Understanding Deductive Logic

We explain Deductive Logic by St. George William Joseph Stock, a book that explains how to use deductive logic and reason in simple terms.

By doing this we’ll give you an introduction to terms, logic, reasoning, the structure of a basic deductive argument called a syllogism, mood, figure, premisses, predicates, and the other basic parts of logic. In other words, like Stock, we’ll cover the basics of “what we can know and how we can know it” (AKA epistemology) as it pertains to deductive reasoning specifically.

While we can’t summarize the whole book, as the book itself is essentially a summary of deductive logic and reason, we can lay out the basics, explain the gist, and point you toward the book (which was written in the late 1800’s and is now in the public domain and is thus free to read online).

ReadDeductive Logic by St. George William Joseph Stock. For more reading on logic see philosophypages.com.

TIP: Deductive reasoning forms the basis of all reason. If you understand deductive reasoning, you’ll understand the foundation of logic and reason and be well on your way to understanding the other reasoning types. See our page on the different types of reasoning to go beyond deductive reasoning. You can also check out the other important reasoning type inductive reasoning (inductive reasoning deals with probability, deductive deals withe certainty, this page is all about deductive reasoning, not inductive).

Who is St. George William Joseph Stock? One might ask the question “wait, who is St. George William Joseph Stock?! Was he a saint?” Those are great questions. To the first question, there doesn’t seem to be much of an answer. To the second, considering he co-authored the NSFW Whippingham Papers, one would have to assume “no, he was not a saint.” With that said, while Stock was not a famous author nor is much written about him, he none-the-less wrote an excellent (and at this point in history free) work on deductive logic… and this is the focus of our page. See: Who was St. George William Joseph Stock? Also, if you are interested in learning more, consider checking out Logic Gallery, ARISTOTLE TO THE 21st CENTURY, E-edition by David Marans.

What is deductive reasoning? Deduction or Deductive Reasoning, which is a reasoning method that deals with certain conclusions (logically certain inferences). It reasons from certain rules and facts “down” to logically certain conclusions that necessarily follow the premises of an argument. It is often called top-down reasoning because it generally starts with a generalization and reasons toward specifics. It is a type of analysis that is closely related to rationalism, as looks at what is logically and necessarily true about a given system. Deductive reasoning method is rooted in Greek thought, in the Socratic method specifically, and is a close cousin to the other main reasoning type inductive reasoning. As noted above, this page covers deductive reasoning, but not induction (although if you understand deduction you are well on your way to understanding induction; and in fact, deduction is probably the best starting point given the fact that it was defined first).

A Summary of Deductive Logic As Presented By St. George William Joseph Stock

Below are some excerpts and explanations of the main points of Deductive Logic by St. George William Joseph Stock to help illustrate the basics of deductive logic and reason.

First, let’s do a quick overview of the book (which is structured in the same way logic itself is structured, into the sections terms, propositions, and inferences):

• The Introduction: Offers an overview of logic in general with a focus on deductive logic (only touching on inductive logic quickly and not discussing adductive logic at all, because it was written before that concept was popularized).
• Part I: Discusses terms, which as you will find out are essentially 1/3rd of the foundation of deductive logic, with other 2/3rds being logical propositions and reasoned inferences. This section discusses the different types of terms and their place in propositions as subjects and predicates.
• Part II: Discusses logic and propositions. This section also includes an examination into the different types of propositions like major and minor premisses.
• Part III: Discusses reasoning and inferences. Part III is long because everything up to that point builds up to it. Stock goes into the many different types of inferences that can be deduced from comparing terms.

An Overview of Terms, Logic, and Reason

The first part of Stock’s book focuses on introducing the reader to logic and reason (to set the stage for working with basic logical arguments like Categorical Syllogisms; see examples, see also Categorical Propositions).

Essentially everything Stock teaches about logic and reason can be summed up as:

Logic and reason is the art/science of comparing terms and propositions and drawing logical inferences, this can all be reduced to some simple rule-sets, the basic elements work like this (as illustrated by the following three points written in the style of a Syllogism-ish; just like much of Stock’s book):

1. There are terms or concepts based on comparing attributes and conceptualizing “things”; like Socrates, men, or mortality.
2. There are logical judgements or propositions (statements) based on comparing terms and concepts; like Socrates (subject) is a (modality; the relation) man (predicate), and all men are mortal.
3. Then there is reasoned inferences or conclusions based on comparing logical judgements and propositions; like since Socrates is a man and since all men are mortal, therefore Socrates is mortal.

TIP: A term can be a word or a phrase. It is a single idea of any sort, anything that can be conceptualized: “Mr. Smith” or “Unicorns with five legs” or “literally everything in the universe, minus five tacos, but with double the zebras.”

The Structure of  Syllogisms (Categorical Syllogisms Specifically), Major and Minor Premisses, Major, Minor, and Middle Terms, and the General Structure of a Deductive Argument

Consider the classic example of a logical deductive argument presented using the structure of Categorical Syllogism (a logical argument with 1. exactly three propositions, a major and minor premise and conclusion, and 2. three terms, called middle, major, and minor terms, are each used exactly twice):

NOTE: The syllogism below is the classic syllogism, but isn’t a pure Categorical Syllogism. This is because Socrates is a person, not a category. If the minor term was “All Greeks” then it would be a pure Categorical Syllogism. Terms that represent categories of things are categorical terms. Categorical propositions are propositions made from categorical terms that state the relationship between categories. And a categorical syllogism is an argument like this made from categorical propositions and terms. Meanwhile categorical cats are… different categories of cats (where one category is jellical cats <— Cats joke; … because categories are also called “cats” for short 😀 ).[1]

1. Major Premise: All men (subject term; middle term) are mortal (predicate term; major term). (judgement using the terms “all men” and “mortal” where “are” tells us their relation; we can reasonably assume all men are mortal using inductive reasoning).
2. Minor Premise: Socrates (subject term; minor term) is a man (predicate term; still the middle term). (judgement; we look and see he is a man).
3. Conclusion: Therefore, Socrates (subject term; minor term) is mortal (predicate term; major term). (inference; we draw the logical conclusion Socrates is mortal).

Here is the same syllogism as a proper categorical syllogism (where each term is a categorical term):

• Major Premise: All humans (subject term; middle term) are mortal (predicate term; major term).
• Minor Premise: All Greeks (subject term; minor term) are human (predicate term; still the middle term).
• Conclusion: Therefore, All Greeks (subject term; minor term) are mortal (predicate term; major term).

TIP: If any part of our syllogism is not true, then the conclusion is not necessarily true. If a conclusion is true, but the premises aren’t, then the reasoning isn’t valid. If the reasoning is valid, but the conclusion isn’t true… let’s stop there. There are rules for all of this (which is why we want to use categorical terms and not specific ones; so we can create “logical rule-sets” that produce definite truth-values). This subject is covered later and is called “Form and Validity.”

TIP: So our three terms were 1. Men/man, 2. Mortal, and 3. Socrates; each used exactly twice. If any of our terms or premisses are wrong, then our conclusion will be wrong. That is the basis of deductive logic. From there it is all about the different ways we can structure and connect propositions and deductive arguments (which, if we stick with our simple categorical syllogism, we can show to be limited to a set number.

Deductive vs. Inductive

The above is the gist of Logic and Stock’s book, but Stock covers more than just the structure of a syllogism (thereby also covering the basics of deductive logic and reason); so let’s take a look at what Stock says specifically. He says:

§ 4. …Deductive logic… may be defined as The Science of the Formal Laws of Thought.

§ 6. Thought, as here used, is confined to the faculty of comparison. All thought involves comparison, that is to say, a recognition of likeness or unlikeness.

TIP: In other words, deductive logic is the art/science of comparing terms and concepts (and then comparing propositions).

§ 1. LOGIC is divided into two branches, namely—

1. Inductive (to determine the actual truth or falsity of propositions; inductive logic looks at premisses to find probable conclusions),
2. Deductive (to determine the relative truth or falsity of propositions, that is to say, given such and such propositions as true, what others will follow from them; deductive logic uses premisses to find definite conclusions).

TIP: The book focuses on deductive logic only, because we are concerned with definite truth-values, not probable truth here (induction deals with probability; that is a topic for another day, but consider the claim “all men are mortal” required us to compare many instances of men and draw a probable conclusion, this was a form of induction… so induction is underlying our deductive reasoning).

The Three Different Modes of Thought

There are three processes of thought that all relate to each other. The reason we want to phrase these three different ways is because there are at least three parts to thought.

§32. There are three processes of thought (what is happening when we think):

1. Conception
2. Judgement
3. Inference

§ 36. Corresponding to these three processes there are three products of thought (once we have thought we get):

1. The Concept.
2. The Judgement.
3. The Inference.

§ 38. When the three products of thought are expressed in language (when we express our thoughts, they are):

1. The Term.
2. The Proposition.
3. The Inference.

In other words:

There are three categories of logic:

1. Conception -> Concept -> Term. This category can be expressed as Terms / Concepts.
2. Judgement (the process form) -> Judgement (the product form) -> Proposition (language form). This category is Logic / Propositions.
3. Inference (the process form)  -> Inference (the product form) -> Inference (the language form). This category is Reason / Conclusions / Inferences.

TIP: So really there are three basic things to deal with here that share names and get many names, but have specific meanings depending on context. So simple concept, but kind of tricky to master. Don’t worry about mastering it, just get the three basics: terms, logic (propositions), and reason (inferences). Those are the key elements of a deductive argument (plus, as we’ll discuss in a second and eluded to above, “the relation”).

Then this relates to the idea that:

1. The concept is the result of comparing attributes.
2. The judgement is the result of comparing concepts.
3. The inference is the result of comparing judgements.

And likewise (to phrase the same thing in different words):

1. The term is the result of comparing attributes.
2. The proposition is the result of comparing terms.
3. The inference is the result of comparing propositions.

The Laws of Thought

Compare all of that to the idea that the laws of thought are all reducible to the three following axioms, which are known as The Three Fundamental Laws of Thought:

1. The Law of Identity: Whatever is, is; or, in a more precise form, Every A is A.
2. The Law of Contradiction: Nothing can both be and not be; Nothing can be A and not A.
3. The Law of Excluded Middle:  Everything must either be or not be; Everything is either A or not A.

TIP: And here I would also note, especially if we were discussing induction, that we must also consider the “laws of probability” (a thing can be in a state of superposition or can be “likely A” or “likely B.”)

And we have all the tools we need to use to understand and employ logic and reason.

To Sum This All Up

Or, in simple logic:

1. TERMS/CONCEPTS: Observe concepts. ex. men, mortals, Socrates, Plato.
2. PROPOSITIONS/LOGICAL JUDGEMENTS (LOGIC): Make judgements about concepts (logic). ex. All men are Mortal, Socrates is mortal, Plato is mortal.
3. INFERENCES/REASONED CONCLUSIONS (REASON): Compare judgements and make inferences (reason). ex. If Socrates and Plato fight to the death, there can be only one left alive, after-all, all men are mortal, even the great Socrates.

Or, in Stocks words:

§ 56. We have seen that the three products of thought are each one stage in advance of the other, the inference being built upon the proposition, as the proposition is built upon the term. Logic therefore naturally divides itself into three parts.

• The First Part of Logic deals with the Term;
• The Second Part deals with the Proposition;
• The Third Part deals with the Inference.

Part I on Terms

From there Stock goes on to describe details pertaining to the above.

The first whole Book is on terms and how to categorize and compare them (which is important in general, but simple to get with a quick read of Stock; it is just the concept of categorization and relation laid down by Plato, improved by Aristotle, and discussed since then).

The general concept here is that there are different types of terms, some are names, some are categories, some are attributes.

§ 61. A name is a word, or collection of words, which serves as a mark to recall or transmit the idea of a thing, either in itself or through some of its attributes.

… in other words, a term (in logic) is essentially a collection of attributes (in the term “swift galloping horses” swift and galloping are attributes, not words we want to use on their own as terms in logic, “horses”, which are nothing more than a collection of properties, is a term and so is the category “swift galloping horses” and so is the name “Mr. Ed”; if you want to go down that rabbit hole see Part I).

Propositions and the Concept of Categorical

There are different ways to describe propositions, but in general:

§ 207. A Simple Proposition is one in which a predicate is directly affirmed or denied of a subject, e.g. ‘Rain is falling.’

§ 208. A simple proposition is otherwise known as Categorical. TIP: A categorical proposition is simply a statements about the relationship between categories (or categorical terms). In other words, it is a proposition that asserts or denies that all or some of the members of one category (the subject term) are included in another (the predicate term). See modality of copula below for the related standard forms (A, E, I, and O).

§ 209. A Complex Proposition is one in which a statement is made subject to some condition, e.g. ‘If the wind drops, rain will fall.’

§ 210. Hence the complex proposition is also known as Conditional.

§ 211. Every complex proposition consists of two parts—

1. Antecedent;
2. Consequent.

§ 212. The Antecedent is the condition on which another statement is made to depend. It precedes the other in the order of thought, but may either precede or follow it in the order of language. Thus we may say indifferently—’If the wind drops, we shall have rain’ or ‘We shall have rain, if the wind drops.’

§ 213. The Consequent is the statement which is made subject to some condition.

TIP: We won’t discuss every type of proposition, just like we didn’t’ discuss every type of term. See the book for that. I just want to focus on the basics and key concepts.

The Modality of the Copula; Quality and Quantity

This part describes the Copula and the Modality of the Copula.

The important thing to grasp here is that the copula is essentially a word that means “the relation.”

Ex. A (subject term) is not (copula) B (predicate term).

TIP: See Chapter II of Part II for a discussion on the different modes of the copula (the modality of the copula). There are many modes to consider if we consider probabilities, but if we focus just on what we can know for sure, we can boil it down to four key modes. They are: Affirmative, Negative, Universal, and Particular[2]

1. Universal Affirmative (A). All A are B.
2. Universal Negative (E). No A are B.
3. Particular Affirmative (I). Some A are B.
4. Particular Negative (O). Some A are not B.

TIP: § 262. These four kinds of propositions are represented respectively by the symbols A, E, I, O (common symbols used in logic with no inherent meaning, just meaning in context, same with “A” and “B”).

These relations (the copula) are either distributed or undistributed.

§ 275. A term is said to be distributed when it is known to be used in its whole extent, that is, with reference to all the things of which it is a name. When it is not so used, or is not known to be so used, it is called undistributed.

§ 276. When we say ‘All men are mortal,’ the subject is distributed, since it is apparent from the form of the expression that it is used in its whole extent. But when we say ‘Men are miserable’ or ‘Some men are black,’ the subject is undistributed.

§ 277. There is the same ambiguity attaching to the term ‘undistributed’ which we found to underlie the use of the term ‘particular.’ ‘Undistributed’ is applied both to a term whose quantity is undefined, and to one whose quantity is definitely limited to a part of its possible extent.

§ 293. Four Rules for the Distribution of Terms.

1. All universal propositions distribute their subject.
2. No particular propositions distribute their subject,
3. All negative propositions distribute their predicate.
4. No affirmative propositions distribute their predicate.

Classification of Relations

The next section discusses predicates and predicables in relation to propositions.

§ 313. A predicate is something which is stated of a subject.

§ 314. A predicable is something which can be stated of a subject.

The gist here is the idea that we can know truths about a subject based on what class it is in. Or rather, that we can know something about the relation between a subject and a predicate depending on the attributes a class does, doesn’t, or sometimes shares.

1. A Genus is a larger class containing under it smaller classes. Animal is a genus in relation to man and brute.
2. A Species is a smaller class contained under a larger one. Man is a species in relation to animal.
3. Difference is the attribute, or attributes, which distinguish one species from others contained under the same genus. Rationality is the attribute which distinguishes the species, man, from the species, brute.
4. A Property is an attribute which is not contained in the definition of a term, but which flows from it. A Generic Property is one which flows from the genus. A Specific Property is one which flows from the difference. It is a generic property of man that he is mortal, which is a consequence of his animality. It is a specific property of man that he is progressive, which is a consequence of his rationality.
5. An Accident is an attribute, which is neither contained in the definition, nor flows from it. § 319. Accidents are either Separable or Inseparable. A Separable Accident is one which belongs only to some members of a class. An Inseparable Accident is one which belongs to all the members of a class.

Part III Inferences

This part starts off with a discussion on the nature of inferences, like the nature of terms and relations and such, there is a lot of lingo to cover that speaks to how things can be certain, uncertain, likely, unlikely, etc. Like with the other sections, we will hardly cover everything here.

§ 426. To infer is to arrive at some truth, not by direct experience, but as a consequence of some truth or truths already known. If we see a charred circle on the grass, we infer that somebody has been lighting a fire there, though we have not seen anyone do it. This conclusion is arrived at in consequence of our previous experience of the effects of fire.

§ 428. Every inference consists of two parts—

1.  the truth or truths already known;
2. the truth which we arrive at therefrom.

The former is called the Antecedent, the latter the Consequent. But this use of the terms ‘antecedent’ and ‘consequent’ must be carefully distinguished from the use to which they were put previously, to denote the two parts of a complex proposition (in an earlier chapter these terms are used to denote the parts of complex propositions).

§ 429. Strictly speaking, the term inference, as applied to a product of thought, includes both the antecedent and consequent: but it is often used for the consequent to the exclusion of the antecedent. Thus, when we have stated our premisses, we say quite naturally, ‘And the inference I draw is so and so.’

§ 430. Inferences are either Inductive or Deductive. In induction we proceed from the less to the more general; in deduction from the more to the less general, or, at all events, to a truth of not greater generality than the one from which we started. In the former we work up to general principles; in the latter we work down from them, and elicit the particulars which they contain.

§ 431. Hence induction is a real process from the known to the unknown, whereas deduction is no more than the application of previously existing knowledge; or, to put the same thing more technically, in an inductive inference the consequent is not contained in the antecedent, in a deductive inference it is.

The next few chapters go on to describe the different types of inferences.

Syllogism

TIP: We already covered syllogism above, but see: CHAPTER VIII. Of Mediate Inferences or Syllogisms. Here is the example from the book.

§ 547. The following will serve as a typical instance of a syllogism—

Middle term Major term \
Major Premiss. All mammals are warm-blooded | Antecedent
> or
Minor term Middle term | Premisses
Minor Premiss. All whales are mammals /

Minor term Major term \ Consequent or
.’. All whales are warm-blooded > Conclusion.

Figure and Mood

A lot of the discussion in the later part of the book is centered around the mood and figure of syllogisms.

It took a long time for Stock to build up to this, but aside from the idea of terms, propositions, and inferences, I would say that syllogisms, mood, and figure are essentially the foundation of logic.

Mood” describes the order of the propositions we are working with within a syllogism, and  “figure” describes the order of major and minor predicates and premisses (see Chapter IX of Part III).

TIP: There are exactly 256 distinct forms of categorical syllogism: four kinds of major premise multiplied by four kinds of minor premise multiplied by four kinds of conclusion multiplied by four relative positions of the middle term.

In Stock’s words:

§ 558. Syllogisms may differ in two ways—

1. In Mood;
2. In Figure.

§ 559. Mood depends upon the kind of propositions employed. Thus a syllogism consisting of three universal affirmatives, AAA, would be said to differ in mood from one consisting of such propositions as EIO or any other combination that might be made. The syllogism previously given to prove the fallibility of the Pope belongs to the mood AAA. Had we drawn only a particular conclusion, ‘Some Popes are fallible,’ it would have fallen into the mood AAI.

§ 560. Figure depends upon the arrangement of the terms in the propositions. Thus a difference of figure is internal to a difference of mood, that is to say, the same mood can be in any figure.

§ 561. We will now show how many possible varieties there are of mood and figure, irrespective of their logical validity.

§ 562. And first as to mood.

Since every syllogism consists of three propositions, and each of these propositions may be either A, E, I, or O, it is clear that there will be as many possible moods as there can be combinations of four things, taken three together, with no restrictions as to repetition. It will be seen that there are just sixty-four of such combinations. For A may be followed either by itself or by E, I, or O. Let us suppose it to be followed by itself. Then this pair of premisses, AA, may have for its conclusion either A, E, I, or O, thus giving four combinations which commence with AA. In like manner there will be four commencing with AE, four with AI, and four with AO, giving a total of sixteen combinations which commence with A. Similarly there will be sixteen commencing with E, sixteen with I, sixteen with O—in all sixty-four. It is very few, however, of these possible combinations that will be found legitimate, when tested by the rules of syllogism.

§ 563. Next as to figure.

There are four possible varieties of figure in a syllogism, as may be seen by considering the positions that can be occupied by the middle term in the premisses. For as there are only two terms in each premiss, the position occupied by the middle term necessarily determines that of the others. It is clear that the middle term must either occupy the same position in both premisses or not, that is, it must either be subject in both or predicate in both, or else subject in one and predicate in the other. Now, if we are not acquainted with the conclusion of our syllogism, we do not know which is the major and which the minor term, and have therefore no means of distinguishing between one premiss and another; consequently we must Stop here, and say that there are only three different arrangements possible. But, if the Conclusion also be assumed as known, then we are able to distinguish one premiss as the major and the other as the minor; and so we can go further, and lay down that, if the middle term does not hold the same position in both premisses, it must either be subject in the major and predicate in the minor, or else predicate in the major and subject in the minor.

§ 564. Hence there result

The Four Figures.

When the middle term is subject in the major and predicate in the minor, we are said to have the First Figure.

When the middle term is predicate in both premisses, we are said to have the Second Figure.

When the middle term is subject in both premisses, we are said to have the Third Figure.

When the middle term is predicate in the major premiss and subject in the minor, we are said to have the Fourth Figure.

§ 565. Let A be the major term; B the middle. C the minor.

Figure I. Figure II. Figure III. Figure IV.
B—A A—B B—A A—B
C—B C—B B—C B—C
C—A C—A C—A C—A

All these figures are legitimate, though the fourth is comparatively valueless.

§ 566. It will be well to explain by an instance the meaning of the assertion previously made, that a difference of figure is internal to a difference of mood. We will take the mood EIO, and by varying the position of the terms, construct a syllogism in it in each of the four figures.

I.
E No wicked man is happy.
I Some prosperous men are wicked.
O .’. Some prosperous men are not happy.

II.
E No happy man is wicked.
I Some prosperous men are wicked.
O .’. Some prosperous men are not happy.

III.
E No wicked man is happy.
I Some wicked men are prosperous.
O .’. Some prosperous men are not happy.

IV.
E No happy man is wicked.
I Some wicked men are prosperous.
O .’. Some prosperous men are not happy.

§ 567. In the mood we have selected, owing to the peculiar nature of the premisses, both of which admit of simple conversion, it happens that the resulting syllogisms are all valid. But in the great majority of moods no syllogism would be valid at all, and in many moods a syllogism would be valid in one figure and invalid in another. As yet however we are only concerned with the conceivable combinations, apart from the question of their legitimacy.

§ 568. Now since there are four different figures and sixty-four different moods, we obtain in all 256 possible ways of arranging three terms in three propositions, that is, 256 possible forms of syllogism.

Of the General Rules of Syllogism and Complex Syllogisms

The rest of the book essentially works out the rule-sets for the 256 possible forms of syllogism, discusses complex syllogisms (a syllogism composed of complex propositions), and discusses special rules.

I’m not going to summarize this as it is essentially already summarized in the book.

The most important part here is to understand sound and unsound reasoning and valid and invalid reasoning.

There are rules for understanding if something is sound or unsound, pair that with the fact that the 256 figures and moods have constant truth values (making any possible categorical syllogism valid or invalid by its nature), and we can see that this is all reducible to a simple-ish logical rule-set.

Consider:

§ 599. It will be remembered that there were found to be 64 possible moods, each of which might occur in any of the four figures, giving us altogether 256 possible varieties of syllogism. The task now before us is to determine how many of these combinations of mood and figure are legitimate.

§ 600. By the application of the preceding rules we are enabled to reduce the 64 possible moods to 11 valid ones. This may be done by a longer or a shorter method. The longer method, which is perhaps easier of comprehension, is to write down the 64 possible moods, and then strike out such as violate any of the rules of syllogism.

TIP: AAA means a universal major premise, a universal minor premise, and a universal conclusion. So these are our A, E, I, O being used in short form to represent the different types of syllogisms.

-EAA- -EEA- -EIA- -EOA- EAE -EEE- -EIE- -EOE- -EAI- -EEI- -EII- -EOI- EAO -EEO- EIO -EOO-

§ 601. The batches which are crossed are those in which the premisses can yield no conclusion at all, owing to their violating Rule 6 or 9; in the rest the premises are legitimate, but a wrong conclusion is drawn from each of them as are translineated.

§ 602. IEO stands alone, as violating Rule 4. This may require a little explanation.

Since the conclusion is negative, the major term, which is its predicate, must be distributed. But the major premiss, being 1, does not distribute either subject or predicate. Hence IEO must always involve an illicit process of the major.

§ 603. The II moods which have been left valid, after being tested by the syllogistic rules, are as follows—

AAA. AAI. AEE. AEO. AII. AOO. EAE. EAO. EIO. IAI. OAO.

§ 604. We will now arrive at the same result by a shorter and more scientific method. This method consists in first determining what pairs of premisses are valid in accordance with Rules 6 and g, and then examining what conclusions may be legitimately inferred from them in accordance with the other rules of syllogism.

§ 605. The major premiss may be either A, E, I or O. If it is A, the minor also may be either A, E, I or O. If it is E, the minor can only be A or I. If it is I, the minor can only be A or E. If it is O, the minor can only be A. Hence there result 9 valid pairs of premisses.

AA. AE. AI. AO. EA. EI. IA. IE. OA.

Three of these pairs, namely AA, AE, EA, yield two conclusions apiece, one universal and one particular, which do not violate any of the rules of syllogism; one of them, IE, yields no conclusion at all; the remaining five have their conclusion limited to a single proposition, on the principle that the conclusion must follow the weaker part. Hence we arrive at the same result as before, of II legitimate moods—

AAA. AAI. AEE. AEO. EAE. EAO. AII. AOO. EIO. IAI. OAO.

TIP: This is then subject to special rules.

Logical Fallacies

The last section worth pointing out is the one on fallacies.

§ 827. After examining the conditions on which correct thoughts depend, it is expedient to classify some of the most familiar forms of error. It is by the treatment of the Fallacies that logic chiefly vindicates its claim to be considered a practical rather than a speculative science. To explain and give a name to fallacies is like setting up so many sign-posts on the various turns which it is possible to take off the road of truth.

§ 828. By a fallacy is meant a piece of reasoning which appears to establish a conclusion without really doing so. The term applies both to the legitimate deduction of a conclusion from false premisses and to the illegitimate deduction of a conclusion from any premisses. There are errors incidental to conception and judgement, which might well be brought under the name; but the fallacies with which we shall concern ourselves are confined to errors connected with inference.

§ 829. When any inference leads to a false conclusion, the error may have arisen either in the thought itself or in the signs by which the thought is conveyed. The main sources of fallacy then are confined to two—

1. Thought,
2. Language.

Exercises

The book then ends with exercise that will help the reader to ensure they retained the knowledge in the book (this also acts as a good summary once you know the basics).

In Summary

Stock covers everything one needs to know about logic and reason in a really simple and readable way.

The only drawback is that some of what we know about logic was defined after the later 1800’s when Stock wrote his book. With that said, most of the fundamentals have been in place since the days of Plato and Aristotle. So there isn’t much more to learn in terms of a foundation (and to my knowledge not much that needs to be corrected).