This set of slides is split between two videos, to allow you
space to think.
Euler (video)
We start by introducing barbara, the simplest classical syllogism, and the
proposition, "all a are b", known as universal assertion. We introduce the mathematical setting we will use
for our investigation of deduction, briefly
review Euler diagrams, and introduce Venn diagrams.
Once you have watched the video above, check your understanding by
drawing a few Euler diagrams and the corresponding Venn diagrams,
each for a universe with three predicates. How many different Euler
diagrams can you draw for three predicates?
Once you've done that, you're ready for the next video.
Venn (video)
In the second video, we return
to discuss the relationship between Euler and Venn diagrams, and introduce some notation.
Having watched this video, you should be able to count how many different Euler
diagrams there should be for three predicates. Can you draw them all?
You should also check you can recognise and name the following
symbols: ¬ ⋀ ⋁ ⊨ .
Quiz
Which of the following are valid in the universe
represented by this Euler diagram?
a ⊨ b ?Yes, a ⊨ bb ⊨ c ?No, b ⊭ cc ⊨ ¬a ?Yes, c ⊨ ¬aa ⊨ ¬c ?Yes, a ⊨ ¬cb ⊨ ¬c ?No, b ⊭ ¬cb ⊨ a ?No, b ⊭ ¬aLecture-2b-barbara
We use Venn diagrams to show that barbara is sound.
Here are three Venn diagrams, 1, 2, 3, and three Euler diagrams, A,
B, C. There are the two matching pairs, for which the Euler
diagram represents a universe in which every unshaded region in the
Venn diagram is inhabited.
The challenge is to first select the non-matching pair. Then you
can select the two matching pairs. For the non-matching pair, find a
Venn diagram to match the Euler diagram and vice versa.
1 - ANon matching, in the
Venn diagram a ∧
b empty.1 - B ?No, the Venn
diagram has six non-shaded regions. How many regions are there in
the Euler diagram?1 - C ?Yes, even though
they look a bit different, these two match. You can check each of
the six regions.2 - AThese do not match.The Venn diagram
has 7 unshaded regions; the Euler diagram has only 6 regions. 2 - B ?Yes, these match!2 - C ?No, 2 has 7
non-shaded regions, while C has 6 regions. 3 - ANo, 3 has 7
non-shaded regions, while A has 6 regions. If this is the first
region you've selected, Well done! No you have to, work the correct diagrams
to match each of these.
3 - B ?No, in the Venn
diagram ¬a ∧ ¬b ∧ c is shaded, but this region appears
in the Euler diagram.3 - C ?No, in the Venn
diagram ¬a ∧ ¬b ∧ c is shaded, but this region appears
in the Euler diagram.Lecture-2c-universals
Our second syllogism requires a new form of proposition --
universal denial. Introducing negation gives a new syllogism as an
instance of barbara.
Which of the following is equivalent to a ⊨ b? When they
are not equivalent, give a counterexample, by taking a ⊨
b to be, "Every man is mortal" and describing what the
invalid proposition in question would say in English.
b ⊨ aNo, b ⊭ a; every man is
mortal, but it's not the case that every mortal is a man. (Think of
Socrates' pet cat.) ¬a ⊨ ¬bNo, ¬a ⊭ ¬b; Socrates' cat
is not a man, but Socrates' cat is not immortal.a ⊨ ¬bNo, a ⊭ ¬b; it's not the
case that every man is immortal; for example, Socrates is not immortal.¬b ⊨ ¬aYES, ¬b ⊨ ¬a; If every man
is mortal, then every immortal is not a man, or to say it
differently, no immortal is a man.
Lecture-2e-syllogisms
Syllogisms for free! We see how some simple reasoning allows us
to derive three more syllogisms.
You should first use the Venn
diagram method to convince yourself that this syllogism is
sound. You can do this by reasoning about the regions in the Venn
diagram that are empty.
Which of the following rules may be derived from Camestres using
a single predicate contraposition?
Which proposition from Camestres was contrapositioned?
a ⊨ bNoc ⊨ ¬bYesc ⊨ ¬aNo
Which proposition from Camestres was contrapositioned?
a ⊨ bNoc ⊨ ¬bNoc ⊨ ¬aYes
Which proposition from Camestres was contrapositioned?
a ⊨ bNoc ⊨ ¬bYes, but it's not the
only one.c ⊨ ¬aYes, but it's not the
only one.
Which proposition from Camestres was contrapositioned?
a ⊨ bNoc ⊨ ¬bNoc ⊨ ¬aNo
If you've got this far you may be eager for more questions.
Ask me more!
Which of the four syllogisms can not be
derived from Camestres by a sequence of one or more predicate
transpositions?
You should have already derived this one.
You should have already derived this one.
You should be able to reach from Canestres this with two predicate transpositions.
Correct! Can you either show this is not sound, using the Venn
diagram method, or find a wayto reach this from Camestres using a
combination of predicate transpositions and substitutions?
Legacy Lecture from 2019
Legacy recording from a lecture delivered in 2019— you can use
this to check your understanding.
You should listen at least from 28m02 onwards, where I propose a
problem for you to ponder.
Video
Quiz
The video ends with a question that leads to another form of
contraposition: contraposition of propositions. This will
lead us to more syllogisms for free. We encourage you to think about
your answer concerning the rules for sale of alcohol in Scotland,
and try to generalise what you have learnt to our original rule,
Barbara, to derive two more new syllogisms.
Don't worry if this doesn't make sense, or if you've run out of
time.
We will look at this idea in detail next week.
