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Lecture 1

PHL245H1 Lecture Notes - Lecture 1: Propositional Calculus, Atomic Sentence, Deductive Reasoning


Department
Philosophy
Course Code
PHL245H1
Professor
Alexander Koo
Lecture
1

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Unit 1: Arguments
Abstract Study of Argument
What are arguments?
- Arguments are made of statements
- Statements are sentences that are true or false
- Premises support conclusion
Type of arguments
1. Deductive arguments are certain (from general to specific)
2. Inductive arguments are fallible (from specific to general)
Deductive Arguments
What is a good deductive argument?
- Two senses of good are valid and sound
A deductive argument is valid if:
- Wherever the premises are true, the conclusion must be true
- It is impossible for the premises to be true and the conclusion to be false
Validity is not about content and truth; it is all about form and structure
If it is raining, then the sidewalk is wet
It is raining
Therefore, the sidewalk is wet Conclusion
A deductive argument is invalid if:
- It is not valid
- It is possible for the premises to be true and the conclusion to be false
If it is raining, then the sidewalk is wet
The sidewalk is wet
Therefore, it is raining
However, it could be snowing
A deductive argument is sound if:
1. It is valid
2. All the premises are true
Soundness is about content, truth, form and structure
If it is raining, then the sidewalk is wet
It is raining
Therefore, the sidewalk is wet
However, there could be a tree on the sidewalk to make the premises not always true
Inductive Arguments
- Inductive arguments are not valid or sound, but they are strong or cogent
- Cogent inductive arguments can have a false conclusion
This swan is white; this other swan is white
Every swan I have ever seen is white
Therefore, all swans are white
However, obviously there are black swans in the world
Premises
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Arguments and Explanation
- Some words are ambiguous premises and conclusion indicators
- You need to use context to know whether it is an argument or an explanation
- The goal of an argument is to convince you of the conclusion
- The goal of an explanation is to give an account for why the conclusion happened
Argument: BVB will win the game since they are the best team
Explanation: I did’t hand in my assignment since my dog ate my homework
Two Types of Sentences
- Tautology is a sentence that is always true or can never be false
- Contradiction is a sentence that is always false or can never be true
Tautology: It is raining or not raining
Contradiction: It is raining and not raining
Unit 2: Semantics in Sentential Logic
Sentential Logic
Types of statements
1. Atomic statements are statements that have no logic connections
2. Molecular statements are statements that have logic connections
Atomic: Grass is green
Molecular: I’ll have apple or orange
Atomic or molecular: I do’t like cats
Neither: Get out of here!
Logic connectives
1. Binary connectives: and ^, or V, if then , if and only if
2. Unary connectives: no~
Complex statements are all build up by joining statements together using logic connectives
o All connectives can operate on atomic or molecular statements
Symbols for sentential logic
1. Symbols for atomic statements: capital letter P-Z
2. Symbols for logic connectives: ^ V ~
3. Symbols for organization: () []
4. Symbols for sentences: φ ψ
Notation Rules
Official notation rules
1. Use round brackets () around every binary connective
2. Never sue brackets () around unary connectives or atomic statements
Informal notation rules
1. Use round brackets () around binary connectives that otherwise would be ambiguous
2. Never sue brackets () around unary connectives or atomic statements
3. always take precedent as the main connectives
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