Beyond formal validity

class: center, middle, inverse, title-slide

.title[
# Beyond formal validity
]
.subtitle[
## How To Think - Week 9
]
.author[
### Fernando Alvear
]
.institute[
### University of Missouri
]
.date[
### Mar 13
]

---

# What we have learned

- Stating arguments in standard form.
- Translating English sentences into SL.
- Using syntax trees to calculate the truth-value of a sentence in SL.
- Create complete truth-tables for sentences.
- Use truth-tables to determine the validity of an argument in SL.
- Learn some valid argument forms that are commonly used in argumentation.

### What we will learn this week

- Argumentation: basics
- Defending a claim
  - Context-dependency of sentences
  - Evident support
  - Reasons and sub-arguments
  - Thought experiments
  - _Reductio ad absurdum_
- Challenging the reasoning process
  - Invalidity
  - Counterexample

---
# Argumentation: basics
.shadow[
.emphasis[
__Argumentation__: The communicative activity of producing and exchanging reasons in order to support claims or defend/challenge positions, especially in situations of doubt or disagreement (Lewiński & Mohammed 2016)
]
]

An argument _in logic_ is a set of two or more sentences, in which premises are presented as supporting a conclusion.

An argument _as a communicative activity_ is the production and exchange of reasons in order to establish the truth or falsity of a claim.

In this context, the word "claim" is synonymous with the words "conclusion", "view", or "thesis".

---
# Argumentation: basics

There are two important _moves_ in an argument (as a communicative activity):
- __Defend__ (or _propose_) that a claim is true (or false), by providing reasons in favor of its truth (or falsity).
- __Challenge or question__ the reasoning involved in the defense of claims.

---
# Argumentation: basics

When you __defend__ a claim, you typically:
- Provide a valid argument that supports the claim in question.
- Provide reasons for accepting the truth of the premises involved in  your argument.

If you accomplish both tasks, your interlocutor should _accept your claim_.

When you __challenge__ the reasoning involved in the defense of a claim, you typically:
- Challenge the validity of your opponent's argument.
- Provide reasons for rejecting the truth of a premise in your opponent's argument.

If you accomplish any of these tasks, your interlocutor should _provide a new argument_.

---
# Example: argumentation

Suppose John and Anna are offering arguments to each other.

John argues that grades in college should be abolished.
1. Grades reduce the quality of student's thinking.
2. If grades reduce the quality of student's thinking, then they should be abolished.
3. Therefore, grades should be abolished.

Anna argues that grades in college should _not_ be abolished.

1. If grades should be abolished, then they don't offer any significant benefit to students.
2. Grades do offer a significant benefit to students.
3. Therefore, grades should not be abolished.

What are John and Anna doing? Defending? or Challenging?

Both of them are _defending_ claims.

To challenge/question the reasoning involved in the interlocutor's argument, one must evaluate the interlocutor's argument.

---

# Example: argumentation

Let's consider again John's argument.
1. Grades reduce the quality of student's thinking.
2. If grades reduce the quality of student's thinking, then they should be abolished.
3. Therefore, grades should be abolished.

Anna could argue that it's not the case that grades reduce the quality of student's thinking. (Let's say she cites data that supports this.) If so, then John's argument would be unsound (it contains a false premise).

Perhaps Anna could argue against the validity of John's argument, but it's hard to see how this could be (as John's argument is a _modus ponens_).

If Anna's questioning is successful, this doesn't mean that grades should _not_ be abolished. It just means that __John needs to provide a different argument__.

---
# Defending a claim

A good defense of a claim involves presenting a __sound__ argument for it.

Sound argument: valid argument + all its premises are true.

We already have seen how to determine the _validity of arguments_:
- By formalizing the argument in SL and checking its validity via truth-tables.
- By using one of the known valid forms.
  - Modus ponens
  - Modus tollens
  - Disjunctive syllogism
  - Hypothetical syllogism
  
--

How to determine the _truth_ of the premises? In doing this, we should consider these factors:
1. Context-dependency of sentences
2. Evident support
3. Reasons and sub-arguments
4. Thought experiments
5. _Reductio ad absurdum_

---
# Context-dependency

> If I am thinking, then I exist. I am thinking. Therefore, I exist. (Adapted from: René Descartes, Discourse on Method, 1637)

1. If I am thinking, then I exist.
2. I am thinking.
3. Therefore, I exist.

- Is it valid? Are the premises true? 
--

- The argument is valid, as it's an instance of _modus ponens_.
  - Regarding truth, what if someone says: "Premise 2 is clearly false, because Descartes is now dead!" What do you think? 
  
--

Suppose I say: "I am thinking."

Now suppose another person says "I am thinking."

Did we say _the same thing_?

---
# Context-dependency

Some sentences in our language contain context-dependent expressions, in the sense that their meaning is determined not only by convention, but also by facts about the circumstances in which they are _said (that is, by the context of utterance)._

Some context-dependent expressions: 
- Pronouns ‘I’, ‘he’, ‘she’, ‘it’, ‘this’, and ‘that’; 
- Adverbs ‘here’, ‘now’, ‘today’, ‘yesterday’, ‘tomorrow’, and ‘actually’;
- Adjectives ‘my’, ‘his’, ‘her’, ‘present’, ‘past’, and ‘actual’.
- ...

### Some terminology:

__Utterance__ (of a sentence): the intentional act of a speaker saying a sentence at a specific time and place.

__Context__ (of an utterance): the circumstances (time and place) of the utterance.

---
# Context-dependency

1. If I am thinking, then I exist.
2. I am thinking.
3. Therefore, I exist.

Premise (2) was true when Descartes offered this argument. If argumentation is understood as a communicative activity, we should think of the _context_ in which the argument was _uttered._

---
# Context-dependency

> If I am thinking, then I exist. I am thinking. Therefore, I exist. (Adapted from: René Descartes, Discourse on Method, 1637)

1. If I am thinking, then I exist.
2. I am thinking.
3. Therefore, I exist.

- The sentences of this argument are true in their context of utterance.
  - That is, in the circumstance (time and place) in which Descartes offered the argument.
- This is so because the sentence contains context-sensitive expressions, which depend for their meaning on a specific context (time and place).
- Sometimes the context is obvious, sometimes it's not obvious. If it's not obvious, it's legitimate to ask for clarification.

.shadow[
.emphasis[
__Context dependency__: When evaluating the soundness of an argument, we should evaluate whether the premises are true _relative_ to the context of utterance of the argument.
]
]

---
# Evident premises

1. If I am thinking, then I exist.
2. I am thinking.
3. Therefore, I exist.

[Context: Descartes uttering this argument in 1637.]

- The argument's premises seem to be evident (clearly understood as true). 
  - Regarding premise (1), it's _impossible_ that one is thinking at at the same time not existing.
  - Regarding premise (2), if Descartes says that he is thinking, then we should assume it as true. Thinking (about anything) seems to be a basic human activity.
- If the argument's premises are not evident, then it's legitimate to ask for more information before accepting them.

.shadow[
.emphasis[
__Evident premises__: It's possible that the truth of a premise is clearly understood by everyone, in which case no further support is needed.
]
]

---
# Reasons and sub-arguments

> If money is the most important thing in life, then we will pursue it for its own sake. We do not pursue money for its own sake, but rather as a means to achieving something else. Thus, money is not the most important thing in life. (Adapted from Aristotle's _Nicomachean Ethics_)

What is the structure of this argument? Is it valid? Are its premises true?

Symbolization key:

- `\(M\)`: Money is the most important thing in life.
- `\(S\)`: We pursue money for its own sake.

1. `\((M \rightarrow S)\)`
2. `\(\Not S\)`
3. `\(\therefore \Not M\)`

Notice that we didn't translate the sentence "but rather [we pursue money] as a means to achieving something else." If this is not a premise in the argument, what is it?

---
# Reasons and sub-arguments

1. `\((M \rightarrow S)\)`
2. `\(\Not S\)`
3. `\(\therefore \Not M\)`

- It seems that the sentence "but rather [we pursue money] as a means to achieving something else." is presented as a _reason_ to think that it's true that "we don't pursue money for its own sake".

.shadow[
.emphasis[
__Reasons and sub-arguments__: When premises are not evident, we should state our reasons for believing in a premise. If that's not enough, then those reasons can be used to create a sub-argument in favor of that premise.
]
]

---
# Reasons and sub-arguments

1. If money is the most important thing in life, then we will pursue it for its own sake.
2. We either pursue money for its own sake or as a means to achieving something else.
3. If we pursue money for it's own sake, then we would not want to exchange it for other things.
4. We usually want to exchange money for other things.
5. Therefore, we don't pursue money for its own sake. (By _modus ponens_, premises 3 and 4)
6. Therefore, we pursue money as a means to achieving something else. (By _disjunctive syllogism_, premises 2 and 5)
7. Therefore, money is not the most important thing in life. (By _modus tollens_, premises 1 and 6)

This argument contains two sub-arguments (that is, arguments _within_ the main argument).

---
# Thought experiments

Consider this story offered by philosopher Peter Singer (1972).

> On your way to work, you pass a small pond. On hot days, children sometimes play in the pond, which is only about knee-deep. The weather’s cool today, though, and the hour is early, so you are surprised to see a child splashing about in the pond.

---

.pull-left.w55[
> As you get closer, you see that it is a very young child, just a toddler, who is flailing about, unable to stay upright or walk out of the pond. You look for the parents or babysitter, but there is no one else around. The child is unable to keep her head above the water for more than a few seconds at a time. If you don’t wade in and pull him out, he seems likely to drown. Wading in is easy and safe, but you will ruin the new shoes you bought only a few days ago, and get your suit wet and muddy. By the time you hand the child over to someone responsible for her, and change your clothes, you’ll be late for work. What should you do?

]
.pull-right.w40[
<img src="assets/soaked.png" alt="" height="500"/>
]

If you were in this situation, would you be __obligated__ to save the child?

---
# Thought experiments

Now consider this argument offered by Singer.

1. It is in our power to prevent suffering and death from lack of food, shelter, and medical care from happening, without thereby sacrificing our own food, shelter, and medical care.

2. If is in our power to prevent suffering and death from lack of food, shelter, and medical care from happening, without thereby sacrificing our own food, shelter, and medical care, we ought, morally, to do it.

3. Therefore, we are morally obligated to prevent suffering and death from lack of food, shelter, and medical care from happening.

Singer defends premise (1) by saying that we could prevent this kind of suffering by donating money to charities; money that we usually spend on things we don't really need, like drinks, meals out, new clothing, movies, concerts, vacacions, new cars, house renovations, etc.

Singer defends premise (2) by offering the thought-experiment of the drowning child.

---
## Thought experiments and arguments by analogy

Thought experiments usually hide an informal argument called _argument by analogy_:

1. Situation S is similar to situation T in relevant respects.
2. S has some further feature Q.
3. Therefore, T also has the feature Q.

This is not a valid argument, but is a good argument nonetheless. We use it all the time:

1. Tom missing 9 classes is similar to Ana missing 9 classes.
2. As a result, Tom received a bad participation grade.
3. Therefore, Ana should also receive a bad participation grade.

But the truth of the premises don't guarantee the truth of the conclusion. Imagine that Ana had an accident and had to spend three weeks in the hospital.

For a good argument by analogy, the situations have to be similar in _all relevant_ respects.

---
# Drowning child analogy

1. Saving the drowning child is similar to situations in which you can do something to prevent bad from happening without sacrificing things of similar importance.
2. Saving the drowning child is an _obligation._
3. Therefore, if you can do something to prevent bad from happening without sacrificing some other important things, then you are obligated to do so.

Other famous thought-experiments:
- Trolley problem (the moral thing to do is to minimize bad consequences).
- The violinist (in the abortion debate).
- Smith, Jones, and their cousin (euthanasia debate, killing vs. letting die are morally equivalent).
- The original position (we must distribute resources in such a way that those in the lower tiers have the same opportunities to live a fulfilling life)
- ...

---
# Thought experiments

1. It is in our power to prevent suffering and death from lack of food, shelter, and medical care from happening, without thereby sacrificing our own food, shelter, and medical care.

3. Therefore, we are morally obligated to prevent suffering and death from lack of food, shelter, and medical care from happening.

.shadow[
.emphasis[
__Thought experiments__: A thought experiment is a hypothetical situation that is meant to elicit people's intuitions, which can then be used to support a premise.
]
]
---
# Reductio ad absurdum

Reductio ad absurdum: latin for "reduction to absurdity."

You can argue for the truth of `\(P\)` by showing that its negation `\(\Not P\)` leads to something absurd, contradictory, or clearly false.

Argument in support of `\(P\)`:

1. Assume `\(\Not P\)`
2. Argue that `\(\Not P\)` derives an absurdity, or contradiction.
3. Therefore, `\(P\)`.

Argument in support of `\(\Not P\)`:

1. Assume `\(P\)`
2. Argue that `\(P\)` derives an absurdity, or contradiction.
3. Therefore, `\(\Not P\)`.

---
# Reductio ad absurdum: example

Is division by zero possible?

1. Division by zero is possible.
2. If division by zero is possible, then for any number `\(x\)`, `\(x/0\)` is some well-defined quantity `\(Q\)`. (In other words, `\(x/0=Q\)`.) 
3. If `\(x/0=Q\)`, then `\(x=Q \times 0\)`.
4. If `\(x=Q \times 0\)`, then for _any_ number `\(x\)`, `\(x=0\)`.
5. For any number `\(x\)`, `\(x=0\)`. (but this is a contradiction, as it’s impossible that, for any number `\(x\)`, `\(x=0\)`)
6. Therefore, division by zero is not possible.

.shadow[
.emphasis[
__Reductio ad absurdum__: An argument that supports that one's conclusion is true, because, if the opposite were true, something absurd or contradictory would be true.
]
]

---
# Summary

- Argumentation: basics ✓
- Defending a claim
  - Provide a valid argument  ✓
  - Context-dependency of sentences ✓
  - Evident support ✓
  - Reasons and sub-arguments ✓
  - Thought experiments ✓
  - _Reductio ad absurdum_ ✓
- Challenging the reasoning process
  - Challenging validity
  - Counterexamples

---

# Challenging the reasoning process

.shadow[
.emphasis[

To _challenge an opponent's argument_, you can do so through either one of these two ways:

- Challenge the validity of the argument
- Challenge the truth of a premise of the argument

]
]

To challenge the _validity of the argument_, you have to show that the structure of the argument is invalid (that is, that the truth of the premises don't guarantee the truth of the conclusion).

### Fallacies

.pull-left[
Affirming the consequent:

1. `\((A \rightarrow B)\)`
2. `\(B\)`
3. `\(\therefore A\)`
]
.pull-right[
Denying the antecedent:

1. `\((A \rightarrow B)\)`
2. `\(\Not A\)`
3. `\(\therefore \Not B\)`

]

---
# Challenging validity

1. If the economy collapsed, then some banks have failed.
2. Some banks have failed.
3. Therefore, the economy collapsed.

Is this argument valid? Are its premises true?

The argument is invalid. It's an instance of affirming the consequent:

1. `\((A \rightarrow B)\)`
2. `\(B\)`
3. `\(\therefore A\)`

This would have been a valid argument instead:

1. If the economy collapsed, then some banks have failed.
2. The economy collapsed.
3. Therefore, some banks have failed.

---
# Counterexamples

Consider this argument. Is it valid? Are its premises true?

1. If some action `\(x\)` is morally wrong, then `\(x\)` is illegal.
2. Some action `\(x\)` is morally wrong.
3. Therefore, some action `\(x\)` is illegal.

This is a valid argument (modus ponens).

We can challenge this argument by showing that premise 1 is false.

To show that it's false, we need to find an example that shows that the antecedent is true but the consequent is false (Second row of the conditional).

Is there any morally wrong action that is not illegal?

---

# Counterexamples

1. If some action `\(x\)` is morally wrong, then `\(x\)` is illegal.
2. Some action `\(x\)` is morally wrong.
3. Therefore, some action `\(x\)` is illegal.

Some counterexamples to premise 1 (morally wrong but not illegal actions):

- Cheating your partner.
- Failing to fulfill (some) promises.
- Failing to help a person when you are able to do so.
- ...

.shadow[
.emphasis[

A __counterexample__ is an example that contradicts a premise. To create a counterexample, one has to find a situation that makes that premise false. In the case of a conditional, the counterexample is a situation in which the antecedent is true but the consequent is false.

]
]