... and what exactly is logic?

Entailment

• Inferences

• Premises and conclusions

• Arguments

• What is logic?

• Entailment

• Validity

• Counterexamples

Overview

To infer something is to conclude it from information you already have.

Inference

To infer something is to conclude it from information you already have.

Inference

The Portable Document Format (PDF) was invented by Adobe Systems. Therefore, PDFs work with Apple’s TrueType fonts.

Bad inference

The Java programming language license declares that the software should not be used to control a nuclear plant. Since the Susquehanna Steam Electric Station runs a nuclear plant, it should not use Java to run that plant.

Good inference

(conclusion)

(premises)

An argument  is simply premises along with a conclusion.

Arguments

Bad inference argument

Good inference argument

The Portable Document Format (PDF) was invented by Adobe Systems. Therefore, PDFs work with Apple’s TrueType fonts.

The Java programming language license declares that the software should not be used to control a nuclear plant. Since the Susquehanna Steam Electric Station runs a nuclear plant, it should not use Java to run that plant.

We separate the premises and conclusion with a line that means "therefore":

Notation for arguments

1. The Portable Document Format (PDF) was invented by Adobe Systems.

2. PDFs work with Apple’s TrueType fonts.

Bad argument

1. The Java programming language license declares that the software should not be used to control a nuclear plant.

2. The Susquehanna Steam Electric Station runs a nuclear plant.

3. The Susquehanna Steam Electric Station should not use Java to run its plant.

Good argument

Logic: The science of inference

Systems of logic (logics) are theories that distinguish good inferences from bad.

Entailment

The strictest notion of "good inference/argument": truth-preserving inference.

Notation

The double turnstile (⊨) means "entails."

Consider these three statements:

For each pair of statements, we can ask whether it entails the third:

1. The printer is printing.
2. The printer has a color cartridge installed.
3. The printer can't print without a color cartridge installed.

Notation

The double turnstile (⊨) means "entails."

Consider these three statements:

1, 2 ⊭ 3

For each pair of statements, we can ask whether it entails the third:

Notation

The double turnstile (⊨) means "entails."

Consider these three statements:

1, 2 ⊭ 3

1, 3 ⊨ 2

For each pair of statements, we can ask whether it entails the third:

Notation

The double turnstile (⊨) means "entails."

Consider these three statements:

1, 2 ⊭ 3

1, 3 ⊨ 2

2, 3 ⊭ 1

For each pair of statements, we can ask whether it entails the third:

1, 3 ⊨ 2

Careful! This definition trips people up a lot.

Entailment

Premises P₁ ,...,P_n entail  a conclusion C  (in symbols, P₁ ,...,P_n ⊨ C )
if and only if
it is not possible for P₁ ,...,P_n to all be true while C  is false.

Validity

Entailment in adjective form.

Proving invalidity/absence of entailment.

Counterexamples

• Validity means

it is not possible for P₁ ,...,P_n to all be true while C  is false.​

• Invalidity therefore means

it is not possible for P₁ ,...,P_n to all be true while C  is false.

• The simplest way to prove that something's possible is to do it.

• So, to prove that an argument is invalid, we construct a counterexample, which is a situation in which P₁ ,...,P_n are true and C  is false.

Example

• To prove that this argument is invalid, we need to show that it's possible for the premises to all be true while the conclusion is false.

• In other words, we need to try to make this happen.

• If we succeed, we've proved that the argument is invalid.

• If we fail, either the argument is valid or we missed something.

To prove that this argument is invalid, we need to...

• get a computer with at least 20 threads,

• play the game in 4K using only 8 threads,

• livestream something in 1080p using only 20 threads, and

• try and fail play in 4K while streaming in 1080p using only 20 threads.

T

Example

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Two aspects of logic: