"And" and the Truth-Functional Connectives

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Do curso por Duke University

Think Again II: How to Reason Deductively

133 ratings

Duke University

Think Again II: How to Reason Deductively

133 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.

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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

Conheça os instrutores

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

Now I just said that in English, the word and

functions as a propositional connective.

But it doesn't always function that way, even in English.

For instance, think about a sentence that uses the word and

like the sentence, Jack and Jill finally talked.

I can think of at least three different interpretations of that sentence,

three different things that that sentence could mean.

And corresponding to those three different things that the sentence could mean,

three different ways that the word and is getting used in that sentence.

So here's one thing the sentence could mean.

Jack and Jill could be a name of a particular entity, let's

say a fast food company, there could be a fast food company called Jack and Jill.

Or maybe Jack and Jill the fast food company makes some kind of special stew.

And lawyers have been asking Jack and

Jill to disclose the ingredients of the special stew because they find that a lot

of the customers who eat this stew have been getting sick recently.

So lawyers are asking Jack and Jill to disclose the ingredients and Jack and

Jill refuses to do so until finally the spokesperson for Jack and Jill holds

a press conference in which he discloses the ingredients of the special stew.

I might describe that situation by saying Jack and Jill finally talked.

What I mean there is that there's a particular entity, namely the company,

Jack and Jill.

And that company finally talked through its spokesperson.

Finally disclosed the ingredients of its special stew through its spokesperson.

Now, on that interpretation of the sentence, Jack and Jill finally talked,

the word and is not being used as a propositional connective.

It's not connecting two propositions and

forming a larger proposition out of those two smaller propositions.

Here's another way of understanding the sentence Jack and Jill finally talked.

Let's suppose Jack, Jill, and Roger are having a silence contest

to see who among the three of them can be silent the longest.

And all three of them are silent for a long period of time and

while I'm watching the silence contest, you're calling in to me periodically to

find out if anyone of these three contestants finally talked.

And finally, at one point, you call in and you say okay, has anyone talked yet?

And I say well, Jack and Jill finally talked, so Roger won the competition.

Now, there in that sentence, when I say Jack and Jill finally talked,

the word and is being used as a propositional connective,

what I'm saying in effect is that Jack finally talked and Jill finally talked.

So there are two propositions that I'm trying to communicate.

The proposition Jack finally talked, and the proposition Jill finally talked, and

I'm using the word and to connect those two propositions into a larger

proposition Jack and Jill finally talked.

And so Roger won the competition.

There the word and is being used as a propositional connective.

It's just connecting the proposition Jack finally talked and the proposition

Jill finally talked to form a larger proposition, Jack and Jill finally talked.

And there's a third way that the word and

could get used just in that sentence Jack and Jill finally talked.

Suppose Jack and Jill are a couple and recently they've been having a tough time.

They've been angry and resentful, and they haven't been talking to each other about

the sources of their anger and resentment.

Well suppose they decide that they're going to finally

get their grievances out into the open, they're going to talk to each other.

Then I might say, Jack and Jill finally talked.

Now, there, when I use the word and, I'm not just connecting the proposition,

Jack finally talked and Jill finally talked.

What I'm trying to communicate is not just that each of them talked,

I'm trying to communicate that each of them talked to the other.

I'm trying to say that they talked to each other.

So there in that third interpretation the word and

is not working just as a propositional connective.

It's not just connecting the proposition that Jack finally talked and

the proposition Jill finally talked, because I'm trying to get across something

more than just that Jack finally talked and Jill finally talked.

I'm also trying to get across the point that they talked to each other.

That's what I'm trying to communicate there with the sentence Jack and

Jill finally talked.

So, there the word and is not being used as a propositional connective.

So I hope I've made it clear that the word and

can get used in a bunch of different ways in English.

One of the ways it can get used is as a propositional connective,

to connect two propositions into a larger proposition.

But that's not the only way it can get used.

It can get used in other ways as well.

I've tried to give examples of those just now.

What we're going to be concerned about here is just the use of and

as a propositional connective.

And then we'll look at other expressions in English, like or, not, but,

only, if, and so forth, that also get used as propositional connectives.

We'll look at those, and see how those work.

Okay, so now let's consider the propositional connective and.

Notice though that even when the word and is being used

as a propositional connective, it can still get used in different ways.

For instance, suppose I say, I took a shower and got dressed.

Well there, and is being used as a propositional connective.

It's connecting two propositions.

The proposition I took a shower, and the proposition I got dressed.

It is connecting those into a larger proposition, I took a shower and

got dressed.

But there, the word and

is being used to convey a sense of temporal ordering, a sense of time.

The idea is, when I say I took a shower and got dressed,

the idea is I first took a shower and then got dressed, right?

It wouldn't mean the same thing if I said I got dressed and took a shower.

But sometimes when the word and is being used as a propositional connective,

there's no suggestion of temporal ordering.

So for instance, if I say I'm holding the binoculars and looking through them,

there's no suggestion that I'm doing one first and then the other.

I'm just doing both, I'm holding them and looking through them.

There's no suggestion that one is happening before the other.

Now in that second usage,

when the word and doesn't convey any kind of temporal ordering.

Where it just combines two propositions into a larger proposition

without conveying any sense of temporal ordering, then I'll say the word and

is not just functioning as a propositional connective, but

it's functioning as what I'll call a truth-functional connective.

So what's a truth-functional connective?

A truth-functional connective is a propositional connective

that creates new propositions whose truth or

falsity depends on nothing other than the truth or

falsity of the propositions that went into creating them.

Let me give you some examples to illustrate that definition.

So consider again, our example Jack and Jill finally talked.

Where that used to just that Jack finally talked and that Jill finally talked.

Okay, so when is that proposition going to be true?

When is it going to be true that Jack and Jill finally talked?

Well, that's going to be true.

Whenever it's true that Jack finally talked and Jill finally talked.

As long as those two conditions are true, as long as Jack finally talked, and

Jill finally talked, it's also going to be true that Jack and Jill finally talked.

So if it's not true that Jack finally talked,

than it won't be true that Jack and Jill finally talked.

If it's not true that Jill finally talked, then it won't be true that Jack and

Jill finally talked.

But as long as it is true, that Jack finally talked and

that Jill finally talked, it will be true that Jack and Jill finally talked.

So that use of and is a use of and as a truth-functional connective,

because it creates a new proposition, Jack and Jill finally talked.

The truth or falsity of which depends on nothing other than the truth or

falsity of the two propositions that it connects.

There are lots of other examples.

So for instance, consider the use of and

as a truth-functional connective in the proposition.

I'm holding the binoculars and looking through them.

Again, when is that proposition going to be true?

It's going to be true only in these cases where it's true that I'm

holding my binoculars, and it's also true that I'm looking through my binoculars.

If both of those are true, then the whole proposition, I'm holding my binoculars and

looking through them, will also be true.

If either one of those initial propositions is false,

if it's false that I'm holding the binoculars or

if it's false that I'm looking through the binoculars.

Then the whole proposition, I'm holding the binoculars and

looking though them, will also be false.

So there again, the truth or falsity of the whole proposition depends on nothing

other than the truth or falsity of the propositions that go into creating it

using the truth functional connective and.

And that's what shows that and in that usage is a truth functional connective.

It's that the proposition that it creates by joining other propositions

is a proposition whose truth or falsity depends on nothing other than the truth or

falsity of the ingredient propositions that went into creating it.

That's a truth-functional connective.

Now in week four, this week on propositional logic,

we're going to be studying truth functional connectives, and

how the use of truth functional connectives in argument can make those

arguments valid no matter what those arguments are about.

Now, let's move to some examples.

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