flowchart LR A["Arguments"] --> L["Logic"] L --> G["Good Args"] L --> B["Bad Args"]
The Pennsylvania State University
Logic
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"]
What do we observe when people are arguing?
Argument
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.
Proposition
A proposition is a sentence, or something expressed by a sentence, that is capable of being true or false.
Are these propositions?
Do these sentences express propositions?
Argument
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
The proposition that is the conclusion is often marked:
Premises are also marked (less common):
In logic and philosophy, the presentation of an argument is often expressed in a standard form:
What makes an argument good?
Subjective 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.
Both are false but experiments show that people are more likely to believe (1).
Logic does not tell you if a proposition is true, e.g., Science, Experience, Religion?
Logic focuses mostly on entailment:
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;
Deductively 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:
Impossible for P1 and P2 to be T and C to be F.
Impossible for P1 and P2 to be T and C to be F.
Possible for P1 and P2 to be T and C to be F.
%%{init: {'themeVariables': {'pie1': '#2ecc71', 'pie2': '#e74c3c'}}}%%
pie title Results
"Valid" : 85
"Invalid" : 15
Conceptual hurdle: Truth and validity are different!
Conceptual hurdle: Truth and validity are different!
Conceptual hurdle: Truth and validity are different!
Conceptual hurdle: Truth and validity are different!
Two informal tests (no logic needed):
Intuition test
Intuition test: Experience argument and check your intuition, instinct, gut feelings.
%%{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
Imagination 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]
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]
%%{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
Some students won’t accept that the basketball player argument is invalid.
Example: Arguments for “Penn State is a good school” (when you like Penn State).
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
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
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