Rules Variables and Generality

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Del curso dictado por Duke University

Think Again II: How to Reason Deductively

134 calificaciones

Duke University

Think Again II: How to Reason Deductively

134 calificaciones

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.

De la lección

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

Conoce a los instructores

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

In the last lecture, we saw how the word and in English can sometimes be used to

express the truth functional connective conjunction.

And we defined conjunction by means of the truth table for conjunction.

Now you'll notice in the last version of the truth table that was presented

in the last lecture.

The propositions that were conjoined in that truth table were called p and q.

Now you might wonder, what's up with calling propositions p and q?

Why are we using letters to refer to propositions?

Is this some kind of strange algebra?

Well today's lecture answers that question.

Today's lecture is about variables like the variables that occur in algebra.

And similar variables that occur in deductive logic, in

the study of the rules for valid argument, which is exactly what unit two is about.

So, let's begin by talking about rules and

why we use variables when we state rules.

The first point to make about rules is that rules are general.

We're also designed to apply to many different possible cases.

So, for instance, if I say to one of my kids, stop hitting your brother right now.

Well that's not a rule, that's a command, right?

I'm telling one of my kids, stop hitting your brother right now.

But it's not a rule.

It's not general.

It doesn't apply to many different possible people.

It doesn't apply to many different possible situations.

I'm just telling someone what to do, or in this case what not to do.

But if I said you should never under any circumstances hit another person,

well now that's a rule.

It's inherently general.

It applies to many different possible situations and

many different possible people.

So is, you should never hit your brother.

That's also a rule, because again,

it applies to many different possible situations.

So rules are general.

But commands may or may not be general.

So how can we express the generality of a rule?

Well notice, if I say something that's not general.

If I say to one of my kids, stop hitting your brother right now, well,

I'm using the expression right now to indicate that my order,

my command applies to a particular situation,

namely the situation that we're in right now when I issued the order.

But if I say you should never under any circumstances hit another person.

I'm using the pronoun you and

the expression never under any circumstances to indicate that

the rule that I'm stating applies to a multitude of possible situations.

In fact, the pronoun you, although it's a second person pronoun,

it needn't be understood as referring merely to the person I'm addressing.

It could be understood quite generally.

Think about the ten commandments, thou shalt not kill.

Thou is an archaic form of the second person singular, you.

Or in French, tu.

So, in those commandments, the second person pronoun

is used not to refer to the specific person, to some specific person

addressed by the commandment, but to refer to anyone at all.

You, whoever you are should not kill.

And notice, generality comes in degrees.

And rules can be more or less general.

So I could say, you should never under any circumstances hit another person.

Well that's a general rule.

But I could state a rule that's even more general than that.

I could say, you should never under any circumstances

do violence to another person.

Now that's more general,

because I'm not just prohibiting hitting, I'm prohibiting any kind of violence,

whether it's hitting, kicking, biting, smacking, slapping.

I could make a rule that's even more general than that.

I could say you should never under any circumstances do something unkind

to another person, whether the unkind thing in question is a form of violence or

merely an insulting action, a demeaning action, a condescending action.

There are lots of different ways of doing things that are unkind,

even if they're not violent.

So here, we have rules of different levels of generality.

If I say, Walter should not force his dog to kill his cat.

I'm saying something that has some generality to it.

Presumably, what I'm saying doesn't simply apply to the present moment, but

it applies more generally to a multitude of moments, not just the present, but

maybe the recent past, maybe the near future.

But of course, I can say something even more general.

I could say people should not force creatures to kill innocent creatures.

That's more general.

I'm not just talking about Walter.

I'm talking about people quite generally.

I'm not just talking about Walter's dog.

But I'm talking about creatures quite generally.

And I'm not just talking about Walter's cat, I'm talking about innocent creatures,

quite generally.

And I'm saying that a certain relationship

between those three things should not hold.

People should not force creatures to kill innocent creatures.

But now notice, that last sentence is unclear.

In particular it has two interpretations,

there are two ways of understanding that last sentence.

We could label them the plural understanding or

the distributive understanding.

On the plural understanding what I'm saying is that people should not force

a whole bunch of creatures to kill a bunch of innocent creatures.

On the distributive interpretation, I'm saying that for any creature that you

pick, people should not force that creature to kill an innocent creature.

So on the first interpretation of the statement,

I'm talking about groups of creatures.

And on the second interpretation of the statement,

I'm talking about each one of a whole bunch of particular creatures.

I'm making a statement about each one of those particular creatures, whereas on

the first interpretation I'm making a statement about the group of creatures.

Now in ordinary language it's not easy to separate out those two interpretations.

You can see just in the last minute

how hard I had to work in order to distinguish those two interpretations.

But if we use variables

then we can distinguish the two interpretations pretty straightforwardly.

We could say in order to specify the distributive interpretation

of that statement, we could say where x stand for any creature at all and

y stands for any innocent creature at all, people should not force x to kill y.

So here we're using the variables x and y in order to express the distributive

interpretation of people should not force creatures to kill innocent creatures.

So variables can be useful to us in clearly expressing

certain kinds of distributively interpreted rules.

Now, in propositional logic variables play an equally useful role.

So consider an inference that we looked at in the last

lecture from the premise, I'm holding my binoculars and looking through them.

It follows that I'm holding my binoculars.

If it's true that I'm holding my binoculars and looking through them, if

that premise is true, then the conclusion, I'm holding my binoculars, must be true.

So that right there is a valid deductive argument.

There's no possible way for

the premise to be true without the conclusion also being true.

But notice, in that deductive argument,

there's nothing special about the proposition I'm holding my binoculars or

about the proposition I'm looking through my binoculars.

In fact for any argument where the premise is the conjunction of two propositions,

and the conclusion is one of those two conjoined propositions

that argument is also going to be valid, there's also going to be no

possible way for the conjunction of the two propositions.

That is the premise of that argument, to be true, without the conjoined

proposition that is the conclusion of that argument, also being true.

In other words, there's not going to be any possible way for

the premise to be true while the conclusion is false.

So any argument of this form right

here is going to be valid.

Now notice in the example I just used as well as in the truth table from

last lecture, I was using Roman letters like x, or y, or P, or Q as variables.

But there's nothing sacrosanct about using Roman letters as variables.

I could have used anything else so

long as it's easy enough to recognize and easy enough to reproduce.

Here are some examples.

I could have used a dot, or a percentage sign, or a plus sign, or a happy face, or

a dollar sign.

The problem with all of those is that while they're easy enough to produce,

they're not that easy to recognize as variables in propositional logic for

the simple reason that they're all used

to express other things besides variables and proposition logic.

So whereas Roman letters like x, y, P, Q typically aren't

used by themselves to express anything at all other than variables.

So we can use them safely as variables in propositional logic

without running the risk that people won't recognize them for what they are.

That people will think that they're expressing something else.

Notice just as we have notation, Roman letters serving as notation

for variables, we also have notation to express truth functions.

For instance,

in the last lecture I introduced the truth functional connective conjunction.

And normally, we express conjunction using this ampersand right here.

And there are other truth functions as well that we're going to learn about

in upcoming lectures.

One is disjunction which we normally express using an uppercase V.

And another one is negation which we can express using

either one of these two signs that looks a little bit like a minus sign,

from arithmetic, but not exactly the same.

These are two different ways of expressing negation.

Now, again, there's nothing sacrosanct about this notation.

There have been different notational systems that people have used,

there is nothing better about this notational system.

We just have to pick one and stick to it for the sake of consistency.

So this is the one we are picking, not for

any good reason, just because we have to pick one.

So in upcoming lectures, you're going to learn about disjunction and

about negation.

And when we talk about those truth functions we're going to use the notation

that you see here.

See you next time.

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