Week 3

More Syllogisms

Last week we introduced contraposition of predicates:
a ⊨ ¬b is equivalent to b ⊨ ¬a.
This is a basic property of the negation of predicates. This equivalence enabled us to generate four new sound syllogisms, starting from the archetypical syllogism, barbara.

Last week we looked at the negation, ¬a, of a predicate, ¬a.
This week we take things to a different level.

We write a ⊭ b to mean that it is not the case that a ⊨ b.
(Just as we write a ≠ b to mean that it is not the case that a = b. )

the assertion, a ⊨ b, means, every a is b.
its denial, a ⊭ b, means, it is not the case that every a is b
equivalently, a ⊭ b, means, some a is not b.

Denial is just negation at a different level; we call this the meta level.

When we consider the truth and falsity of predicates applied to individuals within our universe, we say we are working at the object level.

When we consider whether a proposition is valid or invalid in some universe, we say we are working at the meta level.

Last week we used object-level contraposition to derive four new syllogisms, starting from Barbara. This week we will use meta-level contraposition to derive two more new syllogisms from each of the five we now have, to give a total of 15 sound syllogisms.

Most of the slides for this week are in a single pdf linked immediately below this paragraph. Slide numbers are given before each video, together with a further link to this pdf, in case you find that more convenient.

Slides
Contraposition of Propositions

Meta-level contraposition takes a sound syllogism and produces two new syllogisms — each assuming the denial the original conclusion, assuming one of its assumptions, and inferring the denial of the other. This contraposition of propositions is object-level contraposition.

This is much clearer in symbolic form. You can see an example below, even before you start playing the video. Before you start the video try to decide, what can you infer to replace the question marks??

slides 4..8

Now you have watched the video, try to decide how many different sound syllogisms you can derive from those you already know.

Once you've tried that, and have an answer, you're ready for the next video.

Particular Propositions

Make a mental note of your answer to the question concerning how many different sound syllogisms you can now derive. We'll come back to that later. For the time being we will take a good look at the new propositions we've introduced with this meat-level negation.

You have already met Aristotle's universal propositions, universal assertion, "every a is b", and universal denial, "no a is b". The negations of these are his particular propositions: particular denial "some a is not b", and particular assertion, "some a is b".

slides 9..10

Having watched this video, you should be able to translate any syllogism into symbolic form.

You should also check you understand the difference between these symbols:
¬ ⊭ .

Aristotle's Square of Opposition

This is how Aristotle organised his propositions. Each proposition Asserts or Denies, and is Universal or Particular. He arranges them as a square using these two dimensions; then the diagonally opposed corners contradict each other.

slides 11..12
Video
Using Venn Diagrams to show a Syllogism is Sound
slides 12..13
Constructing a Counterexample
slides 14..17
Presenting your justifications and counterexamples
slides 18..20
The 15 Sound Syllogisms

We're now ready to complete our collection of sound syllogisms. Using substitution and both kinds of contraposition, we can derive 14 new syllogisms from barabara to give a total of 15 sound syllogisms.

In fact these are all the sound syllogisms. For this course, we just take that as a fact, but some of you may choose to think about proving that there are no other sound syllogims.

slides 21..24
Aristotle's Syllogisms
slides 28,29
Video

We describe the traditional presentation of the syllogisms (slide 28), and introduce Aristotle's existential assumption (slide 29), on the basis of which he derives nine more syllogisms. These are not sound in our sense — but every counterexample to one of these must include an empty predicate.

For each syllogism, the conclusion is a categorical proposition relating a subject s to a predicate p The assumptions are categorical propositions relating p and s to a middle predicate, m. In the traditional presentation of a syllogism, we use s m p as names for the three predicates. A syllogism is always presented as follows: the proposition relating m to the predicate, p, is given as the first assumption; the proposition relating m to the subject, s as the second assumption; the conclusion relates s to p, with s as the subject.

You should be able to convert any syllogism to this standard form. The next section is for interest only; it explains the where the classical names of the syllogisms come from. You will not be expected to name, or explain the names of the syllogisms.

The classical names

The three aeio vowels in the name of each syllogism (each name includes 3 of these vowels) signify the forms of the three categorical propositions.

The names are in a code that tells how the syllogism in question is derived from one of the following four syllogisms: barbara, celarent, darii, ferio. The first letter of the name of each syllogism matches the name of the syllogism it is derived from. When one of the consonants smc follows one of the vowels aeio, it tells us how the corresponding proposition should be changed:
The c in bocardo and baroco corresponds to our contrapositive construction of these rules.

The s in festino relates it to ferio via a predicate contraposition, and the two occurrences of s in fresison show it is derived from ferio using two predicate contrapositions, ferison again uses one predicate contraposition.

The letter m means that we swap s and p — observe that each name with an m ends with s, which represents the contraposition required to put s and p back in the correct order. When we swap s and p we also have to change the order of the premises, but first we must apply any the further contrapositions required if there is another s in the name.

In addition to the fifteen sound syllogisms we have derived, Aristotle listed nine more syllogisms that depend on the so-called existential assumption. Any counter-example to these rules must include an empty predicate — one true of no individuals. We may derive these rules from our sound syllogisms if we make the existential assumption, that each predicate, p, is true of some individual. This corresponds to the assumption that some p is p, which we formalise as p ⊭ ¬p.

The existential assumption

Slide 29 introduces Aristotle's existential assumption, and its formulation in our setting.

It shows the existential assumption formalised as a new rule, with no assumptions and the conclusion that some s is s; s ⊭ ¬s. It uses an instance of darii, together with the existential assumption, to derive some s is p from the additional assumption that all s are p.

In the slide below, we combine this derivation with an instance of ferio to derive Celaront.

You should construct a counter-example to show that celaront is not sound. You can find eight more Aristotelian syllogisms that require the existential assumption in Syllogisms on Wikipedia. You can test your understanding by deriving these from the existential assumption in similar manner.

There are further slides in the deck, that you should use to check your understanding. Please post any questions on Piazza.

slides 25, 26

These are intended as prompts for revision: to give examples of things you should now understand.

slide 27

An example of an argument combining two rules to give a derivation, and then applying contraposition.

slides 31, 32

These are quizzes: not-for-credit problems you can attempt to check your understanding.

Solutions to these will be posted on Thursday.

There is one more video, concerning precedence in Haskell, below. This is here to explain the precedence declarations used in the tutorial exercise for this week. You don't need to understand this at this stage, you can just use the code provided, but some of you will be curious to know more.

Precedence

Here is a short video relevant to part B of this week's tutorial submission. In it we introduce the important idea that an expression such as a ⋀ b ⋁ c is really just a written representation of a tree. This expression, without parentheses, could represent either (a ⋀ b) ⋁ c or a ⋀ (b ⋁ c). To specify which we mean, we either need to add the parentheses, or rely on rule of precedence -- just as in algebra where we use a mnemonic to remember the precedences of the operations such as, + × ÷ . I was taught the mnemonic BODMAS, which stands for Brackets Of Division Multiplication Addition Subtraction. An operator with higher precedence comes before an operator with lower precedence: we compute an operator of higher precedence before an operator of lower precedence. In logic, conventionally, has higher precedence than , just as in arithmetic, × has higher precedence than + .

Slides
Video

In Haskell && has precedence 3, and || has precedence 2, as you can see from the documentation. We define our operations &:& |:| on predicates to have the same precedences and our relations |= , |/= to have precedence 0 ,
infixr 0 |=
infixr 0 |/=
infixr 2 |:|
infixr 3 &:&
so we evaluate any logical operators on either side of the turnstile to give two predicates a , b before we check whether a |= b.