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\usepackage[Mickael]{ammaths}
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\begin{document}
\module{Syllogisms}{02}{01}{3 periods}
%\prereq{}
\object{\begin{itemize}
\item Discover and understand syllogisms.
\item Write original syllogisms.
\item See different types of syllogisms.
\end{itemize}}
\mater{\begin{itemize}
\item Fact sheet about syllogisms.
\item Twelve different syllogisms with the premises and conclusion written on different cards.
\item Answer sheet for each team to write the syllogisms.
\item Beamer with different types of syllogisms.
\end{itemize}}
\modpart{Matching game}{20 mins}
The teacher explains quickly what is a syllogism. Students are then handed out cards with one sentence, part of a
syllogism, on each. They have to commit it to memory, then hide the paper and mingle to find the two other parts of the
same syllogism. Once the team is made up, they must order the sentences correctly and read the complete syllogism to the
class.
\modpart{Devising syllogisms in teams, checking their validity}{35 mins}
Every team of three must devise three original syllogisms and read them to the class. The
concept of validity of a syllogism is introduced and checked for each new one.
\modpart{Construction of a syllogism}{55 mins}
The different types of syllogisms are shown, including the universal or particular, the
positive or negative and the four different figures. Students have to fill their answer sheets
with the subject, middle term, predicate, figure, types of proposition and type of syllogism.
\modpart{Recognise and repair invalid syllogisms (test)}{55 mins}
Students, working in pairs, are handed out a page with some syllogisms, most of them invalid. They have to find out which syllogisms are incorrect and repair them. At the end of the 55 minutes, they hand back their answer sheet to be graded.
\pagebreak
\moddocdis{Syllogisms}{02}{01}{Lesson}
\en\nocouleur
One of the main methods of proof in mathematics is the syllogism. Here are two examples :
\begin{quote}
All multiples of 12 are multiples of 4. \\
All multiples of 4 are even.\\
Therefore, all multiples of 6 are even.
\end{quote}
\begin{quote}
No reptiles have fur.\\
All snakes are reptiles.\\
Therefore, no snakes have fur.
\end{quote}
\defi{}{A syllogism is a logical argument in which one proposition, the conclusion, is implied by two
other propositions, the premises.}
\section*{Truth and validity}
While studying syllogism, and logic more generally, it's important to separate clearly the
notions of truth and validity. Truth is about the real world, the one we're living in, and
the way we perceive it. A statement is true if it corresponds to what we see in the world
around us. Validity is strictly related to the form of the statement, to the correctness of
the logical deductions involved. Below are two examples.
\begin{quote}
All penguins have pink stripes.
All math teachers are penguins.
Therefore, all math teachers have pink stripes.
\end{quote}
This syllogism is valid but none of the three statements involved is true.
\begin{quote}
All equilateral triangles have angles of 60$^\circ$ .
The triangle ABC has an angle of 60$^\circ$ .
Therefore, the triangle ABC is equilateral.
\end{quote}
This syllogism is not valid even if the three statements involved can be true. The problem
here resides in the deduction, which doesn't respect the conditions of the first premise.
\section*{Different types of syllogisms}
In each proposition (the premises and the conclusion), the quantifier can be universal
or particular, and the sentence can be affirmative or negative. Since the medieval times,
letters are used to represent each type of proposition :
\begin{center}
\begin{tabular}{lllllll}
& & & & & Type & Example \\
\hline
A & All & S & are & P & universal affirmative & All humans are mortal.\\
E & All & S & are not & P & universal negative & No humans are perfect.\\
I & Some & S & are & P & particular affirmative & Some humans are healthy.\\
O & Some & S & are not & P & particular negative & Some humans are not clever.\\
\end{tabular}
\end{center}
By definition, S is the subject of the conclusion, P is the predicate of the conclusion, M is
the middle term, the major premise links M with P and the minor premise links M with S.
However, the middle term can be either the subject or the predicate of each premise that
it appears in. This gives rise to another classification of syllogisms known as the figure.
The four figures are :
\begin{center}
\begin{tabular}{lcccc}
& Figure 1 & Figure 2 & Figure 3 & Figure 4\\ \hline
Major premise & M-P & P-M & M-P & P-M \\
Minor premise & S-M & S-M & M-S & M-S\\
Conclusion & S-P & S-P & S-P & S-P\\
\end{tabular}
\end{center}
So the total number of possible types of syllogism is $4 \times 4 \times 4 \times 4 = 256$, as there are
four different types of proposition for each of the three propositions and four different
figures. But most of these forms are invalid (the conclusion does not follow logically from
the premises). There are only 19 valid forms of syllogisms, each one having a mnemonic
name used since the medieval times, where only the vowels are relevant.
\begin{center}
\begin{tabular}{cccc}
Figure 1 & Figure 2 & Figure 3 & Figure 4 \\ \hline
Barbara & Cesare & Darapti & Bramantip\\
Celarent & Camestres & Disamis & Camenes\\
Darii & Festino & Datisi & Dimaris\\
Ferio & Baroco & Felapton & Fesapo\\
& & Bocardo & Fresison\\
& & Ferison & \\
\end{tabular}
\end{center}
\pagebreak
\moddocdis{Syllogisms}{02}{01}{Answer sheet 1}
\begin{flushright}
\begin{tabular}{rp{4cm}}
Members of the team & \\ \cline{2-2}
& \\ \cline{2-2}
& \\ \cline{2-2}
\end{tabular}
\end{flushright}
\subsection*{First syllogism}
\begin{center}
\begin{tabular}{c p{12cm}|p{0.5cm}|}
\cline{3-3}
Major premise & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
Minor premise & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
Conclusion & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
\end{tabular}
\vspace{0.7cm}
\begin{tabular}{l|p{3cm}|l|p{3cm}|l|p{3cm}|}
\cline{2-2}\cline{4-4}\cline{6-6}
\small Subject & \rule{0pt}{20pt}& \small Middle term & & \small Predicate & \\
\cline{2-2}\cline{4-4}\cline{6-6}
\end{tabular}
\vspace{0.7cm}
\begin{tabular}{l|p{1.2cm}|p{1.2cm}|p{1.2cm}|l|p{2cm}|l|p{3.8cm}|}
\cline{2-4}\cline{6-6}\cline{8-8}
\small Structure &&& \rule{0pt}{20pt}& \small Figure & & \small\bf Type & \\
\cline{2-4}\cline{6-6}\cline{8-8}
\end{tabular}
\end{center}
\vspace{2cm}
\subsection*{Original syllogisms}
\begin{center}
\begin{tabular}{c p{12cm}|p{0.5cm}|}
\cline{3-3}
Major premise & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
Minor premise & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
Conclusion & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
\end{tabular}
\vspace{0.7cm}
\begin{tabular}{l|p{3cm}|l|p{3cm}|l|p{3cm}|}
\cline{2-2}\cline{4-4}\cline{6-6}
\small Subject & \rule{0pt}{20pt}& \small Middle term & & \small Predicate & \\
\cline{2-2}\cline{4-4}\cline{6-6}
\end{tabular}
\vspace{0.7cm}
\begin{tabular}{l|p{1.2cm}|p{1.2cm}|p{1.2cm}|l|p{2cm}|l|p{3.8cm}|}
\cline{2-4}\cline{6-6}\cline{8-8}
\small Structure &&& \rule{0pt}{20pt}& \small Figure & & \small\bf Type & \\
\cline{2-4}\cline{6-6}\cline{8-8}
\end{tabular}
\end{center}
\pagebreak
\begin{center}
\begin{tabular}{c p{12cm}|p{0.5cm}|}
\cline{3-3}
Major premise & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
Minor premise & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
Conclusion & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
\end{tabular}
\vspace{0.7cm}
\begin{tabular}{l|p{3cm}|l|p{3cm}|l|p{3cm}|}
\cline{2-2}\cline{4-4}\cline{6-6}
\small Subject & \rule{0pt}{20pt}& \small Middle term & & \small Predicate & \\
\cline{2-2}\cline{4-4}\cline{6-6}
\end{tabular}
\vspace{0.7cm}
\begin{tabular}{l|p{1.2cm}|p{1.2cm}|p{1.2cm}|l|p{2cm}|l|p{3.8cm}|}
\cline{2-4}\cline{6-6}\cline{8-8}
\small Structure &&& \rule{0pt}{20pt}& \small Figure & & \small\bf Type & \\
\cline{2-4}\cline{6-6}\cline{8-8}
\end{tabular}
\end{center}
\vspace{2cm}
\begin{center}
\begin{tabular}{c p{12cm}|p{0.5cm}|}
\cline{3-3}
Major premise & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
Minor premise & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
Conclusion & \rule{0pt}{20pt}&\\
\cline{2-2}\cline{3-3}
\end{tabular}
\vspace{0.7cm}
\begin{tabular}{l|p{3cm}|l|p{3cm}|l|p{3cm}|}
\cline{2-2}\cline{4-4}\cline{6-6}
\small Subject & \rule{0pt}{20pt}& \small Middle term & & \small Predicate & \\
\cline{2-2}\cline{4-4}\cline{6-6}
\end{tabular}
\vspace{0.7cm}
\begin{tabular}{l|p{1.2cm}|p{1.2cm}|p{1.2cm}|l|p{2cm}|l|p{3.8cm}|}
\cline{2-4}\cline{6-6}\cline{8-8}
\small Structure &&& \rule{0pt}{20pt}& \small Figure & & \small\bf Type & \\
\cline{2-4}\cline{6-6}\cline{8-8}
\end{tabular}
\end{center}
\pagebreak
\moddocdis{Syllogisms}{02}{01}{Answer sheet 2}
\newcounter{syll}
\newcommand{\syll}[3]{\stepcounter{syll}\subsection*{Syllogism \arabic{syll} {\normalsize \hspace{1cm} {\sf Valid} $\Square$ \hspace{1cm} {\sf Invalid} $\Square$}} {#1\par #2\par $\therefore$ #3\par\bigskip
}
\par\bigskip
\begin{tabular}{cp{10cm}}
Valid revision & \rule{0pt}{20pt}\\ \cline{2-2}
& \rule{0pt}{20pt}\\ \cline{2-2}
& \rule{0pt}{20pt}\\ \cline{2-2}
\end{tabular}}
\begin{flushright}
\begin{tabular}{rp{4cm}}
Members of the team & \\ \cline{2-2}
& \\ \cline{2-2}
& \\ \cline{2-2}
\end{tabular}
\end{flushright}
\syll{Fish live in water.}{Monkeys are not fish.}{Therefore, monkeys do not live in the water.}%Invalid
\syll{No Frenchmen like plum pudding}{All Englishmen like plum pudding}{Englishmen are not Frenchmen}%Valid
\syll{All lions are fierce}{Some lions do not drink coffee}{Some creatures that drink coffee are not fierce}%Invalid
\syll{No pigs can fly}{All pigs are greedy}{Some greedy creatures cannot fly}%Valid
\syll{Some pillows are soft}{No pokers are soft}{Some pokers are not pillows}%Invalid
\syll{No emperors are dentists}{All dentists are dreaded by children}{No emperors are dreaded by children}%Invalid
\syll{Nothing intelligible ever puzzles me}{Logic puzzles me}{Logic is unintelligible}%Valid
\pagebreak
\syll{If war is declared, the enemy country will be invaded}{War is not declared}{The enemy country will not be invaded}%Invalid
\syll{Ninety-nine Cretans in a hundred are liars}{Epimenides is a Cretan}{Epimenides is a liar}
\syll{No bald creatures need a hairbrush}{No lizards have hair}{No lizards need a hairbrush}
\pagebreak
\moddoc{Cards with parts of syllogisms}
\begin{center}\Large
\begin{tabular}{|c|}
\hline
All men are mortal. \rule{0pt}{20pt}\\[2pt] \hline %1
Socrates is a man.\rule{0pt}{20pt}\\[2pt] \hline
Socrates is mortal.\rule{0pt}{20pt}\\[2pt] \hline
No mammal has feathers.\rule{0pt}{20pt}\\[2pt] \hline %2
All horses are mammals.\rule{0pt}{20pt}\\[2pt] \hline
No horse has feathers.\rule{0pt}{20pt}\\[2pt] \hline
No lazy people pass exams.\rule{0pt}{20pt}\\[2pt] \hline %3
Some students pass exams.\rule{0pt}{20pt}\\[2pt] \hline
Some students are not lazy.\rule{0pt}{20pt}\\[2pt] \hline
All cats are black.\rule{0pt}{20pt}\\[2pt] \hline %4
Victoria is a white cat.\rule{0pt}{20pt}\\[2pt] \hline
Victoria is a black cat.\rule{0pt}{20pt}\\[2pt] \hline
All fruit is nutritious.\rule{0pt}{20pt}\\[2pt] \hline %5
All fruit is tasty.\rule{0pt}{20pt}\\[2pt] \hline
Some tasty things are nutritious.\rule{0pt}{20pt}\\[2pt] \hline
Some cats have no tails.\rule{0pt}{20pt}\\[2pt] \hline %6
All cats are mammals.\rule{0pt}{20pt}\\[2pt] \hline
Some mammals have no tails.\rule{0pt}{20pt}\\[2pt] \hline
Some small birds live on honey.\rule{0pt}{20pt}\\[2pt] \hline %7
All birds that live on honey are colourful.\rule{0pt}{20pt}\\[2pt] \hline
Some colourful birds are small.\rule{0pt}{20pt}\\[2pt] \hline
\end{tabular}
\end{center}
\begin{center}\Large
\begin{tabular}{|c|}
\hline
No homework is fun.\rule{0pt}{20pt}\\[2pt] \hline %8
Some reading is homework.\rule{0pt}{20pt}\\[2pt] \hline
Some reading is not fun.\rule{0pt}{20pt}\\[2pt] \hline
All kittens are playful.\rule{0pt}{20pt}\\[2pt] \hline %9
Some pets are kittens.\rule{0pt}{20pt}\\[2pt] \hline
Some pets are playful.\rule{0pt}{20pt}\\[2pt] \hline
No healthy food is fattening.\rule{0pt}{20pt}\\[2pt] \hline %10
All cakes are fattening.\rule{0pt}{20pt}\\[2pt] \hline
No cakes are healthy.\rule{0pt}{20pt}\\[2pt] \hline
All informative things are useful.\rule{0pt}{20pt}\\[2pt] \hline %11
Some websites are not useful.\rule{0pt}{20pt}\\[2pt] \hline
Some websites are not informative.\rule{0pt}{20pt}\\[2pt] \hline
All the industrious boys in this school have red hair.\rule{0pt}{20pt}\\[2pt] \hline %12
Some of the industrious boys in this school are boarders.\rule{0pt}{20pt}\\[2pt] \hline
Some boarders in this school have red hair.\rule{0pt}{20pt}\\[2pt] \hline
All wicked persons will face judgment\rule{0pt}{20pt}\\[2pt] \hline %12
Some humans are wicked\rule{0pt}{20pt}\\[2pt] \hline
Some humans will face judgment\rule{0pt}{20pt}\\[2pt] \hline
\end{tabular}
\end{center}
\end{document}