1-hour introductory talk that i have presented to some undergraduate math clubs

CLASSICAL AND NONCLASSICAL LOGICS

. CLASSICAL AND NONCLASSICAL LOGICS

Introduction

Classical logic

Multivalued logics

Relevant logics

Constructive logic

AXIOM SYSTEMS

1 / 35

an overview of my book and my course

by Eric SchechterVanderbilt University

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Introduction


. Introduction

Who should take a course in logic?

Logics considered in this talk

We all use many different logics every day

(A slide for teachers) Pedagogical advantages of pluralism

Classical logic

Multivalued logics

Relevant logics

Constructive logic

AXIOM SYSTEMS

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Logic is how we prove things.


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Logic is how we prove things. Some teachers ask me,

Is this the course our department should use for our bridge course,our transition to higher math course, our how to do proofs course?


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No, actually I wouldn’t recommend it for that. And not every mathematicianneeds to take a course in logic. Don’t confuse theory with practice.


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analogy practice (how to do it) theory (why it works)


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cars driving lessons auto mechanics


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pastry cookbooks organic chemistry


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pastry cookbooks organic chemistry

proofs other math courses a course in logic •


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classical

Most introductions to logicstill cover only classical(early 20th century)


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classical

Most introductions to logicstill cover only classical(early 20th century), but mybook and a few others lookat some later logics too.

comparative

constructive

fuzzy

crystal


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classical


comparative

constructive

fuzzy

crystal

basic

Different logicshave differentsets of truths


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classical0 and 1


comparativeintegers

constructiveopen sets

fuzzy[0, 1]

crystal6 sets

basic

Different logicshave differentsets of truths,computed usingdifferent maths.

I’ll begin with evaluations (semantics),


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classical0 and 1


comparativeintegers

constructiveopen sets

fuzzy[0, 1]

crystal6 sets

basic

Different logicshave differentsets of truths,computed usingdifferent maths.

I’ll begin with evaluations (semantics),

and end with axiomatizations (syntactics). •


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Classical logic works well for mathematical proofs, but it does notdescribe how we reason about most nonmathematical matters.


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“I will wear a tie tomorrow” or “I will not wear a tie tomorrow.”


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Classical logic says one of those is already true. So is my free willjust an illusion? We need a logic that can say maybe.


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“There’s an 80% chance of rain this afternoon.”

That’s meaningful information; we plan activities around it. Butthat requires a quantitative logic.


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“If pigs have wings, then it’s raining right now in Pittsburgh”


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“If pigs have wings, then it’s raining right now in Pittsburgh”

— true for a classical logician, but nonsense for anyone else. Ourthoughts are closer to relevant logic. •


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Everyday thought is a mixture of many logics. Classical, introduced byitself, seems unnatural and arbitrary.


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Any abstract idea (e.g., completeness) needs several examples; oneexample (e.g., classical) is hardly enough.


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Reasoning requires questioning, not just memorizing. We must teachdoubt. That’s easier if we have multiple possibilities. For instance, to seethe significance of (¬¬P ) → P , it helps to ask “what happens in logicswhere (¬¬P ) → P isn’t always true?”


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Reasoning requires questioning, not just memorizing. We must teachdoubt. That’s easier if we have multiple possibilities. For instance, to seethe significance of (¬¬P ) → P , it helps to ask “what happens in logicswhere (¬¬P ) → P isn’t always true?”

In the classical-only course, true/false tables are too easy, reducing proofsto mere ritual. An omitted step will hardly be noticed if the studentalready knows that the conclusion is true. (Analogously, in Euclidean-onlygeometry, pictures demonstrate isolated facts.) •

Classical logic


Introduction

. Classical logic

Two-valued logic

Using math to study logic

Multivalued logics

Relevant logics

Constructive logic

AXIOM SYSTEMS

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Two-valued logic

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inputs

p

FT

Two-valued logic

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inputs not

p ¬p

F TT F

Two-valued logic

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inputs not or exclu.middle

p q ¬p p∨q q∨¬q

F F T F TF T T T TT F F T TT T F T T

Two-valued logic

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inputs not or and exclu.middle

contra-diction

p q ¬p p∨q p∧q q∨¬q p∧¬p

F F T F F T FF T T T F T FT F F T F T FT T F T T T F

Two-valued logic

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inputs not or and imply

p q ¬p p∨q p∧q p→q

F F T F F TF T T T F TT F F T F FT T F T T T

Two-valued logic

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FFFF FFFF T F F TTTTFFFF TTTT T T F TTTTT F F T F FT T F T T T

Two-valued logic

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A falsehood implies anything — i.e., if p is false then p → q is true.

Two-valued logic

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A falsehood implies anything — i.e., if p is false then p → q is true.If pigs have wings then it is now raining in Pittsburgh.

Two-valued logic

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inputs not or and imply contra-diction explosion

p q ¬p p∨q p∧q p→q p∧¬p (p∧¬p)→q

F F T F F T F TF T T T F T F TT F F T F F F TT T F T T T F T

A falsehood implies anything — i.e., if p is false then p → q is true.If pigs have wings then it is now raining in Pittsburgh.(p ∧ ¬p) → q (Releventists call this “explosion”)

Two-valued logic

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F F T F F TFFFF TTTT T T F TTTTT F F T F FTTTT TTTT F T T TTTT


Anything implies a truth — i.e., if q is true then p → q is true.

Two-valued logic

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F F T F F TFFFF TTTT T T F TTTTT F F T F FTTTT TTTT F T T TTTT


Anything implies a truth — i.e., if q is true then p → q is true.If the Yankees win the pennant next year then 1 + 1 = 2.

Two-valued logic

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inputs not or and imply exclu.middle

superfluoushypothesis

positiveparadox

p q ¬p p∨q p∧q p→q q∨¬q p→(q∨¬q) q→(p→q)

F F T F F T T T TF T T T F T T T TT F F T F F T T TT T F T T T T T T


Anything implies a truth — i.e., if q is true then p → q is true.If the Yankees win the pennant next year then 1 + 1 = 2.p → (q ∨ ¬q) (“superfluous hypothesis”)q → (p → q) (Releventists call this “positive paradox”)

Two-valued logic

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F F T F F TF T T T F TT F F T F FT T F T T T

relabeling. . . •


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inputs not or and impliesp q ¬ p p ∨ q p ∧ q p → q 0 = false

0 0 1 0 0 1 1 = true0 1 1 1 0 11 0 0 1 0 01 1 0 1 1 1


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0 0 1 0 0 1 1 = true0 1 1 1 0 11 0 0 1 0 01 1 0 1 1 1

1 − p


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0 0 1 0 0 1 1 = true0 1 1 1 0 11 0 0 1 0 01 1 0 1 1 1

1 − p maxp, q


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0 0 1 0 0 1 1 = true0 1 1 1 0 11 0 0 1 0 01 1 0 1 1 1

1 − p maxp, q minp, q


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0 0 1 0 0 1 1 = true0 1 1 1 0 11 0 0 1 0 01 1 0 1 1 1

1 − p maxp, q minp, q min1, 1 − p + q •

Multivalued logics


Introduction

Classical logic

. Multivalued logics

Lukasiewicz’s 3-valued logic

Fuzzy logic: infinitely many values

Example of p → q = min1, 1 − p + q

Tall people continued

Relevant logics

Constructive logic

AXIOM SYSTEMS

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0 = false, 1/2 = maybe, 1 = true. (Maybe I’ll wear a tie tomorrow.)


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p q ¬ p p ∨ q p ∧ q p → q

0 0 1 0 0 1

0 1/2 1 1/2 0 1

0 1 1 1 0 1

1/2 0 1/2 1/2 0 1/2

1/2 1/2 1/2 1/2 1/2 1

1/2 1 1/2 1 1/2 1

1 0 0 1 0 0

1 1/2 0 1 1/2 1/2

1 1 0 1 1 1


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p q ¬ p p ∨ q p ∧ q p → q

0 0 1 0 0 1

0 1/2 1 1/2 0 1

0 1 1 1 0 1

1/2 0 1/2 1/2 0 1/2

1/2 1/2 1/2 1/2 1/2 1

1/2 1 1/2 1 1/2 1

1 0 0 1 0 0

1 1/2 0 1 1/2 1/2

1 1 0 1 1 1

or more simply

¬ p = 1−p,p ∨ q = maxp, q,p ∧ q = minp, q,

p → q = min1, 1 − p + q.


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p q ¬ p p ∨ q p ∧ q p → q

0 0 1 0 0 1

0 1/2 1 1/2 0 1

0 1 1 1 0 1

1/2 0 1/2 1/2 0 1/2

1/2 1/2 1/2 1/2 1/2 1

1/2 1 1/2 1 1/2 1

1 0 0 1 0 0

1 1/2 0 1 1/2 1/2

1 1 0 1 1 1

or more simply


p → q = min1, 1 − p + q.

Note that ¬ 12 = 1

2 . •


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p → q = min1, 1 − p + q

Use those same formulas, but


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p → q = min1, 1 − p + q


instead of 0, 12 , 1,


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p → q = min1, 1 − p + q


instead of 0, 12 , 1, use

1 = the only true value,[0, 1) = false values.


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p → q = min1, 1 − p + q




That’s fuzzy logic.


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p → q = min1, 1 − p + q




That’s fuzzy logic. For instance, [[it will rain today]] = 0.8.


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p → q = min1, 1 − p + q





Don’t confuse these:

Fuzzy thinking means imprecise thinking. That’s bad.

Fuzzy logic means precise thinking about imprecise data. That’s good.


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p → q = min1, 1 − p + q





Don’t confuse these:

Fuzzy thinking means imprecise thinking. That’s bad.

Fuzzy logic means precise thinking about imprecise data. That’s good. Itis used in designing thermostats, clothesdriers, car cruise controls, etc. •


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(*) “Consider two people who differ in height by 1/4 inch. If one of thosepeople is very tall, then the other person is also very tall.”

I’ll show that implication (*) is mostly true, but not completely true.


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Suppose I have 101 students, numbered 0 through 100, and the ith studenthas height 78 − i

4 inches.


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4 inches. (These numbers are admittedly contrived.)


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4 inches. (These numbers are admittedly contrived.)Let pi = “the ith student is very tall.” Then (*) says pi−1 → pi.


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The 0th student is 612 feet tall, so p0 is absolutely true. Thus [[p0]] = 1.


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The 100th student is 412 feet tall, so p100 is absolutely false, and [[p100]] = 0.


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The 100th student is 412 feet tall, so p100 is absolutely false, and [[p100]] = 0.

Interpolating, it seems reasonable to assign [[pi]] = 1 − i100 .

(continued next slide) •


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Then our rule p → q = min1, 1 − p + q


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Then our rule p → q = min1, 1 − p + q yields [[pi−1 → pi]] = 0.99, sothe implication pi−1 → pi is mostly true.


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If pi−1 → pi were completely true, then using [[p0]] = 1 and induction wecould prove [[p100]] = 1. But that’s wrong. Thus, in fuzzy logic we don’t getfree repetitions of arguments, unlike in classical logic.


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In classical logic, if assuming A twice yields B, then assuming A once alsoyields B. That’s the idea of the contraction formula:

(A → (A → B)) → (A → B).


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In classical logic, if assuming A twice yields B, then assuming A once alsoyields B. That’s the idea of the contraction formula:

(A → (A → B)) → (A → B).

But contraction fails in fuzzy logic, e.g. when [[A]] = 1/2 and [[B]] = 0. Moreabout that later. •

Relevant logics


Introduction

Classical logic

Multivalued logics

. Relevant logics

Aristotle’s comparisons

Comparative logic

Irrelevance: Bad taste in reasoning

Crystal logic: sets for values

Crystal implication — admittedly complicated (skip thisslide?)

Relevance Principles

A relevance proof

WHY classical logic allows irrelevance

Constructive logic

AXIOM SYSTEMS

15 / 35


16 / 35

If there are two things both more desirable than something, theone which is more desirable to a greater degree is more desirablethan the one more desirable to a less degree.


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Here’s a more modern and digestible version of the same idea . . .


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(∗)

Suppose that “the coffee is hotter than the punch”


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(∗)

Suppose that “the coffee is hotter than the punch”is more true than “the tea is hotter than the punch.”


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(∗)


Then


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(∗)


Then the coffee is hotter than the tea.


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(∗)



That sounds reasonable. But it translates to[(p → t) → (p → c)

]→ (t → c),


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(∗)




]→ (t → c),

which can fail in classical logic — e.g., when [[t]] = 1 and [[c]] = [[p]] = 0.


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(∗)




]→ (t → c),


Classical logic can’t make sense out of “more true.” In classical logic, either astatement is true, or it isn’t.


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(∗)




]→ (t → c),


Classical logic can’t make sense out of “more true.” In classical logic, either astatement is true, or it isn’t.

For comparisons, we need a different logic. . . . •

Comparative logic

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false values = . . . ,−3,−2,−1, true values = 0, 1, 2, 3, . . .,

¬ p = −p, p ∨ q = maxp, q, p ∧ q = minp, q, p → q = q − p.

Comparative logic

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Here “comparison”[(p → t) → (p → c)

]→ (t → c) is always true.

Comparative logic

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Some irrelevancies are ruled out. For instance, neither p → (q ∨ ¬q) nor(p ∧ ¬p) → q is an always-true formula.

Comparative logic

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On the other hand, this logic affirms some irrelevancies. For instance,“unrelated extremes” (p ∧ ¬p) → (q ∨ ¬q) is always true in this logic.

Comparative logic

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On the other hand, this logic affirms some irrelevancies. For instance,“unrelated extremes” (p ∧ ¬p) → (q ∨ ¬q) is always true in this logic.

Note: A few slides from now I’ll use the fact that, in this logic,

¬ 0 = 0 ∧ 0 = 0 ∨ 0 = 0 → 0 = 0. •


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18 / 35

If today is Friday then 1 + 1 = 2.


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If today is Friday then 1 + 1 = 2. If the earth is flat then today is Friday.


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If today is Friday then 1 + 1 = 2. If the earth is flat then today is Friday. (p ∧ ¬p) → q (“explosion”)


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If today is Friday then 1 + 1 = 2. If the earth is flat then today is Friday. (p ∧ ¬p) → q (“explosion”) (p ∧ ¬p) → (q ∨ ¬q) (“unrelated extremes”)


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— all classically true,


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— all classically true, but the hypothesis and conclusion are unrelated,


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— all classically true, but the hypothesis and conclusion are unrelated,which makes the implication confusing, misleading, or


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— all classically true, but the hypothesis and conclusion are unrelated,which makes the implication confusing, misleading, or just plain ugly.


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Most mathematicians know only classical logic, and would say that

they are permitted to make such tasteless statements, but


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they are permitted to make such tasteless statements, but they voluntarily refrain from doing so;


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they are permitted to make such tasteless statements, but they voluntarily refrain from doing so; they exercise good taste.


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They’re practicing relevant logic without realizing it!


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They’re practicing relevant logic without realizing it!

One logic with particularly strong relevance properties is crystal logic . . . •


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−2,−1, +1, +2

−1, +1, +2

−1, +2 +1, +2

+2

∅

6 semantic values


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−2,−1, +1, +2

−1, +1, +2

−1, +2 +1, +2

+2

∅

6 semantic values

∅ is false; the otherfive sets are true


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−2,−1, +1, +2

−1, +1, +2

−1, +2 +1, +2

+2

∅

6 semantic values


the inclusions yielda nonlinear order


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−2,−1, +1, +2

−1, +1, +2

−1, +2 +1, +2

+2

∅

6 semantic values



∧ is ∩, ∨ is ∪


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−2,−1, +1, +2

−1, +1, +2

−1, +2 +1, +2

+2

∅

(flip)

6 semantic values



∧ is ∩, ∨ is ∪

flip for negation


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−2,−1, +1, +2 = Ω

−1, +1, +2 = τ

−1, +2= λ

= ¬λ

+1, +2= ρ = ¬ρ

+2 = β

∅

(flip)

6 semantic values



∧ is ∩, ∨ is ∪

flip for negationS Ω τ λ ρ β ∅

¬S ∅ β λ ρ τ Ω


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−2,−1, +1, +2 = Ω

−1, +1, +2 = τ

−1, +2= λ

= ¬λ

+1, +2= ρ = ¬ρ

+2 = β

∅

(flip)

6 semantic values



∧ is ∩, ∨ is ∪



“implies” is on next slide


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−2,−1, +1, +2 = Ω

−1, +1, +2 = τ

−1, +2= λ

= ¬λ

+1, +2= ρ = ¬ρ

+2 = β

∅

(flip)

6 semantic values



∧ is ∩, ∨ is ∪



“implies” is on next slide

Note that λ ∨ λ = λ ∧ λ = λ → λ = ¬λ = λand ρ ∨ ρ = ρ ∧ ρ = ρ → ρ = ¬ρ = ρ. •

Crystal implication — admittedly complicated (skip this slide?)

20 / 35


20 / 35

S→T T =

∅ β λ ρ τ Ω

∅ Ω Ω Ω Ω Ω Ωβ ∅ β λ ρ τ Ω

S λ ∅ ∅ λ ∅ λ Ω= ρ ∅ ∅ ∅ ρ ρ Ω

τ ∅ ∅ ∅ ∅ β ΩΩ ∅ ∅ ∅ ∅ ∅ Ω


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S→T T =

∅ β λ ρ τ Ω


S λ ∅ ∅ λ ∅ λ Ω= ρ ∅ ∅ ∅ ρ ρ Ω

τ ∅ ∅ ∅ ∅ β ΩΩ ∅ ∅ ∅ ∅ ∅ Ω

or equivalently

S → T =

Ω if S = ∅ or T = Ω,∅ if S is not a subset of T ,T if S = β,

¬S if T = τ ,S if (S, T ) is (λ, λ) or (ρ, ρ).


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S→T T =

∅ β λ ρ τ Ω


S λ ∅ ∅ λ ∅ λ Ω= ρ ∅ ∅ ∅ ρ ρ Ω

τ ∅ ∅ ∅ ∅ β ΩΩ ∅ ∅ ∅ ∅ ∅ Ω

or equivalently

S → T =



Admittedly, contrived and complicated. But


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S→T T =

∅ β λ ρ τ Ω


S λ ∅ ∅ λ ∅ λ Ω= ρ ∅ ∅ ∅ ρ ρ Ω

τ ∅ ∅ ∅ ∅ β ΩΩ ∅ ∅ ∅ ∅ ∅ Ω

or equivalently

S → T =



Admittedly, contrived and complicated. But crystal logic

• includes my “basic logic” (discussed later), so it’s not bizarre; and


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S→T T =

∅ β λ ρ τ Ω


S λ ∅ ∅ λ ∅ λ Ω= ρ ∅ ∅ ∅ ρ ρ Ω

τ ∅ ∅ ∅ ∅ β ΩΩ ∅ ∅ ∅ ∅ ∅ Ω

or equivalently

S → T =



Admittedly, contrived and complicated. But crystal logic

• includes my “basic logic” (discussed later), so it’s not bizarre; and• prevents irrelevant implications — for instance, (p ∧ ¬p) → (q ∨ ¬q) is

not always true. More generally . . . •


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Suppose that formulas A and B are irrelevant to each other, i.e., they haveno variables in common. Can A → B still be tautological (i.e., always true)?


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(1) In classical logic, A → B is tautological if and only ifat least one of B or ¬A is tautological, as in

p︸︷︷︸

A

→ (q ∨ ¬q)︸︷︷︸

B

or (p ∧ ¬p)︸︷︷︸

A

→ q︸︷︷︸

B

.


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p︸︷︷︸

A

→ (q ∨ ¬q)︸︷︷︸

B

or (p ∧ ¬p)︸︷︷︸

A

→ q︸︷︷︸

B

.

(2) In comparative logic, A → B is a tautology if and only ifboth B and ¬A are tautologies, as in “unrelated extremes”

(p ∧ ¬p)︸︷︷︸

A

→ (q ∨ ¬q)︸︷︷︸

B

.


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p︸︷︷︸

A

→ (q ∨ ¬q)︸︷︷︸

B

or (p ∧ ¬p)︸︷︷︸

A

→ q︸︷︷︸

B

.


(p ∧ ¬p)︸︷︷︸

A

→ (q ∨ ¬q)︸︷︷︸

B

.

(3) In crystal logic, A → B cannot be a tautology.


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p︸︷︷︸

A

→ (q ∨ ¬q)︸︷︷︸

B

or (p ∧ ¬p)︸︷︷︸

A

→ q︸︷︷︸

B

.


(p ∧ ¬p)︸︷︷︸

A

→ (q ∨ ¬q)︸︷︷︸

B

.

(3) In crystal logic, A → B cannot be a tautology.

I’ll prove part of (2). (Its other parts and (1) and (3) are proved similarly.) •

A relevance proof

22 / 35

In comparative logic (i.e., with subtraction for implication), if A and B shareno variables and B is not a tautology, then A → B is not a tautology.

Proof.

A relevance proof

22 / 35


Proof.

Since B is not a tautology, there is some assignment of values to thevariables appearing in B that makes B false —

A relevance proof

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Proof.

Since B is not a tautology, there is some assignment of values to thevariables appearing in B that makes B false — i.e., that makes [[B]] < 0.

A relevance proof

22 / 35


Proof.


Since A and B share no variables,

A relevance proof

22 / 35


Proof.


Since A and B share no variables, we are still free to choose values for allthe variables appearing in A.

A relevance proof

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Proof.


Since A and B share no variables, we are still free to choose values for allthe variables appearing in A. Give them all the value 0.

A relevance proof

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Proof.



Then [[A]] = 0, since 0 ∨ 0 = 0 ∧ 0 = 0 → 0 = ¬ 0 = 0.

A relevance proof

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Proof.



Then [[A]] = 0, since 0 ∨ 0 = 0 ∧ 0 = 0 → 0 = ¬ 0 = 0.

Then [[A→B]] = [[B]] − [[A]] < 0. So A→B isn’t always true. •


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23 / 35

If the earth is flat then today is Friday.

We see irrelevance here because we have background information:


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We see irrelevance here because we have background information: We know

something about the earth and something about the week, and we knowthey’re unrelated.


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But in math we have less background information. For instance:


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Theorem. Let X be a Banach space. Then


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Theorem. Let X be a Banach space. Then (i) every lowersemicontinuous seminorm on X is continuous


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Theorem. Let X be a Banach space. Then (i) every lowersemicontinuous seminorm on X is continuous if and only if


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Theorem. Let X be a Banach space. Then (i) every lowersemicontinuous seminorm on X is continuous if and only if(ii) every weak-star bounded subset of the dual space X∗

is also norm-bounded.


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Theorem. Let X be a Banach space. Then (i) every lowersemicontinuous seminorm on X is continuous if and only if(ii) every weak-star bounded subset of the dual space X∗

is also norm-bounded.

Even to someone who speaks this language, and is familiar with conditions (i)and (ii), it is not obvious that there is any relation between those conditions.In fact, that relation is the whole point of the theorem. •

Constructive logic


Introduction

Classical logic

Multivalued logics

Relevant logics

. Constructive logic

Jarden’s Theorem

Two philosophies of mathematics

Jarden’s logic: P ∨ ¬P (“Excluded Middle”)

Constructive evaluations (complicated; skip this?)

AXIOM SYSTEMS

24 / 35

Jarden’s Theorem

25 / 35

There exist positive irrational numbers a and b such that ab is rational.

Jarden’s Theorem

25 / 35


Prerequisites.

Jarden’s Theorem

25 / 35


Prerequisites.

Definition. A number is rational if it can be written as the ratio of twointegers (like 37/5); otherwise it is irrational.

Jarden’s Theorem

25 / 35


Prerequisites.


Lemma 1.√

2 is irrational.

Jarden’s Theorem

25 / 35


Prerequisites.


Lemma 1.√

2 is irrational. Lemma 2. (pq)r = p(qr).

Jarden’s Theorem

25 / 35


Jarden’s Proof.

Prerequisites.


Lemma 1.√


Jarden’s Theorem

25 / 35


Jarden’s Proof. Consider the number j =√

2√

2.

Prerequisites.


Lemma 1.√


Jarden’s Theorem

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2√

2.

• If j is rational,

•

Prerequisites.


Lemma 1.√


Jarden’s Theorem

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2√

2.

• If j is rational, use a = b =√

2, and we’re done.

•

Prerequisites.


Lemma 1.√


Jarden’s Theorem

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2√

2.



• If j is irrational,

Prerequisites.


Lemma 1.√


Jarden’s Theorem

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2√

2.



• If j is irrational, use a = j and b =√

2. Then

Prerequisites.


Lemma 1.√


Jarden’s Theorem

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2√

2.




2. Then a and b are irrational, and abrief computation shows ab = 2 = rational. 2

Prerequisites.


Lemma 1.√


Jarden’s Theorem

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2√

2.




2. Then a and b are irrational, and abrief computation shows ab = 2 = rational. 2

So a and b exist. But we still don’t know what they are! •

Prerequisites.


Lemma 1.√



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Classical philosophy

Constructive philosophy


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Math is a collection of statements, e.g., “there exist a and b such that. . . ”



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Math is a collection of statements, e.g., “there exist a and b such that. . . ” An existence proof does not need to be accompanied by a construction.



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Math is a collection of statements, e.g., “there exist a and b such that. . . ” An existence proof does not need to be accompanied by a construction. Jarden’s proof is acceptable.



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Math is a collection of procedures — “we can find a and b such that. . . ”


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Math is a collection of procedures — “we can find a and b such that. . . ” A mathematical object is meaningless unless we have a method for

producing that object.


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producing that object. Jarden’s proof is unacceptable.


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But it’s just his proof that is nonconstructive. His theorem can be made constructivevia other proofs — for instance, use Gelfond-Schneider theorem to prove that j isirrational, or more simply just take a =

√2 and b = log2 9.


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But it’s just his proof that is nonconstructive. His theorem can be made constructivevia other proofs — for instance, use Gelfond-Schneider theorem to prove that j isirrational, or more simply just take a =

√2 and b = log2 9.

On the other hand, some mathematical results (such as the Axiom of Choice) areinherently nonconstructive, and rejected altogether by constructivists. •


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Jarden applied it with P = “√

2√

2is rational.”


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2√

2is rational.”

It’s tautologous in two-valued logic, so it’s fine for true/false statements.


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2√

2is rational.”


But it cannot be relied upon as a recipe in constructions. Sometimesthere is a task P that we don’t know how to carry out, and we don’tknow how to carry out the opposite task either.


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2√

2is rational.”



Another example of this idea: Most mathematicians would agree that


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2√

2is rational.”




The Twin Prime Conjecture is true or it is false.


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2√

2is rational.”





But we don’t know which, and perhaps we never will.


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2√

2is rational.”





But we don’t know which, and perhaps we never will. Consequently, someconstructivists might say that it is neither “true” or “false” —


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2√

2is rational.”





But we don’t know which, and perhaps we never will. Consequently, someconstructivists might say that it is neither “true” or “false” — like thestatement

Luke Skywalker’s favorite color is red. •


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An open interval is any set of the form (a, b) = x ∈ R : a < x < b.


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An open interval is any set of the form (a, b) = x ∈ R : a < x < b. An open set (in R) is any union of open intervals.


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An open interval is any set of the form (a, b) = x ∈ R : a < x < b. An open set (in R) is any union of open intervals. The interior of any set S is the largest open set contained in S.

Equivalently, int(S) is the union of all the open intervals contained in S.


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Semantics for constructive logic:


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R is the only true value. All other open subsets of R are false values.


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∧ is ∩, ∨ is ∪, ¬S = int(

R \ S)

, S→T = int(

T ∪ (R \ S))

.


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∧ is ∩, ∨ is ∪, ¬S = int(

R \ S)

, S→T = int(

T ∪ (R \ S))

.

P ∨ ¬P is false (for instance) when [[P ]] = (0, 1) ∪ (1, 2). •

AXIOM SYSTEMS


Introduction

Classical logic

Multivalued logics

Relevant logics

Constructive logic

. AXIOM SYSTEMS

Example of proving a theorem from some axioms



Axioms for classical logic, divided into two parts

Two different approaches to any logic

A few examples of completeness pairings

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Assumptions: A, A → B ` B “detachment”` C → (D → C) “positive paradox”

` [E → (F → G)] → [(E → F ) → (E → G)] “self-distribution”


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Theorem: ` X → X (“identity”).

Proof of theorem:


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Proof of theorem:

But wait! Why do we need to prove X → X? Isn’t it obviously true?


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Proof of theorem:

But wait! Why do we need to prove X → X? Isn’t it obviously true?

Only if we assume that the symbol “→ ” has some meaning closeto the usual meaning of “implies.” But we don’t want to assumethat. In axiomatic logic, we start with no meaning at all forsymbols such as “→ ”; they’re just symbols. They obtain only themeanings given to them by our axioms.


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But wait2! Why do we assume the complicated formula “self-distribution”and prove the simple formula “identity”? Wouldn’t it make more sense to gothe other way?


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But wait2! Why do we assume the complicated formula “self-distribution”and prove the simple formula “identity”? Wouldn’t it make more sense to gothe other way?

Many choices of axioms are possible; we’ll discuss those soon. Butsome choices work better than others. For instance, we find that

detachment, positive paradox, self-dist. ⇒ identity, butdetachment, positive paradox, identity 6⇒ self-dist.


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Proof of theorem:


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Proof of theorem:

# formula justification

(1) X → (X → X) positive paradox with C = D = X


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Proof of theorem:



(2) X → [(X → X) → X] pos.pdx. with C = X, D = X→X


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Proof of theorem:




(3) X → [(X → X) → X] → self-distribution with[X→(X→X)]→(X→X) E = G = X, F = X → X


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Proof of theorem:





(4) [X→(X→X)]→(X→X) detach. with A = (2), A→B = (3)


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Proof of theorem:





(4) [X→(X→X)]→(X→X) detach. with A = (2), A→B = (3)

(5) X → X detach. with A = (1), A→B = (4) •


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A, A→B ` B, A, B ` A∧B(A ∧ B) → A, A → (A ∨ B)(A ∧ B) → B, B → (A ∨ B)A → A, (A→¬B)→(B→¬A)[A → (B → C)] → [B → (A → C)](A → B) → [(C → A) → (C → B)][(A → B) ∧ (A → C)] → [A → (B ∧ C)][(B → A) ∧ (C → A)] → [(B ∨ C) → A][A ∧ (B ∨ C)] → [(A ∧ B) ∨ C]

“Basic” logic. This is theuncontroversial, “vanilla”part. Most logics satisfythese axioms. They are nu-merous, but each is fairlysimple by itself.


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A → (B → A) positive paradox[A→(A→B)]→(A→B) contraction

(¬¬A) → A double negation

Non-basic axioms. Addjust some of these spices toget nonclassical logics.


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A → (B → A) positive paradox[A→(A→B)]→(A→B) contraction

(¬¬A) → A double negation

Non-basic axioms. Addjust some of these spices toget nonclassical logics.

A book on just classical logic uses a shorter list of stronger axioms. •


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Evaluations (semantics) Axioms (syntactics)


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The concrete approach.


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The concrete approach. Formulasare evaluated independently of oneanother. They take values (or“meanings”) in 0, 1, [0, 1], Z, orsome other set.


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The concrete approach. Formulasare evaluated independently of oneanother. They take values (or“meanings”) in 0, 1, [0, 1], Z, orsome other set. An always-true

formula is called a tautology.


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The abstract approach.


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The abstract approach. We studywhich formulas generate whichother formulas, without regard towhat they might “mean.”


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The abstract approach. We studywhich formulas generate whichother formulas, without regard towhat they might “mean.” Aformula that can be proved from

the axioms is called a theorem.


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A completeness pairing is a matching of some evaluation system with someaxiom system, such that


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tautologies = theorems,


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tautologies = theorems, hence

every statement has an abstract proof or a concrete counterexample.


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tautologies = theorems, hence

every statement has an abstract proof or a concrete counterexample.

But such pairings are hard to find, and harder to prove. •


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name values: axioms: basic, plus . . .

classical 0, 1 positive paradox, double negation, contraction

Lukasiewicz 0, 12 , 1

positive paradox, double negation,((A→B)→B)→(A∨B), and(A→(A→¬A))→(A→¬A),

fuzzy [0, 1]positive paradox, double negation, and((A → B) → B) → (A ∨ B)

comparative integers((A→B)→B)→A,(A→A) ↔ ¬(A→A)

crystal 6 setscontraction, double negation, A ∨ (A → B),and ((¬A)∧B) → (((¬A) → A)∨ (A → B))

constructive open sets positive paradox, contraction, and explosion


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name values: axioms: basic, plus . . .

classical 0, 1 positive paradox, double negation, contraction√

Lukasiewicz 0, 12 , 1

positive paradox, double negation,((A→B)→B)→(A∨B), and(A→(A→¬A))→(A→¬A),

√

fuzzy [0, 1]positive paradox, double negation, and((A → B) → B) → (A ∨ B)

h

comparative integers((A→B)→B)→A,(A→A) ↔ ¬(A→A)

h

crystal 6 setscontraction, double negation, A ∨ (A → B),and ((¬A)∧B) → (((¬A) → A)∨ (A → B))

h

constructive open sets positive paradox, contraction, and explosion√

√= proved in my book; h = too hard to prove in my book.

1-hour introductory talk that i have presented to some undergraduate math clubs

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