Using Truth Tables to Show Validity

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来自 Duke University 的课程

Think Again II: How to Reason Deductively

147 个评分

Duke University

Think Again II: How to Reason Deductively

147 个评分

How to Reason Deductively Think Again: How to Reason and Argue Reasoning is important. This series of four short courses will teach you how to do it well. You will learn simple but vital rules to follow in thinking about any topic at all and common and tempting mistakes to avoid in reasoning. We will discuss how to identify, analyze, and evaluate arguments by other people (including politicians, used car salesmen, and teachers) and how to construct arguments of your own in order to help you decide what to believe or what to do. These skills will be useful in dealing with whatever matters most to you. Courses at a Glance: All four courses in this series are offered through sessions which run every four weeks. We suggest sticking to the weekly schedule to the best of your ability. If for whatever reason you fall behind, feel free to re-enroll in the next session.We also suggest that you start each course close to the beginning of a month in order to increase the number of peers in the discussion forums who are working on the same material as you are. While each course can be taken independently, we suggest you take the four courses in order. Course 1 - Think Again I: How to Understand Arguments Course 2 - Think Again II: How to Reason Deductively Course 3 - Think Again III: How to Reason Inductively Course 4 - Think Again IV: How to Avoid Fallacies About This Course in the Series: Imagine that a friend denies that modus ponens is a valid form of argument. Can you prove that it is valid without using modus ponens itself and thereby assuming that it is valid? If so, how? If not, what does this show about the validity of modus ponens? How can phrases like "and", "or", "if", and "not" work as "truth-functional connectives"? In this course, you will learn how to evaluate deductive arguments for validity. In particular, you will learn new ways of representing the information that is contained in the premises of a deductive argument. Using these new representational devices (devices that we call "truth tables" and "Venn diagrams"), we will be able to apply rules to determine whether or not a particular deductive argument is valid. Suggested Readings: Students who want more detailed explanations or additional exercises or who want to explore these topics in more depth should consult Understanding Arguments: An Introduction to Informal Logic, Ninth Edition, Concise, Chapters 6 and 7 by Walter Sinnott-Armstrong and Robert Fogelin. Course Format: Each week will be divided into multiple video segments that can be viewed separately or in groups. There will be short ungraded quizzes after each segment (to check comprehension) and a longer graded quiz at the end of the course.

从本节课中

Propositional Logic and Truth Tables

<p><b>CONTENT</b>: This week we will teach you how such phrases as “and”, “or”, “if”, and “not” can work to guarantee the validity or invalidity of the deductive arguments in which they occur. It will also teach you to understand the functioning of these phrases using a device called a “truth-table”, which shows how the truth or falsity of propositions that use these phrases depends upon the truth or falsity of the propositions contained within it. We highly recommend that you practice the skills that you will learn in this week by doing the puzzles at betapuzzle.sonjara.com.</p><p><b>LEARNING OUTCOMES</b>: By the end of this week’s material you will be able to :<ul><li>define what a deductive argument is</li><li>define what it is for a deductive argument to be valid</li><li>identify truth-functional operators and connectives</li><li>build a truth-table for any truth-functional operator or connective</li></ul></p><p><b>OPTIONAL READING</b>: If you want more examples or more detailed discussions of these topics, we recommend <em>Understanding Arguments, Ninth Edition</em>, Chapter 6. </p>

Introduction to Deductive Logic10:24

Propositions and Propositional Connectives4:42

"And" and the Truth-Functional Connectives10:52

Using Truth Tables to Show Validity5:02

Rules Variables and Generality12:39

Disjunction10:38

与讲师见面

Dr. Walter Sinnott-Armstrong
Professor
Philosophy
Dr. Ram Neta
Professor
Philosophy

In the last lecture,

we learned about the truth functional connective that we call conjunction.

The truth functional connective that, in English at least,

is sometimes expressed by certain uses of the word and.

And we learned about the truth table for conjunction.

Now, one thing I'd like to do in today's lecture is show how we can

use the truth table for conjunction.

To show that certain kinds of inferences that use conjunction are going to be

valid.

Let me give you an example.

Consider an inference that starts off with two premises,

doesn't matter what the premises are, just any two premises, call them P and Q.

because it doesn't matter what they are.

And then the conclusion of the inference is simply going to be the conjunction of

those two premises, whatever they were.

So the inference can be sketched as follows, P, Q, conclusion P and Q.

Now we can call that a conjunction introduction inference,

because the conclusion of the inference uses a conjunction.

It introduces a conjunction that wasn't present in either of the two

premisses of the inference.

Now, notice if you look at the first line of the truth table for

conjunction, you can see that any conjunction

introduction inference is going to have to be valid.

Here's how you can see that.

If you look at the the first line of the truth table for conjunction,

that considers the situation in which P is true and Q is true.

And in any such situation, according to that truth table, the conjunction P and

Q is going to have to be true.

Now, let's take that point and apply it to the inference from the premise P and

the premise Q, to the conclusion P and Q.

What does that first line of the truth table for

conjunction tell you about that inference?

Well, it tells you that in any situation in which the premises of that inference,

P and Q, are both true in any situation in which those premises are both true,

the conclusion, P and Q, is going to have to be true.

But that's just what it is for the inference to be valid.

Recall that for an inference to be valid is just for it to be such that there is no

possible situation, where the premises are true and the conclusion is false.

So from the first line of the truth table for conjunction,

you can see that conjunction introduction inferences are all going to be valid.

You could also see, by looking at the truth table for

conjunction that another kind of inference is always going to be valid.

So, consider an inference that starts with only one premise.

A premise that conjoin's two propositions.

Again, never mind what the two propositions are.

They could be anything.

Call them P & Q.

because it doesn't matter what they are.

So consider an argument that starts with that one premise, P&Q, and

it moves to a conclusion that consists simply of one of

those two propositions that is conjoined in the premise.

So the conclusion of the argument will be either the proposition P or

the proposition Q.

Now an argument like that we can call an conjunction elimination argument,

because the conclusion of the argument

eliminates a conjunction that occurs in the premise of the argument.

Right? The premise is a conjunction of two

propositions, P & Q.

And the conclusion is simply one of those two propositions,

not conjoint to anything else.

So the conclusion is either the proposition P or it's a proposition Q.

Now, is that argument going to be valid?

Well, if you look at the truth table for conjunction,

you'll see that it is going to be valid.

In any possible situation, in which of the premise of that argument P and Q is true.

Both of the two propositions that are conjoined in that premise,

both the proposition P and the proposition Q, are also going to be true.

>> So any situation in which the premise of a conjunctional elimination argument is

true, is going to be a situation in which the conclusion of that

conjunctional elimination argument is true.

>> Therefore, all conjunction elimination arguments are valid,

no matter what they're about.

And we can see that just by looking at the truth table for conjunction.

Now, conjunction introduction and conjunction elimination arguments are not

the most interesting kinds of arguments there are to be sure.

But I just wanted to give a simple example for

now of how we can use the truth table for a truth functional connective.

In this case the truth functional connective conjunction.

How we can use the truth table for

that connective to discover that certain kinds of arguments are going to be valid.

In the next lecture, we'll show how we can use other kinds of truth tables for

other connectives, for other true functional connectives

to show that certain other kinds of arguments are valid.

See you next time.

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