A talk I gave at the Yonsei University, Seoul in July 21st, 2015.
The aim was to show my background contribution to the CORCON (Correctness by Construction) research project.
I have to thank Prof. Byunghan Kim and Dr Gyesik Lee for their kind hospitality.
A slide of the talk in Logic and Engineering of Natural Language Semantics (LENLS) 12:
we analyse Friedman-Sheared's truth theory FS from the behavior of truth predicate by a proof theoretic semantics methodology.
A slide of the talk in Logic and Engineering of Natural Language Semantics (LENLS) 12:
we analyse Friedman-Sheared's truth theory FS from the behavior of truth predicate by a proof theoretic semantics methodology.
Correspondence and Isomorphism Theorems for Intuitionistic fuzzy subgroupsjournal ijrtem
ABSTRACT:The aim of this paper is basically to study the First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism theorem, Correspondence Theorem etc. of intuitionistic fuzzy/vague sub groups of a crisp group.
KEY WORDS: Intutionistic fuzzy or Vague Subset, Intutionistic fuzzy Image, Intutionistic fuzzy Inverse Image, Intutionistic fuzzy/vague sub (normal) group, Correspondence Theorem, First (Second, Third) Isomorphism Theorem.
Sentient Arithmetic and Godel's Incompleteness TheoremsKannan Nambiar
For me, there is only one logic that we rational human beings are able to accept and appreciate, and that is the mathematical logic of ZF theory. But in the last century we found that ZF theory is not in a position to provide all that we want, and went in search of a new mode of thinking and got one which we called meta mathematics. My question is: if we can put the unambiguous logic of ZF theory on paper, why can't we do the same with meta mathematics. This paper is my feeble attempt in that direction.
Gödel’s completeness (1930) nd incompleteness (1931) theorems: A new reading ...Vasil Penchev
(T2) Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of inf(T1) Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation
Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity
That is nothing else than right a new reading at issue and comparative interpretation of Gödel’s papers meant here
inity
The most utilized example of those generalizations is the separable complex Hilbert space: it is able to equate the possibility of pure existence to the probability of statistical ensemble
(T3) Any generalization of Peano arithmetic consistent to infinity, e.g. the separable complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
In this talk, logically distributive categories are introduced to provide a sound and complete semantics to multi-sorted, first-order, intuitionistic-based logical theories. The peculiar aspect is that no universe is required to interpret terms, making the semantics really point-free.
Correspondence and Isomorphism Theorems for Intuitionistic fuzzy subgroupsjournal ijrtem
ABSTRACT:The aim of this paper is basically to study the First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism theorem, Correspondence Theorem etc. of intuitionistic fuzzy/vague sub groups of a crisp group.
KEY WORDS: Intutionistic fuzzy or Vague Subset, Intutionistic fuzzy Image, Intutionistic fuzzy Inverse Image, Intutionistic fuzzy/vague sub (normal) group, Correspondence Theorem, First (Second, Third) Isomorphism Theorem.
Sentient Arithmetic and Godel's Incompleteness TheoremsKannan Nambiar
For me, there is only one logic that we rational human beings are able to accept and appreciate, and that is the mathematical logic of ZF theory. But in the last century we found that ZF theory is not in a position to provide all that we want, and went in search of a new mode of thinking and got one which we called meta mathematics. My question is: if we can put the unambiguous logic of ZF theory on paper, why can't we do the same with meta mathematics. This paper is my feeble attempt in that direction.
Gödel’s completeness (1930) nd incompleteness (1931) theorems: A new reading ...Vasil Penchev
(T2) Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of inf(T1) Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation
Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity
That is nothing else than right a new reading at issue and comparative interpretation of Gödel’s papers meant here
inity
The most utilized example of those generalizations is the separable complex Hilbert space: it is able to equate the possibility of pure existence to the probability of statistical ensemble
(T3) Any generalization of Peano arithmetic consistent to infinity, e.g. the separable complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
In this talk, logically distributive categories are introduced to provide a sound and complete semantics to multi-sorted, first-order, intuitionistic-based logical theories. The peculiar aspect is that no universe is required to interpret terms, making the semantics really point-free.
I am kind of confused about quantifiers. I am not sure how to transl.pdfAMITPANCHAL154
I am kind of confused about quantifiers. I am not sure how to translate them into english
correctly so a quick Guide to how or the answer will work. Thanks!
***** (A = universal) , (E = existential), (V = or), (^ = and) I do not know how to input the real
symbol)
1.) AxAyAz([x+y]+z = x+([y+z])
2.) Here x, y are students C(x) is having a computer and F(x,y) indicates that x and y are friends.
Ax(C(x) vEy[C(y)^F(x,y)]
3.) Suppose that x and y are real numbers
a. AxAy([x>0 ^ y>0] ------> [x+y>0])
b. AxAy([x<0 ^ 4<0] [x(y)>0])
Solution
There are several logical symbols in the alphabet, which vary by author but usually
include: The quantifier symbols and The logical connectives: for conjunction, for disjunction, for
implication, for biconditional, for negation. Occasionally other logical connective symbols are
included. Some authors use , or Cpq, instead of , and , or Epq, instead of , especially in contexts
where is used for other purposes. Moreover, the horseshoe may replace ; the triple-bar may
replace , and a tilde (~), Np, or Fpq, may replace ; ||, or Apq may replace ; and &, or Kpq, may
replace , especially if these symbols are not available for technical reasons. Parentheses,
brackets, and other punctuation symbols. The choice of such symbols varies depending on
context. An infinite set of variables, often denoted by lowercase letters at the end of the alphabet
x, y, z, … . Subscripts are often used to distinguish variables: x0, x1, x2, … . An equality symbol
(sometimes, identity symbol) =; see the section on equality below. It should be noted that not all
of these symbols are required - only one of the quantifiers, negation and conjunction, variables,
brackets and equality suffice. There are numerous minor variations that may define additional
logical symbols: Sometimes the truth constants T, Vpq, or , for \"true\" and F, Opq, or , for
\"false\" are included. Without any such logical operators of valence 0, these two constants can
only be expressed using quantifiers. Sometimes additional logical connectives are included, such
as the Sheffer stroke, Dpq (NAND), and exclusive or, Jpq. [edit]Non-logical symbols The non-
logical symbols represent predicates (relations), functions and constants on the domain of
discourse. It used to be standard practice to use a fixed, infinite set of non-logical symbols for all
purposes. A more recent practice is to use different non-logical symbols according to the
application one has in mind. Therefore it has become necessary to name the set of all non-logical
symbols used in a particular application. This choice is made via a signature.[2] The traditional
approach is to have only one, infinite, set of non-logical symbols (one signature) for all
applications. Consequently, under the traditional approach there is only one language of first-
order logic.[3] This approach is still common, especially in philosophically oriented books. For
every integer n = 0 there is a collection of n-ary, or n-place, predicate .
Introduction to complexity theory that solves your assignment problem it contains about complexity class,deterministic class,big- O notation ,proof by mathematical induction, L-Space ,N-Space and characteristics functions of set and so on
Tim Maudlin: New Foundations for Physical GeometryArun Gupta
New Foundations for Physical Geometry
Original URL: http://www.unil.ch/webdav/site/philo/shared/summer_school_2013/NYU.ppt
Tim Maudlin
NYU
Physics & Philosophy of Time
July 25, 2013
In this article we present a brief history and some applications of semirings, the structure of compact monothetic c semirings. The classification of these semirings be based on known description of discrete cyclic semirings and compact monothetic semirings. Boris Tanana "Compact Monothetic C-semirings" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-2 , February 2021, URL: https://www.ijtsrd.com/papers/ijtsrd38612.pdf Paper Url: https://www.ijtsrd.com/mathemetics/algebra/38612/compact-monothetic-csemirings/boris-tanana
Similar to Point-free foundation of Mathematics (20)
The famous Kruskal's tree theorem states that the collection of finite trees labelled over a well quasi order and ordered by homeomorphic embedding, forms a well quasi order. Its intended mathematical meaning is that the collection of finite, connected and acyclic graphs labelled over a well quasi order is a well quasi order when it is ordered by the graph minor relation.
Oppositely, the standard proof(s) shows the property to hold for trees in the Computer Science's sense together with an ad-hoc, inductive notion of embedding. The mathematical result follows as a consequence in a somewhat unsatisfactory way.
In this talk, a variant of the standard proof will be illustrated explaining how the Computer Science and the graph-theoretical statements are strictly coupled, thus explaining why the double statement is justified and necessary.
The Graph Minor Theorem: a walk on the wild side of graphsMarco Benini
The Graph Minor Theorem says that the collection of finite graphs
ordered by the minor relation is a well quasi order. This apparently
innocent statement hides a monstrous proof: the original result by
Robertson and Seymour is about 500 pages and twenty articles, in which a
new and deep branch of Graph Theory has been developed.
The theorem is famous and full of consequences both on the theoretical side
of Mathematics and in applications, e.g., to Computer Science. But there
is no concise proof available, although many attempts have been made.
In this talk, arising from one such failed attempts, an analysis of the
Graph Minor Theorem is presented. Why is it so hard?
Assuming to use the by-now standard Nash-Williams's approach to prove it,we will
illustrate a number of methods which allow to solve or circumvent some
of the difficulties. Finally, we will show that the core of this line of
thought lies in a coherence question which is common to many parts of
Mathematics: elsewhere it has been solved, although we were unable to
adapt those solutions to the present framework. So, there is hope for a
short proof of the Graph Minor Theorem but it will not be elementary.
Analysing the categorical structure of well quasi orders, two proofs of the Higman's lemma are shown, emphasising the structural content of this property of well quasi order in relation to exponentiation.
As a side effect, a variant of the Lemma is found, which says that the finite sequences of elements on A, ordered by embedding, is well-founded if and only if, A is well-founded.
Numerical Analysis and Epistemology of InformationMarco Benini
The slides of my presentation at the workshop "Philosophical Aspects of Computer Science", European Centre for Living Technology, University “Ca’ Foscari”, Venice, March 2015.
L'occhio del biologo: elementi di fotografiaMarco Benini
The slides of the course "L'occhio del biologo", Alta Formazione, Università degli Studi dell'Insubria.
It is a small course on the fundamentals of photography oriented towards the scientific photography in a biological laboratory.
Marie Skłodowska Curie Intra-European FellowshipMarco Benini
A brief report of my experience as a Marie Curie Research Fellow in Leeds to illustrate to my colleagues what means to participate in such a program.
I have to acknowledge the kind invitation of the Research Office of the Università degli Studi dell'Insubria and the Rector delegate to research, Prof. Umberto Piarulli.
By analysing the explanation of the classical heapsort algorithm via the method of levels of abstraction mainly due to Floridi, we give a concrete and precise example of how to deal with algorithmic knowledge. To do so, we introduce a concept already implicit in the method, the ‘gradient of explanations’. Analogously to the gradient of abstractions, a gradient of explanations is a sequence of discrete levels of explanation each one refining the previous, varying formalisation, and thus providing progressive evidence for hidden information. Because of this sequential and coherent uncovering of the information that explains a level of abstraction—the heapsort algorithm in our guiding example—the notion of gradient of explanations allows to precisely classify purposes in writing software according to the informal criterion of ‘depth’, and to give a precise meaning to the notion of ‘concreteness’.
This talk aims at introducing, through a very simple example, a way to represent data types in the λ-calculus, and thus, in functional programming languages, so that the structure of the data types itself becomes a parameter.
This very simple technical trick allows to reconsider programming as a way to express morphisms between models of a logical theory. As an application, it allows to realise a way to perform anonymous computations.
From a philosophical point of view, the presented approach shows how it is possible to conceive a real programming system where properties like correctness of programs can be proved, but data cannot be inspected, not even in principle.
CORCON2014: Does programming really need data structures?Marco Benini
This talk tries to suggest how computer programming can be conceptually simplified by using abstract mathematics, in particular categorical semantics, so to achieve the 'correctness by construction' paradigm paying no price in term of efficiency.
Also, it introduces an alternative point of view on what is a program and how to conceive data structures, namely as computable morphisms between models of a logical theory.
A description of the formal model behind Constructive Adpositional Grammars.
Presented at Proof Theory and Constructive Mathematics Seminar, School of Mathematics, University of Leeds (2011).
Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard ...Marco Benini
A novel approach to giving an interpretation of logic inside category theory. This work has been developed as part of my sabbatical Marie Curie fellowship in Leeds.
Presented at the Logic Seminar, School of Mathematics, University of Leeds (2012).
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
PRESENTATION ABOUT PRINCIPLE OF COSMATIC EVALUATION
Point-free foundation of Mathematics
1. Point-free foundation of Mathematics
Dr M Benini
Università degli Studi dell’Insubria
Correctness by Construction research project
visiting Yonsei University
marco.benini@uninsubria.it
July 21st, 2015
2. Introduction
This seminar aims at introducing an alternative foundation of Mathematics.
Is it possible to define logical theories without assuming the existence
of elements? E.g., is it possible to speak of mathematical analysis
without assuming real numbers to exist?
This talk will positively answer to the above question by providing a sound
and complete semantics for multi-sorted, first-order, intuitionistic-based
logical theories.
Since classical theories can be obtained by adding all the instances of the
Law of Excluded Middle to the theory, in fact, the following results
immediately transpose to the classical case.
(2 of 18)
3. Introduction
Actually, there is more:
the semantics does not interpret terms as elements of some universe, but
rather as the glue which keeps together the meaning of formulae; so,
soundness has to be interpreted as coherence of the system, and
completeness as the ability to express enough structure to capture the
essence of the theory;
the semantics allows to directly interpret the propositions-as-types
correspondence so that each theory is naturally equipped with a
computational meaning, where classically true facts are miracles;
the semantics allows for a classifying model, that is, a model such that
every other model of the theory can be obtained by applying a suitable
functor to it;
most other semantics for these logical systems can be mapped to the one
presented: Heyting categories, elementary toposes, Kripke models,
hyperdoctrines, and Grothendieck toposes.
In this talk, we will focus on the first aspect only.
(3 of 18)
4. Introduction
Most of this talk is devoted to introduce a single definition, logical
categories, which identifies the models of our logical systems.
These models are suitable categories, equipped with an interpretation of
formulae and a number of requirements on their structure.
Although the propositional part has already been studied by, e.g., Paul
Taylor, the first-order extension is novel.
(4 of 18)
5. Logical categories
Let 〈S,F,R〉 be a first-order signature, with
S the set of sort symbols,
F the set of function symbols, of the form f : s1 ×···×sn → s0, with si ∈ S
for all 0 ≤ i ≤ n,
R the set of relation symbols, of the form r : s1 ×···×sn, with si ∈ S for
all 1 ≤ i ≤ n.
Also, let T be a theory on 〈S,F,R〉, i.e., a collection of axioms.
A logical category will be a pair 〈C,Σ〉 where C is a suitable category and Σ a
family of functors modelling the substitution of terms in interpreted
formulae.
Informally, the objects of C will denote formulae while arrows will denote
proofs where the codomain is the theorem and the domain is the
assumption(s).
(5 of 18)
6. Logical categories
Definition 1 (Prelogical category)
Fixed a language where T is the discrete category of its terms, a prelogical
category is a pair 〈C,Σ〉 such that
1. C is a category with finite products, finite coproducts and exponentiation;
2. for each formula A and each variable x :s, ΣA(x :s): T → C is a functor,
called the substitution functor, such that, for every variable y :s ,
ΣA(x :s)(x :s) = ΣA(y :s )(y :s );
3. for each x :s variable, t :s term, A and B formulae,
3.1 Σ⊥(x :s)(t :s) = 0, the initial object in C;
3.2 Σ (x :s)(t :s) = 1, the terminal object in C;
3.3 ΣA∧B(x :s)(t :s) = ΣA(x :s)(t :s)×ΣB(x :s)(t :s), the binary product in C;
3.4 ΣA∨B(x :s)(t :s) = ΣA(x :s)(t :s)+ΣB(x :s)(t :s), the binary coproduct in C;
3.5 ΣA⊃B(x :s)(t :s) = ΣB(x :s)(t :s)ΣA(x :s)(t :s), the exponential object in C;
4. for any formula A, any variable x :s, and any term t :s,
ΣA(x :s)(t :s) = ΣA[t/x](x :s)(x :s).
(6 of 18)
7. Logical categories
The substitution functors model the operation of substituting a variable with
a term in an interpreted formula. This behaviour is fixed by the fourth
condition: it says how the syntactical act of substituting a term for a
variable in formula is performed within the semantics.
These functors allow to define the notion of interpretation, associating every
formula A to an object in the category: MA = ΣA(x :s)(x :s). This definition
is well-founded, thanks to the second condition.
So propositional intuitionistic logic is readily interpreted, as shown in
P. Taylor, Practical foundations of mathematics, Cambridge University
Press, 1999. Of course, the problem lies in quantified formulae. . .
(7 of 18)
8. Logical categories
A star S over a formula A and a variable x :s is a subcategory of C such
that:
1. its objects are the vertexes v of the cones on ΣA(x :s) such that v = MB
for some formula B and x :s ∈ FV(B), i.e., x is not free in B;
2. there is an object MC in S, its centre, such that all the morphisms in S
are either identities, or arrows in the category of cones on ΣA(x :s) with
MC as codomain.
MB
µ
//
θt
**
MC
πttt
ΣX (x)(t) t∈T
Dually, one may define the notion of costar.
Intuitively, the centre of a star is the natural candidate to interpret the
formula ∀x :s.A, with the arrows in S standing for instances of the
introduction rule, while the projections in the cone whose vertex is the
centre, denote instances of the elimination rule. Dually, costars are used to
provide a meaning to ∃x :s.A.
(8 of 18)
9. Logical categories
Definition 2 (Logical category)
A prelogical category 〈C,Σ〉 is called logical when, for each formula A and
each variable x :s,
1. there is a star on A and x :s, denoted by C∀x :s.A, having a terminal
object; we require M∀x :s.A to be the centre of C∀x :s.A;
2. there is a costar on A and x :s, denoted by C∃x :s.A, having an initial
object; we require M∃x :s.A to be the centre of C∃x :s.A;
It is possible to prove that each formula in any theory T in the first-order
language can be interpreted in this class of categories. Also, any proof can
be naturally represented by a morphism in this class of categories.
This interpretation is both sound and complete: we say that a formula is
true when there is an arrow from 1 to its interpretation. Thus, a model for a
theory T is such that it makes true any axiom in T. Then, every provable
formula in a theory T is true in every model of T. And, vice versa, any
formula which is true in any model for T, is provable from T.
(9 of 18)
10. Logical categories
Soundness is not surprising: since arrows denote proofs, each arrow which is
required by the structure, denote an inference rule. For example, the
projections of the product denote the ∧-elimination rules, while the universal
arrow of the product is the ∧-introduction rule.
Completeness proceeds by showing that the syntactical category over a
theory T, having formulae-in-context as objects and proofs as arrows (well,
modulo an equivalence relation that takes care of multiple assumptions, like
in P. Johnstone’s Sketches of an Elephant), is a logical category which is
also a model for T. Moreover, this model is a classifying model: any other
model for T can be derived by applying to the syntactical category a suitable
logical functor, i.e., a functor that preserves finite limits and finite colimits.
These proofs are long, complex, and quite technical. Too boring for a talk. . .
(10 of 18)
11. The role of terms
How terms get interpreted?
Variables are used to identify the required subcategories C∀x : s.A and
C∃x : s.A;
variables are also used to apply the ΣA(x :s) functor properly;
all terms contribute to the substitution process, which induces the
structure used by the semantics.
Thus, it is really the substitution process, formalised in the ΣA(x :s)
functors, that matters: terms are just the glue that enable us to construct
the C∀x : s.A and C∃x : s.A subcategories.
It is clear the topological inspiration of the whole construction. In particular,
it is evident that terms are not interpreted in some universe, and their role is
limited to link together formulae in subcategories that control how
quantifiers are interpreted.
(11 of 18)
12. Abstract syntax
There is no need to consider the set of terms as the only way to provide a
language for the “elements” of the theory: any groupoid will do. Since in a
groupoid G, all the arrows are isomorphisms, the substitution functor forces
them to become instances of the substitution by equals principle.
Variables are not needed, as well: the notion of variable can be defined by
posing that x ∈ ObjG is a variable when there is a formula A and an object
t ∈ ObjG for which the substitution functor ΣA(x)(t) = MA, i.e., a variable is
something which may be substituted obtaining something different. Also a
variable is free in a formula when it may be substituted in the formula to
obtain a different one. When the variables in a groupoid G are infinite, or
there are none, the purely propositional case, the soundness and
completeness results can be proven as before.
Also, fixed a logical category C, we can regard its objects as abstract
formulae, with no harm in the above approach to semantics: in this case, C
becomes the classifying model for the theory whose language is given by
itself, and whose axioms are its true formulae.
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13. Inconsistent theories
A theory T is inconsistent when it allows to derive falsity. However, in our
semantics, T has a model as well.
A closer look to each model of T reveals that they are categorically “trivial”
in the sense that the initial and the terminal objects are isomorphic.
This provides a way to show that a theory is consistent. However, this is not
ultimately easier or different than finding an internal contradiction in the
theory.
Actually, it does make sense in a purely computational view that an
inconsistent theory has a model: it means that, although the specification of
a program is ultimately wrong, there are pieces of code which are perfectly
sound. And every computation does something, even if not useful!
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14. Alternative semantics
Most semantics for first-order intuitionistic logic and theories can be reduced
to elementary toposes via suitable mappings. This is the case for Heyting
categories, for Grothendieck toposes, and, in an abstract sense, with
hyperdoctrines. Since algebraic semantics and Kripke semantics can be
reduced to Heyting categories and Grothendieck toposes, respectively, we
can safely say that elementary toposes provide a good framework to
compare semantics.
It is possible to show that any model in any elementary topos can be
transformed in a model M in a logical category in such a way that exactly
the true formulae in the topos are true in the M. The converse does not
hold in a direct way: in fact, assuming to work in a strong set theory, one
can show that a topos model can be reconstructed from M by means of a
Kan extension.
The difficult part is that one should assume enough power in the set theory
to literally build up a collection of elements big enough to contain the
interpretations of terms and of every other element which is implied to exist
by the theory.
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15. State of the Art
At the moment, these results have been submitted for a possible publication.
Also, they have been presented in a few workshops and in a number of
seminars within the CORCON project.
My current research on this topic follows three main lines:
1. to extend the idea beyond first-order systems. Specifically, I am trying to
provide a point-free semantics to constructive type theory, with a look
towards homotopy type theory. This work has made some progress, but
the results are not yet obtained and not everything is stable.
2. to apply these results to concrete theories. In this respect, I am
developing a presentation of the theory of well quasi-orders along these
lines, using as a logical category the category of quasi-orders, coupled
with substitution functors acting on singletons as variables, and finite
chains as terms. Some results have been obtained, and they will be
presented in an upcoming workshop in Hamburg, Germany, in September.
3. to clean up the framework, to understand exactly how much power is
required to get the results, and to find the minimal fragment of category
theory/set theory required to develop the point-free semantics.
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16. Projects
This work has been supported by two research projects:
Abstract Mathematics for Actual Computation: Hilbert’s Program
in the 21st Century, 2014-2016, John Templeton Foundation, Core
Funding. Partner: University of Leeds (UK)
Correctness by Construction, 2014-2017, FP7-PEOPLE-2013-IRSES,
gr.n. PIRSES-GA-2013-612638. Partner: University of Leeds (UK),
University of Strathclyde (UK), Swansea University (UK), Stockholms
Universitet (SE), Universitaet Siegen (DE), Ludwig-Maximilians
Universitaet Muenchen (DE), Università degli Studi di Padova (IT),
Università degli Studi di Genova (IT), Japan Advanced Institute of
Science and Technology (JP), University of Canterbury (NZ), The
Australian National University (AU), Institute of Mathematical Sciences
(IN), Carnegie Mellon University (US), Hankyong National University
(KR), Kyoto University (JP), National Institute of Informatics (JP),
Tohoku University (JP), University of Gothenburg (SE), University of
Ljubljana (SI)
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17. My visit in Korea
Since Dr Lee is a partner in the CORCON project, my visit is the chance to
share the achievement we have made.
The next theory to develop in point-free terms will be real analysis: in this
respect, I will try to propose a research project to the European Union next
year. It will be a joint effort with numerical analysts, starting from the very
simple idea that a real number in numerical analysis is actually a point-free
entity, denoted by its value and its error. A clean and neat logical framework
to deal with this idea in a direct way may be of benefit both for logicians,
since it explains how numerical computations on reals are designed, and for
numerical analysts, since it is much closer to the way they have to think
when solving problems numerically.
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18. Conclusion
Much more has been done on
logical categories. But, still, I
am in the beginning of the
exploration of this
foundational setting.
So, any comment or
suggestion is welcome!
Questions?
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