arguments and validity

PHIL 220. Introduction to Logic

Gabriel Uzquiano

why logic?

We reason to persuade someone to accept a claim on the basis of something they already accept.

One minute. The point of the puzzle is to provide some illustration of how we reason effectively in certain circumstances. We want to make explicit what we do when we reason well.

two coin tosses

You and I toss a coin. We learn the outcome of our coin toss but we have no access to the other coin. We each make a guess as to the outcome of the other coin toss. We win if at least one of us makes guesses correctly the outcome of the other coin toss. We are allowed to agree on a common strategy even before the coins are tossed.

What strategy guarantees a win, no matter how the coins land?

The coin tosses are independent from each other. How can we make one of us guesses correctly the outcome of the other coin toss?

H

T

H

T

Let them sit with it. Take a show of hands on whether a guaranteed strategy exists at all before revealing that it does. The illustration at coinflips.com.au is worth two minutes if the room is stuck.

one strategy is guaranteed to win

• I make my guess match the outcome of my coin.
• You make your guess differ from the outcome of your coin.

H

T

H

T

Take a moment: why does the strategy work?

Draw it on the chalkboard here — press C and sketch the two cases, MATCH and DIFFER. Do not put the reconstruction up until someone has said the key thing: the two cases are exhaustive.

why it works

Once the two coins are tossed, they either match or they do not.

If they match, my guess is guaranteed to succeed:

• your coin landed exactly as mine did.

If they do not match, your guess is guaranteed to succeed:

• your coin landed differently from mine.

Either way, one of our guesses will succeed, and we win.

The same case, made explicit

1. Either the outcomes of our tosses match or they do not.
2. If they match, then since I guess the outcome I observed in my coin, I am guaranteed to succeed.
3. If I succeed, then at least one of us does, and we win.
4. If they do not match, then since you guess a different outcome from the one you observed, you are guaranteed to succeed.
5. If you succeed, then at least one of us does, and we win.

6. So at least one of us succeeds, and we win the game.

This is the slide the whole opening builds to. Reveal one premise at a time and leave them all up: the argument has to be visible as a whole for the last step to land. Ask before revealing 6: what is still missing? (Nothing — that is the point.)

arguments

From a body of statements — the premises — to a statement they are meant to support — the conclusion.

premises and conclusion

A gift card is better than a postcard. A postcard is better than nothing. Therefore, a gift card is better than nothing.

premises

1. A gift card is better than a postcard.
2. A postcard is better than nothing.

conclusion, marked by ‘therefore’

3. A gift card is better than nothing.

Contrast with the companion case: “A gift card is better than nothing. Nothing is better than a yacht. So a gift card is better than a yacht.” Same shape, absurd result — worth putting on the board by hand rather than on a slide.

the coin toss argument

Either the outcomes of our tosses match or they do not. If they match, I’m guaranteed to succeed. If they do not, you are guaranteed to succeed. Either way, at least one of us is guaranteed to succeed.

premises

1. Either the outcomes of our tosses match or they do not.
2. If they match, I’m guaranteed to succeed.
3. If they do not, you are guaranteed to succeed.

conclusion, marked by ‘so’

4. At least one of us is guaranteed to succeed.

Connect explicitly back to the previous slide: the ‘gift card’ example had two premises; here we have a three-premise argument.

definition

An argument is a sequence of declarative sentences, the premises, and a declarative sentence marked as the conclusion.

There is no fixed order. ‘So’, ‘thus’, ‘therefore’, ‘it follows that’ mark the conclusion wherever it appears.

We use declarative sentences to make statements, which are true or false. Contrast interrogative and imperative sentences, which ask and command.

three types of sentence

declarative (statements)

It is raining outside.

No one can change the past.

✓ true or false

interrogative (questions)

Is it raining outside?

Can anyone change the past?

✗ neither true nor false

imperative (commands)

Bring an umbrella!

Do not attempt to change the past.

✗ neither true nor false

Only declarative sentences can serve as premises or conclusion in an argument.

One beat per column — reveal the verdict row last so students predict it first. The rain / past examples echo what they have already seen, which makes the contrast land faster than fresh examples would.

how to extract an argument from a paragraph

Time travel is impossible. For if time travel were possible, then a time traveler could change the past. Nothing, however, can change the past.

Which sentence is the conclusion?

1. If time travel were possible, then a time traveler could change the past.
2. Nothing can change the past.

3. Time travel is impossible.

Notice that the conclusion is the very first sentence. The last two sentences, marked by ‘for’, are premises offered in support of the conclusion.

The conclusion comes first here, and ‘for’ marks what supports it. This is the skill they will need all term.

how to extract an argument from a paragraph

If I speak both English and Spanish, then I’m bilingual. Therefore, I’m bilingual. For after all, I speak both English and Spanish.

Which sentence is the conclusion?

1. If I speak both English and Spanish, then I’m bilingual.
2. I speak both English and Spanish.

3. I’m bilingual.

Notice that the conclusion, marked by ‘therefore’, is the second sentence. The last sentence is a premise that supports it, not a further conclusion.

The conclusion appears in the middle here, marked by ‘therefore’. The last sentence (‘for after all’) is a premise that supports it, not a further conclusion. Contrast with the time-travel example where the conclusion came first — the point is that position never determines the role.

validity

Some arguments seem risk-free in that the truth of the premises is sufficient to guarantee the truth of the conclusion.

which of these are risk-free?

Whales live underwater. Few mammals live underwater. Therefore, whales are not mammals.

People are now carrying umbrellas on the street. Therefore, it must be raining.

All Things Considered is a radio program. All radio programs are broadcast at least once a week. So All Things Considered is broadcast at least once a week.

No one carries an umbrella on the street unless it is raining. Someone is now carrying an umbrella on the street. Therefore, it is raining.

risk-free because of a pattern

If there is gas in the tank, the engine will start. There is gas in the tank. So the engine will start.

1. If \(p\), then \(q\)
2. \(p\)

3. Therefore, \(q\)

Either the butler did it or the gardener did it. The butler did not do it. So the gardener did it.

1. \(p\) or \(q\)
2. Not \(p\)

3. Therefore, \(q\)

We use \(p\) and \(q\) as placeholders for English declarative sentences. Notice that the pattern never takes you from true premises to a false conclusion, no matter what \(p\) and \(q\) may stand for.

validity

An argument is valid if, and only if, it is an instance of a valid argument form.

What is a valid argument form, you ask?

The course will develop a toolkit designed to answer this question. In the meantime, note that …

1. A valid argument form has no instances with true premises and a false conclusion.
2. A valid argument form preserves truth regardless of subject matter.

So to show that an argument form is invalid, it suffices to produce one instance with true premises and a false conclusion.

one instance suffices

1. \(p\) or \(q\)

2. \(p\) and \(q\)

1. William Shakespeare wrote Hamlet or Mark Twain wrote Hamlet.

2. William Shakespeare wrote Hamlet and Mark Twain wrote Hamlet.

1. The premise is true, since William Shakespeare wrote Hamlet.

2. The conclusion is false, since Mark Twain did not write Hamlet.

One such instance suffices to rule out the validity of the argument form.

Insist on this: we did not need to survey every instance. One suffices. That asymmetry between the justification of validity and and that of invalidity will play a role throughout the course.

truth tables

The truth value of premises and conclusion is determined by that of \(p\) and \(q\):

\(p\)	\(q\)	\(p\) or \(q\)	\(p\) and \(q\)
T	T	T	T
T	F	T	F
F	T	T	F
F	F	F	F

Choose something true for \(p\) and something false for \(q\). Then the premises will be true and the conclusion will false, e.g., it is false that Shakespeare and Twain wrote Hamlet, but true that one did.

Likewise:

1. Los Angeles is in California or New York is in California.

3. Los Angeles is in California and New York is in California.

Next time

We will look for the form of an argument, and we will build a language in which a wide family argument forms can be written down precisely.

Remind them: Quiz 1 follows Problem Session 1 on Monday 8/31.

The slides cover material from Chapter 1 of Introduction to Logic.