Arguments 2: Arguments Continued: Proofs

Arguments 2: Arguments Continued: Proofs

Arguments with Other Connectives
• We know understand enough truth-functional connectives to consider several more complex arguments.
• Here are two examples.
1. Jones is both Swiss and Italian. Jones can buy land in Switzerland just in case Jones is Swiss. Jones can buy land in Switzerland.
2. If the blade is vorpal, then it can kill a bandersnatch. If the blade can kill a bandersnatch, then it is mimsy. The blade is either a vorpal blade or a snicket one. The blade is not snicket. Thus, the blade is mimsy.
• A Key:
• P: Jones is Swiss.
• Q: Jones is Italian.
• R: Jones can buy land in Switzerland.
• S: The blade is vorpal.
• U: The blade can kill a bandersnatch.
• V: The blade is mimsy.
• X: The blade is snicket.
• Using this key we can develop the following symbolization of the arguments.
1. P & Q
R <--> P
------
R
2. S --> U
U --> V
S v X
~X
------
V
• These will have the following truth tables, showing that they are both valid arguments. (The rows which describe kinds of situations in which the premises are true are in bold; note that the conclusion is true for these kinds of situations.)

PQRP&QR <--> PR
TTT T T T
TTF T F F
TFT F T T
TFF F F F
FTT F F T
FTF F T F
FFT F F T
FFF F T F

SUVX S --> U U --> V S v X~XV
TTTT T T T FT
TTTF T T T TT
TTFT T F T FF
TTFF T F T TF
TFTT F T T FT
TFTF F T T TT
TFFT F T T FF
TFFF F T T TF
FTTT T T T FT
FTTF T T F TT
FTFT T F T FF
FTFF T F F TF
FFTT T T T FT
FFTF T T F TT
FFFT T T T FF
FFFF T T F TF

Inference Rules
• As the last arguments above show, it is very cumbersome to make truth tables. It would be helpful if there were a way to check propositional arguments in some more direct way. We can do this with inference rules.
• Look back at our definition of "if...then..." or "-->". Note that from this table, we can see that if P is true, and P-->Q is true, the Q is true. This is the case regardless of what P and Q stand for. Instead of looking always to a truth table, we can codify this insight into a rule. Traditionally, this rule is called "modus ponens." Let P and Q be any two propositions.

1. Modus Ponens
P --> Q
P
-----
Q
• This is called an inference rule. We write it here to mean that whenever you have lines of the form of those above the line in an argument, and are confidant that they are true, then you know that the statement below the line must be true.
• Look back at the truth tables defining the other connectives, and you will be able to satisfy yourself that the following are also acceptable inference rules. Let P and Q be any two propositions.

1. Modus Ponens
P --> Q
P
-----
Q

2. Modus Tollens
P --> Q
~Q
-----
~P

3. Chain Rule
P --> Q
Q --> R
-----
P --> R

4. Double Negation
~~P
-----
P

5. Disjunctive Syllogism
P v Q
~P
-----
Q

P
-----
P v Q

7. Simplification
P & Q
-----
P

8. Conjuction
P
Q
-----
P & Q

9. Biconditional Simplification
P <--> Q
-----
P --> Q
Q --> R

Constructing Proofs
• We demonstrate an argument is valid with a proof. A proof is a list of claims, where each claim is either a premise (and so assumed) or is derived from earlier lines using an inference rule. If one of these derived lines is our conclusion, we will have given a valid argument of that conclusion.
• Here are three complete examples of moving from English language arguments to completed proof.
• English language arguments:
1. The levee will break just in case the water rises twenty feet. The water won't rise twenty feet or it drain in the canal. The water won't drain in the canal but it will rain. The levee won't break.
2. Tom can snowboard but Victor can't. If he had practiced, then Victor could snowboard. Victor did not practice.
3. Either it's not true that fish aren't vertebrates, or they are warm blooded. Fish have scales and are not warm blooded. If they are vertebrates, then they are chordates. Fish are chordates.
• A Key:
• P: The levee will break.
• Q: The water will rise twenty feet.
• R: The water will drain in the canal.
• S: It will rain.
• T: Tom can snowboard.
• U: Victor had practiced the snowboard.
• V: Victor can snowboard.
• W: Fish are vertebrates.
• X: Fish are warm blooded.
• Y: Fish have scales.
• Z: Fish are chordates.

• Symbolization of the arguments:

1. P <--> Q
~Q v R
~R & S
------
~P

2. T & ~V
U --> V
------
~U

3. ~~W v Y
X & ~Y
W --> Z
------
Z

• Three proofs:
1:
1. P <--> Q
2. ~Q v R
3. ~R & S
4. ~R Simplification, 3
5. ~Q Disjunctive Syllogism, 4, 2
6. P --> Q Biconditional to Conditional, 1
7. ~P Modus tollens, 6, 5

2:
1. T & ~V
2. U --> V
3. ~V Simplification, 1
4. ~U Modus Tollens, 3, 2

3:
1. ~~W v Y
2. X & ~Y
3. W --> Z
4. ~Y Simplification, 2
5. ~~W Disjunctive syllogism, 4, 1
6. W Double negation, 5
7. Z Modus ponens, 6, 3
What to look for in real-world arguments
• Our brief excursion into propositional logic illustrates both the rigor of our notion of a valid argument, and also shows that there are rigorous methods to test arguments for validity.
• Real world arguments are typically however more complex than we can model well with our propositional logic. We would need for example a tool to rigorously describe generalizations -- that is, claims with "all" or "some" in them. And we would need even more complex tools to be able to make sense of talk about possibility and so on. However, armed with a rigorous understanding of validity, and some experience at testing for validity, we can use common sense to evaluate complex arguments, and will be more likely to see their benefits and faults now that we know what are good argument is.
• With this in mind, it will be useful to note the common additional complications that occur in natural language arguments. It will also be useful to identify common instances of bad reasoning, so that we become good at recognizing and ignoring them. We will do the first task now, and the second in the next set of notes.
• To evaluate an argument, we must be able to:
• Identify the conclusion
• Identify the relevant premises
• Determine whether we accept the premises
• Determine whether the argument is valid
• Premises are not typically identified as such, but they can be indicated with phrases like the following:
• Since
• Because
• For
• In view of
• This is implied by
• Conclusions are also often not identified, but some phrases may indicate the conclusion:
• Thus
• Therefore
• Hence
• Consequently
• So
• Accordingly
• The greatest variation in everyday arguments is in their structure. An idealized argument is like an ordered list:
Premise 1
Premise 2
Premise 3
Premise 4
....
Conclusion
• Real world arguments diverge from this in several ways
• They are not ordered (the conclusion may come first, in the middle, etc.)
• There may be missing premises (an enthymeme)
• There may be too many premises
• Some premises are irrelevant
• Some premises are independent
• If the steps of an argument are not properly ordered, we need to recognize the conclusion and the premises.
• If an argument is missing premises, you have to see what additional premises would make the argument valid. Such an argument is called an enthymeme, and most arguments that you hear are enthymemes, because we skip the tedium of repeating truths that everyone knows.
• An enthymeme example:
Jones is very smart. She got a perfect score on her GREs.
In our idealized form this is:
Jones got a perfect score on her GREs.
Jones is very smart
But as it stands this is not a valid argument. It seems, however, to assume that we would agree with an implicit claim:
Jones got a perfect score on her GREs.
Anyone who got a perfect score on her GREs is very smart.
Jones is very smart.
An enthymeme must be interpreted with some charity.
• Arguments may also have more premises than are strictly necessary for at least two reasons.
1. Irrelevant premises. These may serve some other purpose than establishing an argument, but still be included in the argument. Here is an example:

Jones is that woman with the long black hair.
Jones won the state trials.
Smith might have won them, but Smith was sick.
Smith had that stomach flu that was going around.
Whoever wins the state trials is very likely to win the national trials.
Jones is very likely to win the national trials.
Several of these premises may be of interest to the speakers, but they don't have any significant relationship to the conclusion.
2. Independent premises. Sometimes several arguments are combined. The premises of these different arguments are independent of the premises of other arguments. Here is an example:
All cephalopods lack endoskeletons.
No invertebrate has internal bones.
Every octopus is an invertebrate.
Octopi are cephalopods.
No octopus has a skeleton.
In this example, there are two arguments for the same conclusion combined.

Evaluating an argument
We need to thus ask several things when evaluating an ordinary language argument
1. What is the conclusion?
2. Is the conclusion significantly different from the premise(s)? (Is the argument non-trivial?)
3. What are the relevant premises?
4. Is there more than one argument?
5. For each argument, is it meant to be deductive?
6. If the argument is meant to be deductive, is it valid?
7. If the argument is valid, should you believe all of the premises of the argument?