Biconditionals

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From the course by Duke University

Think Again II: How to Reason Deductively

134 ratings

Duke University

Think Again II: How to Reason Deductively

134 ratings

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.

From the lesson

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>

The Conditional14:27

Conditionals in Ordinary Language3:33

Biconditionals7:24

Meet the Instructors

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

Today we're going to talk about a new propositional connective,

a truth functional connective that I'll call the biconditional.

Now what's a biconditional?

In order to explain what a biconditional, let me start off

my telling you the following story, and this is a true story, I should add.

When I was in eighth grade I remember

my first class in the morning was a Math class.

And there was a boy who sat next to me in Math class.

The teacher called him Bob.

My third class later in the morning was a Science class and there was a boy who

sat next to me in the Science class and the teacher called him George.

And I was struck by the fact that Bob and

George, who sat next to me in Math and Science class respectively,

had the same last name and they also bore an uncanny resemblance to each other.

Well one day, I asked Bob if he had a brother named George and

it turned out that he didn't have a brother named George.

In fact, Bob was George.

It was one and the same boy and

they just went by two different names to the two different teachers.

But suppose that all you knew from my description of

the situation was that there was a boy who sat next to me in Math class named bob and

a boy who sat next to me in Science class named George.

And you asked me one day, was Bob born in the United States?

Now I might say to you, well Bob was born in the United States if,

and only if, George was.

Now, there I'm using the phrase if and only if.

Now remember how the phrase if works.

If I say if to connect two propositions, P if Q,

that's equivalent to saying if Q then P.

But now remember how only if works, if I say P only if Q,

that's equivalent to saying if P than Q.

So when I use the phrase if and only if,

I'm conjoining the conditional if P than Q.

And the conditional if Q then P, and so I get what I'll call a biconditional.

A biconditional is a propositional connector that connects two

propositions into a larger proposition.

And the larger proposition is true just in case the two propositions.

That are part of it, have the same truth value.

In other words, the larger proposition, P if and

only if Q is going to be true just in case P and Q are both true or

P and Q are both false.

So.

I could say George was born in the United States if and only if Bob was.

Under what circumstances is that proposition going to be proved?

Well, it will be true if George and Bob were both born in the United States.

Or, well, I guess it's misleading to say both because really there's only

one boy that we're talking about.

George was born in the United States and Bob was born in the United States.

It'll be true when both of those propositions are true.

But it'll also be true when George was not born in the United States and

Bob was not born in the United States.

In other words, it'll be true when both of those two propositions are false.

What makes true the proposition George was born in the United States if and

only if Bob was.

Is simply that the two propositions that are part of it George was born in

the United States and Bob was born in the United States.

Have the same truth value whatever that true value is weather it's the truth value

true or the truth value false.

As long as those two propositions have the same truth value It's going to

be true that George was born in the United States if and only if Bob was.

And so the biconditional connecting those two propositions, George was born

in the U.S. and Bob was in the US, that biconditional is going to be true.

So we can state the truth table for

the truth functional connective which is the biconditional as follows.

The biconditional connects, any two propositions, let's call them P and

Q, it doesn't matter what they are.

When P is true and Q is true, then the biconditional,

P if and only if Q is going to be true.

When P is true and Q is false, then the biconditional P if and

only if Q is going to be false.

When P is false and Q is true then the biconditional P if and

only if Q is going to be false.

And finally, if P is false and Q is false then the biconditional P if and

only if Q is going to be true.

So, that's the truth table for the biconditional.

Now, in the last couple of lectures I described both the conditional and

the bi-conditional as truth functional connectives.

And I've given some reason to think that they are truth functional connectives.

But you might worry that there's some examples that suggest that

both the conditional and the biconditional are not truth functional connectives.

For instance, consider the sentence,

if 2 + 2 = 4 then Pierre is the capital of South Dakota.

Now there I am using if then to express the conditional.

Right? I am saying if it's true that 2 +2 = 4,

then it's true that Pierre is the capital of South Dakota.

Now according to the truth table for

the conditional that conditional has gotta be true.

Because the first proposition what we'll call the antecedent,

the proposition that occurs right after the if,

that antecedent is true, it's true that 2+2=4.

And also the consequent, the proposition that occurs right after the event,

the consequent is also true.

It's true that Pierre is the capital of South Dakota.

So, according to the truth table for the conditional, it's going to be true that if

2+2 = 4, then Pierre is the capital of South Dakota.

Now, you might worry wait a second,

this is a very strange consequence of the truth table for the conditional.

Is it really true that if 2+2=4, then Pierre is the capital of South Dakota?

And that's a very baffling thing to say.

Now I want to say I completely agree with this objection.

It is a baffling thing to say that if 2+2=4

then Pierre is the capital of South Dakota.

But just because it's a baffling thing to say, doesn't mean it's not true.

Look, it's baffling that Pierre is the capital of South Dakota.

And baffling as that may be,

it's still true that Pierre is the capital of South Dakota.

Some things are baffling even though they are true.

And this is another example of that general kind of thing.

It's baffling, nonetheless true.

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