Lecture Notes by Christopher Lay

Los Angeles Pierce College

Department of History, Philosophy, and Sociology

 

 

 

 

Mathew Van Cleave's 2016 Introduction to Logic and Critical Thinking

 

https://open.umn.edu/opentextbooks/BookDetail.aspx?bookId=457

 

 

 

 

Review for Chapter 2: "Formal Methods of Evaluating Arguments"

 

 

 

 

 

 

 

 

 

 

 

Formal Evaluation and Precision and Objectivity

 

"The goal of a formal method of evaluation is to eliminate any imprecision or lack of objectivity in evaluating arguments." 

 

 

 

 

 

 

 

 

 

 

Achieving Precision and Objectivity with the Removal of Imagination and Meaning

 

Methods of formal evaluation can be applied "without having to utilize imagination at all." 

 

Methods of formal evaluation can be applied "without really having to understand the meanings of the concepts used in the argument." 

 

 

 

 

 

 

 

 

 

Form, not Content

 

A formal method of evaluation "evaluate[s] an argument by its form, rather than its content." 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Categorical Logic

 

"Categorical logic is the logic that deals with the logical relationship between categorical statements." 

 

 

 

 

 

 

 

 

 

 

 

 

Syllogisms 

 

Don't forget, syllogisms are arguments with two premises and one conclusion.  

 

 

 

 

 

 

 

 

 

 

 

 

Categorical Statements

 

"A categorical statement is simply a statement about a category or type of thing."

 

 

 

 

 

 

 

 

 

 

Logical Terms in Categorical Logic

 

"In categorical logic, the logical terms ... are the terms 'all' and 'some.'

 

 

 

 

 

 

 

 

 

 

Placeholders / Symbols

 

"In ... categorical logic we will use capital letters to stand for categories of things in the world ... ."

 

 

 

 

 

 

 

 

 

Nouns and Noun Phrases

 

"Notice that the categories are nouns or noun phrases." 

 

"In categorical logic, the capital letters stand for noun phrases that denote categories of things in the world—for example, 'cars' or 'things that are man-made' or 'mammals' or 'things that are red.'" 

 

 

 

 

 

 

 

 

 

 

 

 

 

The Four Categorical Forms' Placeholders / Symbols  

 

"The way we will represent [the four, and only four,] categorical forms generally are with an 'S' (which stands for 'subject term') and a 'P' (which stands for 'predicate term')."

 

 

 

 

 

 

 

 

 

 

 

 

The Four Categorical Statement Forms

 

"[A]ny categorical statement can be translated into one of these four forms."

 

All S are P (Universal Affirmative)

 

No S are P (Universal Negative)

 

Some S are P (Particular Affirmative)

 

Some S are Not P (Particular Negative)

 

Let's go over the last three in turn (as we've already gone over the first one). 

 

 

 

 

 

 

 

 

 

 

("We do not want any of our categories to contain a negation." (Do not forget this point.)  "[T]he negation is contained in the form." (Do not forget this point.))   

 

 

 

 

 

 

 

 

 

 

 

 

 

On Translating into Categorical Statement Forms

 

"There is no simple way of doing it other than asking yourself whether your translation accurately captures the meaning of the original English sentence."

 

 

 

 

 

 

 

 

 


 

Determining Validity for Categorical Syllogisms 

 

(The following comes from Chapter Eight: "Patterns of Deductive Thinking" in Burton F. Porter's The Voice of Reason: Fundamentals of Critical Thinking, Oxford University Press, 2002.)  

 

 

A subject or a predicate is called a "middle term" "because it appears twice in the premises." 

 

A subject or a predicate is distributed if it covers every member of the class (subjects and predicates "are distributed, [if] they cover every member of the class."  

 

1) "At least one of the premises must be affirmative." 

 

2) "If a premise is negative then the conclusion must also be negative, and if the conclusion is negative then a premise must be negative." 

 

3) "The middle term must be distributed at least once." 

 

4) "Any [subject or predicate] distributed in the conclusion must also be distributed [somewhere in] a premise."

 

 

 

 

 

 

 

 

 

Hypothetical Syllogisms

 

We will learn that there are two valid hypothetical syllogisms, affirming the antecedent and denying the consequent.  

 

We will learn that there are two invalid hypothetical syllogisms, denying the antecedent and affirming the consequent.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hypothetical Statements 

 

A hypothetical statement is an if/then statement.  

 

 

 

 

 

 

 

 

 

 

Hypothetical's Parts 

 

The part of the hypothetical that comes after the "if" and before the "then" is called the antecedent

 

The part of the hypothetical that comes after "then" is called the consequent.  

 

 

 

 

 

 

 

 

 

Hypothetical Syllogism's Parts 

 

Hypothetical syllogisms start with a hypothetical statement in the first premise.  

 

In the second premise you will find one part of that hypothetical.  

 

In the conclusion you will find the other part of that hypothetical.  

 

 

 

 

 

 

 

 

 

Handling Negations in the First Premise, and Handling Affirmations and Denials in the Second Premises and the Conclusions of Hypothetical Syllogisms 

 

Unlike categorical syllogisms, when it comes to the first premise, the hypothetical statement itself, negations are parts of the contents of the antecedent or the consequent and so are not considered parts of the form of the argument.  

 

For the second premise and the conclusion, affirmations and denials are treated as parts of the form of the argument, not the content.  

 

 

 

 

 

 

 

 

 

 

 

Formal, Hypothetical Syllogistic Validity 

 

There are two valid hypothetical syllogisms, affirming the antecedent and denying the consequent.  

 

This may be helpful: when determining validity here, you are to see that the premises are to be taken as true, even if they are not actually true.  

 

Moreover, consider the first premise, the hypothetical statement, to be a kind of rule.  

 

 

 

 

 

 

 

 

Affirming the Antecedent 

 

1) Hypothetical statement. 

2) Affirmation of antecedent from hypothetical statement in 1), above.  

C) Affirmation of consequent from hypothetical statement in 1), above.  

 

 

 

 

 

 

 

Denying the Consequent 

 

1) Hypothetical statement. 

2) Denial of consequent from hypothetical statement in 1), above.  

C) Denial of antecedent from hypothetical statement in 1), above.  

 

 

 

 

 

 

 

 

 

Formal, Hypothetical Syllogistic Fallacies 

 

There are two invalid hypothetical syllogisms, denying the antecedent and affirming the consequent.