Good and Bad Inferences

Entailment

  • Inferences

  • Premises and conclusions

  • Arguments

  • Entailment

  • Validity

  • Counterexamples

Overview

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

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

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

Entailment

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

If the premises are true, is the conclusion guaranteed to be true?

premises

entail

conclusion

premises

don't entail

conclusion

yes

no

Notation and examples

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

Notation and examples

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

Entailment is hypothetical

It cares about possibility, not truth.

Statement 3 is false, but it can still entail things, because entailment only cares about what happens if the premises are true, not about whether they are actually true or false.

1, 3 ⊨ 2

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

Careful! This definition trips people up a lot.

Entailment definition

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

Validity

if and only if
P1 ,...,Pn ⊨ C .

P1

⋮

Pn

C   
 

An argument

is valid

Entailment in adjective form.

How to prove invalidity/absence of entailment.

Counterexamples

  • Validity means

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

  • Invalidity therefore means

it is not possible for P1 ,...,Pn 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 ,...,Pn are true and C  is false.

Counterexample example

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.

In this case, the computer is the counterexample.

If I have 8 threads, then I can play in 4K.

If I have 20 threads, then I can livestream in 1080p.

If I have 20 threads, then I have 8 threads.

If I have 20 threads, then I can play in 4K while livestreaming in 1080p.

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

Proof theory (next topic)

Model theory

  • Focuses on the structure of phrases and sentences (syntax).

  • Defines "good inference/argument" in terms of patterns of reasoning.

  • Easy to prove that an inference/ argument is good.

  • Notation: P1 ,...,Pn ⊢ C

  • Focuses on the meanings of words, phrases, and sentences (semantics).

  • Defines "good inference/argument" in terms of entailment/validity.

  • Easy to prove that an inference/ argument is bad.

  • Notation: P1 ,...,Pn ⊨ C

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

⊨ and ⊢

  • In practice, mathematicians usually call ⊢ entailment  and don't use ⊨.

  • This is relatively harmless.

  • When learning about logics, however, this distinction is valuable, because it highlights the two very different methods we can use to distinguish good reasoning from bad.

Soundness and completeness

Sidebar

  • Distinguishing ⊢ and ⊨ is also part of understanding Godel's Incompleteness Theorems, which are two of the most important results in logic.

  • Whenever we develop a model theory and a proof theory for a specific kind of reasoning, we hope that

    • everything we can prove is valid   (i.e., ⊢  ⟹  ⊨, known as soundness)  and

    • everything that's valid can be proved (⊨  ⟹  ⊢, known as completeness).

  • In fact, however, we can only have one or the other for most logics we care about, including type theory and any logic for any part of mathematics.

  • Additionally, for all these logics, we can never truly prove soundness.

 

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

⊨ and ⊢

⊢  ⟹  ⊨

⊨  ⟹  ⊢

soundness:

completeness:

  • It turns out that it's rarely possible to have both soundness and completeness:

    • It's possible for first-order logic—logic for and, or, not, if, all, some, and is.

    • It's not possible for type theory or any mathematics.

  • .