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
The Portable Document Format (PDF) was invented by Adobe Systems.
PDFs work with Apple’s TrueType fonts.
Bad argument
The Java programming language license declares that the software should not be used to control a nuclear plant.
The Susquehanna Steam Electric Station runs a nuclear plant.
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 ,...,Pnentail a conclusion C
if and only if
it is not possible for P1 ,...,Pn to all be true while C is false.
The double turnstile (⊨) means "entails":
Notation for entailment
P1 ,...,Pn⊨C
if and only if
it is not possible for P1 ,...,Pn to all be true while C is false.
Testing for entailment
Entailments are inferences that preserve truth.
If the premises are true, is the conclusion guaranteed to be true?
premises
do entail
conclusion
premises
don't entail
conclusion
yes
no
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 notpossible 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
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.
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 ⊢entailmentand 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.