Dani Díaz
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diazcarrete.bsky.social
Dani Díaz
@diazcarrete.bsky.social
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diazcarrete.bsky.social
semella que Ruibal non gustaba de pezas de inspiración mitolóxica
a autora En A volta de Edipo, a outra peza galardoada, presentanos a Edipo de regreso a unha Tebas asolada pola peste por mor dunha conxura de Creonte. A visión apocalip-tica do mundo achégase á que anos máis tarde aparecerá na Traxicomedia do vento de Tebas de Manuel Lourenzo, gaña-dora do VI Concurso, onde o perverso personaxe pretenderá espallar a sífile entre os tebanos. Nas actas, o xurado explicou que "adoutou como criterio, tanto como os méritos literarios intrínsecos das obras, a posi-bilidade de que se axusten ás circunstancias culturais e sociais que nos requiren un teatro coa suma maior de posibilidades de ser representado diante de auditorios preferentemente popula-res". O voto en contra de Euloxio R. Ruibal deu pé a unha cer-ta polémica, posto que descualificou a peza de Ánxeles Penas por "non ter acercamento á realidade actual". A autora defen-deríase nunha longa carta enviada a Abrente na que reivindi-caba a representabilidade da obra (parece ser que tamén posta en cuestión) e o valor intemporal do mito. Respecto ó premio
December 2, 2025 at 11:26 PM
diazcarrete.bsky.social
a morte do autor
Un tema de máxima actualidade por aquel entón e que non estivo ausente dos debates dese ano foi a cuestión da au-toría -non esquezamos que a falla de dramaturgos era unha queixa reiterada no teatro galego, cuestión que sería abor-dada polos cataláns de forma radical, pois consideraban ab-surdo laiarse xustamente desa carencia cando o papel do autor estaba en crise en todo Occidente. Como solución, re-negaban das "pseudocreaciones colectivas" que xulgaban "una irresponsabilización en la que nadie quiere figurar como responsable" e apostaban pola alternativa que atopara o Teatre Lliure para paliar ese déficit: as traduccións de tex-tos de autores estranxeiros. O problema da traducción abor-daríase nesta Mostra desde un punto de vista ideolóxico e político, sobre todo a partir da montaxe de Pic-nic por Arte-llo e de Lenta raigame por Andrómena.
December 1, 2025 at 6:54 PM
diazcarrete.bsky.social
lo que me recuerda... ExpoConvento en A Coruña del 3 al 8 de Diciembre! www.paxinasgalegas.es/fiestas/expo...
poster para"XV feria del dulce de convento ciudad de la coruña. ExpoConvento 3-8 diciembre 2025 iglesia padres capuchinos"
December 1, 2025 at 11:06 AM
diazcarrete.bsky.social
Foto tomada dende o camiño que vai dende San Diego á praia de Oza, xa preto de desembocar na entrada a Oza. A esquerda vese o edificio abandonado dos varaoiros. á dereita a ponte que pasa por enriba do camiño. O día é soleado.
December 1, 2025 at 8:04 AM
diazcarrete.bsky.social
San Francisco Javier en Lugo es de los 60, pero bueno, incluyámosla también.
www.diocesisdelugo.org/san-francisc...
"A arte de Neira Brochs na igrexa lucense de san Francisco Xabier" www.anosavoz.com/a-arte-de-ne... dialnet.unirioja.es/servlet/arti...
Fachada de la iglesia San Francisco Javier en Lugo, con sus característicos forjados en la entrada.
November 29, 2025 at 8:26 PM
diazcarrete.bsky.social
"Typewriters, Typography, and Office Machines of the 20th Century. Various ephemera related to typewriters, typography, and life in the analog office"
archive.org/details/type...
"Operation and maintenance manual for portable typewriter" booklet cover in English. Showing hand in white reaching for some abstracted keys in the space above. The shape of a small green plane crosses the hand orizontally, following a red arrow. Booklet cover for "Woodstock Typewriters". A vertical band shows a somewhat abstract drawing of a city with skycrapers. Swedish poster for the "Alpina" typewriter, showing the machine and a smiling woman. Swedish cover for a "Continental Klein" typewriter brochure, showing a seated woman in a red dress with the typewriter on her lap, typing.
November 29, 2025 at 4:43 PM
diazcarrete.bsky.social
El pensar que aquel tierno pedazo de carne que a tales peligros había de estar sujeto, me era mi hijo, ponía las carnes de gallina. —Le pondremos vacuna. —Cuando sea mayorcito... —Y lo llevaremos siempre calzado, porque no se corte los pies.
November 29, 2025 at 10:39 AM
diazcarrete.bsky.social
matinar sobre quen fixo o tigre

contra

matinar sobre quen fixo a formiga
Olla esa formiga, ise argueiro mouro… ¿Qué delicados dedos de alfareiro puideron modelar tan pequechiño corazón, que latexa agora baixo dos altos árboles?
November 29, 2025 at 9:49 AM
diazcarrete.bsky.social
Cruel to auction an elderly infirm tyger like that. I hope it escapes.
"The Tiger" by Nael, Age 6, also known as The Tiger Is Out, refers to a poem written by a 6-year-old child about a tiger escaping a cage. The tiger He destroyed his cage Yes YES The tiger is out
November 29, 2025 at 9:36 AM
diazcarrete.bsky.social
Split keyboards make this decision for you. Most (but not all) of them put the "6" key on the left, perhaps for ergonomics reasons, perhaps because ease of manufacturing.
Left half of a split keyboard showing the "6" number key. Left half of a split keyboard showing the "6" number key. Right half of a KINESIS split keyboard showing the "6" number key. Right half of a split keyboard showing the "6" number key. The half also includes a cool red trackball.
November 29, 2025 at 9:18 AM
diazcarrete.bsky.social
fell in this internet rabbit hole by investigating whether touch typers who type "6" with the left hand should be excommunicated
deskthority.net/viewtopic.ph...
A collage of which-finger-for-which-key touch typing diagrams, all of them showing "6" being assigned to the left hand. A text at the bottom says: "(Again, there is a similar variety of diagrams that put 6 on the right hand.)"
November 29, 2025 at 9:00 AM
diazcarrete.bsky.social
this... doesn't seem very ergonomic
Drawing from a 19th century touch typing manual, showing a woman typing ("operator sitting in correct position for rapid writing"). From a modern point of view, it doesn't seem very ergonomic because the woman has to look down at the typewriter, and there are no arm rests. That said, wrist position seems good, they're kept straight, in a neutral position, not awkwardly bent up or down.
November 29, 2025 at 8:54 AM
diazcarrete.bsky.social
"Practical Typewriting: By the All-finger Method, which Leads to Operation by Touch"—Bates Torrey archive.org/details/prac...
"How to Become Expert in Typewriting: A Complete Instructor"—Lovisa Ellen Barnes archive.org/details/howt...

en.wikipedia.org/wiki/Touch_t...
On July 25, 1888, an American man, Frank Edward McGurrin, who was reportedly the only person using touch typing at the time, won a decisive victory over Louis Traub (operating Caligraph with eight-finger method) in a typing contest held in Cincinnati. The results were displayed on the front pages of many newspapers. [2] McGurrin won US$500 (equivalent to $13,304 in 2019 USD) and popularized the new typing method. Whether McGurrin was actually the first person to touch type or simply the first to be popularly noticed, is disputed. Speeds attained by other typists in other typing competitions at the time suggest that they must have been using similar systems. [3] In 1889, Bates Torrey coined the words "writing by touch" in his article. [4] In 1890, Lovisa Ellen Bullard Barnes defined the words "write by touch" in her book as follows:[5][6] To learn to write by touch, that is, with only an occasional glance at the key-board, sit directly in front of the machine. Keep the hands as nearly as possible in one position over the key-board. diagram from a 19th century touch typing manual, showing with hand should be used with each key. diagram from a 19th century touch typing manual, showing certain key combinations that require a lot of finger travel.
November 29, 2025 at 8:48 AM
diazcarrete.bsky.social
destroying the present to reach for a valorized past
By the eighth century, the arena had been converted into a fortification, with four stone towers, three of which are still standing. Also in the Middle Ages, the outsides of the arches were filled in, and houses were built inside the arena and up against its outer walls, so that in time, over many years, a fortified village was created within the arena, complete with streets, a public square, and two chapels, one containing the remains of St. Genesius, patron saint of notaries. The village continued in existence until the 1800s, described in the Murray guidebook, in harsh terms, as being "filled within and choked up without by an accumulation of mean hovels, occupied by the poorest and worst part of the population of the town to the number of 2000." Then, in an upsurge of interest in the city's Roman heritage, the decision was made to raze the houses and clear out the village. This began in 1823 and was mostly accomplished by 1844; when Stendhal visited the city in 1837, there were still a few of what he calls "poor dwellings" inside the walls. Despite these remaining houses, the first event to take place in the arena after the clearing was a race of the bulls in 1830, held in celebration of the conquest of Algiers.
November 28, 2025 at 1:47 PM
diazcarrete.bsky.social
"A brief introduction to Higher Order Logic and the HOL proof assistant"
people.disim.univaq.it/monica/MFI/H...
The HOL logic is a version of (classical) higher order logic based on Church's simple theory of types [1]. The interface to the logic is the functional programming language ML [2]. The approach to theorem proving in HOL is based on the LCF methodology [9] for interactive and secure theorem proving by mechanising the logic in a strongly typed language like ML. Theorems are represented by abstract data types and the user can interact with the theorem prover by ML procedures which operate on such data types. Theorem proving tools are ML functions and user-defined ML programs can only perform valid logical inferences. Typically, a purely definitional approach is taken when using the HOL logic. This means that new entities are only introduced by means of (primitive and derived) definition mechanisms which allow one to extend the logic in a sound way. Propositions and theorems are then derived from definitions and/or previously proved theorems by formal proof. This guarantees that inconsistencies are not introduced into the logic. Theories are extended in a sound way by deriving new theorems by formal proof. To prove a theorem in a theory, one must apply a sequence of proof steps to either axioms or previously proved theorems using ML programs called inference rules (forward proof). The core of the deductive system in HOL is made up of eight primitive inference rules, from which all other rules are derived. Among the primitive inference rules is ẞ-conversion, implemented by the ML function BETA_CONV, which reduces a β-redex (λχ. κ) v to the term uv/x] by returning the theorem (λχ. κ) υ = [v/x] (by renaming variables where necessary to avoid free variable capture). #BETA_CONV "(\x y. x + y)y";; 1- (\x y. x + y)y = (\y'. y + y') theorems into other theorems, e.g. the rule BETA_RULE which, given a theorem Γt, returns the theorem resulting from reducing all the ẞ-redexes, at any depth, in the conclusion t. #let thm = ASSUME "f = ((\x y. x + y) y)";; thm = f = (\x y. x + y)y | f = (\x y. x + y)y #BETA_RULE thm;; f = (\x y. x + y)y | f = (\y'. y + y') BETA_RULE is defined in terms of BETA_CONV using the ML functions DEPTH_CONV, that applies a conversion repeatedly to all subterms in a bottom-up order, and CONV_RULE, which transforms a conversion into an inference rule.
November 25, 2025 at 10:09 PM
diazcarrete.bsky.social
"también mi historia es también un Sendero, así quisiera yo"
Para un momento. Antes, cuando la nombraste, una mano-de-serpiente era algo Ahora es un paraje. Y explícame también eso de rodilla-de-quién, ya que te has parado. en la conversación. Está bien. Ya te hablé del Sendero. El Sendero es como una serpiente, se enrosca alrededor de toda Belaire Pequeña con la cabeza en el centro y la punta de la cola cerca de la puerta de los cuerda Bucle, pero sólo alguien que conozca Belaire Pequeña puede ver por dónde va. A cualquier otro puede parecerle que se abre en todas direcciones. Así pues, cuando corres por el Sendero, pero descubres que no es más que una serie de cuartos intercomunicados que se enros-can en un pequeño laberinto y que no tiene salida y siempre vuelve al Sendero... ahí tienes una mano-de-serpiente. Asoma de la serpiente que es el Sendero como una multitud de dedos diminutos. Y además los llamamos manos-de-serpiente porque las ser-pientes no tienen manos, como si dijéramos que el Sendero es uno solo. Pero una mano-de-serpiente es también algo más: también mi historia es también un Sendero, así quisiera yo, y por tanto ha de tener manos-de-serpiente. Las manos-de-serpiente son a veces lo mejor de lah la historia, si la historia es larga. Rodilla-de-quién. Yo nunca he sido muy bueno en rodilla-de-quién, pero como cualquier otro chico de Belaire llevaba siempre conmigo la pelota y las pinzas: parte del equipo de todo muchacho. Mi
November 25, 2025 at 8:10 PM
diazcarrete.bsky.social
HOL Foundations
www21.in.tum.de/teaching/pro...
HOL's logic and novelties The language corresponds to that of the LCF formalism with the difference, that types were interpreted as sets instead of Scott Domains Higher-order logic admits quantification over sets or predicates, that are nested arbitrarily deep Example of a third-order term: VQ3R ∈ Q3f3x∃y: R(f(x)) → R(y) Two theories form the basis of HOL (bool, ind) The theory bool Contains: Primitive type 'bool' Four axioms for higher-order logic - Three primitive constants (Equality, Implication and Choice) and some more useful but less important constants With these three constants we can define T (truth), (falsity), ¬ (negation), Λ (conjunction), v (disjunction), ∀ (universal quantification), ∃ (existential quantification) and ! (unique existence quantification) Inference rules in HOL HOL uses eight inference rules: - ASSUME: Assumption Introduction REFL: Reflexivity BETA CONV: Beta-conversion SUBST: Substitution ABS: Abstraction INST_TYPE: Type Instantiation DISCH: Discharging an assumption MP: Modus Ponens The LCF approach in ML Logical inference rules are implemented as functions Modus Ponens as an example: Fp In ML: val MP:thm → thm → thm MP(Γ + p ⇒ q)(∆ ├p) = (ΓU∆Εq)
November 24, 2025 at 8:59 PM
diazcarrete.bsky.social
www.di.uminho.pt/~jno/mfes/08...
in DSL parlance wouldn't the logical framework approach be more of a "deep" encoding? 🤔
Direct encoding. Each logical construction have a counterpart in the type theory. Theorem proving consists of the (interactive) construction of a proof-term, which can be easily checked independently. Examples: Coq based on the Calculus of Inductive Constructions Agda based on Martin-Lof's type theory Lego based on the Extended Calculus of Constructions Nuprl - based on extensional Martin-Lof's type theory Encoding of logic in type theory Shallow encoding (Logical Frameworks). The type theory is used as a logical framework, a meta system for encoding a specific logic one wants to work with. The encoding of a logic L is done by choosing an appropriate context IL, in which the language of L and the proof rules are declared. Usually, the proof-assistants based on this kind of encoding do not produce standard proof-objects, just proof-scripts. Examples: HOL, based on the Church's simple type theory. This is a classical higher-order logic. Isabelle, based on intuitionistic simple type theory (used as the meta logic). Various logics (FOL, HOL, sequent calculi,...) are described.
November 24, 2025 at 8:54 PM
diazcarrete.bsky.social
The de Bruijn criterion vs the LCF architecture
lawrencecpaulson.github.io/2022/01/05/L...
How the kernel works in a LCF theorem prover
cs.stackexchange.com/questions/12...
Isabelle Tutorial: System, HOL and Proofs
www.lri.fr/~wolff/tutor...
And so, this software architecture ensures that the logical rules of the calculus are always applied correctly. The design is reminiscent of operating systems based on a kernel that has the sole right to perform unsafe operations. However, a sceptic must trust this kernel: the proof assistant does not deliver any sort of certificate that could be checked independently. The LCF architecture refers to precisely this: a proof assistant that securely encapsulates the rules of the logical calculus within an abstract data type. It must be coded in a strongly typed language providing a suitable notion of abstract type. Confusion about the meaning of "LCF" is widespread because Edinburgh LCF introduced many other features common to modern-day proof assistants, in particular a subgoaling proof style based on backward-chaining reasoning steps and a system of "proof tactics" for accomplishing this. The LCF architecture is an alternative to the de Bruijn criterion, not an implementation of it. The use of abstract types in the LCF architecture has absolutely nothing to do with the well-known propositions-as-types correspondence. In developing the successor system, Edinburgh LCF, Milner had the idea that instead of saving whole proofs, the system should just remember the results of proofs, namely theorems. The steps of a proof would be performed but not recorded, like a mathematics lecturer using a small blackboard who rubs out earlier parts of proofs to make space for later ones. To ensure that theorems could only be created by proof, Milner had the brilliant idea of using an abstract data type whose predefined values were instances of axioms and whose operations were inference rules. Strict typechecking then ensured that the only values that could be created were those that could be obtained from axioms by applying a sequence of inference rules - namely theorems. (As told by Michael Gordon in "From LCF to HOL: a short history", Proof, Language, and Interaction: Essays in Honour of Robin Milner 169-185. Quote on p. 170. Available for free here.) Also here is a quotation of from Paulson book on ML. About abstract datatypes he says: A modular structure makes a program easier to understand. Better still, the modules ought to serve as interchangeable parts: replacing one module by an improved version should not require changing the rest of the. program. Standard ml's abstract types and functors can help us meet this objective too. A module may reveal its internal details. When the module is replaced, other parts of the program that depend upon such details will fail. ML provides several ways of declaring an abstract type and related operations, while hiding the type's representation. So you cannot instantiate an abstract datatype you can use it (its fields + functions). The matter on how you implement this feature is very of dependent of the language. If you want to do it in the
November 24, 2025 at 8:24 PM
diazcarrete.bsky.social
Formalising Mathematics In Simple Type Theory
"Despite the considerable interest in new dependent type theories, simple type theory [...] is sufficient to formalise serious topics in mathematics."
arxiv.org/abs/1804.07860
Higher-order logic is based on the work of Church [10], which can be seen as a simplified version of the type theory of Whitehead and Russell. But while they were exponents of logicism, today's HOL Light and Isabelle/HOL users clearly aren't, or at least, keep their views secret. Someone wishing to formalise mathematics today has three main options: Higher-order logic (also known as simple type theory), where types are built inductively from certain base types, and variables have fixed types. Generalising this system through polymorphism adds considerable additional expressiveness. Dependent type theories, where types are parameterised by terms, embodying the propositions-as-types principle. This approach was first realised in NG de Bruijn's AUTOMATH [12]. Such systems are frequently but not necessarily con-structive: AUTOMATH was mainly used to formalise classical mathematics. Set theories can be extremely expressive. The Mizar system has demonstrated that set theory can be a foundation for mathematics in practice as well as in theory [2]. Recent work by Zhan [55] confirms this point independently, with a high degree of automation. All three options have merits. While this paper focuses on higher-order logic, I make no claim that this formalism is the best foundation for mathematics. It is certainly less expressive than the other two. And a mathematician can burst free of any for- 2 Higher-Order Logic on the Computer A succinct way to describe higher-order logic is as a predicate calculus with simple types, including functions and sets, the latter seen as truth-valued functions. Logical types evolved rapidly during the 20th century. For Whitehead and Rus sell, types were a device to forestall the paradoxes, in particular by enforcing the dis-tinction between sets and individuals, But they had no notation for types and never wrote them in formulas. They even proved (the modern equivalent of) VV, con-cealing the type symbols that prevent Russell's paradox here [14]. Their omission of type symbols, which they termed typical ambiguity, was a precursor to today's polymorphism. It seems that they preferred to keep types out of sight. Church [10] provided a type notation including a type t of individuals and a separate type o of truth values, with which one could express sets of individuals (having type ot), sets of sets of individuals (type o(or)) etc., analogously to the cumulative hierarchy of sets, but only to finite levels. Church assigned all individuals the same type. The need to reason about n-dimensional spaces threatens to introduce infinite duplication. Simple type theory does not allow dependent types, and yet the param-eter n (the dimension) is surely a natural number. The theory of Euclidean spaces concerns R" for any n, and it might appear that such theorems cannot even be stated in higher-order logic. John Harrison found an ingenious solution [23]: to represent the dimension by a type of the required cardinality. It is easy to define types in higher-order logic having any specified finite number of elements. Then R" can be represented by the type n→ real, where the dimension n is a type. Through polymor-phism, n can be a variable, and the existence of sum and product operations on types even allow basic arithmetic to be performed on dimensions. It must be admitted that things start to get ugly at this point. Other drawbacks include the need to write R as R¹ in order to access topological results in the one-dimensional case. Neverthe-
November 24, 2025 at 8:17 PM
diazcarrete.bsky.social
"Gödel’s system T as a precursor of modern type theory"
inria.hal.science/hal-04046289v1
3.1 Proofs are algorithms The main idea of modern type theory is that the proofs should be objects of the theory. In set theory, or in early type theories, the number 2 is an object, the set of even numbers is an object, the function multiplying its argument by 2 is an object, but the proof that 2 is an element of the set of even numbers is not. In modern type theories, in contrast, it is. Modern type theories also agree on the fact that a proof is not just any object of the theory, but that it is an algorithm, through the Brouwer-Heyting-Kolmogorov interpretation. In particular, a proof of a formula A B is an algorithm mapping proofs of A to proofs of B. In this respect, modern type theory changes the perspective on proofs. Proofs were considered, by the axiomatic method, as the very first notion of mathematics. Now proofs are defined in term of a more primitive notion: the notion of algorithm. This means that an elementary theory of algorithm has to be developed before, and without, the notion of proof. We can see that the system T, already in Gödel's formulation anticipates these ideas: first the idea that the notion of algorithm is more fundamental than that of proof, then that proofs must be objects of the theory, and finally that they should be algorithms. All these ideas, if anticipated, are not yet expressed in their full clarity. For instance, the fact that algorithms are more fundamental than proofs is anticipated as this interpretation allows to reduce the existence of a proof to the existence of an algorithm, but it is not completely implemented because the formula FIFA still requires a proof. The same holds for the fact that proofs should be objects of the theory. Indeed, neither arithmetic, nor the system T can express its own proofs, but the proofs of one theory are expressed in the other. This is why the extensions of this interpretation to simple type theory are important, as they are a first step towards a theory that expresses its own proofs. 3.2 Computation rules Another property of the system T that anticipates modern type theory is that it is the first extension of simple type theory where most of the computable functions can be expressed. It is well-known that Church's initial goal with A-calculus was to provide a formalization of mathematics where all computable functions could be computed, but that the discovery of the Kleene-Rosser para-dox lead to split the project: the A-calculus to express all computable functions and Church's theory of type, to formalize mathematics. We know, by a result of Schwichtenberg, that the computational power of the simply typed 3 rule is much weaker than the untyped one. Strangely enough, the fact that simple type theory could be extended in such a way that all primitive recursive functions could be computed does not seem to have been put forward before the system T. Of course, both in the system T and in modern type theory, this computa-tional power is a prerequisite for proofs to be expressed as algorithms.
November 24, 2025 at 8:13 PM
diazcarrete.bsky.social
discord.com/channels/280...
Keep in mind, the type theory isn't being used to construct and check your proofs in this system. (editado) The type theory just forms the language of terms you're writing proofs about. Type checking is about whether a term is well-formed, not that a proof is correct.
November 24, 2025 at 8:09 PM
diazcarrete.bsky.social
"in the [propositions-as-types] view the logic is about arbitrary props and the lambda terms corresponds to proofs of those props. In the latter view, the logic is about lambda terms, i.e. you prove facts about lambda terms, e.g. whether two are equal."
math.stackexchange.com/questions/27...
In the latter view, used by e.g. the HOL family of theorem provers and described in The Seven Virtues of Simple Type Theory, the simply typed lambda terms are just terms in an equational logic. In this case, a proof is a proof in this logic which can be classical or intuitionistic or whatever. Typically it is classical and a Boolean type B is added with axioms to ensure that it has only two values, F and T. We can then identify propositions with terms of type B, and more generally, predicates with terms of type (71→(12→ (TnB))). Due to the use of higher-order terms (lambda terms), just a handful of rules and axioms for equality and Booleans produces a fairly powerful higher-order logic. See sections 2.3.1 and 2.4.3 of the HOL4 Logic manual for details of the deductive system in the case of HOL4 (which actually uses the polymorphic lambda calculus, not the simply typed lambda calculus as that is much more convenient). So in HOL4, the proofs would be deductions in this deductive system. Summarizing, in the former view the logic is about arbitrary propositions and the lambda terms corresponds to proofs of those propositions. In the latter view, the logic is about lambda terms, i.e. you prove facts about lambda terms, e.g. whether two are equal.
November 24, 2025 at 8:07 PM
diazcarrete.bsky.social
"Simply typed lambda calculus and simple type theory are not equivalent."

"there are two ways to relate logic to type theory"

mathoverflow.net/questions/32...
Simple type theory is simply-typed A-calculus and additionally at least: the type of truth values o, with the corresponding term formers (constants and T, connectives, quantifiers at every type) the type of natural numbers 1, with the corresponding term formers (zero, succcesor, primitive recursion into arbitrary types) equations governing the term formers and substitution What I think is confusing you is the fact that there are two ways to relate logic to type theory: 1. The Curry-Howard correspondence relates the propositional calculus to the simply-typed X-calculus by an interpreation of propositional formulas as types. 2. Higher-order logic embeds into simple type theory by an interpretation of logical formulas as terms of the type o of truth values. There is a difference of levels, which makes all the difference. I recommend taking heed of Peter Lumsdaine's advice: the terminology is not standard, and there are many variants of everything. Products or no products? Is equality up to ẞ or up to βη-equivalence? Do we assume any basic types, and how many? And so on, and so on. The only safe way is to always make precise references to precisely given formal systems, and not to assume anything. - Andrej Bauer Jan 30,
November 24, 2025 at 8:05 PM
diazcarrete.bsky.social
Church's Type Theory a. k. a. simple type theory
plato.stanford.edu/entries/type...
"The seven virtues of simple type theory"
www.sciencedirect.com/science/arti...
"Hidden within this hierarchy [of predicates?] is the full power of higher-order predicate logic."
4. Expressivity On the surface, the expressivity of STT appears to be rather modest. The formulas of STT are limited to variables, constants, and definite descriptions of type *, equations between expressions of the same type, and applications of predicates; there are no propositional connectives nor quantifiers. However, STT can be used to reason about an infinite hierarchy of higher-order functions constructed over a domain of individuals and a domain of truth values. Since sets can be represented by predicates, the hierarchy includes a subhierarchy of higher-order sets constructed from individuals and truth values. Hidden within this hierarchy is the full power of higher-order predicate logic. 8.5. Polymorphism An operator is polymorphic if it can be applied to expressions of more than one type. For instance, the equality operator (=) in STT is polymorphic because it can be applied to any two expressions of the same type. One of the greatest shortcomings of STT is that it has no mechanism for introducing new polymorphic operators. For example, recall the constant compose defined in Section 4 for expressing the composition of two functions of type (1→1). Let us rename compose to compose to show what kind of expressions it can be applied to. If we wanted to compose two functions of other types we would need to define in exactly the same way other composition constants of the form composes. Since there are an infinite number of function types in STT, there is potentially an infinite number of distinct composition constants that might be needed. It would be far more efficient if we could define a single polymorphic composition operator.
November 24, 2025 at 8:01 PM