SlideShare a Scribd company logo
Common language and modal logic NL
William Heartspring
Ver 1.0, July 14, 2019
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 1 / 27
Table of Contents
1 Common language and modal logic NL
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 2 / 27
Table of Contents
1 Common language and modal logic NL
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 3 / 27
Common language in formal logic
Ideally, logic should be able to analyze both common language and
mathematics.
But: common language too much filled with paradoxes.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 4 / 27
Goal of Modal logic NL
Formal logic that resolves those paradoxes but still allow sentences in
common language to be analyzed. That is, unrestricted comprehension
(recursion) allowed.
But we also would like to preserve powerfulness of classical logic. Example:
proof by contradiction.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 5 / 27
Strategy
If a sentence implies a contradictory or paradoxical sentence (a
paradoxical sentence can be considered a generalized curry sentence),
then the sentence must be considered false.
But in classical logic, the definition of a paradox can be said as:
“Sentence φ causes a contradiction. Thus, φ is false, so ¬φ must be
true. But ¬φ also causes a contradiction. So what should I do?”
Classical logic disallows both φ and ¬φ to be false.
Even for Curry’s paradox, the paradox is intensified in classical logic
because both φ and ¬φ cannot be false.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 6 / 27
Digression: Curry’s paradox in classical logic
A ↔ (A → B)
That is, sentence A says A → B. This sentence allowed by unrestricted
comprehension. Classical logic says A → B and A must both be true.
Miracle: we can prove any sentence by forming an appropriate curry
sentence for each target sentence (B). But a curry sentence exists for ¬B
as well, so paradoxical.
Initially, it may seem that there is nothing wrong with assigning false to A
and (A → B), so a paradox may be resolved.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 7 / 27
Digression: Curry’s paradox in classical logic
But classical logic condemns us to assign truth value true to A and
(A → B). Thus not a problem of deduction or proof in classical logic.
This is an internal problem in classical logic.
A → (A → B) means ¬A ∨ (A → B). Take ¬A as true.
(A → B) → A means ¬(A → B) ∨ A. Take ¬(A → B), since ¬A was
assumed.
¬(A → B) means A ∧ ¬B. But cannot be , since ¬A was assumed!
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 8 / 27
Implication problem
The main problem is that A → B is assumed to be ¬A ∨ B. Why does this
hold in classical logic?
If A is false, then B is proved by the principle of explosion. Thus ¬A to
take care of this circumstance.
If A is true, then B must be true if A → B is true. That is B.
Thus, ¬A ∨ B.
Thus, if falsity of A does not imply ¬A, then we short-circuit the issue in
Curry’s paradox. Of course this is just one consideration out of many
required, but still a good guiding light.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 9 / 27
Strategy
Let’s continue the strategy.
Refine the law of excluded middle. However, we would like to use the
law of excluded middle for non-paradoxical situations from classical
logic perspectives.
Requires a definition of a paradox. p is considered a paradoxical
sentence only when p ↔ ¬p holds. If this does not hold and p leads
to a paradoxical sentence: only implies that p is false.
Would like to assign false to both p and ¬p only when p is a
paradoxical sentence.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 10 / 27
Strategy: refining the law of excluded middle
The easiest na¨ıve thought would be to add:
¬(p ↔ ¬p) → (p ∨ ¬p)
But: Some model of a theory in this new logic would exist that affirms p
being a paradoxical sentence, even if there exists no proof that p is a
paradoxical sentence.
In other words, a classical logic proof that relied on the law of excluded
middle would not work in the new logic!
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 11 / 27
Strategy: refining the law of excluded middle
Thus, modal logic is required. More precisely, provability logic that
borrows modal logic.
¬ (p ↔ ¬p) → (p ∨ ¬p)
That is, if a paradox is unprovable, then the law of excluded middle must
hold. ( means provable)
But comes the heavy cost which the new logic accepts: completeness no
longer holds. If p holds in all models, p still may not be provable (in finite
deduction steps).
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 12 / 27
Strategy: “Post-proofs”
But if p holds in all models, we still would like to have some concept of
proof of p.
That is, if we assume unprovability of a paradox, then p is provable by
invoking the law of excluded middle. This unprovability of a paradox may
never be proved, but then this “post-proof” effectively justifies the use of
classical logic proofs based on the law of excluded middle.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 13 / 27
Back to Curry’s paraodx
A ↔ (A → B)
Proving A and A → B requires the rule:
(p → (q → r)) → ((p → q) → (p → r))
In classical logic, not possible to render A as false. Now we can, as far as a
rule exists that overwrites the above rule:
(p → (p → r)) → ((p → ⊥) ∧ ((p → r) → ⊥))
⊥ represents a contradiction. The above rule is called Then-4.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 14 / 27
Back to Curry’s paradox
What about a generalized Curry’s paradox? Still such a generalized curry
sentence implies some form of a curry sentence or a contradictory
sentence. As far as a rule exists that render any sentence leading to a
contradiction false, no problem.
p ↔ q can be unprovable, even if true. Matters because if p ↔ q holds,
then effectively p → (q → r) is p → (p → r).
Here, no reason to impose provability or unprovabiliy requirements. Better
to leave things this way for a good match with classical logic. In classical
logic, there may exist a model with p ↔ q, when a model with ¬(p ↔ q)
exists.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 15 / 27
Strategy
Strategy-wise, we are done.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 16 / 27
NL axioms excluding NL3
NL1: (p → ⊥) → ¬ p
NL2: ¬ p ∨ p.
NL4: p ∨ q → (p ∨ q)
NL5: p ∧ q ↔ (p ∧ q)
These axioms just confirm the usual point about provability logic. So why
are they there? Because if not thought in terms of provability, one may
not get to confirm how ¬ is related to , or how ¬ is supposed to
behave for modal operators from purely modal logic perspectives. Also,
one may not get to confirm how behaves in response to ∨ or ∧ inside
the scope of the modal operator.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 17 / 27
NL axioms: NL4
( p ∨ ¬ p)
does not mean
p ∨ ¬ p
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 18 / 27
NL axioms: NL3
(¬ p → ⊥) → p
( p → ⊥) → ¬ p
This is the real important new axiom. Allows us to prove unprovability or
provability of modal sentences by assuming them first, and then deriving a
contradiction. This heavily critical, given that unprovability of a paradox
may be unprovable. Gives us an important proof power.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 19 / 27
Non-tautologous new axioms
New non-tautologous axioms: Refined Not-1’, Then-4 and NL-3.
Features of classical logic that do not survive:
The law of excluded middle. But refined in Refined Not-1’ axiom, so
in non-paradoxical circumstances from perspectives of classical logic,
modal logic NL (new logic) equivalent to classical logic.
(p → q) no longer ¬p ∨ q. Falsity of p does not mean ¬p, thus
¬p ∨ q no longer always valid. But valid if the law of excluded middle
is allowed by the new logic.
Law of noncontradiction is not to be expressed as ¬(p ∧ ¬p). Since
De Morgan’s law holds, that form means the law of excluded middle.
It is to be expressed as (p ∧ ¬p) → ⊥.
Proof by contradiction is allowed but it is considered a post-proof,
having assumed unprovability of a paradoxical situation. Thus,
technically these post-proofs requiring infinite deduction steps.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 20 / 27
Common features shared between NL and classical logic
De Morgan’s laws
Principle of explosion
Double negation elimination and introduction
Propositional logic axioms and rules, except for Then-4 overwriting
some axioms/rules.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 21 / 27
Common language in formal logic
Now that modal logic NL is constructed:
What is common language from perspectives of formal logic? Roughly,
one may define as featuring unrestricted comprehension (or recursion).
Example:
Liar paradox “this sentence is false” φ ↔ ¬φ
Curry’s paradox: “truth of this sentence implies B is true.”
A ↔ (A → B)
Barber paradox: Barber Alice shaves all of those and only those who
do not shave themselves.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 22 / 27
More paradoxes of common language
Liar/Barber/Russell’s/knower/Fitch’s paradox: basically they are of
the same type. Type 1 paradox. Basically of the form p ↔ ¬p.
Curry’s paradox: Type 2 paradox. Of the form p ↔ (p → r).
Generalized Curry’s paradox allows to consider both type 1 paradox
and type 2 paradox in the same footing - but we do not need to
discuss this here, as we have covered both types of paradoxes.
No-no paradox. In a way, not a paradox, but an asymmetry problem.
Somewhat boring examples, but then reality filled with unrestricted
recursion of language demonstrated above.
Can we simply ban unrestricted comprehension/recursion in language? No.
Examples in lambda calculus demonstrate this clearly.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 23 / 27
No-no paradox
Statement A by Plato: What Socrates says in Statement B is false.
Statement B by Socrates: What Plato says in Statement A is false.
In classical logic, only possible scenarios are either A true and B false, or A
false and B true.
But A and B only discuss each other and are symmetrical! Should not the
solution be symmetrical as well?
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 24 / 27
Truth in common language
What if we interpret the word “true” in common language as provable,
and “false” in common language as unprovable?
A : ¬ B
B : ¬ A
The resulting solution is symmetrical! A and B are true, in terms of formal
logic, and ¬ A and ¬ B.
Seems a much more reasonable solution than what usual interpretations of
“true” in common language into formal logic imply.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 25 / 27
Truth in common language
In fact, in formal logic, because of Tarski’s undefinability/impossibility
theorem, there exists no suitable truth predicate that allows us to say:
A : T(S)
where T(S) can be translated as “S is true.” We can only say S or ¬S,
but that worked as far as we implicitly understand ¬S as “S is false” in
classical logic. Works most of time, though in few cases can be dangerous.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 26 / 27
Truth in common language
In the new logic (modal logic NL), there are cases when P and ¬P are
both false - thus difficult to say that ¬P represents P is false.
May benefit if we get to translate p in formal modal logic as “p is true”
in common language.
William Heartspring modal logic NL July 14, 2019 (ver 1.0) 27 / 27

More Related Content

PDF
Unit I discrete mathematics lecture notes
PDF
Gödel’s incompleteness theorems
PPTX
Logic, contrapositive, converse, Discrete Mathematics, conjunction, negation
PDF
Mathematical Logic
PDF
Conflicts between relevance-sensitive revision
PPT
Mathematical Logic - Part 1
PDF
Argumentation and Machine Learning: When the Whole is Greater than the Sum of...
PPTX
Propositional logic
Unit I discrete mathematics lecture notes
Gödel’s incompleteness theorems
Logic, contrapositive, converse, Discrete Mathematics, conjunction, negation
Mathematical Logic
Conflicts between relevance-sensitive revision
Mathematical Logic - Part 1
Argumentation and Machine Learning: When the Whole is Greater than the Sum of...
Propositional logic

What's hot (16)

PDF
Logic paper
PDF
congruence lattices of algebras
PDF
Contradiction
PDF
Completeness: From henkin's Proposition to Quantum Computer
PDF
Linear Logic via Logical Dependencies (Mirantao2018)
DOCX
PDF
Entrega2_MALGTN_DEFINITVA
PPTX
Conjunction And Disjunction
PDF
Mathematical Logic
PPT
Mathematical Logic Part 2
PPT
Analytic Trigonometry
PDF
ESSLLI2016 DTS Lecture Day 5-2: Proof-theoretic Turn
PPTX
The logic
PPT
Truth tables
PDF
Propositional logic
PPTX
Ai lecture 11(unit03)
Logic paper
congruence lattices of algebras
Contradiction
Completeness: From henkin's Proposition to Quantum Computer
Linear Logic via Logical Dependencies (Mirantao2018)
Entrega2_MALGTN_DEFINITVA
Conjunction And Disjunction
Mathematical Logic
Mathematical Logic Part 2
Analytic Trigonometry
ESSLLI2016 DTS Lecture Day 5-2: Proof-theoretic Turn
The logic
Truth tables
Propositional logic
Ai lecture 11(unit03)
Ad

Similar to Slides: Common language and modal logic NL (20)

PDF
S2 1
PPTX
True but Unprovable
PDF
PPT
Introduction to Logic Powerpoint presentation.ppt
PPT
Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكرو
PPT
Propositional Logic and Pridicate logic
PPT
Discrete mathematics Chapter1 presentation.ppt
PPT
LogicLogicLogicLogicLogicLogicLogicLogicLogic
PDF
Ai lecture 07(unit03)
PPT
good teaching skills and other beautiful things
PDF
Constructive Modalities
PDF
Math nha ae la math do nha, nen xem xet coi cung hay lam a
PDF
PPT
Best presentation about discrete structure
PPT
desmath(1).ppt
PPT
02-boolean.ppt
PPTX
DMS UNIT-1 ppt.pptx
PPTX
Chapter1p1
PPTX
Propositional logic in Discretes tructures.pptx
PPTX
d79c6256b9bdac53_20231124_093457Lp9AB.pptx
S2 1
True but Unprovable
Introduction to Logic Powerpoint presentation.ppt
Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكرو
Propositional Logic and Pridicate logic
Discrete mathematics Chapter1 presentation.ppt
LogicLogicLogicLogicLogicLogicLogicLogicLogic
Ai lecture 07(unit03)
good teaching skills and other beautiful things
Constructive Modalities
Math nha ae la math do nha, nen xem xet coi cung hay lam a
Best presentation about discrete structure
desmath(1).ppt
02-boolean.ppt
DMS UNIT-1 ppt.pptx
Chapter1p1
Propositional logic in Discretes tructures.pptx
d79c6256b9bdac53_20231124_093457Lp9AB.pptx
Ad

Recently uploaded (20)

DOCX
"Project Management: Ultimate Guide to Tools, Techniques, and Strategies (2025)"
PPTX
worship songs, in any order, compilation
PDF
Tunisia's Founding Father(s) Pitch-Deck 2022.pdf
PPTX
Relationship Management Presentation In Banking.pptx
PPTX
Lesson-7-Gas. -Exchange_074636.pptx
PDF
Swiggy’s Playbook: UX, Logistics & Monetization
PDF
Presentation1 [Autosaved].pdf diagnosiss
PPTX
water for all cao bang - a charity project
PPTX
Project and change Managment: short video sequences for IBA
PPT
The Effect of Human Resource Management Practice on Organizational Performanc...
PDF
oil_refinery_presentation_v1 sllfmfls.pdf
PPTX
fundraisepro pitch deck elegant and modern
PDF
Microsoft-365-Administrator-s-Guide_.pdf
PDF
Instagram's Product Secrets Unveiled with this PPT
PPTX
INTERNATIONAL LABOUR ORAGNISATION PPT ON SOCIAL SCIENCE
PPTX
Self management and self evaluation presentation
PPTX
Impressionism_PostImpressionism_Presentation.pptx
PDF
COLEAD A2F approach and Theory of Change
PPTX
FINAL TEST 3C_OCTAVIA RAMADHANI SANTOSO-1.pptx
PPTX
_ISO_Presentation_ISO 9001 and 45001.pptx
"Project Management: Ultimate Guide to Tools, Techniques, and Strategies (2025)"
worship songs, in any order, compilation
Tunisia's Founding Father(s) Pitch-Deck 2022.pdf
Relationship Management Presentation In Banking.pptx
Lesson-7-Gas. -Exchange_074636.pptx
Swiggy’s Playbook: UX, Logistics & Monetization
Presentation1 [Autosaved].pdf diagnosiss
water for all cao bang - a charity project
Project and change Managment: short video sequences for IBA
The Effect of Human Resource Management Practice on Organizational Performanc...
oil_refinery_presentation_v1 sllfmfls.pdf
fundraisepro pitch deck elegant and modern
Microsoft-365-Administrator-s-Guide_.pdf
Instagram's Product Secrets Unveiled with this PPT
INTERNATIONAL LABOUR ORAGNISATION PPT ON SOCIAL SCIENCE
Self management and self evaluation presentation
Impressionism_PostImpressionism_Presentation.pptx
COLEAD A2F approach and Theory of Change
FINAL TEST 3C_OCTAVIA RAMADHANI SANTOSO-1.pptx
_ISO_Presentation_ISO 9001 and 45001.pptx

Slides: Common language and modal logic NL

  • 1. Common language and modal logic NL William Heartspring Ver 1.0, July 14, 2019 William Heartspring modal logic NL July 14, 2019 (ver 1.0) 1 / 27
  • 2. Table of Contents 1 Common language and modal logic NL William Heartspring modal logic NL July 14, 2019 (ver 1.0) 2 / 27
  • 3. Table of Contents 1 Common language and modal logic NL William Heartspring modal logic NL July 14, 2019 (ver 1.0) 3 / 27
  • 4. Common language in formal logic Ideally, logic should be able to analyze both common language and mathematics. But: common language too much filled with paradoxes. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 4 / 27
  • 5. Goal of Modal logic NL Formal logic that resolves those paradoxes but still allow sentences in common language to be analyzed. That is, unrestricted comprehension (recursion) allowed. But we also would like to preserve powerfulness of classical logic. Example: proof by contradiction. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 5 / 27
  • 6. Strategy If a sentence implies a contradictory or paradoxical sentence (a paradoxical sentence can be considered a generalized curry sentence), then the sentence must be considered false. But in classical logic, the definition of a paradox can be said as: “Sentence φ causes a contradiction. Thus, φ is false, so ¬φ must be true. But ¬φ also causes a contradiction. So what should I do?” Classical logic disallows both φ and ¬φ to be false. Even for Curry’s paradox, the paradox is intensified in classical logic because both φ and ¬φ cannot be false. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 6 / 27
  • 7. Digression: Curry’s paradox in classical logic A ↔ (A → B) That is, sentence A says A → B. This sentence allowed by unrestricted comprehension. Classical logic says A → B and A must both be true. Miracle: we can prove any sentence by forming an appropriate curry sentence for each target sentence (B). But a curry sentence exists for ¬B as well, so paradoxical. Initially, it may seem that there is nothing wrong with assigning false to A and (A → B), so a paradox may be resolved. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 7 / 27
  • 8. Digression: Curry’s paradox in classical logic But classical logic condemns us to assign truth value true to A and (A → B). Thus not a problem of deduction or proof in classical logic. This is an internal problem in classical logic. A → (A → B) means ¬A ∨ (A → B). Take ¬A as true. (A → B) → A means ¬(A → B) ∨ A. Take ¬(A → B), since ¬A was assumed. ¬(A → B) means A ∧ ¬B. But cannot be , since ¬A was assumed! William Heartspring modal logic NL July 14, 2019 (ver 1.0) 8 / 27
  • 9. Implication problem The main problem is that A → B is assumed to be ¬A ∨ B. Why does this hold in classical logic? If A is false, then B is proved by the principle of explosion. Thus ¬A to take care of this circumstance. If A is true, then B must be true if A → B is true. That is B. Thus, ¬A ∨ B. Thus, if falsity of A does not imply ¬A, then we short-circuit the issue in Curry’s paradox. Of course this is just one consideration out of many required, but still a good guiding light. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 9 / 27
  • 10. Strategy Let’s continue the strategy. Refine the law of excluded middle. However, we would like to use the law of excluded middle for non-paradoxical situations from classical logic perspectives. Requires a definition of a paradox. p is considered a paradoxical sentence only when p ↔ ¬p holds. If this does not hold and p leads to a paradoxical sentence: only implies that p is false. Would like to assign false to both p and ¬p only when p is a paradoxical sentence. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 10 / 27
  • 11. Strategy: refining the law of excluded middle The easiest na¨ıve thought would be to add: ¬(p ↔ ¬p) → (p ∨ ¬p) But: Some model of a theory in this new logic would exist that affirms p being a paradoxical sentence, even if there exists no proof that p is a paradoxical sentence. In other words, a classical logic proof that relied on the law of excluded middle would not work in the new logic! William Heartspring modal logic NL July 14, 2019 (ver 1.0) 11 / 27
  • 12. Strategy: refining the law of excluded middle Thus, modal logic is required. More precisely, provability logic that borrows modal logic. ¬ (p ↔ ¬p) → (p ∨ ¬p) That is, if a paradox is unprovable, then the law of excluded middle must hold. ( means provable) But comes the heavy cost which the new logic accepts: completeness no longer holds. If p holds in all models, p still may not be provable (in finite deduction steps). William Heartspring modal logic NL July 14, 2019 (ver 1.0) 12 / 27
  • 13. Strategy: “Post-proofs” But if p holds in all models, we still would like to have some concept of proof of p. That is, if we assume unprovability of a paradox, then p is provable by invoking the law of excluded middle. This unprovability of a paradox may never be proved, but then this “post-proof” effectively justifies the use of classical logic proofs based on the law of excluded middle. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 13 / 27
  • 14. Back to Curry’s paraodx A ↔ (A → B) Proving A and A → B requires the rule: (p → (q → r)) → ((p → q) → (p → r)) In classical logic, not possible to render A as false. Now we can, as far as a rule exists that overwrites the above rule: (p → (p → r)) → ((p → ⊥) ∧ ((p → r) → ⊥)) ⊥ represents a contradiction. The above rule is called Then-4. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 14 / 27
  • 15. Back to Curry’s paradox What about a generalized Curry’s paradox? Still such a generalized curry sentence implies some form of a curry sentence or a contradictory sentence. As far as a rule exists that render any sentence leading to a contradiction false, no problem. p ↔ q can be unprovable, even if true. Matters because if p ↔ q holds, then effectively p → (q → r) is p → (p → r). Here, no reason to impose provability or unprovabiliy requirements. Better to leave things this way for a good match with classical logic. In classical logic, there may exist a model with p ↔ q, when a model with ¬(p ↔ q) exists. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 15 / 27
  • 16. Strategy Strategy-wise, we are done. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 16 / 27
  • 17. NL axioms excluding NL3 NL1: (p → ⊥) → ¬ p NL2: ¬ p ∨ p. NL4: p ∨ q → (p ∨ q) NL5: p ∧ q ↔ (p ∧ q) These axioms just confirm the usual point about provability logic. So why are they there? Because if not thought in terms of provability, one may not get to confirm how ¬ is related to , or how ¬ is supposed to behave for modal operators from purely modal logic perspectives. Also, one may not get to confirm how behaves in response to ∨ or ∧ inside the scope of the modal operator. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 17 / 27
  • 18. NL axioms: NL4 ( p ∨ ¬ p) does not mean p ∨ ¬ p William Heartspring modal logic NL July 14, 2019 (ver 1.0) 18 / 27
  • 19. NL axioms: NL3 (¬ p → ⊥) → p ( p → ⊥) → ¬ p This is the real important new axiom. Allows us to prove unprovability or provability of modal sentences by assuming them first, and then deriving a contradiction. This heavily critical, given that unprovability of a paradox may be unprovable. Gives us an important proof power. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 19 / 27
  • 20. Non-tautologous new axioms New non-tautologous axioms: Refined Not-1’, Then-4 and NL-3. Features of classical logic that do not survive: The law of excluded middle. But refined in Refined Not-1’ axiom, so in non-paradoxical circumstances from perspectives of classical logic, modal logic NL (new logic) equivalent to classical logic. (p → q) no longer ¬p ∨ q. Falsity of p does not mean ¬p, thus ¬p ∨ q no longer always valid. But valid if the law of excluded middle is allowed by the new logic. Law of noncontradiction is not to be expressed as ¬(p ∧ ¬p). Since De Morgan’s law holds, that form means the law of excluded middle. It is to be expressed as (p ∧ ¬p) → ⊥. Proof by contradiction is allowed but it is considered a post-proof, having assumed unprovability of a paradoxical situation. Thus, technically these post-proofs requiring infinite deduction steps. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 20 / 27
  • 21. Common features shared between NL and classical logic De Morgan’s laws Principle of explosion Double negation elimination and introduction Propositional logic axioms and rules, except for Then-4 overwriting some axioms/rules. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 21 / 27
  • 22. Common language in formal logic Now that modal logic NL is constructed: What is common language from perspectives of formal logic? Roughly, one may define as featuring unrestricted comprehension (or recursion). Example: Liar paradox “this sentence is false” φ ↔ ¬φ Curry’s paradox: “truth of this sentence implies B is true.” A ↔ (A → B) Barber paradox: Barber Alice shaves all of those and only those who do not shave themselves. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 22 / 27
  • 23. More paradoxes of common language Liar/Barber/Russell’s/knower/Fitch’s paradox: basically they are of the same type. Type 1 paradox. Basically of the form p ↔ ¬p. Curry’s paradox: Type 2 paradox. Of the form p ↔ (p → r). Generalized Curry’s paradox allows to consider both type 1 paradox and type 2 paradox in the same footing - but we do not need to discuss this here, as we have covered both types of paradoxes. No-no paradox. In a way, not a paradox, but an asymmetry problem. Somewhat boring examples, but then reality filled with unrestricted recursion of language demonstrated above. Can we simply ban unrestricted comprehension/recursion in language? No. Examples in lambda calculus demonstrate this clearly. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 23 / 27
  • 24. No-no paradox Statement A by Plato: What Socrates says in Statement B is false. Statement B by Socrates: What Plato says in Statement A is false. In classical logic, only possible scenarios are either A true and B false, or A false and B true. But A and B only discuss each other and are symmetrical! Should not the solution be symmetrical as well? William Heartspring modal logic NL July 14, 2019 (ver 1.0) 24 / 27
  • 25. Truth in common language What if we interpret the word “true” in common language as provable, and “false” in common language as unprovable? A : ¬ B B : ¬ A The resulting solution is symmetrical! A and B are true, in terms of formal logic, and ¬ A and ¬ B. Seems a much more reasonable solution than what usual interpretations of “true” in common language into formal logic imply. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 25 / 27
  • 26. Truth in common language In fact, in formal logic, because of Tarski’s undefinability/impossibility theorem, there exists no suitable truth predicate that allows us to say: A : T(S) where T(S) can be translated as “S is true.” We can only say S or ¬S, but that worked as far as we implicitly understand ¬S as “S is false” in classical logic. Works most of time, though in few cases can be dangerous. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 26 / 27
  • 27. Truth in common language In the new logic (modal logic NL), there are cases when P and ¬P are both false - thus difficult to say that ¬P represents P is false. May benefit if we get to translate p in formal modal logic as “p is true” in common language. William Heartspring modal logic NL July 14, 2019 (ver 1.0) 27 / 27