Modal logic #1146

spcfox · 2024-05-31T20:47:37Z

Module containing the formalization of normal modal logics. It includes:

Syntax and Kripke semantics of modal logics
Definition of modal logics K and S5
Constructive proofs of soundness and consistency
Classical proofs of completeness and finite approximability

The theorems proved are easily generalized to other normal modal logics.

The code is currently being refactored.

- Add index for variables in formulas - Add soundness for decidable valuation

- Add priorities

- Add me to contributors

EgbertRijke

This is a very nice pull request!

I am very impressed with this pull request. It covers a very substantial amount of modal logic, and many nontrivial theorems. The formalization is well thought-out and most of the coding style is very good. So all in all, this seems to be excellent work!

Since it is a larger pull request, I will review it in parts. I've looked over a couple of files and left a few comments on those. I will continue reviewing it over the coming days.

EgbertRijke · 2024-06-01T07:14:57Z

src/foundation-core/decidable-propositions.lagda.md

+equiv-empty-is-decidable-prop :
+  {l : Level} {A : UU l} → is-decidable-prop A → ¬ A → A ≃ empty
+equiv-empty-is-decidable-prop {l} {A} (is-p , _) contra =
+  equiv-iff (A , is-p) empty-Prop contra ex-falso


This already in the library as equiv-is-empty. Note that any map into the empty type (or indeed into any empty type) is an equivalence.

EgbertRijke · 2024-06-01T07:25:08Z

src/foundation-core/decidable-propositions.lagda.md

+equiv-unit-is-decidable-prop :
+  {l : Level} {A : UU l} → is-decidable-prop A → A → A ≃ unit
+equiv-unit-is-decidable-prop {l} {A} (is-p , _) a =
+  equiv-iff (A , is-p) unit-Prop (λ _ → star) (λ _ → a)


A few pointers:

Any proposition equipped with an element is contractible. This is shown here: is-proof-irrelevant-is-prop. The notion of proof irrelevance means A -> is-contr A in agda-unimath.

Any contractible type is equivalent to the unit type: equiv-unit-is-contr.

EgbertRijke · 2024-06-01T07:26:28Z

src/foundation-core/decidable-propositions.lagda.md

+is-small-prop-is-decidable-prop :
+  {l1 : Level} (l2 : Level) (A : UU l1) → is-decidable-prop A → is-small l2 A
+is-small-prop-is-decidable-prop l2 A H
+  with equiv-empty-or-unit-is-decidable-prop H
+... | inl e = is-small-equiv unit e (raise-unit l2 , compute-raise-unit l2)
+... | inr e = is-small-equiv empty e (raise-empty l2 , compute-raise-empty l2)


EgbertRijke · 2024-06-01T07:39:44Z

src/foundation-core/injective-maps.lagda.md

+  injection-comp :
+    injection A B → injection B C → injection A C
+  injection-comp (f , is-inj-f) (g , is-inj-g) =
+    g ∘ f , is-injective-comp is-inj-f is-inj-g


Would you mind renaming this to comp-injection? This would fit better with the naming scheme we currently have, which goes comp-pointed-map, comp-hom-Group, and so on.

Note that I just found an exception to this naming scheme, which is that we use equiv-comp. That's probably a mistake on our part, which we missed because we usually use the infix notation for composition of equivalences.

EgbertRijke · 2024-06-01T07:50:45Z

src/foundation/binary-relations-transitive-closures.lagda.md

+# Transitive Closures
+
+```agda
+module foundation.binary-relations-transitive-closures where


Would you mind renaming this file to transitive-closures-binary-relations?

We usually name the concept of interest first, and then disambiguate later. For instance, we have several files about the action on identifications files all named action-on-identifications-disambiguation.

EgbertRijke · 2024-06-01T09:49:17Z

src/modal-logic/formulas.lagda.md

+The formula of modal logic is defined inductively as follows:
+
+- A variable is a formula.
+- ⊥ is a formula.
+- If `a` and `b` are formulas, then `a → b` is a formula.
+- If `a` is a formula, then □`a` is a formula.


Suggested change

The formula of modal logic is defined inductively as follows:

- A variable is a formula.

- ⊥ is a formula.

- If `a` and `b` are formulas, then `a → b` is a formula.

- If `a` is a formula, then □`a` is a formula.

The type of {{#concept "formulas of modal logic" Agda=modal-formula}} is defined inductively as follows:

- A variable is a formula of modal logic.

- `⊥` is a formula of modal logic.

- If `a` and `b` are formulas of modal logic, then `a → b` is a formula of modal logic.

- If `a` is a formula of modal logic, then `□ a` is a formula of modal logic.

EgbertRijke · 2024-06-01T09:49:42Z

src/modal-logic/formulas.lagda.md

+
+## Definition
+
+### Inductive definition of formulas


Suggested change

### Inductive definition of formulas

### Inductive definition of formulas of modal logic

EgbertRijke · 2024-06-01T09:51:17Z

src/modal-logic/formulas.lagda.md

@@ -0,0 +1,200 @@
+# Modal logic formulas


Suggested change

# Modal logic formulas

# Formulas of modal logic

I would make "Formulas" the first word, so that it doesn't confuse people that something starting with the "M" is listed among the files starting with "F" in an otherwise alphabetically ordered list.

EgbertRijke · 2024-06-01T09:52:48Z

src/modal-logic/formulas.lagda.md

+### If formulas are equal, then their subformulas are equal
+
+```agda
+  eq-implication-left : {a b c d : modal-formula i} → a →ₘ b ＝ c →ₘ d → a ＝ c
+  eq-implication-left refl = refl
+
+  eq-implication-right : {a b c d : modal-formula i} → a →ₘ b ＝ c →ₘ d → b ＝ d
+  eq-implication-right refl = refl
+
+  eq-box : {a b : modal-formula i} → □ₘ a ＝ □ₘ b → a ＝ b
+  eq-box refl = refl
+
+  eq-diamond : {a b : modal-formula i} → ◇ₘ a ＝ ◇ₘ b → a ＝ b
+  eq-diamond refl = refl
+```


I would place this under ## Properties

EgbertRijke · 2024-06-01T09:53:29Z

src/modal-logic/formulas.lagda.md

+  is-prop-Eq-modal-formula (□ₘ a) (□ₘ c) = is-prop-Eq-modal-formula a c
+```
+
+### A formula is a set


Suggested change

### A formula is a set

### The type of formulas of modal logic is a set

EgbertRijke

Here is another batch of comments :)

EgbertRijke · 2024-06-01T11:13:10Z

src/modal-logic/deduction.lagda.md

@@ -0,0 +1,338 @@
+# Modal logic deduction


Suggested change

# Modal logic deduction

# Deductions in modal logic

I would make the first word of this title "Deductions" so that it starts with the D, because the file titles will be listed by the file names in alphabetic ordering.

EgbertRijke · 2024-06-01T11:24:54Z

src/modal-logic/deduction.lagda.md

+### Modal logic predicate
+


A sentence explaining this definition informally would be nice. For instance:

Suggested change

### Modal logic predicate

### The predicate of being a modal logic

A {{#concept "modal logic" Agda=is-modal-logic}} is a theory of modal logic that contains its own deductive closure.

EgbertRijke · 2024-06-01T11:33:38Z

src/modal-logic/deduction.lagda.md

+### Closure of a theory under the deduction system
+
+```agda
+  modal-logic-closure : modal-theory l2 i → modal-theory (l1 ⊔ l2) i
+  modal-logic-closure axioms a = trunc-Prop (axioms ⊢ₘ a)
+```


We try to avoid modules that cross over sections, just because it is harder for the reader to keep track of what is assumed and sections seem a natural breaking point. This approach requires the arguments to be given a few more times, but we find it improves readability.

It also improves maintainability to break modules more often, because it is easier to move code around if it's not in a module that spans a large portion of the file.

Suggested change

### Closure of a theory under the deduction system

```agda

modal-logic-closure : modal-theory l2 i → modal-theory (l1 ⊔ l2) i

modal-logic-closure axioms a = trunc-Prop (axioms ⊢ₘ a)

```

### The deductive closure of a theory

A modal formula is in the deductive closure of a theory if there [exists](foundation.existence.md) a deduction of the formula from the theory.

```agda

module _

{l1 l2 : Level} {i : Set l1}

where

modal-logic-closure : modal-theory l2 i → modal-theory (l1 ⊔ l2) i

modal-logic-closure axioms a = trunc-Prop (axioms ⊢ₘ a)

```

I would also suggest:

Renaming modal-logic-closure to deductive-closure-modal-theory

Adding an entry is-in-deductive-closure-modal-theory instead of using is-in-subtype (modal-logic-closure theory) later on.

EgbertRijke · 2024-06-01T11:40:26Z

src/modal-logic/deduction.lagda.md

+### TODO: Title
+
+```agda
+  is-in-modal-logic-closure-deduction :


If my previous renaming suggestions are accepted, then this one could be named is-in-deductive-closure-is-in-modal-theory.

EgbertRijke · 2024-06-01T11:44:54Z

src/modal-logic/deduction.lagda.md

+  is-consistent-modal-logic-Prop : modal-theory l2 i → Prop l2
+  is-consistent-modal-logic-Prop = neg-Prop ∘ is-contradictory-modal-logic-Prop
+
+  is-consistent-modal-logic : modal-theory l2 i → UU l2
+  is-consistent-modal-logic = type-Prop ∘ is-consistent-modal-logic-Prop


It would make sense if the consistent and inconsistent modal theories each get their own sections ### Inconsistent modal theories and ### Consistent modal theories.

EgbertRijke · 2024-06-01T11:46:15Z

src/modal-logic/deduction.lagda.md

+  is-modal-logic = type-Prop ∘ is-modal-logic-Prop
+```
+
+### TODO: Title


Suggested change

### TODO: Title

### Any modal theory is contained in its own deductive closure

EgbertRijke · 2024-06-01T11:51:28Z

src/modal-logic/deduction.lagda.md

+
+  is-contradictory-modal-logic-Prop : modal-theory l2 i → Prop l2
+  is-contradictory-modal-logic-Prop logic = logic ⊥ₘ
+
+  is-contradictory-modal-logic : modal-theory l2 i → UU l2
+  is-contradictory-modal-logic = type-Prop ∘ is-contradictory-modal-logic-Prop


I would name this is-inconsistent-modal-theory-Prop, and the next one is-inconsistent-modal-theory.

The point is that it is a predicate on modal theories, which isn't currently reflected in the name. I personally find inconsistent reading a bit better than contradictory, and it is also more consistent with your use of consistent of the name for consistent modal theories.

I would also put this in a new section.

One question I have about your definition of contradictory modal logics is that currently it says that a theory is contradictory if it contains falsehood. However, theories are not deductively closed, so wouldn't it be more natural to say that modal theories are inconsistent if their deductive closure contains falsehood?

Suggested change

is-contradictory-modal-logic-Prop : modal-theory l2 i → Prop l2

is-contradictory-modal-logic-Prop logic = logic ⊥ₘ

is-contradictory-modal-logic : modal-theory l2 i → UU l2

is-contradictory-modal-logic = type-Prop ∘ is-contradictory-modal-logic-Prop

```

### Inconsistent modal theories

```agda

is-inconsistent-modal-theory-Prop : modal-theory l2 i → Prop l2

is-inconsistent-modal-theory-Prop logic = logic ⊥ₘ

is-inconsistent-modal-theory : modal-theory l2 i → UU l2

is-inconsistent-modal-theory = type-Prop ∘ is-contradictory-modal-logic-Prop

However, theories are not deductively closed, so wouldn't it be more natural to say that modal theories are inconsistent if their deductive closure contains falsehood?

On the one hand, this would be correct from a definitional point of view. But on the other hand, in proofs this definition is already used in relation to closures, therefore it is convenient to use it in this form. I'll think about how best to determine

fredrik-bakke · 2024-06-01T12:38:11Z

Hi Viktor and welcome to agda-unimath! This is very impressive stuff, thanks for the big contribution!

fredrik-bakke · 2024-06-01T12:46:48Z

src/foundation/propositional-resizing.lagda.md

+
+```agda
+module _
+  {l1 l2 : Level} ((Ω , prop-resize) : propositional-resizing l1 l2)


I don't wish to dogpile on you, but I noticed this one while browsing through the code and thought you might not know about it. Agda implements pattern-matching in module declarations and type declarations via let-bindings. These don't interact well with Agda's interactive mode, leading to unreadable goal expressions down the line. Therefore we advise against using pattern matching in places such as this one. The principled approach is to define named projection maps for propositional resizing and then use those instead.

Oh, ok, I will fix this. I agree, It's really hard to read in interactive mode

EgbertRijke

Here is another batch of comments :)

EgbertRijke · 2024-06-01T15:53:54Z

src/foundation/intersections-subtypes.lagda.md

+  {l1 l2 l3 : Level} {X : UU l1} (P : subtype l2 X) (Q : subtype l3 X)
+  where
+
+  subtype-intersection-left : P ∩ Q ⊆ P


A better name for this will be left-inclusion-intersection-subtype. Note that we build our names exactly the other way around: We first state the name of the operation left-inclusion and then disambiguate it by intersection-subtype.

EgbertRijke · 2024-06-01T15:54:18Z

src/foundation/intersections-subtypes.lagda.md

+  subtype-intersection-left : P ∩ Q ⊆ P
+  subtype-intersection-left _ = pr1
+
+  subtype-intersection-right : P ∩ Q ⊆ Q


A better name for this will be right-inclusion-intersection-subtype

EgbertRijke · 2024-06-01T15:57:21Z

src/foundation/intersections-subtypes.lagda.md

+  subtype-intersection-right : P ∩ Q ⊆ Q
+  subtype-intersection-right _ = pr2
+
+  subtype-both-intersection :


Maybe a better name would be inclusion-intersection-inclusion-subtype, which would read as the inclusion in the intersection obtained from the inclusion(s) into the subtypes.

Perhaps there are even better suggestions.

EgbertRijke · 2024-06-01T15:59:47Z

src/foundation/intersections-subtypes.lagda.md

+  {l1 : Level} {X : UU l1}
+  where
+
+  is-reflexivity-intersection :


This would say that intersection is an idempotent operation, so perhaps is-idempotent-intersection-subtype would be a more descriptive name.

See also https://en.wikipedia.org/wiki/Idempotence

EgbertRijke · 2024-06-01T16:01:10Z

src/foundation/intersections-subtypes.lagda.md

+        ( refl-leq-subtype P)
+        ( refl-leq-subtype P))
+
+  is-commutative-subtype-intersection :


I would call this inclusion-is-commutative-intersection-subtype

EgbertRijke · 2024-06-01T19:54:03Z

src/lists/concatenation-lists.lagda.md

+in-concat-left _ l2 (is-in-tail a x l1 in-l1) =
+  is-in-tail a x (concat-list l1 l2) (in-concat-left l1 l2 in-l1)
+
+in-concat-right :


I would call this in-concat-list-in-right-list.

EgbertRijke · 2024-06-01T19:55:36Z

src/lists/concatenation-lists.lagda.md

+in-concat-right (cons x l1) l2 in-l2 =
+  is-in-tail _ x (concat-list l1 l2) (in-concat-right l1 l2 in-l2)
+
+in-concat-list :


This one is more tricky to name, but in-concat-list reads too much like the predicate "to be in a concatenated list". Perhaps we could consider calling it cases-in-concat-list.

EgbertRijke · 2024-06-01T20:01:51Z

src/lists/functoriality-lists.lagda.md

@@ -257,3 +257,27 @@ map-snoc-list :
 map-snoc-list f nil a = refl
 map-snoc-list f (cons b x) a = ap (cons (f b)) (map-snoc-list f x a)
 ```
+
+### TODO: maybe another file


Suggested change

### TODO: maybe another file

### Lists obtained from maps of elements of lists

Consider a list `l : list A` of elements of `A`, and consider a function `f : (a : A) → a ∈ l → B`, which assigns to each element of `l` a value in `B`. Then we obtain a list of elements of `B` whose elements are given by `f`.

This operation is a generalization of `map-list`, because we only need to assign values in `B` to the elements of the list.

I think this entry is fine in this file.

EgbertRijke · 2024-06-01T20:03:24Z

src/lists/functoriality-lists.lagda.md

+### TODO: maybe another file
+
+```agda
+dependent-map-list :


Given what this function does, perhaps map-element-list is more descriptive. Feel free to disregard this suggestion, if you think dependent-map-list is better.

EgbertRijke · 2024-06-01T20:13:26Z

src/lists/lists-subtypes.lagda.md

+
+## Idea
+
+TODO


Perhaps the "image" of a list would be a good name for this concept. What you describe here is equivalent to the image of the projection map

(Sigma (a : A), a ∈ l) → A

where the first time is the type of all elements of l. If you agree with this suggestion, we could change the file name to images-lists.lagda.md and the title to # Images of lists.

For the ## Idea section, I have the following suggestion:

Suggested change

TODO

For any [list](lists.lists.md) `l : list A` of elements of `A`, the {{#concept "image" Disambiguation="lists"}} of `l` is the least [subtype](foundation.subtypes.md) of `A` that contains all elements of `l`. In other words, it is the subtype of all elements of `A` that are merely elements of `l`, in the sense that the [proposition](foundation-core.propositions.md)

```text

║a ∈ l║

```

holds, where we applied the [propositional truncation](foundation.propositional-truncations.md) to the elementhood relation of lists.

Changing the concept name to "image of lists" would also trigger some name changes in your entries below. The basic name would be image-list. Let me know if you want me to go through all of them and make suggestions, or whether you like to come up with names yourself.

EgbertRijke

Here is another couple of comments.

I find myself making suggestions that are dependent on previous suggestions, so perhaps it is best for me to let you work on my current suggestions first, before continuing the review.

Your work is definitely very good, and it would be an amazing addition to the library. The suggestions I made are mostly about what names to choose for certain entries and how to integrate it with the rest of the library better, but the quality of your work is without doubt excellent.

EgbertRijke · 2024-06-02T08:54:04Z

src/modal-logic/deduction.lagda.md

+The deduction system of modal logic is defined inductively as follows:
+
+a ∈ T ------ (AX) T ⊢ a
+
+T ⊢ a → b T ⊢ a ------------------ (MP) T ⊢ b
+
+T ⊢ a ------ (NEC) T ⊢ □a
+
+where `T` is a set of formulas and `a`, `b` are formulas.
+
+Modal logic generated by a set of axioms is the smallest set of formulas that
+contains the axioms and is closed under the deduction system.


Suggested change

The deduction system of modal logic is defined inductively as follows:

a ∈ T ------ (AX) T ⊢ a

T ⊢ a → b T ⊢ a ------------------ (MP) T ⊢ b

T ⊢ a ------ (NEC) T ⊢ □a

where `T` is a set of formulas and `a`, `b` are formulas.

Modal logic generated by a set of axioms is the smallest set of formulas that

contains the axioms and is closed under the deduction system.

A {{#concept "theory" Disambiguation="modal logic" Agda=modal-theory}} of modal logic is a [subtype](foundation.subtypes.md) `T : modal-formula i → Prop` of the type of [formulas](modal-logic.formulas.md).

Given a theory, a {{#concept "deduction" Disambiguation="modal logic" Agda=_⊢ₘ_}} an element of the type `_⊢ₘ_` inductively generated by:

* Axioms: formulas in the theory.

* Modus ponens: Given deductions `T ⊢ₘ a →ₘ b` and `T ⊢ₘ a` hold, we obtain a deduction `T ⊢ₘ b`.

* Necessity: Given a deduction `T ⊢ₘ a` we obtain a deduction `T ⊢ₘ □a`.

In other words, deductions can be formed by combining the following inference rules:

```text

a ∈ T

----------- (AX)

T ⊢ a

T ⊢ a → b T ⊢ a

----------------------- (MP)

T ⊢ b

T ⊢ a

----------- (NEC)

T ⊢ □a

```

The {{#concept "modal logic"}} generated by a set of axioms is the smallest set of formulas that

contains the axioms and is closed under the deduction system, i.e., the modal logic generated by a set of formulas is the deductive closure of that set of formulas.

EgbertRijke · 2024-06-02T08:54:34Z

src/modal-logic/deduction.lagda.md

+Modal logic generated by a set of axioms is the smallest set of formulas that
+contains the axioms and is closed under the deduction system.
+
+## Definition


Suggested change

## Definition

## Definitions

EgbertRijke · 2024-06-02T08:54:50Z

src/modal-logic/deduction.lagda.md

+
+## Definition
+
+### Theory of modal formulas


Suggested change

### Theory of modal formulas

### Theories of modal formulas

EgbertRijke · 2024-06-02T08:55:18Z

src/modal-logic/deduction.lagda.md

+  modal-theory = subtype l2 (modal-formula i)
+```
+
+### Deduction system


Suggested change

### Deduction system

### Deductions of modal logic

EgbertRijke · 2024-06-02T08:55:55Z

src/modal-logic/deduction.lagda.md

+    modal-nec : {a : modal-formula i} → axioms ⊢ₘ a → axioms ⊢ₘ □ₘ a
+```
+
+### Closure of a theory under the deduction system


Suggested change

### Closure of a theory under the deduction system

### The deductive closure of a theory of modal logic

EgbertRijke · 2024-06-02T09:27:45Z

src/modal-logic/axioms.lagda.md

+  ax-1-parameter :
+    (h : modal-formula i → modal-formula i) → is-injective h → modal-theory l i
+  pr1 (ax-1-parameter h inj f) = Σ (modal-formula i) (λ a → f ＝ h a)
+  pr2 (ax-1-parameter h inj f) (a , refl) =
+    is-prop-is-contr
+      ( is-contr-Σ-is-prop a refl
+        ( λ b → is-set-modal-formula (h a) (h b))
+        ( λ x → inj))
+      ( a , refl)


This operation takes an injective map and returns a modal theory. Perhaps an informal sentence explaining this would be helpful. A few things to note: For this construction to work, h could be any injection X → modal-formula i where X is an arbitrary type. This observation leads me to think that a more descriptive name for this construction would be modal-theory-injection-modal-formula, to say that it is the modal theory obtained from an injection into the type of modal formulas.

One thing to note is that we have a lot more infrastructure for embeddings than for injective maps. A map between sets is injective if and only if it is an embedding, but because the concept of embeddings is better behaved for arbitrary types (types that are not sets), we generally prefer to work with embeddings.

For instance, the fact that any embedding induces a subtype is in the library as fiber-emb-Prop. Admittedly, this name is rather cryptic and could be improved on our part (a better name would be subtype-emb).

The fact that injective maps between sets are embeddings is is-emb-is-injective, and the fact that injective maps between sets are propositional maps is is-prop-map-is-injective.

Your construction here is essentially combining those constructions without referring to them, so it would be better to use them explicitly.

I wouldn't ask you to change your current approach, which is absolutely fine, but it could be even better if you tried using embeddings more. One benefit of using embeddings is that a potential future generalization to modal theories with an arbitrary type of variables could be easier. We could be less reliant on the fact that modal theories are sets.

One further minor difference between the fibers of an embedding and your type Σ (modal-formula i) (λ a → f ＝ h a), is that your type is fiber' h f, where fiber' can be found here. In general, it is good to use existing definitions, so that you can also use the existing infrastructure for those definitions. In this case I would also suggest that using fiber would be better than using fiber', because the library has a lot more stuff about the main definition of fibers.

EgbertRijke · 2024-06-02T09:47:52Z

src/modal-logic/axioms.lagda.md

+  ax-2-parameters :
+    (h : modal-formula i → modal-formula i → modal-formula i) →
+    ({x x' y y' : modal-formula i} → h x y ＝ h x' y' → x ＝ x') →
+    ({x x' y y' : modal-formula i} → h x y ＝ h x' y' → y ＝ y') →
+    modal-theory l i
+  pr1 (ax-2-parameters h inj-1 inj-2 f) =
+    Σ (modal-formula i) (λ a → Σ (modal-formula i) (λ b → f ＝ h a b))
+  pr2 (ax-2-parameters h inj-1 inj-2 f) (a , b , refl) =
+    is-prop-is-contr
+      ( is-contr-Σ-is-prop a (b , refl)
+        ( λ x → is-prop-type-Prop (ax-1-parameter (h x) inj-2 (h a b)))
+        ( λ x (y , e) → inj-1 e))
+      ( a , b , refl)


Given the previous discussion, perhaps modal-theory-injection-pair-modal-formula could be more descriptive? It would describe the modal theory obtained from the injection of pairs of modal formulas.

EgbertRijke · 2024-06-02T09:48:35Z

src/modal-logic/axioms.lagda.md

+  ax-3-parameters :
+    (h :
+      modal-formula i → modal-formula i → modal-formula i → modal-formula i) →
+    ({x x' y y' z z' : modal-formula i} → h x y z ＝ h x' y' z' → x ＝ x') →
+    ({x x' y y' z z' : modal-formula i} → h x y z ＝ h x' y' z' → y ＝ y') →
+    ({x x' y y' z z' : modal-formula i} → h x y z ＝ h x' y' z' → z ＝ z') →
+    modal-theory l i
+  pr1 (ax-3-parameters h inj-1 inj-2 inj-3 f) =
+    Σ ( modal-formula i)
+      ( λ a →
+        ( Σ (modal-formula i)
+          ( λ b → Σ (modal-formula i) ( λ c → f ＝ h a b c))))
+  pr2 (ax-3-parameters h inj-1 inj-2 inj-3 f) (a , b , c , refl) =
+    is-prop-is-contr
+      ( is-contr-Σ-is-prop a (b , c , refl)
+        ( λ x → is-prop-type-Prop (ax-2-parameters (h x) inj-2 inj-3 (h a b c)))
+        ( λ x (y , z , e) → inj-1 e))
+      ( a , b , c , refl)


Similarly here: perhaps a more descriptive name would be modal-theory-injection-triple-modal-formula.

src/order-theory/maximal-elements-posets.lagda.md

Subset of #1146 #1170 --------- Co-authored-by: spcfox Co-authored-by: Egbert Rijke <e.m.rijke@gmail.com>

spcfox and others added 30 commits May 31, 2024 20:37

first syntax and semantics draft

8a92a52

add soundness

63b0e15

- Change Hilbert system

5f89c97

- Add index for variables in formulas - Add soundness for decidable valuation

Kripke fram should be not empty

49f2cf6

Refactor soundness

517fc91

Add finite model

6053173

simplify finite-is-classical proof

d1744d9

- Rewrite valuate to Propositions

1f8f53f

- Add priorities

Small fixes, and refactor kripke-frame

661adc6

Refactor modal logic

3b3ba68

- Refactor

957e112

- Add me to contributors

Change operators levels and associativity

341456a

Refactor universe levels

9db88ca

Small fixes

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Refactor

0d22369

Add logics as sets of formulas

21d74ce

Remove old K definition

8ecf60e

Fix index file

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Add K soundness

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Refactor kripke-models

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Add finite models

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Experiments with axioms definitions

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Add completeness definition

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Refactor axioms

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Add uncompleted proof of completeness K

e17134d

Run pre-commit

c1ae1ff

Proof canonical model theorem

b358d85

fix pre-commit

9e3bb01

Prove the second point of canonical model theorem

fd1cc31

Prove completeness K

39bccd5

spcfox added 15 commits May 31, 2024 20:38

Refactor subtypes

873449a

Refactor formulas and module names

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Refactor var

662bc0f

Fix filtration lemma and deduction theorem

29eed33

Fix filtratiom lemma file name

b2d85c4

Add recurser for equivalence classes and refactor

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Prove inductor for equivalence classes

b978ed1

Replace zorn lemma in canonical model theorem

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Refactor kripke semantics

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Fix filtration definition

0df9fd4

Complete eliminators for equivalence class

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Refactor finite approximability

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Refactor

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Refactor filtrations

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Fix import in finite-approximability

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fredrik-bakke mentioned this pull request Oct 19, 2024

Order theory from @spcfox's modal logic #1205

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Order theory from @spcfox's modal logic (#1205)

7403e40

Subset of #1146 #1170 --------- Co-authored-by: spcfox Co-authored-by: Egbert Rijke <e.m.rijke@gmail.com>

	### Inductive definition of formulas
	### Inductive definition of formulas of modal logic

	### A formula is a set
	### The type of formulas of modal logic is a set

-### Modal logic predicate
+### The predicate of being a modal logic
+A {{#concept "modal logic" Agda=is-modal-logic}} is a theory of modal logic that contains its own deductive closure.

	### TODO: Title
	### Any modal theory is contained in its own deductive closure

-  is-contradictory-modal-logic-Prop : modal-theory l2 i → Prop l2
-  is-contradictory-modal-logic-Prop logic = logic ⊥ₘ
-  is-contradictory-modal-logic : modal-theory l2 i → UU l2
-  is-contradictory-modal-logic = type-Prop ∘ is-contradictory-modal-logic-Prop
+```
+### Inconsistent modal theories
+```agda
+  is-inconsistent-modal-theory-Prop : modal-theory l2 i → Prop l2
+  is-inconsistent-modal-theory-Prop logic = logic ⊥ₘ
+  is-inconsistent-modal-theory : modal-theory l2 i → UU l2
+  is-inconsistent-modal-theory = type-Prop ∘ is-contradictory-modal-logic-Prop

-### TODO: maybe another file
+### Lists obtained from maps of elements of lists
+Consider a list `l : list A` of elements of `A`, and consider a function `f : (a : A) → a ∈ l → B`, which assigns to each element of `l` a value in `B`. Then we obtain a list of elements of `B` whose elements are given by `f`.
+This operation is a generalization of `map-list`, because we only need to assign values in `B` to the elements of the list.

-TODO
+For any [list](lists.lists.md) `l : list A` of elements of `A`, the {{#concept "image" Disambiguation="lists"}} of `l` is the least [subtype](foundation.subtypes.md) of `A` that contains all elements of `l`. In other words, it is the subtype of all elements of `A` that are merely elements of `l`, in the sense that the [proposition](foundation-core.propositions.md)
+```text
+  ║a ∈ l║
+```
+holds, where we applied the [propositional truncation](foundation.propositional-truncations.md) to the elementhood relation of lists.

	### Closure of a theory under the deduction system
	### The deductive closure of a theory of modal logic

Modal logic #1146

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Modal logic #1146

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spcfox commented May 31, 2024

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fredrik-bakke commented Jun 1, 2024

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spcfox Jun 1, 2024 •

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