Logic and Philosophy

An Integrated Introduction

William H. Brenner

• Publisher: University of Notre Dame Press
• Year: 1993
• ISBN:
  9780268158989
• Topic: Philosophy
• Subtopic: Logic in Philosophy

The dual purpose of this volume—to provide a distinctively philosophical introduction to logic, as well as a logic-oriented approach to philosophy—makes this book a unique and worthwhile primary text for logic and/or philosophy courses. Logic and Philosophy covers a variety of elementary formal and informal types of reasoning, including a chapter on traditional logic that culminates in a treatment of Aristotle's philosophy of science; a truth-functional logic chapter that examines Wittgenstein's philosophy of language, logic, and mysticism; and sections on induction, analogy, and fallacies that incorporate material on mind-body dualism, pseudoscience, the "raven paradox, " and proofs of God.

Throughout the book Brenner highlights passages and ideas from various prominent philosophers, and discusses at some length the work of Plato, Aristotle, Descartes, Kant, and Wittgenstein.

APPENDIX 1

Rules of the Syllogism

In the Middle Ages a set of “rules of the syllogism” was added to Aristotle’s system of deductive logic. These rules provide a quick way of distinguishing valid from invalid categorical syllogisms. But they presuppose another set of rules, rules that pertain to the terms in a standard-form categorical statement, namely the rules of distribution:

1. The universal affirmative distributes its subject. That is, in a universal affirmative what is said about the subject is said about each and every item contained in the subject category. Thus, to say “All Scots are patriotic” is to say something about each and every Scot: that she or he is patriotic.

2. The universal negative distributes both terms. In “No S are P,” something is said about everything in the S category, namely that it is separate from, or “outside,” everything in the P category.

3. The particular affirmative distributes neither term. In “Some S are P,” nothing is said about the entire category of either S or P. Thus, in “Some roses are red,” nothing is said either about all roses or about all red things. In other words, nothing is said distributively of either roses or red things.

4. The particular negative distributes its predicate. For example, “Some Buddhists are not theists” says something about each and every theist, namely that he or she is not a certain Buddhist (or certain Buddhists). In other words, there is no place in the entire category of theists for some Buddhist or Buddhists. (You’re not alone if you find this rule harder to grasp than the others!)

SUMMARY

(Underlined, distributed; not underlined, undistributed)

All S are P. No S are P.

Some S are P. Some S are not P.

Here, now are the four rules of the syllogism:

1. A valid syllogism distributes its middle term at least once. Failure to obey this rule is known as the fallacy of undistributed middle.

2. No valid syllogism has a term distributed in the conclusion which is not also distributed in a premise. To violate this rule is to commit the fallacy of illicit process.

3. For any valid syllogism a premise is negative if and only if the conclusion is negative. The following syllogism violates this and the preceding two rules.

Some M are P.

Some S are M.

∴ Some S are not P.

Explanation

Particular affirmative propositions distribute no term; thus, M is undistributed in both premises and Rule 1 is violated.

As it occurs in the conclusion, P is distributed because it is the predicate of a negative proposition; as it occurs in a premise, P is undistributed because no term is distributed in a particular affirmative. Thus, Rule 2 is violated.

There is a negative conclusion but no negative premise; therefore, Rule 3 is violated.

4. No valid syllogism has two negative premises.

Note well that these rules are designed for standard-form categorical syllogisms, that is, for syllogisms made up of exactly three terms contained in three standard-form categorical statements. (The “standard forms of categorical statement” are given in a chart early in Chapter 2.) If applied to non-standard syllogisms, they may lead to erroneous results—as in the following case:

No animals are immortals.

No humans are non-animal.

∴ All humans are mortals.

This syllogism appears to violate Rules 3 and 4. But it is not in standard form, having more than three terms (animals, non-animal, immortals, mortals, humans). Translating it into standard form shows that in reality no rule is violated:

All animals are mortals.

All humans are animals.

∴ All humans are mortals.

There is no longer even an appearance of violating the last two rules. And the first two are obeyed as well: the first because “animals” (the middle term) is distributed in the first premise; the second because the term distributed in the conclusion (“humans”) is also distributed in a premise. So the syllogism obeys all four rules and is therefore valid.*

In translating the first form of the syllogism into the second, we made use of the rule that a categorical statement and its obverse are equivalent (↔), and thus necessarily the same in truth or falsity:

No S are non-P is the obverse of All S are P.

“No animals are immortal” ↔ “All animals are mortal.”

(The prefix “im-” is often used in place of “non-.”)

The rules for obverting any categorical proposition are:

1. change its “quality”—e.g., change a universal negative form of statement into a universal affirmative;

2. negate its predicate—put a “non-,” an “im-” or an “un-” before the predicate term.

Note how they apply in the following cases:

All S are P ↔ No S are non-P. E.g.,

“All humans are animals” ↔ “No humans are non-animals.”

No S are P ↔ All S are non-P. E.g.,

“No plastics are conductors” ↔ “All plastics are non-conductors.”

Some S are P ↔ Some S are not non-P. E.g.,

“Some cats are friendly” “Some cats are not unfriendly.”

Some S are not P ↔ Some S are non-P. E.g.,

“Some liquids are not conductors” ↔ “Some liquids are non-conductors.”

For purposes of translating into standard form, it is important to remember these obversion patterns, together with the rule that every categorical statement is equivalent to its own obverse.

EXERCISES

[*answers in Appendix 4, page 198]

1. Give an original example of each of the four categorical statements. Then give the obverse of each. (Use standard form.)

2.* Fill in each black with either “distributed” or “undistributed”:

All universal propositions have__________ subjects.

All particular propositions have__________ subjects.

All affirmative propositions have__________ predicates.

All negative propositions have__________ predicates.

3.* Which of the following would violate Rule 2? A syllogism with a term distributed in: (a) a premise but not the conclusion; (b) the conclusion but not a premise.

4. Construct a syllogism that commits the fallacy of undistributed middle. Construct another one that commits the fallacy of illicit process.

5. For each of the following, is any rule or rule of the syllogism violated? Explain.

(a)* All communists are socialists, and all far-left liberals are socialists; therefore, all far-left liberals are communists.

(b)* No philosophers are sophists. No politicians are philosophers. Hence, no politicians are sophists.

(c)* No animals with acidic urine are non-carnivorous. Some rabbits have acidic urine. So, some rabbits must be carnivorous!

(d) All animals are non-divine, and all humans are animals. Therefore, no humans are divine.

(e) All computers are mechanical. No computers are persons. Consequently, no persons are mechanical.

(f) Some New Yorkers are not unfriendly. All Staten Islanders are New Yorkers. So, some Staten Islanders are friendly.

(g) No dogs devour glass; only devourers of glass defecate diamonds; therefore, no dogs defecate diamonds.

6. For good practice in deductive reasoning (and for good, clean fun), deduce the following “syllogistic t...