatic Logic, and looks at each premise for its Truth Value – whether the premise is supportable through evidence or prior true sub-premises. This has produced another level of logic: the testing of premises for adherence to axioms including First Principles. First Principles are the most basic axioms that underlie all logic; they will be covered under Philosophical Logic.

The Source of Logic
Logical processes are attributed to Aristotle for having developed propositional logic, and the processes have been developed further by Gottlob Frege, who contributed quantificational logic. There have been a multitude of contributions from many others including Boole, De Morgan, Peirce, Russell, Tarski etc.

But the actual source of logic is not human derived. Logic has dual natural sources: first, the law-driven structure of the universe which have been observed for the eons of human existence; second the rational faculties that are found in the human mind, before (a priori) and independent of the mind being subjected to an educational environment.

According to John Locke, at birth the human mind, while not containing any naturally included factual information concerning the material universe, does contain the elements of rationality: the human mind comes with inborn faculties including these capabilities:

a) Apprehension (information gathering);

b) Comparison (relating new information to prior information as it is stored in the memory);

c) Differentiation (finding differences between new information input and stored information);

e) Judgment (determining whether the new information is similar to or the same as stored information);

f) Comprehension (deriving meaning).

These two rational structures, the law-ordered universe and the rational mind, allow us to perform the necessary intellectual functions to create a valid view of the universe and its constituents, and to determine the truth value of such an analysis.

Defining Evidence:
Whether a statement qualifies as proper evidence is a serious and somewhat complex subject; this is somewhat outside the realm of logic, which is more interested in valid formats and procedures. But since it is necessary for determining a truth value, evidence will be covered, and its several definitions will be provided.

Defining Truth
The basic materialist definition of truth is this:

A statement is true if it conforms to reality.
(Correspondence Theory of Truth)

This statement has a limited utility if one is uncertain of the definition of reality, and where its limits do or do not lie. Further, one’s view of reality might not correspond to another’s view of reality. This leads to a squishy understanding of truth, whether it exists, and whether it is merely dependent on one’s personal viewpoints.

Possibly the best definition of truth was provided by George Boole, a very simple mathematical statement which we will cover later. It will be shown that logic is a binary function (having only two states – True and False): any amount of falseness in an argument, even deeply embedded, renders the argument false (even if the argument has a valid structure). Conversely, true arguments contain only truth, no falseness.

Getting through the layers of premises and presuppositions to reach the axiomatic foundation will be covered.

Intellectual Integrity.
It is presupposed that honesty is an underlying characteristic of those who deal in logic. Yet that need not be the case. For every argument pro and con, one conclusion must be incorrect. While the incorrect use of logic might be intentional (in order to sell something, for example) it is not the intent behind the argument that concerns a logician. A logician is merely trying to find out if an argument is valid.

The analysis of process, taken by itself, is not enough to guarantee that truth is being generated by an argument. Also needed is a commitment to investigate the sub-premises and unstated presuppositions that underlay the premises. This is necessary in order to ferret out falseness which might not be visible within the stated argument but which nonetheless ultimately negates the argument when the fallacy is ultimately discovered. Such fallacies might exist several layers below the visible argument. This means that there is a requirement for analyzing more than just the visible argument. We will see some examples.

Perhaps this makes logic appear difficult. I hope not, because logic is a natural inclination of the human mind.

Please ask any and all questions you might have, and make any comments too.

Logic Class #1; Introduction

Welcome to Logic Class. There are some things I'd like you to know before we get started. This class will cover college level material, but it will be in a conversational language, at least as straightforward as I can manage to make it. Logic is not a difficult subject. Also, it might not be quite as complete and rigorous (no tests), but it will be more complete in some regards, which will become more apparent as we go along. Any material that is not normally included in textbooks will be annotated.

Logic is a natural function and is something we do frequently. We all already are logicians; but some have more skill than others. What this class will help with is to put a foundation and structure to what we already do. If I fail at this, please, please challenge me to do it better. The material will be derived primarily from college texts and from philosophers as appropriate; these will be credited. I will add my own comments as we go.

Don't worry about big, two-dollar words. They actually describe simple things; mostly they are just names. Besides, a dollar doesn't go that far these days.

I think that there are four main categories for this class to pursue:

1) Put a structure forward for analyzing arguments, propositions and assertions for values such as validity, correctness and truth;

2) Give examples for clarity;

3) Develop robust participation through discussion in the comments;

4) Use these principles for analyzing complete truth statements.

I hope that you will ask questions and make comments, and that everyone who is interested will engage in the conversation.

What is logic? The Definition.
There are a great many books that claim to be logical or to teach “critical thinking”, many of which are actually selling an ideology which is then supported with rationalization. There are many ideologies that claim logic as their focus; but these can be saturated with errors. But there are principles for logical thinking that we can access.

Logic has a specific history and is a "science" which is based upon specific principles. Logic is taught in college courses, and there are college texts available.

Here are some definitions from several college texts on Logic:

”To study logic is to study argument. Argument is the stuff of logic…" "The central problem which worries the logician is just this: how, in general, can we tell good arguments from bad ones?”
"Logic"; Paul Tomassi, Routledge, 2004; p2.

”The core of logic has always been the study of inference.”
"The Art of Reasoning"; David Kelley, W.W.Norton & C0, 1988; p2.

”Logic is the study of the methods and principles used to distinguish good (correct) from bad (incorrect) reasoning.”
"Introduction to Logic", 5th Ed.; Irving M. Copi, McMillan & Co, 1978, p3.

"The distinction between correct and incorrect reasoning is the central problem with which logic deals.
"Introduction to Logic", 5th Ed.; Irving M. Copi, McMillan & Co, 1978, p5.

”Every argument confronted raises this question: Does the conclusion reached follow from the premises used or assumed? There are objective criteria with which that question can be answered; in the study of logic we seek to discover and apply those criteria.”... ”But where judgments that must be relied upon are to be made, their most solid foundation will be correct reasoning. With the methods of and techniques of logic we can distinguish reliably between sound and faulty reasoning.”
"Introduction to Logic", 12th Ed; Irving M. Copi & Carl Cohen, p4, 5; p3.

Logic is not an unsupported, fluke invention of man, nor is it merely a pragmatic program for organizing mental objects. I have come across people who have stated just those opinions of logic. One person claimed that logic could be bent to prove anything whatsoever. That's not logic, it is illogic.

Logic is a reflection of the order that is visible in the immutable laws of the universe, and their effect on our ability comprehend. If there were no universal order there could be no order to thinking about it. It is the ordered functioning of the universe that allows ordered thinking.

Certain things about the orderly nature of the universe can be observed; there are rules for existence that never are violated, at least in the non-quantum, macro universe which is the one in which we live and think. These observations of characteristics of the universe can reveal that even though the most basic rules are not provable, they are indisputable within our limits of observation and they are useful in describing order.

These useful observations regarding order in the universe cover two separate arenas: the basic properties of existence, and the basic properties of validity and truth. They can then be developed into “principles” which guide our thinking in certain categories.

When these principles are violated, an incorrect view of the subject at hand is likely. So following these principles is necessary in order to achieve valid thinking.

Informal vs. Formal Logic
Formal logic assigns symbols to premises and then manipulates the symbols using a mathematics of logic. This allows the form of logic to be analyzed without any confusing interference from the meaning contained in the words of the premises. It allows for increased complexity to be reduced in order to be more easily managed without error.

Informal logic addresses premises directly, with a minimum of mathematical manipulation or symbols. We will use informal logic in this series of classes, so that much of the time the meaning is preserved and visible during the analysis.

Pragmatic Logic and Propositional Logic
I have defined an additional split in logic, Pragmatic Logic and Philosophical Logic. Pragmatic Logic is a procedure for producing a conclusion. The conclusion is based on accurate use of prior premises (assertions) which support the conclusion. This procedure is intended to produce conclusions that are necessarily valid, given that the premises are used in the prescribed fashion. Testing for this type of logic includes looking for informal fallacies.

However, Pragmatic Logic looks only at the process, and ignores the possibility that one or more of the premises might be wrong. If the process is correct, Pragmatic Logic is happy. Basic Propositional Logic, which exams statements that propose a conclusion, fall under Pragmatic Logic.

These terms and more will be defined as we proceed: proposition, argument, premise, valid, correct, truth.

Philosophical Logic and The First Principles
Philosophical Logic goes beyond Pragmatic Logic, and looks at each premise for its Truth Value – whether the premise is supportable through evidence or prior true sub-premises. This has produced another level of logic: the testing of premises for adherence to axioms including First Principles. First Principles are the most basic axioms that underlie all logic; they will be covered under Philosophical Logic.

The Source of Logic
Logical processes are attributed to Aristotle for having developed propositional logic, and the processes have been developed further by Gottlob Frege, who contributed quantificational logic. There have been a multitude of contributions from many others including Boole, De Morgan, Peirce, Russell, Tarski etc.

But the actual source of logic is not human derived. Logic has dual natural sources: first, the law-driven structure of the universe which have been observed for the eons of human existence; second the rational faculties that are found in the human mind, before (a priori) and independent of the mind being subjected to an educational environment.

According to John Locke, at birth the human mind, while not containing any naturally included factual information concerning the material universe, does contain the elements of rationality: the human mind comes with inborn faculties including these capabilities:

a) Apprehension (information gathering;

b) Comparison (relating new information to prior information as it is stored in the memory);

c) Differentiation (finding differences between new information input and stored information);

e) Judgment (determining whether the new information is similar to or the same as stored information);

f) Comprehension (deriving meaning).

These two rational structures, the law-ordered universe and the rational mind, allow us to perform the necessary intellectual functions to create a valid view of the universe and its constituents, and to determine the truth value of such an analysis.

Defining Evidence:
Whether a statement qualifies as proper evidence is a serious and somewhat complex subject; this is somewhat outside the realm of logic, which is more interested in valid formats and procedures. But since it is necessary for determining a truth value, evidence will be covered, and its several definitions will be provided.

Defining Truth
The basic materialist definition of truth is this:

A statement is true if it conforms to reality.
(Correspondence Theory of Truth)

This statement has a limited utility if one is uncertain of the definition of reality, and where its limits do or do not lie. Further, one’s view of reality might not correspond to another’s view of reality. This leads to a squishy understanding of truth, whether it exists, and whether it is merely dependent on one’s personal viewpoints.

Possibly the best definition of truth was provided by George Boole, a very simple mathematical statement which we will cover later. It will be shown that logic is a binary function (having only two states – True and False): any amount of falseness in an argument, even deeply embedded, renders the argument false (even if the argument has a valid structure). Conversely, true arguments contain only truth, no falseness.

Getting through the layers of premises and presuppositions to reach the axiomatic foundation will be covered.

Intellectual Integrity.
It is presupposed that honesty is an underlying characteristic of those who deal in logic. Yet that need not be the case. For every argument pro and con, one conclusion must be incorrect. While the incorrect use of logic might be intentional (in order to sell something, for example) it is not the intent behind the argument that concerns a logician. A logician is merely trying to find out if an argument is valid.

The analysis of process, taken by itself, is not enough to guarantee that truth is being generated by an argument. Also needed is a commitment to investigate the sub-premises and unstated presuppositions that underlay the premises. This is necessary in order to ferret out falseness which might not be visible within the stated argument but which nonetheless ultimately negates the argument when the fallacy is ultimately discovered. Such fallacies might exist several layers below the visible argument. This means that there is a requirement for analyzing more than just the visible argument. We will see some examples.

Perhaps this makes logic appear difficult. I hope not, because logic is a natural inclination of the human mind.

Please ask any and all questions you might have, and make any comments too.

Starting Up Logic Class

I will be posting the first lesson (introduction) by Thursday of this week, if all else holds up. I will put the lessons in the right hand column for reference at any time.