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### Modal Reasoning = Metric Reasoning, via Lawvere [article]

Ugo Dal Lago, Francesco Gavazzo
<span title="2021-03-05">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove such a relation to be a congruence, this way obtaining a compositional technique for reasoning about modal and coeffectful behaviours.  ...  bisimilarity (which we dub modal applicative bisimilarity).  ...  The following result states that modal reasoning is indeed equal to metric reasoning, via Lawvere. Theorem 2. Let V and W be a quantale and a monoidal Kripke frame, respectively.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2103.03871v1">arXiv:2103.03871v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/x6prnsruf5fdtpyrhgv5fzk4vu">fatcat:x6prnsruf5fdtpyrhgv5fzk4vu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210309000859/https://arxiv.org/pdf/2103.03871v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a5/5b/a55b1fa7c997168682f7d956562fbe89b684900f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2103.03871v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Up-To Techniques for Behavioural Metrics via Fibrations

Filippo Bonchi, Barbara König, Daniela Petrisan, Michael Wagner
<span title="2018-08-13">2018</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/iv4yo5vao5ctfjfushi4akt5xi" style="color: black;">International Conference on Concurrency Theory</a> </i> &nbsp;
We focus on the fibrations of predicates and relations valued in a quantale, for which pseudo-metric spaces are an example.  ...  We introduce up-to techniques for behavioural metrics between systems modelled as coalgebras and we provide abstract results to prove their soundness in a compositional way.  ...  C O N C U R 2 0 1 8 17:4 Up-To Techniques for Behavioural Metrics via Fibrations Notice that we use the reversed order on [0, 1], for technical reasons (see Section 4) .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.concur.2018.17">doi:10.4230/lipics.concur.2018.17</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/concur/Bonchi0P18.html">dblp:conf/concur/Bonchi0P18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/n6dyzhli2zgatgwndph4nnmyxi">fatcat:n6dyzhli2zgatgwndph4nnmyxi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220119174603/https://drops.dagstuhl.de/opus/volltexte/2018/9555/pdf/LIPIcs-CONCUR-2018-17.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c0/57/c057b42f04f94dfe4fe84542e640c21e4dbf3d91.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.concur.2018.17"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

<span title="2012-12-24">2012</span> <i title="OpenEdition"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z5zp3gljsrhpbpvwdox3z2iaeq" style="color: black;">European Journal of Pragmatism and American Philosophy</a> </i> &nbsp;
Generality is the law of cohesiveness among parts beyond any individual and any possibility of metrically measuring it; modality means plasticity, namely the fact that a continuum is not tied to actualities  ...  So we can easily find in this movement generality, modality, and transitivity. 38 Our new paradigm implies that a synthetic judgment (and reasoning) is a judgment (and reasoning) that recognizes -with  ...  Finally, we indicate some main features that a systematical synthetic/horotic reasoning should enforce, in order to fulfill its historic sequence. AUTHORS  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4000/ejpap.741">doi:10.4000/ejpap.741</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3tpipzsutncp7jlwkla7bcqxrm">fatcat:3tpipzsutncp7jlwkla7bcqxrm</a> </span>
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### Probabilistic systems coalgebraically: A survey

Ana Sokolova
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
It is interesting to note that, for different reasons, for both discrete and continuous probabilistic systems it may be more convenient to work with behavioral equivalence than with bisimilarity.  ...  The value of the results on coalgebraic modal logics is inverse proportional to the space devoted here, the main reason for such a treatment being that more information on coalgebraic modal logics can  ...  Behavioral distances via finality: metric vs. measurable spaces The work on behavioral pseudometrics for probabilistic systems by van Breugel and Worrell; cf. e.g.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2011.05.008">doi:10.1016/j.tcs.2011.05.008</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/21998490">pmid:21998490</a> <a target="_blank" rel="external noopener" href="https://pubmed.ncbi.nlm.nih.gov/PMC3185909/">pmcid:PMC3185909</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z4iljxlfofc5jbzyg3bwxk4g5y">fatcat:z4iljxlfofc5jbzyg3bwxk4g5y</a> </span>
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### Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory [article]

David Jaz Myers
<span title="2022-05-31">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The reason for this divergence between formalism and intuition is that the points of spaces cannot have internal symmetries in traditional, set-level foundations.  ...  We will use the metric topology. Definition 5.1.4.  ...  Then the ball B(x, 1 n ) is a metrically open subset containing x but not y. Next, we will show that R is regular. Let U be metrically open and let x ∈ U .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2205.15887v1">arXiv:2205.15887v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/etop2t4cm5gllpome5jsmd4jiq">fatcat:etop2t4cm5gllpome5jsmd4jiq</a> </span>
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### Foundational Frameworks [chapter]

Geoffrey Hellman*
<span title="">2011</span> <i title="Springer Netherlands"> The Western Ontario Series in Philosophy of Science </i> &nbsp;
Finally, we sketch a variant of a modal-structural core system, incorporating elements of predicativism and the systems of reverse mathematics, and consider how it fares with respect to the desiderata  ...  ∀x n (R(x 1 ...x n ) ↔ Φ),where Φ lacks free 'R' and is modal free.  ...  Presumably that would be by understanding a variety of examples, e.g. of familiar spaces, topological, metric, etc., or algebraic structures along with the relevant types of morphisms.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-94-007-0431-2_1">doi:10.1007/978-94-007-0431-2_1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6jmyppvj4zd5zhisaeujmrwfiu">fatcat:6jmyppvj4zd5zhisaeujmrwfiu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20100729021146/http://www.tc.umn.edu/~hellm001/Publications/FoundationalFrameworks.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c5/45/c545fe9cfd99f3c2ccdb57b7f4923e5e0bdce8ce.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-94-007-0431-2_1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

### Logical Foundations of Quantitative Equality (long version) [article]

Francesco Dagnino, Fabio Pasquali
<span title="2021-10-11">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Main examples are based on metric spaces and Lipschitz maps and on quantitative realisability.  ...  To overcome this issue, we consider the extension of Linear Logic with graded modalities and use them to write a resource sensitive substitution rule that keeps equality quantitative.  ...  Quantitative equality via graded modalities Graded modalities indexed by a semiring R are used in many linear calculi and type theories to enable resource sensitive reasoning [40, 9, 20, 37, 1, 14, 36  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2110.05388v1">arXiv:2110.05388v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ru46cprs2berzpdcmkyigltmhu">fatcat:ru46cprs2berzpdcmkyigltmhu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20211013105820/https://arxiv.org/pdf/2110.05388v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/cf/5a/cf5aab1458a34b0474002bf3059378bee3e72d05.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2110.05388v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### An Internal Language for Categories Enriched over Generalised Metric Spaces [article]

Fredrik Dahlqvist, Renato Neves
<span title="2021-10-27">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In the case of (ultra)metric equations, we get an internal language for autonomous categories enriched over (ultra)metric spaces.  ...  In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others.  ...  We start by recalling basics concepts of V-categories, which are central in a field initiated by Lawvere in [24] and can be intuitively seen as generalised metric spaces [38, 17, 39] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2105.08473v2">arXiv:2105.08473v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zhdydzm27rhdfmxvfnzsqlohty">fatcat:zhdydzm27rhdfmxvfnzsqlohty</a> </span>
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### On the Fusion of Coalgebraic Logics [chapter]

Fredrik Dahlqvist, Dirk Pattinson
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Fusion is arguably the simplest way to combine modal logics.  ...  For normal modal logics with Kripke semantics, many properties such as completeness and decidability are known to transfer from the component logics to their fusion.  ...  Moreover, if we view the algebraic models of a Λ L -logic L as models of a Lawvere theory T L (the theory of Λ L modal algebras) and the algebraic models of a Λ R -logic R as models of a Lawvere theory  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-22944-2_12">doi:10.1007/978-3-642-22944-2_12</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zbxnlinhvne7han6mp64t6fxya">fatcat:zbxnlinhvne7han6mp64t6fxya</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170816020848/http://www.research.ed.ac.uk/portal/files/19256125/Dahlqvist_Pattinson_2011_On_the_Fusion_of_Coalgebraic_Logics.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b8/d1/b8d1ee5dcd98e291a511f32554e341e93083f51a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-22944-2_12"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

### A cost-aware logical framework [article]

Yue Niu
<span title="2021-10-08">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Lastly we substantiate the soundness of quantitative reasoning in calf by means of a model construction.  ...  Taking inspiration from recent work that reconstructs traditional aspects of programming languages in terms of a modal account of phase distinctions, we argue that the cost structure of programs motivates  ...  In the context of functorial semantics [Lawvere 1963 ], the universal property states that one may define a model calf → ℰ by simply specifying the image of the constants of Σ calf in ℰ 5 .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2107.04663v2">arXiv:2107.04663v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3bit6tz77ff65drvkyiid5zgne">fatcat:3bit6tz77ff65drvkyiid5zgne</a> </span>
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### Axiomatic Method and Category Theory [article]

Andrei Rodin
<span title="2012-09-25">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In [161] develops this theory further and, in particular, provides the categories of space and quantity with metrics.  ...  How Lawvere manages to do this?  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1210.1478v1">arXiv:1210.1478v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iz72tr2f2jcjdikdpg5flh6hli">fatcat:iz72tr2f2jcjdikdpg5flh6hli</a> </span>
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### Relation lifting, a survey

Alexander Kurz, Jiří Velebil
<span title="">2016</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/75odiq25o5gtzcy4h7mdcztb5y" style="color: black;">Journal of Logical and Algebraic Methods in Programming</a> </i> &nbsp;
As applications, it is shown how different notions of bisimulation, simulation and modal logics do arise.  ...  This approach to metric spaces via enriched category theory was pioneered by Lawvere [35] .  ...  By varying the base category from sets to preorders or even to (generalised) metric spaces one goes from bisimulation via simulation to metric simulation.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jlamp.2015.08.002">doi:10.1016/j.jlamp.2015.08.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zbyc3ptpxvfxrc4cqxkzhhpjqi">fatcat:zbyc3ptpxvfxrc4cqxkzhhpjqi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922002915/https://lra.le.ac.uk/bitstream/2381/38793/2/kv-relation-lifting.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/57/9b/579b8e4dd2e9bf4bbb9ec53c720d2caa4a982c94.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jlamp.2015.08.002"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

### Symbolic Conscious Experience

Venkata Rayudu Posina
<span title="2017-01-01">2017</span> <i title="Christ University Bangalore"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/45tftztudjef5os22di7udgp3i" style="color: black;">Tattva - Journal of Philosophy</a> </i> &nbsp;
The [undefined] primordial conscious experience of early life differentiates, via a thousand cuts such as the spectator-spectacle subdivision and the schism between spatial intuition and symbolic reason  ...  The understanding of mathematical structures in terms of presentation-representation (Lawvere, 2004) in particular, and of languages in terms of syntax-semantics in general can be adopted as a thinking  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.12726/tjp.17.1">doi:10.12726/tjp.17.1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/x2yrihecuzalpieag2ulvtl3pm">fatcat:x2yrihecuzalpieag2ulvtl3pm</a> </span>
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### Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions [article]

Paul Wild, Lutz Schröder
<span title="2022-05-19">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Our central result then shows that in fact all fuzzy lax extensions are Kantorovich extensions for a suitable set of quantitative modalities, the so-called Moss modalities.  ...  In systems involving quantitative data, such as probabilistic, fuzzy, or metric systems, behavioural distances provide a more fine-grained comparison of states than two-valued notions of behavioural equivalence  ...  space in the sense of Lawvere [Law73] .  ...
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### Generating the algebraic theory of C(X): the case of partially ordered compact spaces [article]

Dirk Hofmann, Renato Neves, Pedro Nora
<span title="2017-06-16">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
of Lawvere [Law73] .  ...  In [KKV04] the authors argue that the category BooSp of Boolean spaces and continuous maps "is an interesting base category for coalgebras"; among other reasons, due to the connection with modal logic  ...
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