Validity

class: center, middle, inverse, title-slide

.title[
# Validity
]
.date[
### MSA 2025
]

---

# What we have learned

- Stating arguments in standard form.
- Translating English sentences into SL.
- Calculate the truth-value of a sentence in SL from a given truth-value assignment (by using syntax trees).
- Create complete truth-tables for sentences.

## What we will learn this week

- Use truth-tables to determine the validity of an argument in SL.
- Learn some valid argument forms that are commonly used in argumentation.

---
# Determining the validity of an argument

> An argument is logically valid if and only if _it is not possible for all the premises to be true and the conclusion false_. An argument is logically invalid if and only if it is not logically valid.

An argument is valid when the truth of the premises __guarantees__ the truth of the conclusion.

---
class: medium-font

# Determining the validity of an argument

Steps to determine the validity of an argument:
1. We imagine/assume for the moment that the premises are true.
2. We examine the conclusion and ask: "Assuming that the premises are true, **can the conclusion be false**?"
  - If no (conclusion can't be false), then the argument is _valid_.
  - If yes (conclusion can be false), then the argument is _invalid_.

.pull-left[

1. If James is in Missouri, then James is in the US
2. James is in the US.
3. Therefore, James is in Missouri.

]

.pull-right[

1. James is in Missouri.
2. If James is in Missouri, then James is in France.
3. Therefore, James is in France.

]

---
# Determining the validity of an argument

.pull-left[

1. James is in Missouri.
2. If James is in Missouri, then James is in France.
3. Therefore, James is in France.

]

.pull-right[

1. If James is in Missouri, then James is in the US
2. James is in the US.
3. Therefore, James is in Missouri.

]

Can we use truth-tables to check for validity? Yes!

Let's first translate these arguments into SL. To do this, we create a translation key.

`\(M\)`: James is in Missouri. 
`\(F\)`: James is in France.
`\(U\)`: James is in the US.

.pull-left[

1. `\(M\)`
2. `\((M \rightarrow F)\)`
3. `\(\therefore F\)`

]

.pull-right[

1. `\((M \rightarrow U)\)`
2. `\(U\)`
3. `\(\therefore M\)`

]

---
## Determining the validity of an argument

Let's consider the first argument.

1. `\(M\)`
2. `\((M \rightarrow F)\)`
3. `\(\therefore F\)`

Let's put these sentences in a truth-table, like this:

---
class: medium-font

## Determining the validity of an argument

To check for validity, we said earlier that we should do two things:
1. Imagine that the premises are true.
2. Ask whether the conclusion can be false. If yes, then argument is invalid, if no, argument is valid.

We can do something similar with truth-tables:
1. Look at the rows in which the premises are true.
2. For each of these rows, check whether the conclusion is false. If it's false in a row, the whole argument is invalid. If there's no row in which the premises are true and the conclusion false, the argument is valid.

---
class: medium-font

## Determining the validity of an argument

1. Look at the rows in which the premises are true.
2. For each of these rows, check whether the conclusion is false. If it's false in a row, the whole argument is invalid. If there's no row in which the premises are true and the conclusion false, the argument is valid.

In which row or rows are the _premises_ true?

Only in the first row.

Is the conclusion true in that row? Yes.

Since there are no other rows in which the premises are true and the conclusion false, we say this argument is valid.

---
## Determining the validity of an argument

Let's consider the second argument.

1. `\((M \rightarrow U)\)`
2. `\(U\)`
3. `\(\therefore M\)`

Step 1: Mark the rows in which the premises are true.

Step 2: Check if the conclusion in those rows is true as well. If it's true in all rows, the argument is valid, otherwise, it's invalid.

---
## Determining the validity of an argument

Since in the third row the premises are true but the conclusion is false, the argument is _invalid._

---
# Valid argument forms
- Modus ponens
- Modus tollens
- Hypothetical syllogism
- Disjunctive syllogism

# Fallacies
- Affirming the consequent
- Denying the antecedent

---
# Valid form: Modus ponens

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

---
# Modus ponens: example

1. James is in Missouri.
2. If James is in Missouri, then James is in France.
3. Therefore, James is in France.

`\(M\)`: James is in Missouri. 
`\(F\)`: James is in France.
`\(U\)`: James is in the US.

1. `\(M\)`
2. `\((M \rightarrow F)\)`
3. `\(\therefore F\)`

---
# Fallacy: Affirming the consequent

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

---

# Affirming the consequent: example

1. If James is in Missouri, then James is in the US
2. James is in the US.
3. Therefore, James is in Missouri.

`\(M\)`: James is in Missouri. 
`\(U\)`: James is in the US.

1. `\((M \rightarrow U)\)`
2. `\(U\)`
3. `\(\therefore M\)`

---
# Valid form: Modus tollens

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

---

# Modus tollens: example

1. If James is in Missouri, then James is in the US.
2. James is not in the US.
3. Therefore, James is not in Missouri

`\(M\)`: James is in Missouri. 
`\(U\)`: James is in the US.

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

---
# Fallacy: Denying the antecedent

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

---
#Denying the antecedent: example

1. If James is in Missouri, then James is in the US.
2. James is not in Missouri.
3. Therefore, James is not in the US.

`\(M\)`: James is in Missouri. 
`\(U\)`: James is in the US.

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

---
# Summary: valid forms and fallacies

.pull-left[
### Valid forms

Modus ponens:

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

Modus tollens:

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

]

.pull-right[
### Fallacies

Affirming the consequent:

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

Denying the antecedent:

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

]

---
# Example

Are these valid arguments? Does this correspond to a valid form or a fallacy?

1. If the moon is up, the birds are singing.
2. The moon is not up.
3. Therefore, the birds are not singing.

---

# Valid argument forms
- Modus ponens ✓
- Modus tollens ✓
- Hypothetical syllogism ←
- Disjunctive syllogism ←

# Fallacies
- Affirming the consequent ✓
- Denying the antecedent ✓

---
# Valid form: Hypothetical syllogism

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

---
# Valid form: Disjunctive syllogism

Two forms:
1. `\((A \vee B)\)`
2. `\(\Not A\)`
3. `\(\therefore B\)`

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

---
class: medium-font

### Valid forms
.pull-left[
Modus ponens:

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

Modus tollens:

1. `\((A \rightarrow B)\)`
2. `\(\Not B\)`
3. `\(\therefore \Not A\)`
]
.pull-right[

Disjunctive syllogism:
1. `\((A \vee B)\)`
2. `\(\Not A\)`
3. `\(\therefore B\)`

Hypothetical syllogism:
1. `\((A \rightarrow B)\)`
2. `\((B \rightarrow C)\)`
3. `\(\therefore (A \rightarrow C)\)`
]

### 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. `\(\therefore \Not B\)`

]

---
# Solutions

---
<table class="truth">
  <tbody>
    <tr><td></td>
      <th>A</th>
      <th>B</th>
      <th class="dv"></th>
      <th></th>
      <th>[</th>
      <th>A</th>
      <th>∨</th>
      <th>(</th>
      <th>B</th>
      <th>→</th>
      <th>A</th>
      <th>)]</th>
      <th></th>
      <th class="dv"></th>
      <th></th>
      <th>A</th>
      <th class="dv"></th>
      <th></th>
      <th>(</th>
      <th>~</th>
      <th>A</th>
      <th>→</th>
      <th>~</th>
      <th>B</th>
      <th>)</th>
    </tr>
    <tr><td>→</td>
      <td>T</td>
      <td>T</td>
      <td class="dv"></td>
      <td></td>
      <td></td>
      <td>T</td>
      <td class="mc">T</td>
      <td></td>
      <td>T</td>
      <td>T</td>
      <td>T</td>
      <td></td>
      <td></td>
      <td class="dv"></td>
      <td></td>
      <td class="mc">T</td>
      <td class="dv"></td>
      <td></td>
      <td></td>
      <td>F</td>
      <td>T</td>
      <td class="mc">T</td>
      <td>F</td>
      <td>T</td>
      <td></td><td>✓</td>
    </tr>
    <tr><td>→</td>
      <td>T</td>
      <td>F</td>
      <td class="dv"></td>
      <td></td>
      <td></td>
      <td>T</td>
      <td class="mc">T</td>
      <td></td>
      <td>F</td>
      <td>T</td>
      <td>T</td>
      <td></td>
      <td></td>
      <td class="dv"></td>
      <td></td>
      <td class="mc">T</td>
      <td class="dv"></td>
      <td></td>
      <td></td>
      <td>F</td>
      <td>T</td>
      <td class="mc">T</td>
      <td>T</td>
      <td>F</td>
      <td></td><td>✓</td>
    </tr>
    <tr><td></td>
      <td>F</td>
      <td>T</td>
      <td class="dv"></td>
      <td></td>
      <td></td>
      <td>F</td>
      <td class="mc">F</td>
      <td></td>
      <td>T</td>
      <td>F</td>
      <td>F</td>
      <td></td>
      <td></td>
      <td class="dv"></td>
      <td></td>
      <td class="mc">F</td>
      <td class="dv"></td>
      <td></td>
      <td></td>
      <td>T</td>
      <td>F</td>
      <td class="mc">F</td>
      <td>F</td>
      <td>T</td>
      <td></td>
    </tr>
    <tr><td></td>
      <td>F</td>
      <td>F</td>
      <td class="dv"></td>
      <td></td>
      <td></td>
      <td>F</td>
      <td class="mc">T</td>
      <td></td>
      <td>F</td>
      <td>T</td>
      <td>F</td>
      <td></td>
      <td></td>
      <td class="dv"></td>
      <td></td>
      <td class="mc">F</td>
      <td class="dv"></td>
      <td></td>
      <td></td>
      <td>T</td>
      <td>F</td>
      <td class="mc">T</td>
      <td>T</td>
      <td>F</td>
      <td></td>
    </tr>
  </tbody>
</table>

---
# Problem 1

- Formalize an argument that involves philosophical premises and conclusions and determine whether it is valid or not.

> If the correct moral action is to flip the switch in the original trolley problem, then utilitarianism is true and Kant's ethics is false. The correct moral action is to flip the switch in the original trolley problem. Therefore, Kant's ethics is true.

---
# Problem 2

- Formalize an argument that involves philosophical premises and conclusions and determine whether it is valid or not.

---
# Problem 3

- Formalize an argument that involves philosophical premises and conclusions and determine whether it is valid or not.

> Either Utilitarianism or Kant's Ethics is the correct moral theory. But utilitarianism is clearly false as it recommends harvesting someone's organs to save more people. Therefore, Kant's ethics is the correct moral theory.