Good and Bad Inferences

Entailment

• Inferences

• Premises and conclusions

• Arguments

• Entailment

• Validity

• Counterexamples

Overview

To infer something is to conclude it on the basis of 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

To infer something is to conclude it on the basis of 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

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

Bad inference argument

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 argument

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

The strictest notion of "good inference/argument":

Entailment

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

The double turnstile (⊨) means "entails":

Notation for entailment

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

Testing for entailment

Entailments are inferences that preserve truth.

Validity

Entailment in adjective form.

How to prove invalidity/absence of entailment.

Counterexamples

• Validity means

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

• Therefore, invalidity 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.

Counterexample example

This argument is invalid:

To prove it's 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.

In this case, the computer is the counterexample.

Entailment/validity is a semantic concept.

Semantics (as opposed to syntax)

• Whether an argument is valid depends on truth and possibility.

• Truth and possibility depend on what the premises and conclusions mean.

• This makes proving validity/entailment difficult—you have to make sure you think of every possible way things could turn out.

Two ways to define "good inference/argument":

Semantics and syntax

Math frequently blurs the line between these two, and that's mostly ok.

⊨ and ⊢

Math frequently blurs the line between these two, and that's mostly ok.

⊨ and ⊢