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

by OC736321

Department

PhilosophyCourse Code

PHL245H1Professor

Alexander KooLecture

1This

**preview**shows pages 1-2. to view the full**7 pages of the document.**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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