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\subsection*{Community of Philosophical Inquiry}
\addcontentsline{toc}{subsection}{Community of Philosophical Inquiry, by Anne Brel Cloutier}

{\scshape Anne Brel Cloutier}\index{Cloutier, Anne Brel}\\
{\scshape Philosophy doctorate student, Institute of cognitive sciences, Université du Québec à Montréal, Montreal, QC, Canada}\\
{\scshape annebrelcloutier@gmail.com}\\

According to Piaget, the first psychologist to study reasoning from a
logician point of view, children are not born logical and logical
reasoning only appears progressively up to adolescence. His theory of
the development of rationality (Piaget,1964) was criticised for
diverse reasons. Several studies demonstrated that children have some
degree of logical understanding at a very young age (Pears \& Bryant,
1990) and that adults are not optimally logical (Wason,1969).

David Moshman, a professor of educational psychology at the University
of Nebraska-Lincoln, offers a new reading of Piaget's work by
understanding the development of rationality at a metalogical level.
Following his pluralist rational constructivism theory (Moshman,
2004), logical reasoning develops through the increase of metalogical
understanding. In order to have a consciousness on ones inference, it
is necessary to make it explicit and that process occurs during peer
interaction. I argue that Community of Philosophical Inquiry (CPI)
used in Philosophy for Children (P4C), if practiced with a special
attention on its metacognitive aspects, can constitute the perfect
didactic to put into practice Moshman's theory. Furthermore, adding
some explicit notions of logic and reflections on logical thinking
could transform the CPI method into a logic lesson for children and
learners of all ages.

First, I will introduce David Moshman's theory. I will then present
CPI as the practice of dialogue developed by the logician and
pedagogue Matthew Lipman (2003) and how this method puts into practice
Moshman's theory through intellectual moves performed by the children
themselves. My research consists in linking the metacognitive and
metalogical strategies with those moves in order to foster the
development of logical understanding, transforming CPI in CLI ---
Community of Logical Inquiry. I am using Michel Sasseville and Mathieu
Gagnon's work in the observation of CPI (Sasseville \& Gagnon, 2012)
to link most common behaviours to metacognitive and metalogical
strategies. We will proceed to a close examination of some of those
behaviours and see how it can link to a metalogical approach of the
development of rationality.

Philosophical discussions allow children to starts from concrete
examples of their day-to-day experiences and thoughts, and then
generalize to wider thoughts, constructing their own theory of mind.
In P4C, not only we commonly witness participants expressing rational
and logical thoughts, but also the metalogical aspects of the CPI
methodology has multiple underlying strategies that could foster the
development of their logical reasoning. We will discuss how these
strategies consist in metacognitive and metalogic strategies that
adults could also greatly benefice from. The claims I endorse put
forward the possibility to build a toolbox for the learning of logical
thinking in schools. This work could help teachers' work in providing
them the tools they need to develop better teaching methods that they
can put into practice in their classroom. Since metacognitive
strategies have been proven efficient for all levels learners, this
approach could have a major impact in scholar system, in teachers'
training and also in a broader social scale.\\

{\bf References}


\item Beaulac, G. et Robert, S.\ (2011). «Les théories de l'éducation
  à l'ère des sciences cognitives: le cas de l'enseignement de la
  pensée critique et de la logique». Les Ateliers de l'éthique. 5(2).

\item Lipman, M.\ (2003). Thinking in education, New York: Cambridge
  University Press.

\item Moshman, D. (2004). «From inference to reasoning: the
  construction of rationality». Thinking and Reasoning, 10, 221-239.

\item Pears, R. \& Bryant, P., «Transitive inferences by young
  children about spatial position», British Journal of Psychology,
  1990, 81, 497-510.

\item Piaget, J., «Cognitive Development in Children: Piaget,
  Development and Learning», in Journal of research in science
  teaching, originally published in Volume 2, Number 3, pp. 176-186

\item Sasseville, M. \& Gagnon M. (2012). Penser ensemble à l'école,
  Des outils pour l'observation d'une communauté de recherche
  philosophique en action. Québec:PUL.

\item Schraw, G. \& Moshman, D. (1995). «Metacognitive theories».
  Educational Psychology Review.7(4),351-371.

\item Wason, P. C. (1969) «Regression in reasoning?». British Journal
  of Psychology, 60(4) 471-480.



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\subsection*{On the concreteness of certain categories}
\addcontentsline{toc}{subsection}{On the concreteness of certain categories, by Ivan Di Liberti}

{\scshape Ivan Di Liberti}\index{Di Liberti, Ivan}\\
{\scshape Department of Mathematics and Statistics, Masaryk University, Czech Republic}\\
{\scshape diliberti@math.muni.cz}\\

$K$ is concrete when there is a faithful funtor $F: K \to
\mathbf{Set}$. People say that concrete categories are those such that
one can think their objects as some sets and their arrows as some
functions preserving a structure. For many years there was no natural
example of non concrete categories. Freyd proved in [1] that the
homotopy category of topological spaces is not concrete. In the
seminar we will see the main ideas of the proof.

{\bf References}


\item J.P. Freyd, On the concreteness of certain categories.

\item J.P. Freyd, Concreteness.

\item J.P. Freyd, Homotopy is not concrete.

\item F. Loregian and I. Di Liberti, Homotopical Algebra is not



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\subsection*{Visualization as Restructuring and thus a Source of Logical Paradox}
\addcontentsline{toc}{subsection}{Visualization as Restructuring and thus a Source of Logical Paradox, by Andrius Kulikauskas}

{\scshape Andrius Kulikauskas}\index{Kulikauskas, Andrius}\\
{\scshape Department of Philosophy and Cultural Studies, Vilnius Gediminas Technical University}\\
{\scshape ms@ms.lt}\\

We survey and systematize the ways our minds organize and visualize
thoughts. We then observe their relevance in explaining different
kinds of logical paradox. We also show where they arise in math.

We were inspired by educator Kestas Augutis's vision that every high
school student write three books (a chronicle, a thesaurus, and an
encyclopedia) so as to master three kinds of thinking (sequential,
hierarchical, and network). We thus collected dozens of examples of
how we organize our thoughts. Surprisingly, we never use sequences,
hierarchies or networks in isolation. Instead, we use them in pairs:


\item Evolution: A hierarchy (of variations) is restructured with a
  sequence (of times).

\item Atlas: A network (of adjacency relations) is restructured with a
  hierarchy (of global and local views).

\item Handbook: A sequence (of instructions) is restructured with a
  network (of loops and branches).

\item Chronicle: A sequence (of events in time) is restructured with a
  hierarchy (of time periods).

\item Catalog: A hierarchy (of concepts) is restructured with a
  network (of cross-links).

\item Odyssey: A network (of states) is restructured with a sequence
  (of steps).


In general, a first, large, comprehensive structure grows so robust
that we restructure it with a second, smaller, different structure of
multiple vantage points.

In a separate investigation, we listed and grouped paradoxes. This
yielded the following six themes:


\item Concepts may be inexact. (The paradox of an evolution.) We can't
  specify exactly at what point in the womb a child becomes conscious,
  or at what point in evolution two species diverge.

\item The whole is not the sum of the parts. (The paradox of an
  atlas.) If we replace all of the parts of an automobile, and then
  build a copy with all of the old parts, which is the original?

\item Our attention affects what we observe. (The paradox of a
  handbook.) Achilles can never catch a tortoise if we keep measuring
  the distance between them.

\item There may be a limited contradiction. (The paradox of a
  chronicle.) How can we reliably learn from one who has ever made a

\item We cannot make explicit all relevant assumptions. (The paradox
  of a catalog.) $10+4$ may equal 2 if we happen to be thinking about
  a clock.

\item We can choose differently in the same circumstances. (The
  paradox of an odyssey.) \\I am lying when I say `I am lying.'\,''


Each type of paradox brings to light the fundamental gap between the
(seemingly infinite) primary comprehensive structure and the
(manifestly finite) secondary structure which organizes our vantage
points. Our mind visualizes a qualitative but illusory relationship
between the two structures.

These same six restructurings arose in a broader investigation which
yielded 24 ways of figuring things out in mathematics. We identify the
six restructurings with six axioms of set theory: Pairing,
Extensionality, Well-ordering, Power set, Union and Regularity.


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\subsection*{Elementary introduction to pasting}
\addcontentsline{toc}{subsection}{Elementary introduction to pasting, by Fernando Lucatelli Nunes}

{\scshape Fernando Lucatelli Nunes}\index{Nunes, Fernando Lucatelli}\\
{\scshape Centre for Mathematics, University of Coimbra}\\
{\scshape flnlucatelli@gmail.com }\\

The operation of pasting of 2-cells is part of the foundations of
2-category theory [4]. It was introduced by Bénabou in [1] and, then,
further explored by Kelly and Street [2]. However its associative
property, fundamental aspect that makes it useful to prove theorems,
was not proved (or even properly stated) before [4].

The main purpose of the talk is to give some elementary aspects of
pasting, giving examples within basic category theory in order to
motivate its day-to-day use even in 1-dimensional category theory.
These examples intend to demonstrate that, once we assume pasting is
well-defined, pasting gives nice ways of understanding and dealing
with proofs diagrammatically. For instance, the whiskering and
interchange law come for free in proofs using pasting of 2-cells.

If time permits, we finish giving a brief discussion on results that
gives another perspective on the well definition/associativity of the
operation pasting, relating it with presentation of 2-categories,
deficiency of presentations and, hence, topology [3].

{\bf References}


\item J. Bénabou. Introduction to Bicategories, in ``Lecture Notes in
  Mathematics, Vol 47'', pp.~1--77, Springer-Verlag, 1967.

\item G.M. Kelly and R.H. Street. Review of the elements of
  2-categories, in ``Lectures Notes in Mathematics, Vol 420'',
  pp.~75--103, Springer-Verlag, New York/Berlin, 1974.

\item F. Lucatelli Nunes. Freely generated n-categories, coinserters
  and presentations of low dimensional categories. Arxiv: 1704.04474.

\item A.J. Power. A 2-categorical pasting theorem. Journal of Algebra
  129, 439--445, 1990.



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\subsection*{Subjectivism and inferential reasoning on teaching practice}
\addcontentsline{toc}{subsection}{Subjectivism and inferential reasoning on teaching practice, by Laura Rifo}

{\scshape Laura Rifo}\index{Rifo, Laura}\\
{\scshape UNICAMP, Brazil}\\
{\scshape laurarifo@gmail.com}\\

In this work, we analyze well-succeeded strategies and challenges of
teaching principles of decision theory as developed by DeGroot,
Lindley and Blackwell, for students in secondary school. Among other
things, we emphasize the aspects of probability and conditional
probability under the subjectivistic interpretation and inferential
reasoning based on a Bayesian learning approach.

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