## Saturday, April 20, 2013

### What an argument is, in formal logic

I. What An Argument Is.

By "argument" I do not mean an emotionally charged, verbal fight. That's a different use of the word. The notion of argument discussed here has to do with providing evidence or support for the truth of propositions.  Arguments are expressed or stated by people to establish or support the truth of a proposition.
An argument has parts.  Its parts are propositions.  Propositions are expressed by declarative sentences.  Declarative sentences, roughly, are sentences one can produce, in writing, speaking, typing, or by other means, to "declare" that something is so and so.  For example, the sentence, "Sacramento is the capital of California", can be used by someone to declare that Sacramento is the capital of California.  The sentence, "The snow is white", can be used to declare that some patch of snow is white.  But so can the German sentence, "Der Schnee ist weiss".  The German sentence declares the very same thing, that the snow is white.
What both sentences declare is what someone "says" or "means" when she speaks, writes, or otherwise produces those sentences.  Now, here's an interesting point.  Suppose a German friend says to you, while you are stumbling through snow in Tahoe, "The snow is white".  You are looking for a place to sit down and have a picnic.  Then he turns and says to his German friends, "Der Schnee ist weiss."  He's said the very same thing with both sentences.  What is said or meant in producing a declarative sentence is a proposition.  So, he's used two very different sentences to express the same proposition.  What this shows is that a sentence and a proposition are different things.  How is that?  If what he said or meant on each occasion was the sentence, then he'd have said or meant two different things, since the two sentences are very different.  Sentences are concrete, physical items, whether they are marks on a piece of paper, or certain collections of sound waves, or patterns of light emissions on a monitor, and language speakers use them to say or mean things.
Importantly for our purposes, propositions are the things that are true or false.  Marks on a paper or sound waves in the air are not the sort of things that are true or false.  The proposition that Sacramento is the capital of California, i.e., the proposition expressed by the sentence, "Sacramento is the capital of California", is true, since that is in fact the way things are.  If Stockton were the capital, then the proposition that Sacramento is the capital would be false.  If someone speaks or writes the sentence, "The CSUS campus is north of the American River", then he or she has said something false. That is, the proposition expressed by that sentence is false, and it is false because that is not the way things really are.
Not all sentences can be used to say something that is either true or false.  So, not all sentences express propositions.  For example, if a lawyer in a courtroom says, "Did you kill Margaret?", the lawyer has not said something that is either true or false.  The lawyer has not expressed a proposition; he's asked a question.  His interrogative sentence does not declare that something is so and so. Neither does the mother express a proposition when she says to her child, "Johnny, shut the door".  The imperative sentence, "Shut the door", does not declare that anything is so and so. It presupposes, but does not assert, that the door is in fact open.  If Johnny were to say, "That's false", his mother should painfully pinch his ear, lest he persist in the bad habit of confusing commands with assertions.
The following three sentences express propositions. They say something that is either true or false.  So, we have a set of propositions expressed.

Dogs bark.
Los Angeles is in California.
The Dorian invasions of ancient Greece were very violent.

But is an argument expressed?  No. Why not?  Read the account of what an argument is in the box above.  While we have a set of propositions expressed by these sentences, there's no indication that one of the propositions is taken by someone to be supported by the rest.  Or, another way to put the point is that we have no indication that someone believes any of the propositions of the set support the truth of any other.  Suppose, on the other hand, Maria says,

Catherine is a student at State University.  All students at State University own computers.  So, Catherine owns a computer.

Do we have an argument? Yes. We have a set of three propositions.  We have one which is taken by someone, Maria, to be supported by the rest.  The "so" at the beginning of the third sentence indicates that Maria takes the proposition expressed by that sentence to be supported by the first two.  But what does it mean to say that Maria "takes" the first two to "support" the third?  It means she believes the two propositions, that Catherine is a student at State and that all students at State own computers, are both true, and that given they are true, they provide reasons or evidence for believing it is also true that Catherine owns a computer.  So, Maria has expressed an argument.  The premises of the argument are the propositions that Catherine is a student at State and that all students at State own computers. The conclusion is the proposition that Catherine owns a computer.
Suppose Jonathan, who's always a bit spacey, says the following.

Wow, man, peace and harmony will come to Earth, since the Moon is full, the   mountains are covered in snow, and Giecko just insured me for hundreds less.

Is Jonathan expressing an argument?  Do we have a set of propositions?  Yes.  Proposition 1: peace and harmony will come to Earth. Proposition 2: the Moon is full. Proposition 3: the mountains are covered in snow.  Proposition 4: Giecko insured Jonathan for hundreds less.  Is one of them such that someone regards it as being supported by the rest?  Yes.  Jonathan takes Proposition 1 to be supported by 2, 3, and 4.  This is shown by Jonathan's use of "since". So, Jonathan has expressed an argument.  Proposition 1 is the conclusion. Propositions 2, 3, and 4 are the premises.  How's it known that Proposition 1 is the conclusion?  Jonathan's use of "since" indicates that he takes the propositions that are stated after it to support Proposition 1.  (I'll discuss common premise and conclusion indicator words like "since" and "so" below.)
Now, Jonathan's argument is very bad.  It's bad, because the fact that the moon is full, the mountains are covered in snow, and Giecko insured Jonathan for less, does not give us any reason at all to think its true that peace and harmony will come to Earth.  An argument doesn't have to be a good argument to be an argument.  There are bad arguments.
Arguments expressed in everyday, professional, and even academic contexts are usually expressed informally, like Maria and Jonathan expressed their arguments above.  When an argument is expressed informally, it's not always clear which propositions the person takes to be the conclusion nor which the premises, since often he or she does not explicitly indicate which are which. Further, a writer or speaker will often mix in with the statement of an argument sentences that do not really play a role in supporting the conclusion.  To clearly identify the argument a person intends to make and to weed out irrelevant information, we "reconstruct" the argument by stating it in what is called "standard form".  Maria's argument expressed in standard form appears like this.

1. All students at State University own computers.
2. Catherine is a student at State University.
---------------------------------
3. Catherine owns a computer.

The horizontal line is called an "inference bar". It indicates that the proposition expressed by the sentence below the line is regarded as supported by the propositions expressed by the sentences written above the line.  The sentences above the inference bar express the propositions that are the premises of the argument.  When reconstructing an argument, only the propositions intended by the arguer to support the conclusion are expressed above the inference bar.  Usually, but not always, the premises and conclusion are numbered for easy reference.  There's no standard numbering system.

Certain words that people use when expressing arguments informally in writing or speaking will usually indicate what propositions they regard as premises and conclusions.

When you see or hear such words, look for the argument being expressed.  It's usually easiest to identify the conclusion first.  Once identified, then seek out the premises by finding premise indicator words.  (Note: not all expressions of argument have premise and conclusion indicator words.)

Since an argument is employed by someone with the intention of providing support for the truth of a proposition, an argument is good if it does in fact support the truth of the conclusion and bad if it does not.  Now, whether an argument supports the truth of the conclusion depends on what kind of argument is presented.  There are two kinds of argument: deductive and inductive.  A deductive argument is one that the arguer takes to be such that if the premises are true, then the conclusion must be true.  On the other hand, an inductive argument is one that that the arguer takes to be such that if the premises are true, then it is likely, or probable, that the conclusion is true.  In this course, as in most philosophy courses, we'll focus almost entirely on deductive argument, and so it is deductive argument that I will discuss almost exclusively below.
But, before I do that, I'll show you quickly the difference between deductive and inductive arguments. The first argument below is a deductive argument.  Suppose Donald Rumsfeld, the U.S. Secretary of Defense, says the following at a Pentagon press conference.

Either Saddam shipped his WMD's out of Iraq, or they are still there. Well, they are not there now.  So it follows that Saddam shipped his WMD's out of Iraq.

Stated in standard form, the argument is this.

i.   Either Saddam shipped his WMD's out of Iraq, or Saddam's WMD's are in Iraq.
ii.  Saddam's WMD's are not in Iraq.
-----------
iii.  Saddam shipped his WMD's out of Iraq.

Why is this a deductive argument? Because Rumsfeld takes it that if both premises are true, then it must be true that Saddam shipped his WMD's out of Iraq.  If both premises are true, but yet it is possible that the conclusion is false, Rumsfeld would concede the argument is bad.
The following is an inductive argument.

P1. Most Athenians during the ancient period ate fish daily.
P2. Socrates was an Athenian during the ancient period.
-----------
C. Socrates ate fish daily.

Someone who offers this argument does not suppose that it must be true that Socrates ate fish daily, if the premises are true.  He or she is only claiming that it is likely or probably true he did, given most Athenians during the ancient period ate fish daily and Socrates was an Athenian during this period.  He or she would readily admit it's possible that Socrates did not eat fish daily, but insist nevertheless the argument is still a good one.  If the premises are true, he or she would say that we have very good reason for thinking that Socrates ate fish daily, even if it doesn't follow that he did.

The following concepts apply only to deductive arguments.  A good deductive argument is called a "sound" argument.  A bad deductive argument is called an "unsound" argument.  So, sound argument establishes that its conclusion must be true, and an unsound argument fails to do so.

A sound argument must satisfy both conditions (i) and (ii) at the same time.  An argument that satisfies (i) and (ii) automatically is a good deductive argument. Nothing more is needed.
To understand what a sound argument is, we must understand what conditions (i) and (ii) amount to and how they are related.  Let's start with condition (i): deductive validity.  Students invariably have the most trouble with this concept.  Study the account below very carefully and work diligently through the practice examples.

An argument that is not valid is an argument that is invalid. An argument that is not valid is just one that fails to satisfy the account of argument validity above. An argument that fails to satisfy the above account is one with respect to which it is possible all the premises are true but yet the conclusion is false.  A more informal way to think of validity is like this. An argument is valid just in case the conclusion "follows" from the premises, and invalid just in case the conclusion does not follow from the premises.  Consider these two propositions.

All human beings are mortal.
Socrates is a human being.

??

When you think about both of the above statements, what comes to mind?  I hope it is this.

Socrates is mortal.

If the proposition that Socrates is mortal "popped" into your mind, so to speak, you were recognizing that the proposition that Socrates is mortal follows from the other two propositions.  This is another way of saying that you were judging that the following argument,

All human beings are mortal
Socrates is a human being
---------
Socrates is mortal,

is a valid argument.  If the premises are true, the conclusion must be true--it follows. On the other hand, I'll wager that when you considered the above two propositions, the following did not come to mind.

Dan Rather left CBS under allegations of political bias.

The above proposition does not follow from the first two.  That is, the argument, if there were someone to express it, that

All human beings are mortal
Socrates is a human being
--------
Dan rather left CBS under allegations of bias,

would be invalid. The conclusion does not follow.  It's possible the premises are true but the conclusion is false. Let's consider some more examples.

Example #1.

1. If the President is a space alien, then the Vice President is the Pope.
2. The President is a space alien.
----------
3. The Vice President is the Pope.

Is the argument valid or invalid?  Don't guess intuitively.  Study the account of argument validity in the box and ask yourself whether this argument satisfies that account or not.  Now, forget for the moment about whether the premises and conclusion are actually true or false. When it comes to the validity of an argument, the actual truth or falsity of the premises or conclusion is irrelevant.  Ask yourself this when you judge whether an argument is valid.  If the premises are true, must the conclusion be true?  That is, suppose the world is such that the premises are in fact true, even if they are not really.  Then, ask yourself whether or not it would have to be true in that world that the conclusion is true. If the (correct) answer is, Yes, then the argument is valid, and if, No, then the argument is invalid.  In the above case, even though both premises are false, and even though the conclusion is false, the argument is valid.
Notice something interesting about Example #1.  Both premises and the conclusion are actually false.  Nevertheless, the argument is valid.  What the above example shows is that an argument can be valid even if all of its propositions, premises and conclusion, are actually false.  Again, the actual truth or falsity of the premises and conclusion is irrelevant as to whether or not the argument is valid or not.

Example #2

Every skier loves some snowboarder.
Erin is a snowboarder.
------------
Every skier loves Erin.

Valid or Invalid? Don't guess.  Go back to the account of argument validity, read it, and then ask yourself whether this argument does or does not satisfy that account.  It's only when you practice making judgments about argument validity according to the concept that you'll learn to do it well. If you rely on quick, intuitive judgments, you're liable to make mistakes in more difficult cases.  Psychological experimentation shows that certain subtle but invalid argument forms consistently fool very high percentages of college students.  It does so because they judge validity intuitively.  Now, the account says that an argument is valid, if and only if, if the premises are true, the conclusion must be true. Or, to say the same thing, an argument is valid just in case it is not possible the premises are true but the conclusion is false.  So, ask yourself this.  Suppose it is true that every skier, every last one of them, loves some snowboarder.  And, suppose it is true that Erin is a snowboarder. Must it then be true that every skier loves Erin?  No.  It's possible that both premises are true, but the conclusion is false.  It's possible that every skier loves some snowboarder, but that every skier loves some different snowboarder, say Kyle or Jennifer.  Or, it's possible every skier loves some snowboarder but half of the skiers love Kyle and the other half love Jennifer.  Or yet, supposing there are as many snowboarders as skiers, it's possible each skier loves a snowboarder that is different from the snowboarder any other skier loves.  So, the argument is invalid.

Example #3

Suppose Samantha, an economics major, says the following.

"Well, given that the U.S. budget deficit for 2005 is over 200 billion dollars, and the average U.S. unemployment rate for 2005 is just a bit over 5.0%, we can conclude that U.S. federal tax revenues increased in 2005."

The argument is expressed informally. Before evaluating it, let's express it in standard form, so that we can more clearly identify what the argument is exactly.  The first thing to notice is that the quote above contains a premises indicator, "given that", and a conclusion indicator, "conclude that". So, the argument, expressed in standard form, will look like this.

P1. The U.S. budget deficit for 2005 is over 200 billion dollars.
P2. The average U.S. unemployment rate for 2005 is a bit over 5.0%.
-------------
C.  U.S. federal tax revenues increased in 2005.

Valid or invalid?  Is it possible that the premises are true but the conclusion is false?  Quite so.  The argument is invalid.  Notice however that all of the propositions of the argument are true!  Even the conclusion is true!  But given the argument is invalid, the premises do not support the conclusion.
Given the argument is invalid, it's a bad argument.  This may seem strange. But, keep this in mind.  One's aim in stating an argument is not to state a true proposition, the conclusion.  One's aim is to provide evidence or support for the truth of the conclusion.  Samantha's argument is bad, not because the conclusion is false, but because the reasons she gave for thinking the conclusion is true, do not show that it is true.  There may well be other evidence for thinking it is true that federal tax receipts increased in 2005, but her argument doesn't provide that evidence.

Validity has to do with the way that the premises are logically related to the conclusion.  For one or some propositions to support or establish a conclusion, not only must they be true, as condition (ii) of argument soundness states, but they must also be related logically in the right way to the conclusion.  The point is sometimes hard to grasp, because nearly always people assert premises that roughly seem like they are related in the right way.  The following propositions are true.

George W. Bush is President of the U.S.
November follows October.
Toyota is the largest car manufacturer in the world.

But now suppose I say, "Since Bush is President, November follows October, and Toyota is the largest car manufacturer, it follows that Labrador retrievers are either black, yellow or chocolate."  Stated in standard form, the argument is,

1. George W. Bush is President of the U.S.
2. November follows October.
3. Toyota is the largest car manufacturer in the world.
------------------
4. Labrador retrievers are either black, yellow, or chocolate.

It's a bad argument.  But why!?  The conclusion is true!  Again, the premises are not logically related to the conclusion in the right way, and hence, they cannot logically support the conclusion, even though all the premises are true.
If an argument is invalid, you need not know anything more to judge that it is unsound, and hence that it does not establish its conclusion.  But if valid, you need to determine also whether it satisfies condition (ii) of the concept of argument soundness.  Condition (ii) says that all of the premises of a sound argument are true. How do you determine whether a premise is true or false?  Well, easy rules can't be provided.  The evidence for and against the truth of a proposition varies depending on the kind of proposition at issue.  Sometimes scientific studies, experiments, theories, etc., will provide the evidence we can use to judge whether a premise is true or false. Sometimes common sense is all that is needed.  If the proposition that human beings are language-users is a premise in an argument, you can judge it is true simply based on your thorough and life-long experience interacting with other human beings.  Sometimes propositions are conceptual truths.  For example, the proposition that all bachelors are unmarried is conceptually true.  Being unmarried is part of the concept of being a bachelor.  Sometimes simple observation or ordinary experience is all that is needed.  For instance, suppose little Johnny tells his Mom, "I didn't break the neighbor's window! I was at Josh's all day. Call his Mom and ask her!"  Little Johnny is expressing the following argument.

(1) I was at Josh's all day.
(2) If I was at Josh's all day, I didn't break the window.
--------
(3) I didn't break the window.

Johnny's mother judges correctly that the argument is valid--and pats Johnny on the back for his good logical skills.  But now she must determine whether the premises are true.  Well, she doesn’t need to haul in a team of physicists from M.I.T. to make a good judgment about whether (1) is true. She can simply call up Josh's mother and ask her whether he was there at the time the window next door was broken or not.  And, she can rely on her ordinary understanding of the way the world works to judge that (2) is true.  If Johhny was at Josh's all day, then he wouldn't have been in a position to bring any force against the window.  Josh's house is on the other side of town.
If you judge that an argument is both valid and has all true premises, then you've judged that the argument is sound, and that it does establish the truth of its conclusion.  There's nothing more you need do to judge that it is a good argument.

III.  Some Properties of Argument Soundness

If you believe that an argument is both (i) valid and (ii) has all true premises, then you are rationally committed to belief in the truth of the conclusion.  That is, if you believe that an argument is both valid and has all true premises, but nevertheless you believe the conclusion is false, or even that it is possibly false, then you are holding inconsistent beliefs.  Inconsistent beliefs are beliefs that cannot be true at the same time.  Think about what you judge when you judge that an argument is valid.  You judge that (i) if the premises are true, then the conclusion must be true. But now when you go a step further and judge that (ii) the premises are in fact true, then given your belief that (i) and (ii) hold, you've got to conclude that the conclusion must be true.  If you judge that an argument is sound, you're essentially thinking thorough an argument of your own, having the following premises.

(1) If the premises of this argument are true, then this conclusion must be true. (It's valid)
(2)  The premises of this argument are true.

Now, look at both, and ask yourself what must follow if both are true?

(3) This conclusion must be true.

This feature of soundness is interesting for what it tells us when we believe that the conclusion of an argument is actually false.  If one judges that the conclusion of an argument is false, then one must hold that the argument is unsound.  After all, the conclusion of a sound argument must be true. If an argument is unsound, then it must fail to satisfy either or both of (i) and (ii).  Either the argument is invalid, or it has at least one premise that is false, or both.

Another interesting property of soundness is this.  To say that an argument is unsound is to say that the premises do not support the truth of the conclusion.  It doesn't mean that the conclusion is false.  Many unsound arguments have true conclusions, including the following.

If Lance Armstrong is the president of Russia, then Shaquille O’Neal is an NBA center.
Lance Armstrong is the president of Russia.
---------------------------
Shaquille O’Neal is an NBA center.

The argument is valid.  But, both premises are false, and so the argument is unsound.  There may be other arguments which do successfully establish this conclusion, but it's not this one.  To say that an argument is unsound is not to say that its conclusion is false; it is just to say that the particular argument used in the attempt to establish the conclusion as true fails to do so.

I've been speaking so far as if an argument that is sound is good and an argument that is unsound is bad and that's all there is to it. You might call the concepts of good and bad used here "absolute" concepts. (This is not a standard term). Usually, though, it will be difficult to say for certain that the premises of an argument are true.  We might in some cases judge that the argument is valid and that its premises are plausible or very plausible, and then nevertheless call such an argument a good argument.  In such cases we'll mean that the argument provides sufficient evidence or reason for thinking the conclusion is true, even though we aren't certain that the premises are true.  Now, it will depend on the context and what the argument is about as to whether we will call an argument good even if we aren't certain that its premises are true.  For instance, in mathematics, a proof, which is just a particular form of deductive argument, is bad if the steps in the proof, or the premises, are only plausibly true.  But in everyday contexts, we might consider an argument very good even if we can only judge that its premises are plausibly true.  Consider again Johnny's argument.

(1) I was at Josh's all day.
(2) If I was at Josh's all day, I didn't break the window.
--------
(3) I didn't break the window.

If Josh's Mom tells Johnny's Mom that Johnny was at her place all day, then Johnny's Mom is going to think this is a good argument, good enough to think that the premises provide sufficient evidence for the truth of the conclusion.  Nevertheless, she doesn't know for certain that both premises are true.  She can't say for certain that Johnny did not set up a tricky mechanical device that propelled a rock into the neighbor's window when remotely activated from Josh's house.  She can't say for certain that Johnny does not possess paranormal powers, so that just by thinking about the window breaking, it breaks.  But given she knows Johnny spends all day playing baseball, rather than tinkering around with remote activation devices, and given that she's never seen Johnny exhibit any paranormal powers, she's certainly safe judging that Premise (2) is true.  If Johnny's father came home, listened to Johnny's argument, and then said, "That's a bad argument, since it's not certain the premises are true," Johnny would be right to protest.
So, here's another sense in which an argument is good.  A deductive argument is good if it is valid and the premises are plausibly true, to such an extent that in the context, it is sufficient grounds for believing the conclusion is true.  The concept of good is here a relative concept because whether an argument is sufficient for believing the conclusion is relative to the context and what the argument is about.  If there are high stakes involved in believing the truth of a conclusion, we'll want the premises to be very plausible, or nearly certain, or even certain, before we accept that the argument provides a sufficient basis for believing the truth of the conclusion.  If less weighty matters are involved, like in Johnny's case, we may like his mother accept that argument as sufficient reason for believing that Johnny did not break the window.
Approach the matter in this way. The concept of argument soundness spelled out above, the "absolute" concept, you might say, presents an ideal.  An argument is better the closer it is to the ideal, and worse the further away it is from the ideal. You should aim to express sound arguments in your thinking generally, in class and in your papers--to get as close as you can to the ideal.  And in evaluating the arguments of others, even if we can't say the premises are certainly true, we may nevertheless judge that they've presented fairly good arguments, depending on the circumstances in which they are expressed.
But here are two qualifications.  Validity itself is not relative to a context.  An argument in any context is either valid or invalid. Secondly, while in cases where we cannot say we know for certain that a premise is true but may nevertheless think an argument a good one, if we know a premise is false, then the argument is done for.  If a premise is false, the argument automatically fails to support its conclusion; a deductive argument with a false premise gives you no reason at all to believe the conclusion is true.

IV. Argument Criticism

If someone offers an argument you do not think supports the conclusion, you may want to show or establish that the arguer has failed to support the conclusion. Doing so is called "argument criticism".  An argument fails to be a good argument, if it is either invalid or has a false premise.  So, showing that an argument fails to establish its conclusion requires showing either that it is invalid or that it has a false premise.  There is no other way to show that an (deductive) argument fails.   When in class an argument is up on the board for evaluation, there are only two things one should say initially in criticizing the argument.  "The argument is invalid".  "The argument has a false premise."  From there, we can go on to explore why you think so.

How do you show that an argument is invalid?  Logic provides mechanical techniques for doing so.  In other words, a machine or computer programmed in the right way can determine whether an argument is valid or invalid, so long as the sentences stating the argument can be accurately translated into a form the machine can read.  Discussing these techniques would be too involved for this brief discussion.  But, intuitively, one can assess whether an argument is valid or invalid simply by imagining a world in which the premises are true and thinking carefully about whether the conclusion would also have to be true in that world.  If you can think of a possible way the premises are true but the conclusion is false, then you have the means to judge that the argument is invalid.  Beyond that, most (deductive) argumentation will fall into a limited number of common argument patterns.  If you know well which common argument patterns are valid and invalid, you can immediately detect whether any argument having one of those patterns is valid or invalid. I'll introduce a few of the common argument patterns towards the end of this document.
Showing that a premise is false or that any proposition generally is false, is somewhat complicated logically.  But, getting the hang of this will benefit you in argumentative essay writing, in public debate, or in any other milieu in which argument is required.  Study the following carefully.
For any proposition, there is another proposition called its "negation".  For example, the negation of

(1) Al Gore is the U.S. President,
is,
(2) Al Gore is not the U.S. President.

Another way to state the negation of (1) is like such:

(3) It is not the case that Al Gore is the U.S. President.

Proposition (3) says the same thing as (2).  The negation of any proposition, p, can be formed simply by adding "it is not the case that" in front of the sentence expressing the proposition, as was done in the case of (3).
Here's the key thing.  Propositions and their negations are logically related in a very important and interesting way. Where a proposition, p, is true, its negation must be false, and where p is false, its negation must be true.  So, for example, if it's true that

Al Gore is the U.S. President,

then, the negation,

Al Gore is not the U.S. President,

must be false.  And, if it is true that Al Gore is not the U.S. President, then it must be false that Al Gore is the U.S President.  So, again, where a proposition is true, its negation must be false, and where a proposition is false, it's negation must be true.
O.K.  So, what does this have to with showing that a premise is false and why is it so important?  I'm going to introduce two terms to make understanding what comes next easier. An "objection" to a premise, or any proposition, is just the assertion or claim that the premise, or proposition, is false.  Suppose Sam claims that,

U.S. federal tax receipts increased in 2005.

Jose thinks this is false, and says,

U.S. federal tax receipts did not increase in 2005.

Jose has objected to the claim made by Sam.  The important thing at the moment is this.  To assert that a proposition is false, to object to it, is the same as to assert that the negation of the proposition is true.  Jose asserts as true a negation when he says, "U.S. federal tax receipts did not increase in 2005", and he asserts the negation of the claim made by Sam.
Now, suppose Sam asserted the proposition that U.S. federal tax receipts increased in 2005 as a premise in an argument, say the following.

(1) If U.S. federal tax reciepts increased in 2005, then the Dow Jones stocks increased in value by 10% in value in 2005.
(2) U.S. federal tax receipts increased in 2005.
-------------
(3) The Dow Jones socks increased in value by 10% in 2005.

Jose evaluates the argument. He judges that premise (2) is false, and so concludes that the argument fails to establish its conclusion.  Now, so far, Jose has only judged that it is unsound. If Jose wants to show that the argument is unsound--to engage in argument criticism--then he needs to show that premise (2) is false.  How does he do it?  First, he must object to premise (2).  He must assert the negation of (2). The negation of (2) is

(X) U.S. federal tax receipts did not increase in 2005.

Let's name this proposition (X), for easy reference.  So far, though, Jose has merely objected to (2).  Objecting to the truth of a proposition is merely announcing that you believe it is false. But merely announcing that you believe something is false does not show that it is false.  So, Jose needs to do something more than merely assert (X).
Here is where the interesting logical relationship between a proposition and its negation comes into play.  If a proposition is true, its negation must be false; and if the negation of a proposition is true, the proposition it negates must be false.  That means that if (2) is true, then (X) must be false; and if (X) is true, then (2) must be false, since (X) is the negation of (2).  Given this, if Jose can offer reasons or evidence that will show (X) is true, he'll have shown at the very same time that (2) is false.  A "rational objection", as I will call it, is an objection made with reasons or evidence supporting the objection.
What is the thing by which someone attempts to show that a proposition is true?  An argument.  So, to show that premise (2) is false, Jose must offer an argument that has (X) as its conclusion.  Jose needs to offer an argument having the following structure.

(1) Premise.
(2) Premise.
. . .
(n) Premise. (However many premises the argument may need.)
----------
(X) It is not the case that U.S. federal tax receipts increased in 2005.

If Jose offers an argument having this structure that is valid and has true premises, Jose will have shown that (X) is true. But if (X) is true, then (2) must be false, since (X) is the negation of (2).  So, if Jose's argument is sound, then (2) must be false, and hence Sam's argument must be unsound.
That is the logic of an effective and rational objection.  There's no other logic to it.  If
an attempt to criticize an argument on the grounds that it has a false premise does not include the assertion of the negation of the premise and an argument for the truth of the negation of the premise, then the criticism fails.  When we evaluate arguments in class, I want you to try your best to offer criticism of the arguments we discuss in this way.  Assert the negation of a premise you think is false.  Offer as good an argument as you can having the negation as a conclusion.

V.  Common valid and invalid argument patterns.

Each of the following common and valid patterns of argument--the ones in the left hand column below--you'll see often during this course.  The patterns in the right hand column are deceptively like the valid patterns in the right hand column, but they are invalid.  Some of these patterns, valid and invalid, appeared in the above discussion: go back and try to identify them for practice after studying them.  The letters "p" and "q" and "r" used below you can think of as "placeholders" for propositions.  In other words, in any given pattern below, if you replace the same letter in all its occurrences with the same declarative sentence expressing a proposition, then you'll end up with an "instance" of that pattern.  All the instances of the patterns in the left hand column will be valid arguments, no matter what sentences you replace the letters with.  All the instances of patterns in the right hand column will be invalid, no matter what sentences you replace the letters with.  The symbol, "~", called the "tilde", symbolizes negation.  You can read "~" simply as, "it is not the case that", and hence, ~p, is the proposition that it is not the case that p.  The common names for the patterns, where there are such, are printed right above the patterns.

Valid Invalid

Modus Ponens Affirming the Consequent

If p, then q If p, then q
p q
-------- ---------
q p

Modus Tollens Denying the Antecedent

If p, then q If p, then q
~q ~p
----------- ------------
~p ~q

Disjunctive Syllogism (No standard name)

p or q p or q p or q           p or q
~p or ~q p   or q
------- ------- ------ -------
q p ~q ~p

The Dilemma

p or q
If p, then r
If q, then r
--------
r

Let's test a pattern. Let's start with the modus ponens pattern.  Let's replace both occurrences of "p" in that pattern with any old declarative sentence, say, "All dogs bark", and let's replace both occurrences of "q" with any old declarative sentence, say, "Either Jupiter or Mars is larger than the earth."  We then get the following instance of the modus ponens pattern.

If all dogs bark, then either Jupiter or Mars is larger than the earth.
All dogs bark.
------------
Either Jupiter or Mars is larger than the earth.

Sure enough, we have a valid argument.  If the premises are true, the conclusion must be true.  (It is not sound of course; both premises are false.)
Now, here's a tricky logical matter.  You can replace all the occurrences of "p" in the above pattern with any proposition, no matter how complex.  For example, since negations are propositions, you can replace "p" with a negation, ~r, and you will still have a modus ponens argument.  Consider the following two sentences.

It is not the case that cats fly.
If all dogs bark, then some dogs bark.

Both sentences express propositions.  Let's replace "p" in the modus ponens pattern with the first, and "q", with the second.  We end up with the following instance of the modus ponens pattern.

If (it is not the case that cats fly), then (if all dogs bark, then some dogs bark).
It is not the case that cats fly.
------------
If all dogs bark, then some dogs bark.

Even though the propositions involved in the argument are more complex, this is still a modus ponens argument, and so it is also valid.  Even though the second premise is a negation, it is still a modus ponens argument, and not a modus tollens argument.

VI.  Arguments, Subarguments, and Argument Annotation

Often an arguer will attempt to establish a conclusion by introducing several arguments at the same time.  Call the argument below "Argument A".

Argument A (1)  If p, then r.
(2)  If q, then ~r
(3)  q
(4)  ~r
-----------
(5)  ~p

Argument A contains both a modus tollens and a modus ponens argument.  The modus ponens argument, call it "Argument B", is the following.

Argument B (2) If q, then ~r
(3) q
---------
(4) ~r

and the modus tollens argument, call it "Argument C", is the following.

Argument C (1)  If p, then r.
(4)  ~r
---------
(5) ~p

If you think carefully about Argument A, you'll see that it is an independent argument for the truth of a premise in the argument for the ultimate conclusion, (5).  The separate argument is Argument B, which has (2) and (3) as premises supporting as its conclusion, (4).  But (4), while a conclusion in Argument B, is also a premise in the argument for (5), i.e., Argument C.
The logical structure of Argument A looks like this, where an arrow is equivalent to the horizontal line called previously an inference bar.  The arrow points to the conclusion.

(1)
(2) & (3) ---->  (4)
|
|
\/
(5)

Premises (2) and (3) support (4), and (1) and (4) support (5).  Premise (1) has no support offered for it in the argument. In even more complicated cases, a subargument for either (2) or (3) could have been offered as well, so that the structure of the reasoning would then be,

(x)
(y)
|
\/ (1)
(2) & (3) ----> (4)
|
\/
(5)

Stretches of writing expressing argument will often have this complicated structure.  And, the arguer may well fail to indicate which propositions support which propositions--which are conclusions and which are premises.  This makes it difficult to properly understand the reasoning of the arguer. For example, we may mistake the argument for (5) as

(x)
(1)
(2)
-------
(5)

But if we do, we'll have failed to accurately understand the argument, and when we turn to evaluate the argument, we may think that the argument for (5) is invalid, when the writer has presented a valid argument for (5), i.e., Argument C.  So, one thing you want to pay special attention to is determining which proposition is meant to support which, and one thing you want to avoid is automatically assuming that all the propositions stated are part of the same one argument for the main conclusion.
Strictly, our arguer has presented only one argument for the main conclusion, (5). That is Argument C, with premises (1) and (4).   Why aren't (2) and (3), or also (x) and (y), a part of the argument for (5)?  An argument is a set of propositions, one of which, the conclusion, is taken to be supported by the rest, the premises.  Supposing the arguer knows what he is doing, he will not take (2) and (3) to support (5).  It's (1) and (4) that support (5).  So if the arguer does not take (2) and (3) to support (5), they will not be part of the argument for (5).  Why is that?  Because if (1) and (4) only are true, (5) must be true, given Argument C is valid. In fact, (2) and (3) could actually be false and Argument C could nevertheless be a sound argument!  Recall, the conclusion of an unsound argument may in fact be true, even though the argument given for it may have premises that are all false.  So, Argument B could be unsound given both (2) and (3) are false, but nevertheless (4) might still be true. Well, if (2) and (3) could be false, yet Argument C could still be sound, then (2) and (3) are not supporting the main conclusion, (5), and hence are not part of the argument for (5).
Why then call Argument A an argument at all?  It's really a matter of convention.   Traditionally, many have called all the claims and arguments that someone expresses at one time in support of an ultimate conclusion "an argument".  I will at times in the class as well.  Given this convention, some will distinguish between a "main" argument and a "main" or "ultimate" conclusion, and "subarguments" and "subconclusions".  For example, in Argument A, the main conclusion is (5).  Premises (2) and (3) are the premises of a subargument having (4) as its conclusion.  We can call (4) a "subconclusion", even though (4) is at the same time a premise in the argument for (5), that is, argument (C).
Often in reconstructing an argument in standard form that consists of subarguments, the statements of the premises are "annotated" to explain the structure of the reasoning, as below.

(1)  If p, then r. premise
(2)  If q, then ~r premise
(3)  q premise
(4)  ~r (2), (3), modus ponens (M.P.)
-----------
(5)  ~p (1), (4), modus tollens (M.T.)

The lines of the argument that are annotated by "premise" express propositions that are not supported themselves by the arguer.  Sometimes the propositions of an argument that are not themselves supported in the argument are also called "assumptions".  That just means that in the particular argument given, they are not supported. (It doesn't mean that they cannot be supported.)  Notice also that in calling (1), (2), and (3) a premise, but not (4), we are using the term "premise" in a different way than originally introduced.  Before, (1) and (4) would be called premises for (5).  Here, only those propositions that are unsupported by argument are called premises.  Lastly, in the annotation of (4), two propositions are cited, (2) and (3), as well as the pattern of argument, or the rule of inference, by which (4) is inferred from (2) and (3).  The pattern of argument is cited to insure the reader sees that the argument having (4) as its conclusion is a valid argument.  Given the main conclusion, (5), is argued for, the propositions that support it are also cited, as well as the pattern of argument by which (5) was inferred.

VII. Study Guide

1. The exam will contain about 10-12 questions, and you'll be given about 10 minutes to complete the exam.  The questions will be true/false and multiple choice questions.  I will provide the test sheet and the scantron forms.  Bring a #2 pencil.

2. The items in the boxes above are especially important. For several of the questions, I will present a number of different definitions for a term, and then ask you to identify which one correctly states the definition that goes with the term.   Needless to say, I will write the many different definitions in such a way that if you don’t understand the concept well, they will appear to be indistinguishable.  With respect to the "lessons", I will not ask you to identify which lesson is #6 or #4, but you still need to know the lessons well to answer some of the questions.

3. I will ask you to judge whether sentences express propositions or not, whether sets of sentences express arguments or not, whether arguments are valid or not, and whether arguments are sound or not.

4.  I will ask specific questions about what an objection is, what a rational objection is, and what counts as an effective criticism showing that a premise of an argument is false.

5. I will ask you to identify the modus ponens, modus tollens, and disjunctive syllogism, and dilemma argument patterns, in their symbolic forms, as well as the patterns that look like them but are invalid, in their symbolic forms.  Additionally, I will ask you to identify instances of these patterns stated with English sentences, both the valid and invalid forms.

6. There may be other sorts of exam questions about material on this document, but the exam will focus primarily on the matters discussed above in 1-5.

*This document was written specifically to provide you with the most basic and most important logical concepts that are used or presupposed in the lecture presentations and in the reading material.  Also, you'll need to master the concepts above to understand the paper assignments and write good papers. So, the more quickly and the more effectively you can master the material in this document the better you will understand the lectures, the reading, and know how to complete the paper assignments.  I'd recommend studying this carefully for the exam, and then studying it again before working on each paper assignment.

Concept: Argument

An argument is a set of propositions, one of which, the conclusion, is taken to be supported by the remaining propositions, the premises.

Concept: Standard Form

An argument is expressed in standard form when the conclusion is written below a horizontal line, called an "inference bar", and the premises (only) are written above the horizontal line.

premise indicators             conclusion indicators
for, since, because thus, hence, therefore
for the reason that, consequently, so, in
given that conclusion, it follows that

Concept: Soundness

A (deductive) argument is sound, if, and only if, (i) it is valid, and (ii) all of its premises are true.

Concept: Validity

A (deductive) argument is valid, if, and only if, if the premises are true, then conclusion must be true. (I.e., it is not possible its premises are true and its conclusion is false.)

Lesson #1:  An argument can be valid even if all of its premises and its conclusion are at the same time actually false.

Lesson #2:  An argument can be invalid even though every proposition of the argument, all the premises and the conclusion, are actually true.

Lesson #3 : if you judge that the conclusion of an argument is false, then you are rationally committed to believing either (a) that the argument is invalid or (b) that it has at least one false premise.

Lesson #4:  If an argument is unsound, the conclusion may still be true.  I.e., to say that an argument is unsound is not to say its conclusion is false.

Lesson #5 : To show that a deductive argument does not establish its conclusion, you must show that either it is invalid or that it has at least one false premise.

Lesson #6: To show that a premise of an argument is false, you must offer an argument for thinking that the negation of the premise is true.