CH1 - Slides

Author

Affiliation

David W. Agler

The Pennsylvania State University

Abstract

Test

What is Logic?

NoteLogic

Logic is a science that aims to specify criteria and methods for the identification and construction of good and bad arguments.

flowchart LR
  A["Arguments"] --> L["Logic"]
  L --> G["Good Args"]
  L --> B["Bad Args"]

Argument vs. Arguing?

What do we observe when people are arguing?

Argument defined

NoteArgument

An argument is a series of propositions in which a certain proposition—the conclusion—is represented as following from another set of propositions—the premises or assumptions.

What is a proposition?

NoteProposition

A proposition is a sentence, or something expressed by a sentence, that is capable of being true or false.

Propositions: Examples

1. The sky is blue.
2. Tek is 6’0 tall.
3. If there are three cookies, then one cookie is missing.

Propositions: Not Propositions

1. A rock on the ground
2. The blueness in Tek’s shirt
3. The sentence: do you have any water?

Proposition: Two Tests

1. Naturalness Test
2. Cognitive Attitude Test

Exercise

Are these propositions?

1. Be a yardstick of quality.
2. Let’s Go Pens!
3. How may I help you?
4. Let the dog out.
   
5. In a fixed rate par bond, the issuer issues the bond at par value.

Finer Points

1. Propositions are abstract.
2. Questions expressing propositions.
3. Declarative sentences not expressing propositions
4. Two different sentences \(\to\) One proposition

Finer Points (Continued)

1. One sentence but different propositions
2. A sentence can express a proposition even if you don’t know if it is true
3. Propositions expressed by things other than sentences

Exercise

Do these sentences express propositions?

1. God does not exist.
2. I know that God exists.
3. You are beautiful.
4. It is morally wrong to eat meat.
5. I bet you five dollars.

Premises and the Conclusion

NoteArgument

An argument is a series of propositions in which a certain proposition—the conclusion—is represented as following from another set of propositions—the premises or assumptions.

graph TD
    A["Argument<br/>(composed of)"]
    P["Propositions"]
    B["Premises<br/>(supporting)"]
    C["Conclusion<br/>(supported)"]
    
    A --> P
    P --> B
    P --> C

Set of Propositions vs. Arguments

1. Narratives / Stories
2. Lists
3. Sequence of events

Conclusion Indicators

The proposition that is the conclusion is often marked:

• “In conclusion,”
• “Therefore,”
• “It follows that”

Premise Indicators

Premises are also marked (less common):

• C follow because P is the case.
• Since P is the case, C is the case.
• P suggests C.

Argument Standard Form

In logic and philosophy, the presentation of an argument is often expressed in a standard form:

• P1: Tek is a lawyer.
• P2: Lawyers have passed the Bar Exam.
• C: Therefore, Tek has passed the Bar Exam.

Evaluating Arguments

Evaluating Arguments

What makes an argument good?

1. Subjective Evaluation
2. Objective Evaluation

Subjective Criteria

NoteSubjective Evaluation

Subjective criteria are not properties of the argument itself but arise from the perspective, background, standards, purposes, or responses of the people evaluating or receiving the argument.

Subjective: Bolding Text

1. Hitler was born in 1892.
2. Hitler was born in 1887.

Both are false but experiments show that people are more likely to believe (1).

Subjective: Rhyming and Color

1. Rhyming: “Woes unite foes” vs. “Woes unite enemies”
2. Color: Hitler was born in 1887. vs. Hitler was born in 1887.

Familiarity and Repetition

1. “The body temperature of a chicken …”
2. “The body temperature of a chicken …”
3. “The body temperature of a chicken …”
4. “The body temperature of a chicken …”
5. “The body temperature of a chicken …”

Familiarity and Repetition

• “The body temperature of a chicken is 144 degrees.”
• Repeated exposure to false beliefs (even partial false beliefs) will make you more likely to believe them.

Objective Criteria

1. Truth: Propositions are true
2. Relevance: Premises are relevantly related to C
3. Entailment: C “follows from” Premises

Truth

Logic does not tell you if a proposition is true, e.g., Science, Experience, Religion?

Relevance

• Relevance is a complex topic.
• Let’s keep it intuitive: if premises have nothing to do with the conclusion, then the premises are not relevant.

Entailment

Logic focuses mostly on entailment:

1. What does it mean for C to “follow from” the premises?
2. When does this happen?

Objective Criteria: Diagram

graph TD
  accTitle: Logic and Argument Types
    accDescr: A flowchart showing that logic divides arguments into good and bad arguments. Good arguments have true premises, relevance, and a conclusion that follows. When the conclusion follows necessarily, the argument is valid; when it follows probably, the argument is strong.

    L["Logic"]
    G["Good args"]
    B["Bad args"]
    T["True Premises"]
    R["Relevance"]
    C["C follows"]
    V["C follows necessarily (valid)"]
    P["C follows probably (strong)"]
    
    L --> G
    L --> B
    G --> T
    G --> R
    G --> C
    C --> V
    C --> P

    classDef badArgs fill:#f8d7da,stroke:#dc3545,stroke-width:2px,color:#000;
    class B badArgs;
    classDef goodArgs fill:#90D5FF,stroke-width:2px,color:#000;
    class G goodArgs;

Deductive Validity

NoteDeductively valid

An argument is deductively valid if and only if it is logically impossible for the premises to be true and the conclusion false.

It is impossible for the following:

• P1. True
• P2. True
• C. False

Valid argument: Example 1

• P1. Either Jennifer Lopez or Mario Lopez is the president of the United States of America.
• P2. Mario Lopez is not the president of the USA.
• C Therefore, Jennifer Lopez is the president of the USA.

Impossible for P1 and P2 to be T and C to be F.

Valid argument: Example 2

• P1. All humans are mortal.
• P2. Tek is a human.
• C Therefore, Tek is mortal.

Impossible for P1 and P2 to be T and C to be F.

Invalid Argument

• P1: Jon is tall.
• C: Therefore, Jon is a professional basketball player.

Possible for P1 and P2 to be T and C to be F.

Valid or invalid?

• P1. Some basketball players are millionaires.
• P2. Some millionaires have fancy cars.
• C. Therefore, some basketball players have fancy cars.

Results

%%{init: {'themeVariables': {'pie1': '#2ecc71', 'pie2': '#e74c3c'}}}%%
pie title Results
    "Valid" : 85
    "Invalid" : 15

Valid argument with a false premise

Conceptual hurdle: Truth and validity are different!

• P1: All men are jerks. (FALSE!)
• P2: David is a man.
• C: David is a jerk.

Valid but all false premises

Conceptual hurdle: Truth and validity are different!

• P1: All biological pigs can fly. (FALSE)
• P2: David is a biological big. (FALSE)
• C: David can fly. (FALSE)

Valid with all false premises and true conclusion

Conceptual hurdle: Truth and validity are different!

• P1: The earth has a moon and Paris is in Germany. (FALSE)
• P2: The moon is made of cheese. (FALSE)
• C: The earth has a moon. (TRUE)

Invalid with all true propositions

Conceptual hurdle: Truth and validity are different!

• P1: Snow is white. (TRUE)
• P2: Grass is green. (TRUE)
• C: David is 6’0. (TRUE)

Testing for Validity

Two informal tests (no logic needed):

1. Intuition test
2. Imagination test

Intuition test

NoteIntuition test

Intuition test: Experience argument and check your intuition, instinct, gut feelings.

Bat and ball

• A bat and a ball cost $1.10.
• The bat costs one dollar more than a ball.
• Question: How much does the ball cost?

Results

%%{init: {'themeVariables': {'pie1': '#2ecc71', 'pie2': '#e74c3c'}}}%%
pie title Bat and Ball -- Harvard Students
accTitle: Bat and Ball Results
accDescr: A pie chart showing -- 50 percent of the responses are classified as incorrect and 50 percent as correct (5 cents).
    "Incorrect" : 50
    "5 cents" : 50

%%{init: {'themeVariables': {'pie1': '#2ecc71', 'pie2': '#e74c3c'}}}%%
pie title Bat and Ball -- University Students
accTitle: Bat and Ball Results
accDescr: A pie chart showing -- 80 percent of the responses are classified as incorrect and 5 percent as correct (5 cents).
    "Incorrect" : 80
    "5 cents" : 20

Problems

1. Inconsistent results
2. Biases
3. Large arguments
4. Arguments about abstract stuff
5. Not clear we have a power of intuition (spooky power)

Imagination Test

NoteImagination test

Imagination test: If you can imagine a scenario where the premises are true and the conclusion is false, then the argument is invalid. If you cannot, then it is valid.

flowchart LR
accTitle: Imagination Test for Argument Validity
accDescr: A flowchart showing that arguments are evaluated through imagination by asking whether the premises can be true while the conclusion is false. If yes, the argument is invalid; if no, the argument is valid.
  A[Arguments] --> H[Imagination]
  H --> Q{"Can the premises be true<br>and the conclusion false?"}
  Q -->|Yes| I[Invalid]
  Q -->|No| V[Valid]

Examples

• P1. All smokers love smoking.
• P2. Tek is a smoker.
• C. Therefore, Tek loves smoking.

flowchart LR
accTitle: Imagination Test for Argument Validity
accDescr: A flowchart showing that arguments are evaluated through imagination by asking whether the premises can be true while the conclusion is false. If yes, the argument is invalid; if no, the argument is valid.
  A[Arguments] --> H[Imagination]
  H --> Q{"Can the premises be true<br>and the conclusion false?"}
  Q -->|Yes| I[Invalid]
  Q -->|No| V[Valid]

Results (Imagination Test)

• P1. Some basketball players are millionaires.
• P2. Some millionaires drive fancy cars.
• C. Therefore, some basketball players drive fancy cars.

%%{init: {'themeVariables': {'pie1': '#2ecc71', 'pie2': '#e74c3c'}}}%%
pie title Imagination Test
accTitle: Imagination Test Results
accDescr: A pie chart showing an evenly divided result -- 50 percent of the responses are classified as valid and 50 percent as invalid.
    "Valid" : 50
    "Invalid" : 50

Basketball players

Some students won’t accept that the basketball player argument is invalid.

1. Jon, Liz, and Sam are the only millionaire basketball players but don’t drive fancy cars.
2. Tek and Frank are millionaires who work on Wall Street and drive fancy cars.

• P1. Some bball players are millionaires. True
• P2. Some millionaires drive fancy cars. True
• C. Some basketball players drive fancy cars. False

Problems with Imagination test

1. Terrible results on certain arguments (e.g., Basketball players)
2. Size
3. Limits to imagination
4. Biases
5. Doesn’t work - Basketball players example

Bias

• ARG is invalid if we can imagine a counterexample (Prem = True and C = False)
• If we are biased, then we might refuse to think of a counterexample because we believe the conclusion of the argument.

Example: Arguments for “Penn State is a good school” (when you like Penn State).

Objective Criteria: Combined

graph TD
  accTitle: Logic and Argument Types
    accDescr: A flowchart showing that logic divides arguments into good and bad arguments. Good arguments have true premises, relevance, and a conclusion that follows. When the conclusion follows necessarily, the argument is valid; when it follows probably, the argument is strong.

    L["Logic"]
    G["Good args"]
    B["Bad args"]
    T["True Premises"]
    R["Relevance"]
    C["C follows"]
    V["C follows necessarily (valid)"]
    P["C follows probably (strong)"]
    
    L --> G
    L --> B
    G --> T
    G --> R
    G --> C
    C --> V
    C --> P

    classDef badArgs fill:#f8d7da,stroke:#dc3545,stroke-width:2px,color:#000;
    class B badArgs;
    classDef goodArgs fill:#90D5FF,stroke-width:2px,color:#000;
    class G goodArgs;

Soundness

NoteSoundness

An argument is sound if and only if all of the premises are, in fact, true and the argument is deductively valid.

\[\text{soundness} = \text{validity} + \text{all true premises}\]

Soundness is a stronger notion than validity because an argument can be valid without being sound.

Timeline for PHIL12

timeline
    title Timeline of Phil 12
    section Propositional Logic
        Exam 1: ch1: ch2 (Pollock)
        Exam 2: ch3: ch4
        Exam 3: ch5 (Pollock)
        Exam 4: ch5
    section Predicate Logic
        Exam 5: ch6 (Pollock)
        Exam 5: ch8