Module 1 of “Introduction to non classical

logics”

LOGIC (LM) 2017-18

Hykel Hosni

http://www.filosofia.unimi.it/~hosni/

Dipartimento di Filosofia, Universita degli Studi di Milano

E-mail address: hykel.hosni@unimi.it

Contents

Chapter 1. Review of classical propositional logic 5

1. Language and sentences 5

2. Satisfiability and consequence 18

3. Using sentences to describe models 24

Chapter 2. Classical logic as a model of rational reasoning 27

1. Propositional arguments 27

2. Agents as consequence relations 32

Chapter 3. Nonmonotonic logics 35

1. Gabbay-Makinson constraints on rational expectation 36

2. Preferential semantics 42

3. Key references 47

Chapter 4. Further projects 49

1. Abduction is non-monotonic 49

2. 2,3,4 . . . infinity 50

3

CHAPTER 1

Review of classical propositional logic

We will introduce classical propositional logic as the analysis of

propositional connectives and of their role in determining

the key notions of logical consequence and tautology.

The key concepts of this section are:

• language and sentences

• compositionality

• satisfiability

• logical consequence

• tautology

1. Language and sentences

It is a good approximation to say that sentences are characterised by the fact

that it makes sense to ask whether they are true or false. Without venturing

into slippery philosophical considerations, we shall content ourselves with

the following stipulation

A sentence is a well-formed linguistic expression (in some language, that is,

either natural or artificial) whose characteristic feature consists of admitting

either Yes or No (and exactly one of them) for answers, if put as questions.

To see the idea, which is indeed rather simple, consider the following natural

language expressions:

(1) The Pieta Rondanini is in Florence

(2) The Pieta Rondanini is in Milano

(3) Where is the Pieta Rondanini?

(4) The editor-in-chief of A.C. Milan is vegetarian

A moment’s reflection should suffice to convince you that (1) and (2) make

unambiguously sense. You may not know what kind of thing Pieta Ron-

danini is, but you may possess enough contextual information to make sense

of them. Note that this doesn’t mean assenting to their truth (or their fal-

sity). To do be able to do this, you must know that it is a statue. And

still, you may not know the answer, but you can easily come up with meth-

ods (experiments) which unambiguously settle the question. This captures

5

6 1. REVIEW OF CLASSICAL PROPOSITIONAL LOGIC

the essence of sentences: they can always be turned unambiguously into

YES/NO questions.

(3) is clearly a meaningful expression, which however differs radically from

(1) and (2): it expresses an open-ended question, so it cannot be turned

into a YES/NO question. Hence it cannot be considered a sentence (pretty

obvious, ain’t it, for a question?).

Much less obvious is the case illustrated by (4). We can certainly make sense

of it, we can easily graps its meaning, but we cannot as easily come up with

experiments which allow us to decide YES/NO. Pause for a second and try

to design an experiment which falsifies (4).

We shall cut a very long story short, one which started with contributions

of no less than Gottlob Frege and Bertrand Russell and stipulate that cases

like (4) are ruled out in our analysis because.

We start with a language L, which for our purposes is going to be a finite

set of propositional variables L = {p1, p2, . . . , pn}. Intuitively, elements of Lare thought of as the smallest units for which it makes sense to ask whether

they are true or false. Hence we can design them to be our fundamental

semantic objects.

Example 1.1. Consider the expression

3 + 2 = 6 (1)

It clearly makes sense to ask whether (1) is true or false. But it doesn’t

make sense to ask the same question for

3 + 2. (2)

Aside. This characterisation of sentences is closely related to the notion

of event as it is developed in the theory of probability. This links tightly

logic and probability. Indeed this is one clear step in which we can appreciate

how probability builds on logic. 1

1.1. Propositional connectives and sentences. Those building blocks

can then be used to form increasingly more complex sentences by suitably

applying four propositional connectives,

C = {¬,∧,∨,→} ,

1 See, e.g. Flaminio, T.; Godo, L.; and Hosni, H. “On the logical structure of de

Finetti’s notion of event”. Journal of Applied Logic, 12(3): 279–301. 2014.

1. LANGUAGE AND SENTENCES 7

which read as “negation”, “conjunction”, “disjunction” and “implication”,

respectively.

Their definition (according to Tarski’s “no-arbitrary choices” maxim) will

then follow two desiderata:

(1) produce “more complex” sentences from simpler ones preserving

“meaningfulness”

(2) adhere (to some extent) to the common usage of those words

We postpone the justification of our choice of connectives to the next section,

when we actually define their semantic meaning. For the time being we are

interested in seeing how they can be used to combine the (finite) set of

propositional variables in L to give rise to the infinite set of sentences of L,

denoted by SL.

We start by saying that all propositional variables are sentences, i.e.

SL1 = L.

Suppose now that we have combined several simpler sentences to give rise to

more complicated ones, and have done so n times. The following condition

describes rigorously how we get from n to n+ 1:

SLn+1 = SLn ∪ {¬θ, (θ ∗ ϕ) | θ, ϕ ∈ SLn, ∗ ∈ {∧,∨,→}} .

Note that n is arbitrary here. Finally we add a condition that says that

we can iterate this (recursive) construction for as long as we please, and by

doing so we get an infinite set out of the combination of a finite number of

building blocks (SL0) and four connectives:

SL =⋃n∈NSLn.

More compactly:

Definition 1.1 ( SL). Fix L. The set of sentences of L is defined recursively

by using the connectives in C as follows:

SL1 = LSLn+1 = SLn ∪ {¬θ, (θ ∗ ϕ) | θ, ϕ ∈ SLn, ∗ ∈ {∧,∨,→}}

SL =⋃n∈NSLn.

This definition fives us a rigorous way to describe the building mechanism

of complex sentences from simpler ones. We can indeed imagine the set of

sentences as a series of concentric circles, emanating out of L. There are

two ways to move from this inner circle. We can either pick p ∈ SL1 and

8 1. REVIEW OF CLASSICAL PROPOSITIONAL LOGIC

Figure 1. The sentences of L

stick negation in front of it, giving rise to ¬p which then belongs to SL1.

Otherwise we could have taken two variables in L, say p, q and apply any of

the binary connectives to them to get to an element of SL1, say p ∧ q.

Example 1.2. Suppose p, q ∈ L and p reads as “Alain Delon si refused to

act”, and q reads “Mastroianni acted in Il Bell’Antonio”. We can represent

the sentence

Had Alain Delon not refused, Mastroianni would have not

acted in Il Bell’Antonio

by the sentence

θ = (¬p→ ¬q),where θ ∈ SL2.

Omitting parentheses

The following is a small set of conventions that may be used to reducing

significantly the number of parentheses in our sentences.

(1) Remove the outmost parentheses. i.e instead of writing (¬p→ ¬q),we shall write ¬p→ ¬q etc.

(2) If a sentence contains only one binary connective, remove all the

parentheses by associating to the left. Instead that is of writing

((θ(∧ϕ) ∧ ψ) we write θ ∧ ϕ ∧ ψ.

(3) Prioritise connectives as follows: “→”,“∧”,“∨”, “¬” and remove

parentheses by applying the following rules:

1. LANGUAGE AND SENTENCES 9

(a) ¬ applies to the simplest sentence that follows it;

(b) ∧ applies to the simplest sentences around it;

(c) ∨ applies to the simple sentences around it;

(d) → applies to the simple sentences around it.

In practice convention 1. is always used; convention 2. is used sometimes,

and convention 3. is often used for just one or two connectives. It is in fact

much easier to use parentheses than to work out the priority of connectives.

Example 1.3. To decide whether

θ ∨ ¬ϕ→ ψ (3)

is an element of SL, we need to check whether the rules for omitting paren-

theses can be applied backwards.

The strongest binding connective is negation, so

θ ∨ (¬ϕ)→ ψ

then disjunction is the second-strongest connective

(θ ∨ (¬ϕ))→ ψ

and finally the outmost parentheses

((θ ∨ (¬ϕ))→ ψ)

Remark 1.1. Note that the prioritisation of connectives is analogous to the

prioritisation of arithmetical operations. In this context,

−3 + 2 · 5

is unambiguously processed as

((−3) + (2 · 5)).

Exercise 1.1. Restore the parentheses in ¬q ∨ r ∧ s→ q

Exercise 1.2. Decide whether the following can be taken as sentences. You

need to justify your answers.

(1) there exists a transcendental number

(2) 2+3=9

(3) All natural numbers are red

(4) is global temperature rising?

(5) the global mean temperature is rising

(6) regular exercise increases the strength of the immune system

(7) regular exercise is good for you

(8) regular exercise is boring

(9) regular exercise is regular

10 1. REVIEW OF CLASSICAL PROPOSITIONAL LOGIC

Notation. Lower case Greek letters θ, ϕ, ψ, etc. denote elements of

EL, whereas uupper case Greek letters Γ,∆, ω denote subsets of SL. Hence

θ, ϕ ∈ SL and Γ = {θ, ϕ} ⊆ SL.

Sentences in natural language are very often ambiguous, i.e. allow for non-

unique parsing, or readability. Take for instance the sentence

I’m glad I’m a logician, and so are you.

Uttered by me to you, this admits at least three readings:

(1) I’m glad that I am a logician and you are also glad about that

(2) I’m glad that we are all logicians

(3) I’m glad that I am a logician and that you are glad you are logicians

The following result, of impossible-to-overestimate practical importance, il-

lustrate the advantages of dealing with a recursively defined set of sentences

and with that only: there can never be any form of ambiguity in our logical

expressions.

Theorem 1.1. Unique readibility

Let θ, ϕ1, ϕ2 ∈ SL. Exactly one of the following holds:

θ = p ∈ L;

θ = ¬ϕ1;

θ = (ϕ1 ∧ ϕ2);

θ = (ϕ1 ∨ ϕ2);

θ = (ϕ1 → ϕ2);

where ϕ1, ϕ2 are the unique sub-sentences of θ (i.e. if ϕ1, ϕ2 ∈ ELn then

θ ∈ ELn+1).

Proof. Omitted. �

Exercise 1.3. List all the sub-sentences (aka sub-formulas) of

((p→ q) ∨ (¬(p→ q))

Exercise 1.4. Suppose two switch buttons are located at the two sides of

a long corridor. Suppose that

• p: the RHS switch is “on”

• q: the LHS switch is “on”

• r: light is on

Build an appropriate θ ∈ SL, with L = {p, q, r} describing when the light

is on.

1. LANGUAGE AND SENTENCES 11

1.2. Sentences and semantics. We can now make the vague concept

of “meaningful expression” mathematically precise, and we do so by intro-

ducing the notion of propositional valuation. The key idea is that a valuation

is a function mapping a propositional language L into a set which we call

the set of truth values. Note that the choice of the set of truth values is

a representational choice, and distinct choices lead to distinct non classical

logics. It is one distinctive feature of classical logic to have the binary set

{0, 1} as the codomain of propositional valuations. And it is often the case

that we refer to those values with “false” and “true”, respectively.

Definition 1.2 (Valuation). A two-valued propositional valuation on L 6= ∅is a function

v : L → {0, 1}.

The set of all two-valued propositional valuations is denoted by V.

Example 1.4. Suppose L = {p, q}, V = {v1, v2, v3, v4}, where

v1(p) = 0; v4(q) = 0

v2(p) = 0; v3(q) = 1

v3(p) = 1; v2(q) = 0

v4(p) = 1; v1(q) = 1

Exercise 1.5. (1) Let L = {p, q, r}, describe the set of all valuations

on L.(2) Let L = {p1, p2, . . . , , pn} what is the cardinality of V?

We are now interested in understanding how this basic semantic notion

propagates across the elements of SL. This leads us to the first fundamental

property of classical propositional logic, compositionality.

For a fixed L we can define the truth conditions expressed by the connectives

in C by means of the Boolean tables. The idea is to consider all the possible

valuations on L (left-hand-side column) whilst writing on the r-h-s column

the truth value of the (non-atomic) sentences of interest.

So fix L = {p, q}. The following table sums up the truth-conditions for

classical propositional connectives.

A two-valued propositional valuation which satisfies the above conditions is

called a classical valuation.

Note that a function v : L → {0, 1} is a complete description of a logical

state of affairs.

12 1. REVIEW OF CLASSICAL PROPOSITIONAL LOGIC

p q ¬p p ∧ q p ∨ q p→ q

0 0 1 0 0 1

0 1 1 0 1 1

1 0 0 0 1 0

1 1 0 1 1 1Table 1. The meaning of the connectives in C

Example 1.5. We insisted earlier on that the object of logical (and math-

ematical) reasoning is constituted of sentences, i.e. linguistic expressions

which will command either Yes or No if formulated as questions. So let us

ask whether the following is true

(q ∧ (p→ q))→ p (4)

Well, it depends, and in particular it depends on how we distribute truth val-

ues on the language {p, q}. However we can get an intuition by interpreting

informally p and q as follows

p = you exercise,

q = you are fit.

This allows us to reason by cases. (4) is false if you don’t exercise and you

are fit, but it is true if you exercise, whether or not you are fit.However one

practical purpose of logic is precisely to adjudicate arguments independently

of any informal interpretation.

As a first step let us just copy all the possible truth-value distributions, as

illustrated by Table 2

p q (q ∧ (p → q)) → p

v1 0 0 0 0 0 0

v2 0 1 1 0 1 0

v3 1 0 0 1 0 1

v4 1 1 1 1 1 1Table 2.

Then we work out the truth-value of the innermost connective according to

the boolean table for →, which gives us the result described in Table 3.

We move on computing the truth-value of the second-innermost connective,

according to the boolean table for ∧, as described in Table 4.

1. LANGUAGE AND SENTENCES 13

p q (q ∧ (p → q)) → p

v1 0 0 0 0 1 0 0

v2 0 1 1 0 1 1 0

v3 1 0 0 1 0 0 1

v4 1 1 1 1 1 1 1Table 3.

p q (q ∧ (p → q)) → p

v1 0 0 0 0 0 1 0 0

v2 0 1 1 1 0 1 1 0

v3 1 0 0 0 1 0 0 1

v4 1 1 1 1 1 1 1 1Table 4.

Finally, we can use this latter computation to calculate the truth-value of

the only remaining (and therefore main) connective. To make it more visible

and to signal that this coincides with the truth values taken by the under all

possible distributions of truth-values, we enclose this column within a triple

vertical line, as described in Table 5.

p q (q ∧ (p → q)) → p

v1 0 0 0 0 0 1 0 1 0

v2 0 1 1 1 0 1 1 0 0

v3 1 0 0 0 1 0 0 1 1

v4 1 1 1 1 1 1 1 1 1

Table 5. The distributions of truth-values over {p, q}makes

it clear that the truth of (q ∧ (p→ q))→ p is contingent.

This argument shows quite clearly that boolean tables allow us to analyse the

sentence of interest in a completely formal way, free that is, of any particular

intuitions we may or may not associate to natural language interpretation

of L. It also illustrate a very important point about logic, which deserves

to be well emphasised:

This also shows that logic is at root conditional – conditional that is

on the fact that certain distributions of truth-values are given. This is why

when arguing logically we need to take into account the whole set V, i.e. all

logical distributions.

Exercise 1.6. Write down the boolean table for

14 1. REVIEW OF CLASSICAL PROPOSITIONAL LOGIC

(1) exclusive disjunction.

(2) joint denial (neither . . . nor)

(3) Because

Exercise 1.7. Compute the truth-values of the following under the assump-

tion that p is evaluated to 1 and q is evaluated to 0:

(1) ¬p ∧ q(2) ¬q ∧ p(3) p→ p

(4) q → q

(5) q → ¬q(6) (q ∨ ¬q) ∧ p

Exercise 1.8. Construct appropriate boolean tables for the following sen-

tences

(1) (p→ q) ∨ ¬p(2) ((p→ q)→ s)→ ¬t

Exercise 1.9. Find a sentence for each of the following distribution of

truth-values.

(1) 〈0, 0, 1, 0〉(2) 〈0, 1, 1, 0, 1, 0, 0, 1〉(3) 〈0, 1, 1, 1, 1〉

where the vector notation is chosen to emphasise the natural order of the

valuations’d indexes, i.e. the first coordinate is the value of the sentence

under v1, the second coordinate the value of the sentence under v2, etc.,

with the conventional assumption that truth-values are written down in

lexicographic order.

1.3. Compositionality (truth-functionality). Strictly speaking, boolean

tables can be applied only to propositional variables (why?). However this

restriction turns out to be immaterial, as shown by the next Lemma which

leads to a rather general consideration, and indeed helps us identifying

an extremely important property of classical logic, a property which goes

by the rather odd-sounding label of Principle of compositionality or truth-

functionality which has bee finally made clear by Alfred Tarski in 1933.

This explains why sometimes we refer to the semantics of classical logic as

tarskian semantics.

1. LANGUAGE AND SENTENCES 15

Theorem 1.2. Let θ, ϕ ∈ SL. Valuations v : L → 2 extend uniquely to SLaccording to the tarskian condition:

v(¬θ) = 1 ⇔ v(θ) = 0

v(θ ∧ ϕ) = 1 ⇔ v(θ) = v(ϕ) = 1

v(θ ∨ ϕ) = 0 ⇔ v(θ) = v(ϕ) = 0

v(θ → ϕ) = 0 ⇔ v(θ) = 1 and = v(ϕ) = 0.

Proof. The proof is by induction on the construction of SL. The base

case is immediately settled, for it requires us to check that the Lemma holds

for SL1 = L, so nothing to check here.

We now need to define a useful inductive hypothesis. Here is one.

GIVEN: (INDUCTIVE HYPOTHESIS) Lemma is true

for SLn.

WANTED: : The Lemma holds for SLn+1.

So let θ ∈ SLn+1. By definition

SLn+1 = SLn ∪ {¬θ, (θ ∗ ϕ) | θ, ϕ ∈ SLn, ∗ ∈ {∧,∨,→}} .

If θ ∈ SLn, then we can invoke the inductive hypothesis which guarantees

that v(θ) is well defined.

If θ 6∈ SLn then it must be the case that θ ∈ SLn+1 \SLn and then we have

to check (by Unique readibility) exactly four cases:

(1) θ = ¬ϕ1. Let

v(θ) =

{1 if v(ϕ1) = 0;

0 if v(ϕ1) = 1.

(2) θ = ϕ1 ∧ ϕ2. Let

v(θ) =

{1 if v(ϕ1) = v(ϕ2) = 1;

0 otherwise.

(3) θ = ϕ1 ∨ ϕ2. Let

v(θ) =

{0 if v(ϕ1) = v(ϕ2) = 0;

1 otherwise.

(4) θ = ϕ1 → ϕ2. Let

v(θ) =

{0 if v(ϕ1) = 1 and v(ϕ2) = 0;

1 otherwise.

16 1. REVIEW OF CLASSICAL PROPOSITIONAL LOGIC

In the above, clearly, ϕ1, ϕ2 ∈ SLn. The inductive hypothesis guarantees

that the previous stipulations are well-defined, and this concludes the proof.

�

Remark 1.2. The above discussion (together with Theorem 1.1) guarantees

that the method of boolean tables can be extended to those cases in which

the distribution of truth-values on L is not known, as we now illustrate.

Example 1.6. Let θ, ϕ, ψ ∈ SL. No further knowledge of L is available.

Consider now the sentence

χ = (θ ∧ ϕ)→ ψ.

We can construct a boolean table for χ simply by assuming that it is written

in the language L′ = {p, q, r} where p is identified with θ, q is identified with

ϕ and r is identified with ψ. There is of course nothing special about this

identification. This is an elementary illustration of what we mean when we

say that logic is formal.

Exercise 1.10. Compute the extended boolean table for the following sen-

tences

(1) (θ ∧ ϕ)→ ψ

(2) (θ → ϕ)→ (¬ϕ→ ¬θ)(3) (¬ϕ→ ¬θ)→ (θ → ϕ)

(4) ((θ ∧ ϕ) ∧ ψ)→ (theta ∧ (ϕ ∧ ψ)

(5) θ → (ϕ→ θ)

(6) (θ → ϕ)→ θ

Let us go back to the tarskian conditions of Definition ?? to note that what

they effectively amount to is the fact that the truth-value of any sen-

tence in SL is a fixed function of the truth values of its components.

This is best appreciated from an algebraic point of view. We will spell out

the argument only for the connective of conjunction. The other cases are

similar and are left as exercise.

Let θ, ϕ ∈ SL. We claim that there exists a function

f∧ : {0, 1} × {0, 1} → {0, 1}

such that

v(θ ∧ ϕ) = f∧(v(θ), v(ϕ)) =

{1, if v(θ) = v(ϕ) = 1;

0, otherwise.(5)

By applying the method of extended boolean tables, it is immediate to check

that

min{v(θ), v(ϕ)}

1. LANGUAGE AND SENTENCES 17

, a suitable such f∧ which settles the claim.

Exercise 1.11. Define suitable truth functions for

• negation

• disjunction

Exercise 1.12. Can you define an alternative truth-function f∧ satisfying

(5)?

The above is important enough to deserve the status of a Principle.

Principle 1.1 (Principle of truth-functionality). For all v ∈ V, θ ∈ SLn+1,

ϕ1, ϕ2 ∈ SLn and ∗ ∈ {∧,∨,→} there exist fixed functions f∗ : {0, 1} ×{0, 1} → {0, 1} such that

v(θ) = f∗(v(ϕ1), v(ϕ2)).

Such functions are said to represent the connectives in ∗ if they satisfy the

corresponding tarskian conditions.

Remark 1.3. The principle of truth-functionality is closely related to the

concept of extensionality which originates in set theory and plays a central

role throughout classical logic. The idea is indeed a the very roots of the

classical interpretation of naıve set theory and it can be held responsible

for the view that two sets are the same whenever their elements cannot be

distinguished. It is not, however, a principle which defines logic. There are

in fact many interesting and useful logics which fail it.

Modal logic features certain connectives, called modalities, which are not

truth-functional. For an illustration, compare the following sentences:

(1) triangles have three edges

(2) some politicians are corrupt

Both sentences are true in their relevant interpretations (elementary geom-

etry and society, respectively) but if we apply a necessity operator to them,

the latter presumably becomes false.

An alternative way of expressing the compositionality of a logic consists in

saying that its semantics is deterministic. Again many useful logics possess

non-deterministic semantics, as you will learn in part 3 of this course.

Exercise 1.13. Are any of the following individually sufficient to determine

v(θ)

(1) v(θ ∧ ϕ) = 0 and v(ϕ) = 0 ?

(2) v(θ ∨ ϕ) = 1 and v(ϕ) = 1 ?

18 1. REVIEW OF CLASSICAL PROPOSITIONAL LOGIC

2. Satisfiability and consequence

Recall that a valuation v : L → {0, 1} is a complete description of a logical

state of affairs –analogously, a full description of a row in a boolean table–

and therefore that the set of all valuations on L can be thought of as a

complete distribution of truth-values on L. 2 As anticipated above,

the full generality of logical reasoning is attained when all possible truth

valuations are considered. This Section explains why this is the case.

2.1. Satisfiability. Let us begin with the concept of satisfiability, one

of the most important notions in mathematical logic.

Definition 2.1 (Model). Let V the set of all (classical) valuations on L and

Γ ⊆ SL. Say that v ∈ V is a model of Γ, if v(γ) = 1 for all γ ∈ Γ.

Example 2.1. Let L = {p, q} and Γ = {γ1︷ ︸︸ ︷

p→ q,

γ2︷︸︸︷q }. The set of all valua-

tions on L is:v1 : v(p) = 1; v(q) = 0

v2 : v(p) = 0; v(q) = 1

v3 : v(p) = v(q) = 1

v4 : v(p) = v(q) = 0

It then follows from Definition 2.1 that the set of models of Γ consists of all

valuations mapping γ1 and γ2 to 1, i.e. v2 and v3.

Definition 2.2 (Satisfiability). We say that Γ ⊆ SL is satisfiable if it has

a model.

We usually simplify the notation by writing v(Γ) = 1 to say that v is a

model of Γ.

Exercise 2.1. Write an appropriate boolean table to check the v2 and v4

are models of Γ in the previous example.

Exercise 2.2. Check whether the following are satisfiable. You need to

justify your answer

(1) {θ → ϕ, θ,¬ϕ}(2) {θ → ϕ,ϕ, θ}(3) {θ → ϕ,ϕ,¬θ}(4) {θ ∨ ϕ,¬θ,¬ϕ}(5) {θ ∨ ϕ,¬θ, ϕ}

2The terminology is reminiscent of probability theory for good reasons, as Appendix

?? illustrates.

2. SATISFIABILITY AND CONSEQUENCE 19

(6) {ψ → (θ ∧ ϕ),¬θ, ϕ, ψ}

The notion of satisfiability and the associated notion of logical model can be

seen as a formalisation of the intuitive notion of coherence. For if a set is

unsatisfiable, there is no way in which its elements can be all true, that is

to say they do not cohere.

If we identify rationality with coherence, the mathematical definition of

satisfiability can be used to define what it means to reason irrationally,

namely to move from premisses which are satisfied to conclusions which are

not. Since not all sets are satisfiable, satisfiability can be seen as an asset

which we certainly don’t want rational reasoning to waste. Hence the central

feature of logical consequence is that it should preserve satisfaction.

2.1.1. A practical illustration. Let L = {p, q, r}. Suppose three jurors

must decide whether a defendant is liable (r). According to contract law,

a defendant is liable if and only if they are under the obligations of a valid

contract (p) and they are in breach of it (q).

The so-called doctrinal paradox, arises when Jurors submit the following

judgments:

p q r

Juror A 1 1 1

Juror B 1 0 0

Juror C 0 1 0

Majority 1 1 0

The so-called discursive dilemma, which ignited the interest in the theory of

Judgment Aggregation arises when jurors submit the following judgments:

p q r ↔ (p ∧ q) r

Juror A 1 1 1 1

Juror B 1 0 1 0

Juror C 0 1 1 0

Majority 1 1 1 0

where r ↔ (p ∧ q) is short for (r → (p ∧ q) ∧ (p ∧ q)→ r).

Contract law says that (r → (p ∧ q) ∧ (p ∧ q) → r), and it is implicitly

assumed that all Jurors accept that. Under this assumption, the of the

above becomes apparent. Whilst individually each Juror judges in accord

with the constrains imposed by the definition of propositional connectives as

presented in Table ??, the simple-majority aggregation of their judgments

20 1. REVIEW OF CLASSICAL PROPOSITIONAL LOGIC

by simple majority gives does violate such constraints. Indeed the Majority’s

judgment is inconsistent.

Inconsistency is a central notion in logic. To some extent logic claims its

normative role by freeing reasoners who conform to it from inconsistency.

Hence, many would regard consistency (or coherence) to be the single most

important contribution made by logic to the normative analysis of rational-

ity. We approach the formal definition of (in)consistency via the notion of

satisfiability.

Let v be a propositional valuation on L. We say that v is a model of Γ ⊆ SLif v(γ) = 1 for all γ ∈ Γ (which we abbreviate by writing v(Γ) = 1). We say

that Γ is satisfiable if it has a model, and unsatisfiable otherwise.

This captures a central feature of coherence. For if a set is unsatisfiable,

there is no way in which its elements can be all true, that is to say they do

not cohere.

It is a simple exercise to show that taken individually all jurors submit satisfi-

able judgments, but that the simple-majority aggregation of their individual

judgments is unsatisfiable. Hence incoherent. If coherence is identified with

rationality (and at this level of abstraction, this seems entirely plausible),

the discursive dilemma is a situation in which individual rationality leads,

via simple majority aggregation, to collective irrationality.

2.1.2. Project. The above paradox is the tip of a very interesting ice-

berg, which relates closely mathematical logic and social choice theory. This

project can take several directions:

• The paper The Logic of Group Decisions: Judgment Aggregation

(Pigozzi, G. J Philos Logic (2015) 44: 755. https://doi.org/10.1007/s10992-

015-9357-7) surveys the methodological foundations

• The paper On the logic of preference and judgment aggregation

(Auton Agent Multi-Agent Syst (2011) 22:4–30) explores the im-

plications for artificial intelligence

Choose one of the above papers and illustrate the approach therein con-

tained. A further and very interesting project is to do with exploring

the question as to what is the contribution of the Doctrinal Paradox to

(non)classical logics.

2.2. Classical consequence. The concept of logical consequence is

intrinsically relational: on the assumption that we are accepting the set of

premises Γ, the relation |= tells us which sentences we –as logical agents–

2. SATISFIABILITY AND CONSEQUENCE 21

are forced to accept. This interpretation fits Tarski’s original informal de-

scription without too much stretching:

Let us consider an arbitrary class of sentences Γ and an

arbitrary sentence θ which follows from the sentences of

this class. From the point of view of everyday intuitions it

is clear that it cannot happen that all the sentences of the

class Γ would be true but at the same time the sentence θ

would be false3.

Definition 2.3 (Tarski 1936). We say that θ is a logical consequence of Γ,

written Γ |= θ if and only if ∀v ∈ V, if v(Γ) = 1 then v(θ) = 1.

It is an immediate consequence of our definitions that θ is a logical conse-

quence of Γ if and only if every model of Γ is a model of θ, which we write

as follows

Γ |= θ ⇔MΓ ⊆Mθ.

Example 2.2. Let Γ = {p, p → q} and let θ = q. Is it the case that

Γ |= θ? Compositionality allows us to answer with an effective procedure.

This amounts to saying that we possess a procedure which – in principle –

allows us to answer definitely (either Yes or No) in a finite number of steps.

One such method is provided by truth-tables as follows:

p q p→ q q

0 0 1 0

0 1 1 1

1 0 0 0

1 1 1 1

Each row corresponds to one of the four distinct on L. By Definition 2.3 we

need to check whether there exists a model of Γ which does not satisfy θ.

A quick inspection of the table shows that no such model exists, hence it is

indeed the case that Γ |= θ.

Exercise 2.3. Determine whether Γ |= θ where

(1) Γ = {p, p→ q} and θ = q

(2) Γ = {q, p→ q} and θ = q

(3) Γ = {q, p→ q} and θ = p

As an anticipation of things to come, the method of truth tables

3A. Tarski On the concept of following logically, History and Philosophy of Logic,

Volume 23, Number 3, pp. 155-196, 2002

22 1. REVIEW OF CLASSICAL PROPOSITIONAL LOGIC

(1) is meaningful only for finite languages

(2) is practical only for small languages

(3) it is generally unfeasible

We say that a sentence θ is a tautology, and write |= θ, if and only if every

valuation satisfies θ (alternatively: θ is true under all the 2L valuations

on L, alternatively: if it is mapped to 1 in each row of a suitable truth-

table). A tautology is therefore true “in virtue of its logical structure”, and

therefore we don’t need to mention its models. This explains the notation.

A contradiction is defined similarly, i.e. as a sentence which has no models.

So contradictions are unsatisfiable.

Exercise 2.4. Check whether the following are satisfiable

(1) Γ = {θ ∨ ϕ,¬θ}(2) ∆ = {θ ∨ ϕ,¬θ, ϕ}(3) Λ = {θ → ϕ,ψ → ϕ, (θ ∨ ψ)→ ψ}

Exercise 2.5. Check whether the following are tautologies

(1) θ ∨ ϕ(2) θ → (¬θ → θ)

(3) θ → (¬θ → ϕ)

(4) (θ ∨ ϕ) ∧ (θ ∨ ψ)→ θ ∨ (ϕ ∧ ψ)

(5) (θ ∨ ϕ) ∧ ¬θ → ϕ

(6) (θ → ϕ)→ ((θ → ¬ϕ)→ ¬θ)

Exercise 2.6. Show that the following are equivalent

(1) {γ1, γ2, . . . , γn} |= θ

(2) {γ1, γ2, . . . , γn} ∪ ¬θ is not satisfiable.

Definition 2.4 (Logical equivalence). θ, ϕ ∈ SL are logically equivalent,

written θ ≡ ϕ, if ∀v ∈ V,

v(θ) = v(ϕ).

2. SATISFIABILITY AND CONSEQUENCE 23

Exercise 2.7. Prove that the following are equivalent:

θ ≡ ϕ⇐⇒ ∀v ∈ V, v(θ) = v(ϕ) (6)

⇐⇒ ∀v ∈ V, v(θ) = 1⇔ v(ϕ) = 1 (7)

⇐⇒ ∀v ∈ V, v(θ) = 0⇔ v(ϕ) = 0 (8)

⇐⇒ ∀v ∈ V, v(θ) = 1⇒ v(ϕ) = 1 and ∀v ∈ V, v(ϕ) = 1⇒ v(θ) = 1

(9)

⇐⇒ θ |= ϕ and ϕ |= θ (10)

⇐⇒|= θ → ϕ and |= ϕ→ θ (11)

⇐⇒ ∀v ∈ V, v(θ → ϕ) = 1 and v(ϕ→ θ) = 1 (12)

⇐⇒ ∀v ∈ V, v((θ → ϕ) = 1 ∧ v(ϕ→ θ)) = 1 (13)

⇐⇒|= (θ → ϕ) ∧ (ϕ→ θ) (14)

2.3. Some key properties of |=. The following Proposition captures

a number of central properties of classical consequence.

Proposition 2.1. Let θ, ϕ ∈ SL. Then:

(1) Γ, ϕ |= θ ⇐⇒ Γ |= (ϕ→ θ)

(2) θ |= ¬ϕ ⇐⇒6 ∃v ∈ V, s.t. v(θ) = v(ϕ) = 1;

(3) θ |= ϕ ⇐⇒ ¬ϕ |= ¬θ(4) ∅ |= (θ ∨ ¬θ)(5) Γ |= (θ ∧ ¬θ) =⇒ Γ |= α ∀α ∈ SL.

(6) θ |= ϕ =⇒ θ, ψ |= ϕ, ∀ψ ∈ SL

Proof. (1)

Γ, ϕ |= θ ⇐⇒ ∀v ∈ V if v(γ) = 1∀γ ∈ Γ and v(ϕ) = 1, then v(θ) = 1

⇐⇒6 ∃v ∈ V s.t. v(Γ) = v(ϕ) = 1 and v(θ) = 0

⇐⇒ 6 ∃v ∈ V s.t. v(Γ) = 1 and v(ϕ→ θ) = 0

⇐⇒ Γ |= ϕ→ θ.

(2) similar, exercise

(3) similar, exercise

(4) similar, exercise

(5) similar, exercise

(6) similar, exercise

�

Remark 2.1. Part (1) in Proposition 2.1 allows us to make very precise

the rather obvious semantic analogy between the propositional connective

24 1. REVIEW OF CLASSICAL PROPOSITIONAL LOGIC

→ and the consequence relation |=. Although they sit on distinct levels of

formalisation they isolate the very same constraints on “if, then” This is

particularly easy to visualise in the special case Γ = ∅. Then

ϕ |= θ ⇐⇒ |= ϕ→ θ. (15)

Note also that this extends immediately to the analysis of arguments which

combine several consequence relations.

3. Using sentences to describe models

The material covered in this section has a twofold purpose:

• to provide a very useful technique to handle models that will be

central in our development of nonmonotonic logic

• to show one situation in which the distinction, treasured by so many

textbooks, between syntax and semantics has little raison d’etre

Let ATL be the set of atoms of L, that is the set of sentences of the form

α = pε11 ∧ pε22 ∧ . . . ∧ p

εnn ,

where εi ∈ {0, 1}, i = 1, . . . , n, and for p ∈ L, p1 = p and p0 = ¬p In other

words, the atoms of L are maximal conjunctions of literals from L.

Then, for a valuation v and ε ∈ {0, 1}:

v(pε) = 1⇔{ε = 1 and v(p) = 1 or

ε = 0 and v(¬p) = 1

}⇔ v(p) = ε

Notice that the set ATL is in 1-1 correspondence with the valuations on L.

This implies that there is a unique valuation satisfying v(α) = 1 namely

vα(pi) = εi for 1 ≤ i ≤ n. Conversely, given a valuation v ∈ V there exists

a unique atom α ∈ AT L such that v(α) = 1, namely that α =∧ni=1 p

εii for

which εi = v(pi) for 1 ≤ i ≤ n, i.e. the atom:

α =

n∧i=1

pv(pi)i .

To fix ideas, let L = {p, q}. Then

It is immediate to see that the set {p∧ q, p∧¬q,¬p∧ q,¬p∧¬q} contains all

and only the maximal (i.e. using all propositional variables) and consistent

conjunctions of propositional variables and their negations, and is therefore

the set of atoms of L.

Let us further observe that

3. USING SENTENCES TO DESCRIBE MODELS 25

α1 α2 α3 α4

p q p ∧ q p ∧ ¬q ¬p ∧ q ¬p ∧ ¬q p→ q

v1 0 0 0 0 0 1 -

v2 0 1 0 0 1 0 -

v3 1 0 0 1 0 0 -

v4 1 1 1 0 0 0 -Table 6. Atoms of L = {p, q}

(1) Atoms are in 1-1 correspondence with the set of valuations on L(because each atom is satisfied by exactly one valuation)

(2) Every sentence in SL is logically equivalent to a disjunction of

atoms of L. Inspection of the table above shows that p → q is

logically implied by α1 ∨ α3 ∨ α4 (exercise!)

Those observations lead us naturally to write the set of models of any sen-

tence θ as the set of atoms which logically imply it, i.e.

Mθ = {α ∈ ATL | α |= θ}

= {α ∈ ATL | vα(θ) = 1}.where vα is clearly the unique v corresponding to atom α (see the Table).

Proposition 3.1. For θ, ϕ ∈ SL

(1) θ ≡∨Mθ

(2) Mθ is the unique subset of AT L such that θ ≡∨Mθ

(3) For R ⊆ AT L, M∨R = R

(4) θ ≡ ϕ⇐⇒Mθ ≡Mϕ

(5) Mθ∧ϕ = Mθ ∩Mϕ

(6) Mθ∨ϕ = Mθ ∪Mϕ

(7) M¬θ = AT L −Mθ

(8) θ |= ϕ =⇒Mθ ⊆Mϕ

Proof. Omitted �

CHAPTER 2

Classical logic as a model of rational reasoning

1. Propositional arguments

This section investigates the (limited) powers of classical propositional logic

to model even very abstract concepts of “rational reasoning”, as captured

by the following definition.

Definition 1.1. An abstract propositional argument (or simply argument)

is a claim that a certain sentence θ follows from a given set of premisses

{γ1, . . . γn}. We say that the argument is correct or valid or sound if and

only ifn∧i=1

γi |= θ.

Remark 1.1. Arguments, as investigated in artificial intelligence under the

heading Formal Argumentation Theory, are obtained by making the above

definition interactive. So the above can be seen as the limiting case of a

simple argument that one has against “truth”. This justifies the qualification

“abstract” in the definition.

We often abbreviate the expression “the conclusion θ follows from the set of

premisses {γ1, . . . γn}” by writing

γ1, . . . γn ∴ θ,

where the symbol “∴” reads “therefore”.

Another widely used notation separates the premisses and the conclusion of

a propositional argument by means of a horizontal line:

γ1, . . . γnθ .

This latter is the standard way of presenting arguments in the field of mathe-

matical logic which investigates the properties of mathematical proofs, called

quite self-explanatorily proof theory.

Example 1.1. Suppose you hear the following argument over lunch:

27

28 2. CLASSICAL LOGIC AS A MODEL OF RATIONAL REASONING

AC Milan will win the 2017-18 Serie A title, or Inter Milan

will. AC Milan, however is not going to win the title.

Hence, Inter Milan will.

You can see some of your friends nodding around the table, and yet you find

this a little fishy so you decide to use the propositional calculus to find out

more. Here is what you do.

You begin by fixing the relevant propositional language, in self-explanatory

letters this could be {M, I}, since after careful inspection you realise that

there are only two elementary statements involved in the argument. Then

you are ready to visualise the argument, say as follows

M ∨ I, ¬MI (16)

You are now ready to apply the decision procedure to give a definite answer

to your doubts, i.e. you check wether

{M ∨ I,¬M} |= I.

You decide to apply what you’ve learnt in Exercise 2.6, so you suppose not.

Then ∃v ∈ V such that v(M ∨ I) = v(¬M) = 1 but v(I) = 0. By a

straightforward application of the tarskian conditions to the latter term you

have that ∃v ∈ V such that v(M ∨ I) = v(¬M) = 1 = v(¬I) = 1. But

this contradicts the tarskian conditions, for v(M ∨ I) = v(¬M) = 1 force

v(I) = 1 and therefore v(¬I) = 0. In other words, the hypothesis that

{M ∨ I,¬M} ∪ {¬I} is satisfiable

contradicts the tarskian conditions, and therefore has to be rejected, giving

{M ∨ I,¬M} |= I.

You go back to your table and congratulate those who were nodding at the

above argument, for you have a proof that it is indeed valid. But wait a

minute! – a friend says – you’ve been proving it by contradiction. If you

really want to persuade us, you have to prove that your method of proof is

indeed valid.

Good point! you think as you roll up your sleeves and get back to work.

Here’s how you reason about it. In its simplest rendering, a proof by con-

tradiction is an argument which establishes that a certain sentence (say θ)

is true because the supposition that it is false leads to a contradiction. This

is enough to give you the intuition to formulate arguments by contradiction

1. PROPOSITIONAL ARGUMENTS 29

as follows, for all θ, ϕ ∈ SL:

¬θ → (ϕ ∧ ¬ϕ)

θ (17)

Of course you don’t need to be specific about the contradiction – they’re all

logically equivalent! This time, of course, you’d rather not prove that the

argument is correct by contradiction. Instead you observe that v(ϕ∧¬ϕ) =

0, ∀v ∈ V. Hence the tarskian conditions force you to have v(¬θ) = 0 in

order for the premis of the argument to be satisfied. But then the tarskian

conditions give you v(θ) = 1, as required.

Hence argument (17) –that you used to establish that argument (16) is valid

– is valid. And it is so, eventually, because of the tarskian conditions, i.e.

the unique extension to the whole of SL of the boolean tables.

The fact that the validity of propositional arguments boils down to the

boolean tables has two important consequences we now briefly illustrate in

the remainder of this Section.

In particular we shall begin by making an observation that will allow you

to see the point of your rather smart colleague who is still not persuaded by

your argument to the effect that Inter Milan will win the 2017-18 serie A

title. She is in fact pointing out to you that your argument for the validity

of (16) is rather thin – it does satisfy the definition but it is hardly an

argument that would convince bookmakers. Well, she definitely has a point.

1.1. Not all valid arguments are decision-relevant. Fix L to be

finite, as usual. We already observed several times that boolean tables are

in 1-1 correspondence with the set V. This gives a natural reading of the

rows of boolean tables as an exhaustive distribution of truth-values to the

elements of L. Now, you should recall that the definition of |= quantifies

over all valuations on L. But in many situations we do possess some factual

knowledge or information beyond the purely logical fact that truth-values

are so distributed as the boolean table says.

Definition 1.2. Say that a propositional argument is decision-relevant in

a given context if it is valid and all its premisses are satisfied.

Remark 1.2. Note that decision-relevance is not a formal requirement. It

rather requires that we possess data/information/evidence/knowledge/etc.

about the problem we are modelling as a propositional argument. This

information is precisely what allows us to decide whether the premisses are

all satisfied and by this very fact, it is not information that you can obtain

by means of logical reasoning.

30 2. CLASSICAL LOGIC AS A MODEL OF RATIONAL REASONING

Let us go back to (16). What your smart colleague is rightly objecting to is

the truth of premis M ∨ I. We have (arguably!) no reason to think, believe

and more importantly act as if it was true of serie A this year. In other

words, knowledge about the football season is crucial to determine whether

it is reasonable or not to assume that the premisses of the argument are

true. As a rule of thumb, you may safely assume that the more important

is the matter, the harder is to obtain reliable knowledge about the truth of

the premisses Whatever you’ll get, it will fall short of the kind of certainty

that propositional model grants you.

Note that we need to refer to a particular context (more precisely a domain

of interpretation) in order to evaluate whether a certain valid argument is

also decision-relevant. This is because propositional logic cannot tell you

which sentences are satisfiable. This is where logic meets knowledge and

as a consequence, where logic links with the inevitable uncertainty which

originates with non-logical knowledge. Here is the first vantage point from

which you can appreciate a highlight in the panorama of this course.

The construction of a mathematically sound model is a necessary, but gen-

erally not sufficient, condition to determine whether a certain correct ar-

gument is decision-relevant, i.e. whether you can act as if its conclusion

is true. This requires us to be able to reason logically under the

assumption that we do not possess all the relevant information, or

that the information we possess is not precise enough to determine

“true” and “false”.

In doing this, however, we do not throw away classical logic. For most

purposes in fact the soundness of abstract propositional arguments is a nec-

essary condition for rational reasoning – an incoherent argument will not

get you much mileage.

Exercise 1.1. Consider the following argument:

Either you are a citizen of a country which lies outside

the solar system or you are the CFO of a large investment

bank. You are not the CFO of a large investment bank.

Therefore you are a citizen of a country which lies outside

the solar system.

Write the above as a propositional argument and prove it is valid. Argue

that it is not decision-relevant.

Exercise 1.2. For each of the following propositional arguments prove their

validity or construct an appropriate counterexample.

1. PROPOSITIONAL ARGUMENTS 31

(1) θ ∨ ϕ, θ ∴ ϕ(2) θ ∧ ϕ, θ ∴ ϕ(3) θ ∴ θ ∧ ϕ(4) θ → ψ,ϕ→ ψ, θ ∨ ϕ ∴ ψ(5) θ → ψ,ψ ∴ θ(6) θ ∴ θ

Exercise 1.3. For each argument that you proved to be valid in the previous

exercise decide whether it is also decision-relevant in some context (of your

choice).

1.2. Decidability in theory, and in practice. The decision proce-

dure for propositional arguments rests on the method of boolean tables.

As you should recall, the practical applicability of this method approaches

“zero” pretty quickly, exponentially in fact. To see this consider the follow-

ing situation.

You are interested in checking the argument

{γ1, . . . γn} ∴ θ (18)

where, γi, θ ∈ SL for i = 1, . . . , n and L = {p1, . . . , pk} with fairly large k,

say k = 15. Suppose that the argument is not valid, and that it so happens

that there exists exactly one model of the premisses (which, of course, must

not be a model of the conclusion). It can well be that that unique model

is the one assigning 1 to all distinct propositional variables of L, which

according to the usual convention we adopt on naming valuations will be

v32768. However, you don’t know this yet.

So you go about checking the validity of (18) with your boolean tables,

when you suddenly realise it’s not going to be quick. Yet you keep going,

and start filling the table out. You’ve been working very hard all night and

found no models of the premisses, until you really have to go and leave the

job unfinished at the row which corresponds to v32767. All that work is of

no use to you: you are no closer to knowing whether (18) is valid than you

were before the evening started. Then you come back and find out, at the

very last row of the boolean table, that the argument is indeed not valid.

This nightmare-ish situation is what is known as worst case scenario in the

field which investigates the complexity decidable problems. What makes

it “worst case” is the fact you have to wait until the very end to get your

YES/NO answer. In real life it may well happen that you don’t have the

time, or perhaps the resources to wait until the very end. In those cases, the

fact that you have an algorithm for checking objectively the validity of an

32 2. CLASSICAL LOGIC AS A MODEL OF RATIONAL REASONING

argument is of little practical value. This motivates the following important

consideration

Whilst decidable in theory, the problem of determining whether

a given propositional argument is valid may not be solvable in

practice.

Of course, you might say, things would be radically different in the above

example, if we adopted an anti-lexicographic convention for writing down

the rows of our boolean table. In that case, the row

11 . . .︸︷︷︸32765 times 1

1

would indeed correspond to v1, and we would be immediately in a position

to say that (18) is not a valid argument.

This opens up the question as to whether the problem with the difficulty,

which can arise in practice, to find the solution to a decision problem really

depends on the method of boolean tables. Indeed what this and Exercise

2.6 suggest is that a far better (i.e. more efficient) method would be to try

and find a refutation of the validity of any propositional argument. An

analysis of this fascinating question is well beyond the scope of this course.

However you may want to know that under the widely believed conjecture

that P 6= NP ,1 the systematic search for a refutation of the validity of

arguments is no uniform improvement on the method of boolean tables.

More on this in Part 3 of this course!

2. Agents as consequence relations

Relation (6) in Proposition (2.1) is a special case of a property which is

to inference what compositionality (or bivalence or non-contradiction) is to

representation. That’s why it really deserves to stand out as a

Principle 2.1. (Monotonicity) If Γ |= θ then Γ, ϕ |= θ, ∀ϕ ∈ SL.

The importance of monotonicity cannot be underestimated.Recall that in

our applied logic perspective we are thinking of a consequence relation as

an agent. In particular by writing Γ |= θ we are interpreting the properties

of |= as norms norms regulating which sentences θ an agent must accept

as a consequence of the fact they accept the sets of sentences in Γ. Now,

in this reading, it is apparent that Monotonicity –elevated as a principle–

determines the irrelevance of learning that any additional sentence (w.r.t.

1You may want to read more about this at https://en.wikipedia.org/wiki/List_

of_unsolved_problems_in_mathematics

2. AGENTS AS CONSEQUENCE RELATIONS 33

Γ), say ψ is also to be accepted. But this is just another way of saying that

for agent |=, Γ provides sufficient reasons to accept θ. Hence a monotonic

logical agent never changes their mind.

In our applied logic spirit, this logical obstinacy must immediately ques-

tioned. As stressed above a good way of questioning it is to ask which

reasoning contexts makes Monotonicity desirable, and which don’t.

A moment’s reflection shows that Monotonicity and uncertainty don’t quite

go hand in hand. On the contrary, if we have (as modellers) reason to believe

that logical agents live in an environment which is prone to uncertainty, then

it will not take long to find out that a Monotonic logical agent is blatantly

inept.

Even if we assume that our goal is to model agents who inhabit a perfectly

knowable world (say the Platonic world of mathematical entities), one may

still question the appropriateness of Monotonicity as a norm of rational

reasoning. For even the very best mathematicians make mistakes or simply

do guess work. This nicely put by P. Halmos2

Mathematics is not a deductive science –that’s a cliche.

When you try to prove a theorem, you don’t just list the

hypotheses, and then start to reason. What you do is trial

and error, experimentation, guesswork.

However if we make a little abstraction and aggregate all mathematicians

into a large collective agent which we can call Mathematics, Monotonicity

looks a lot more promising. One thing which always characterised mathe-

matics is the “peer-review” mechanism. One individual claims that a certain

statement is true, and offers a proof for it. This can be scrutinised by other

mathematicians, and if there is a problem with it – a mistake – this really

gets spotted. Otherwise, the statement can safely be added to the body of

mathematical truths which we keep piling up since the Ancient Greeks.

This rather hyperbolic remark is useful to introduce the key motivation

for the study of nonmonotonic logics. The problem isn’t with the idea of

cumulating consequences, but with the universal quantification “∀ϕ ∈ SL”

in the statement of the Principle. As we will shortly, see, in most cases

some form of monotonicity is indeed desirable. For reasoning is costly and

we don’t want to waste its products unless we have good reasons to believe

that they do not serve our purposes well enough.

2P. Halmos (1985), I want to be a Mathematician, Springer-Verlag.

34 2. CLASSICAL LOGIC AS A MODEL OF RATIONAL REASONING

With a slogan, our goal will be that of identifying the conditions under which

monotonicity is applicable.

CHAPTER 3

Nonmonotonic logics

The purpose of this Chapter is to extend the inferential abilities a classical,

monotonic, agent to encompass reasoning under background uncertainty. As

pointed out in the introduction, this is the normal context for rational rea-

soning in idealised contexts. [Please refer to the presentation of this course

for motivating examples.]

In doing this we will lose some desirable properties of classical logic, and in

particular we are forced to abandon the uniform characterisation of “if . . .

then” reasoning which, in classical logic, is captured at the object level, by

the truth-table of →, and at the meta level by the property which we have

been referring to as the (semantic analogue of) the Deduction theorem, see

Remark 2.1 above.

Our aim now is to define a new meta linguistic relation

|∼⊆ SL× SL

which we read out loud as “squiggle” or “snake”, and which we interpret in

slightly distinct ways, depending on the context. Expressions of the kind

θ |∼ ϕ are often referred to as default statements and have been given three

predominant intuitive readings

Stereotypical: reading. θ |∼ ϕ reads as “normally/typically”θ im-

plies ϕ.

Presumptive: reading. θ |∼ ϕ reads as “unless otherwise stated, ϕ

follows from θ.

statistical: reading. θ |∼ ϕ reads as “most θ’s are ϕ’s”.

In broader terms, default statements can be viewed as conversational con-

ventions: when you are asked “do you know what time is it?” you normally

do not reply “Yes”, but you understand that the person asking would like

you to share that information with them. Similarly, competent English

speakers are aware of the convention to the effect that one should not reply

35

36 3. NONMONOTONIC LOGICS

to a question with another question, and yet when hearing “How do you

do?” they know that they should reply with the very same question.

Hence defaluts are about rational expectations. Like all expectations they

may be wrong, but when they fail they must do so in a principled way –

that’s what rationality means in this context. So there may be plenty of

“non examples” or exceptions to default statements, which nonetheless do

not have the status of “counter-examples”, i.e. instances which prove the

statement false. No matter many non-flying birds you will observe, it is still

rational of you to expect that the next bird you’ll see is capable of flying,

i.e. birds typically fly. This is where the robustness of default reasoning

lies: tolerating an arbitrary number of negative examples, and yet carrying

useful, practical information.

On this grounds we shall interpret θ |∼ ϕ as:

It is rational to expect ϕ given that θ.

We are now ready to consider a number of desirable properties for |∼ with

the idea that a nonmonotonic logical agent should be characterised by a

relation satisfying:

• flexibility: reasoning should adjust to new, potentially contradic-

tory information

• principled: the resulting consequence relation should capture the

idea of accepting statements based on “good reasons”

1. Gabbay-Makinson constraints on rational expectation

In a series of seminal papers published in the second half of the 1980’s Dov

Gabbay and David Makinson brought to the general logician’s attention the

interesting work done by Reiter, McCarthy, McDermott and Doyle, Shoham

and others in computer science and artificial intelligence.

Gabbay and Makinson’s approach can be illustrated by considering a set of

constraints that arguments based on “rational expectations” should satisfy.

Here is the list of properties that emerged as being particularly well-justified.

θ |∼ θ(REF)

If you know θ then it is rational of you to expect it.

Note that this is an argument with an emtpy set of premisses, so it is best

thought of as an axiom, but a very plausible one.

1. GABBAY-MAKINSON CONSTRAINTS ON RATIONAL EXPECTATION 37

The remaining conditions will be expressed by ‘rules of inference’ in which

the premisses are taken as knowledge possessed by a rational agent.

θ ≡ ϕ, θ |∼ ψϕ |∼ ψ

(LLE)

If you know that θ ≡ ϕ and it is rational of you to expect ψ given that you

know θ, then you may rationally expect ψ to hold given the information ϕ.

Example 1.1. I expect bananas to be sweet. Since Musa paradisiaca sapien-

tum is just another name for bananas, I should not expect Musa paradisiaca

sapientum not to be sweet.

θ |∼ ϕ, ϕ |= ψ

θ |∼ ψ(RWE)

If you expect ψ given that you know θ and you know that ϕ |= ψ, then you

may rationally expect ψ to hold given the information θ.

Example 1.2. I expect bananas to be sweet. Since sweet fruit contain

sucrose, I should expect bananas to contain saccarose.

Left Logical Equivalence (LLE) and Right Weakening (RWE) characterise

the relation between classical consequence and the nonmonotonic conse-

quence relation that we are defining. They are often referred to as pure or

abstract conditions as they mention no connectives

Next two rules governing the behaviour of classical connectives

θ |∼ ψ, ϕ |∼ ψθ ∨ ϕ |∼ ψ

(∨ L)

If you expect ψ given that you know θ and you expect ψ given that you

know ϕ, then you may rationally expect ψ if you know that either ψ or ϕ

hold.

Example 1.3. I expect the next Cannes Festival to be extremely interesting

(ψ) if Quentin Tarantino presents a film there (θ). I expect the next Cannes

Festival to be extremely interesting (ψ) if Wes Anderson presents a film

there (ϕ). If I know that either Tarantino or Anderson will present a film

at the next Cannes Festival, then it is rational of me to expect it to be very

interesting.

38 3. NONMONOTONIC LOGICS

θ |∼ ϕ, θ |∼ ψθ |∼ ϕ ∧ ψ

(∧ R)

If you expect ψ given that you know θ and you also expect ϕ given that

you know θ, then you may rationally expect ϕ ∧ ψ if you know that either

θ hold.

Example 1.4. I expect the next Cannes Festival to be extremely interesting

(ψ) if Quentin Tarantino presents a film there (θ). I also expect the next

Cannes Festival to be extremely controversial if (ψ) if Quentin Tarantino

presents a film there (ϕ). If I know that Tarantino will present a film at

the next Cannes Festival, then it is rational of me to expect it to be very

interesting and controversial.

Disjunction on the Left (∨ L) captures reasoning by cases, whereas Con-

junction on the Right (R ∧ L) sets the very natural norm according to

which conclusions of rational expectations should be closed under conjunc-

tion. This latter property, which holds for classical logic in virtue of the

Tarskian conditions, is one characterising rule of qualitative uncertain rea-

soning. Probability logic, for instance, fails it as illustrated by the so-called

The lottery paradox.1

We are finally ready to introduce the rule of rational expectation that gives

the name to this subject (although an unfortunate one!).

θ |∼ ψ, θ |∼ ϕθ ∧ ϕ |∼ ψ

(CMO)

If we think of θ as giving a logical agent good reasons to expect ψ, then

θ ∧ ϕ also provides our agent good reasons to expect ψ, for the acceptance

of θ provides already good reasons to expect ϕ.

Example 1.5. Suppose I am a medical doctor. I expect patients with high

cholesterol to be at greater risk of coronary desease. (θ |∼ ψ). I also expect

high cholesterol in patients with unhealthy eating habits (θ |∼ ϕ). Hence, if

I am observing a patient with high cholesterol and unhealthy eating habits

I should rationally expect them to be at higher risk of coronary desease

(θ ∧ ϕ |∼ ψ)

1See the “Epistemology” lecture notes or read, e.g. Makinson, D. (2010). Logical

Questions behind the Lottery and Preface Paradoxes: Lossy Rules for Uncertain Inference.

Synthese.

1. GABBAY-MAKINSON CONSTRAINTS ON RATIONAL EXPECTATION 39

Note that classical logic (i.e. |=) is easily seen to be inadequate to model

the pattern of reasoning captured by Cautious Monotonicity. The reason

clearly lies in the fact a few lucky people have high cholesterol, unhealthy

eating habits, and yet do not suffer from coronary desease.

(CMO) captures the key idea according to which inference based on “good

reasons” require an internal coherence from the logical agent: (CMO) allows

the logical agent to safely use consequences of what they already expect

(given some background knowledge). Hence. (CMO) acts as a “filters”

defining what counts as irrelevant information: ϕ is irrelevant with respect

to deciding whether θ gives good reasons to accept ψ.

As a consequence, “nonmonotonic” logic is really not a good name. “Con-

strained monotonic” logic would be a much better description, but certainly

an uglier one.

Definition 1.1. We define a preferential consequence relation (p.c.r)

any relation |∼ ⊆ SL × SL satisfying REF + (LLE) - (CMO).

Proposition 1.1. |= is a preferential consequence relation.

Proof. The proof proceeds by checking that |= satisfies all the condi-

tions from REF to CMO. �

This amounts to say that preferential consequence relations are extensions

of Tarskian consequence.

The same result is obtained by considering the first of a set of derived rules

of system REF + (LLE) - (CMO), or as it is often referred to, system P.

1.1. Derived rules of system P.

Proposition 1.2.θ |= ϕ

θ |∼ ϕ (SC)

holds for p.c.r’s.

Proof.θ |∼ θ θ |= ϕ

θ |∼ ϕ (RWE)

�

Supraclassicality (SC) captures the intuition according to which if θ gives

the logical agent sufficient reasons to accept ϕ, then it certainly gives the

agent good reasons for so doing. More formally, this allows us to use all

classical tautologies in nonmonotonic reasoning.

40 3. NONMONOTONIC LOGICS

Proposition 1.3.θ |∼ ϕ θ |∼ ϕ→ ψ

θ |∼ ψ (MP )

holds for p.c.r’s

Proof.

θ |∼ ϕ→ ψ θ |∼ ϕθ |∼ ϕ ∧ (ϕ→ ψ)

∧Dϕ ∧ (ϕ→ ψ) |= ψ

θ |∼ ψ |= D

�

Proposition 1.4.θ ∧ ϕ |∼ ψθ |∼ ϕ→ ψ

(∧ →)

holds for p.c.r’s.

Proof.

1. θ ∧ ϕ |∼ ψ (hyp)

2. θ ∧ ϕ |∼ ϕ→ ψ (RWE)

3. θ ∧ ¬ϕ |∼ ϕ→ ψ (SC)

4. (θ ∧ ϕ) ∨ (θ ∧ ¬ϕ) |∼ ϕ→ ψ (∨L 2, 3)

5. θ ≡ (θ ∧ ϕ) ∨ (θ ∧ ¬ϕ)

6. θ |∼ ϕ→ ψ (LLE 4, 5)

�

1.2. Undesirable properties for |∼. So far we have focussed on de-

sirable properties for |∼ (REF+ ∧R − CMO) and their consequences. To

argue that a property is not desirable for |∼ we can prove that it leads to

(MON) and hence, if adopted, it would lead |∼ to collapse on |=.

Proposition 1.5. The converse of (∧ →) implies MON

Proof.θ |∼ ψ ψ |= ϕ→ ψ

θ |∼ ϕ→ ψ(RWE)

θ ∧ ϕ |∼ ψ (∧ →)

�

1. GABBAY-MAKINSON CONSTRAINTS ON RATIONAL EXPECTATION 41

Remark 1.1. The above shows that only one direction of the (semantic

version of) the Deduction Theorem holds for rational consequence relations.

This effectively breaks the classical symmetry between the different levels of

“if . . . then” reasoning that was mentioned in the introduction.

Transitivity

θ |∼ ϕ ϕ |∼ ψθ |∼ ψ (TRN)

is easily seen to be intuitively undesirable:

• Yellow objects are typically highly visible (θ |∼ ϕ)

• Highly visible objects are usually landmarks (ϕ |∼ ψ)

∴ Yellow objects are usually landmarks (θ |∼ ψ)

Proposition 1.6. If |∼ satisfies TRN then it also satisfies MON

Proof.

θ ∧ ψ |= θ

θ ∧ ψ |∼ θ (SC)θ |∼ ϕ

θ ∧ ψ |∼ ϕ (∧R)

�

We can argue similarly against the desirability of Contrapisition:

¬ϕ |∼ ¬θθ |∼ ϕ (CTP )

Let us first observe its intuitive inadequacy with respect to the context of

background ignorance:

• Bright orange things are normally easy to see from a distance(θ |∼ϕ)

• Things which are hard to see from a distance are normally not

bright orange (¬ϕ |∼ ¬θ)

Again we can sharpen our intuition formally by proving that Contraposition

implies MON in the theory of preferential consequence relations.

Proposition 1.7. Contraposition implies MON for |∼.

42 3. NONMONOTONIC LOGICS

Proof.1. θ |∼ ϕ (hyp)

2. ¬ϕ |∼ ¬θ (CTP1)

3. ¬θ |= ¬(θ ∧ ψ) (logic)

4. ¬ϕ |∼ ¬(θ ∧ ψ) (RWE2, 3)

5. θ ∧ ψ |∼ ϕ (CTP4)

�

1.3. Project. The justification for CMO appeals to the intuitively plau-

sible fact that reasoning is very costly and so is information. Hence whilst

full monotonicity is clearly undesirable when reasoning under background

ignorance (i.e. in the whole of practical reasoning), so is anti-monotonicity.

In other words we certainly do not want to revise our stock of rational beliefs

unless we have good reasons do to so. Good reasons for nonmonotonic be-

haviour are provided by exceptional or unexpected or untypical situations.

If, on the other hand, we know that the situation at hand is indeed untypi-

cal, then we may want to give up some of the conclusions which we arrived

at under the assumption that the sityation was typical. But in doing so we

do not want to make unnecessary sacrifices.

A formal account of this methodological approach, which is closely related to

W.V. Quine’s Maxim of Minimal Mutilation is the subject of theory revision,

which is rooted in non monotonic logics. Read the seminal paper

• Makinson, D. (1993). Five faces of Minimality. Studia Logica, 53,

339–379.

and come up with your assessment of this methodology.

2. Preferential semantics

We are now interested in matching the “proof-theoretic” presentation of

|∼ with the kind of analysis of “follows” that led us to the definition of

|=. As further developments of the subject which are beyond the scope of

this course would show, this leads indeed to replicating the soundness and

completeness results which we know are available in classical logic.

The leading idea is to interpret “ϕ typically follows from θ” or “you can

rationally expect that θ if ϕ, as “ϕ is (classically) satisfied in all the most

normal situations in which θ is satisfied”.

The terminology preferential semantics arises from the fact that we in-

tuitively say that we, as modellers, have a preference for assuming that

2. PREFERENTIAL SEMANTICS 43

“worlds” are as normal as possible, and so we rank (i.e. order) them ac-

cordingly. The soundness and completeness results of |∼ (as defined by

rules (REF - CMO)) with respect to the preferential semantics which we

are about to introduce formally, ultimately justifies the name “preferential

consequence relations” which we have used for |∼.

In the following worked example suppose that |∼ is a preferential consequence

relation, which we think about as a logical agent. Suppose further that we

feed in the knowledge base of this logical agent with the following statements:

Example 2.1. You decided to write up an algorithm that helps your com-

pany deal with the preferences of social media users. Among the various

facts that you will want to teach your algorithm are the following:

(1) You can rationally expect young users (Y) to be politically active

(A)

(2) You can rationally expect haters (H) not to be politically active

(3) You can rationally expect haters to be young.

The idea here is that haters are atypical young users, hence it is not rea-

sonable to expect that they posses the properties that typical young people

have. If the intuitive notion of “rational expectation” is formalised by |∼, we

can see that this informal idea can be naturally accommodated by cautious

monotonicity:

Y |∼ A H |∼ ¬A H |∼ YY ∧H |∼ ¬A (CMO)

(19)

Hence our goal now is to formalise the leading idea recalled above. To this we

need to formalise the concept of a “normal situation”. Recall from Section

3 that for a finite propositional language L there is a bijection between the

ofatoms AtL and the set of all classical valuations. This means that the set

AtL can be used to model the intuitive idea of a logical situation.

To see this let L = {Y,A,H}, which is adequate to formalise our running

example. Then the set of atoms of L is given by Table 2.1

As far as language L is concerned Table 2.1 provides a full logical description

of the situation. But of course not all situations, aka atoms, aka valuations,

are equal when it comes to the intended interpretation. For instance the

atoms α1, α2, α3, α4 can be tagged as extremely implausible, for they assert

that the non-existence of young people. Reasoning in this way we may agree

that the following vector

~s = 〈α6, α5, α7, α8, 〉

44 3. NONMONOTONIC LOGICS

Y H A αi, i = 1, . . . , 8

v1 0 0 0 ¬Y ∧ ¬H ∧ ¬Av2 0 0 1 ¬Y ∧ ¬H ∧Av3 0 1 0 ¬Y ∧H ∧ ¬Av4 0 1 1 ¬Y ∧H ∧Av5 1 0 0 Y ∧ ¬H ∧ ¬Av6 1 0 1 Y ∧ ¬H ∧Av7 1 1 0 Y ∧H ∧ ¬Av8 1 1 1 Y ∧H ∧A

formalises the fact that α6 is the most normal/typical situation; α5 is the

second most normal/typical situation . . ., α8 is the lest normal/typical situ-

ation.

We insisted several times on the fact that in the construction of a nonmono-

tonic extension of the inferential machinery of classical logic we use classical

logic extensively. So our formalisation of what a situation is relies essentially

on the classical notion of a model.

This way of writing valuations as sentences (atoms) allows us to say precisely

what we mean when we say that θ normally implies ϕ if ϕ is satisfied by the

most normal situations which satisfy θ.

To this end let us write ~m = m1,m2, . . . ,mk ⊆ ATL with the idea that m1

is the most preferred situation and for i < j, mi is preferred to mj .

Definition 2.1 (Preferential semantics). For ~m = m1,m2, . . . ,mk ⊆ ATL

and θ, ϕ ∈ SL define

θ |∼~m ϕ⇔

{∀i ∈ ~mmi ∩Mθ = ∅ or

∃imi ∩Mθ 6= ∅ e and for the least such i, mi ∩Mθ ⊆Mϕ.

Example 2.2. Let us now go back to our running example. It follows that

M(Y ∧H) = {α ∈ AtL|α |= G ∧H} = {α7, α8}M(I) = {α6, α8}

M(¬I) = {α5, α7}.(20)

It is immediate to see that 7 is the least i such that

∅ 6= αi ∩M(Y ∧H) ⊆M(¬I),

which by definition gives us Y ∧H |∼ ¬I.

This is no happy coincidence, as the next seminal result shows that the

deductive system REF-CMO is sound with respect to |∼~m.

2. PREFERENTIAL SEMANTICS 45

Theorem 2.1. [Lehmann and Magidor 1992] Let ~m = m1,m2, . . . ,mk ⊆ATL. Then |∼~m is a rational consequence relation

Proof. We must check in turn that all the rules defining rational con-

sequence relations are satisfied by the definition 2.1.

Let us consider the case of CMO (other cases are similar and are left as

exercise)

Suppose θ |∼~m ϕ and θ |∼~m ψ. We must show that θ ∧ ψ |∼~m ϕ.

The definition of |∼~m requires us to check two cases. Suppose first that

∀imi ∩Mθ = ∅ . Then ∀i, mi ∩Mθ∧ϕ = mi ∩Mθ ∩Mϕ = ∅ and therefore

θ ∧ ϕ |∼~m ψ, as required.

Otherwise let i be minimal such that

mi ∩Mθ 6= ∅. (21)

From our first hypothesis (i.e. θ |∼~m ϕ) we know that i is minimal such that

mi ∩Mθ ⊆Mϕ, (22)

so that

mi ∩Mθ∧ϕ = mi ∩Mθ ∩Mϕ = mi ∩Mθ 6= ∅. (23)

Note that i is minimal such that mi ∩Mθ∧ϕ 6= ∅. To see this suppose mj ∩Mθ∧ϕ 6= ∅. Then by (21) mj ∩Mθ 6= ∅, so i ≤ j).

From our second hypothesis mi ∩Mθ∧ϕ ⊆Mψ since mi ∩Mθ∧ϕ = mi ∩Mθ.

∴ θ ∧ ϕ |∼~m ψ. This establishes CMO. �

Theorem 2.1 tells us, in complete analogy with its classical counterpart,

that the properties singled out by Definition 2.1 is preserved through the

Gabbay-Makinson rules (or conditions) for |∼. This has a very practical

consequence to the effect that it allows us to prove statements of the form

θ 6|∼ ϕ by proving that θ 6|∼~m ϕ. In other words, it gives us a definition of

what it is not rational to expect.

To grasp the idea, let us resume our running example about haters being

exceptional among young people. We proved that by establishing an instance

of CMOY |∼ A H |∼ ¬A H |∼ Y

Y ∧H |∼ ¬A (CMO)(24)

We can now show that it is not the case that young haters are politically

involved, i.e. that

Y ∧H |∼ I

46 3. NONMONOTONIC LOGICS

cannot be derived from

Y |∼ A H |∼ ¬A H |∼ Y

and hence that it should not be rationally expected to hold.

By Theorem 2.1 it suffices that we find a ~m such that:

(1) Y |∼~m I and

(2) H |∼~m G, but

(3) Y ∧H 6|∼~m I

This would in fact licence us to conclude Y ∧H 6|∼ I by contraposition.

In other words we need to construct an ~m such that

(1) i is min. such that ∅ 6= αi ∩M(Y ) ⊆M(I) and

(2) i is min. such that ∅ 6= αi ∩M(H) ⊆M(I), but

(3) i is min. such that ∅ 6= αi ∩M(Y ∧H) 6⊆M(I).

It can be easily verified that ~m = 〈α6, α7〉 will do because:

(1) 6 is min. such that ∅ 6= αi ∩M(G) and α6 ∈M(I);

(2) 7 is min. such that ∅ 6= αi ∩M(H) and α6 ∈M(G);

(3) 7 is min. such that ∅ 6= αi ∩M(Y ∧H) but α7 6∈M(I),

as required.

More generally, we can use the Soundness Theorem to show that

θ |∼ ϕ¬ϕ |∼ ¬θ (CTP )

θ |∼ ϕ→ ψ

θ ∧ ϕ |∼ ψ (→ ∧)

are not derivable in |∼.

• For (CTP), take L = {p, q} and ~m = {p ∧ q}, {p ∧ ¬q}.• For ((→ ∧)) take L = {p, q, r} and ~m = {p ∧ ¬q ∧ r}, {p ∧ q ∧ ¬r}.

Hence the Soundness theorem allows us to provide a direct argument agains

those rules which we have already identified as undesirable (because they

imply MON)

We conclude this module by mentioning a key result in the field.

Theorem 2.2 (Lehmann e Magidor 1992). Every preferential consequence

relation |∼ on SL is of the form |∼~m for some ~m = m1,m2, . . . ,mk ⊆ ATL

Delving into this beautiful result would take us goes beyond the scope of

this course. See the book by Makinson listed at the end of this section for

more details.

3. KEY REFERENCES 47

3. Key references

For an accessible and comprehensive survey of the field written by one of its

inventors see

• Makinson, D. Bridges from Classical to Nonmonotonic Logic. Lon-

don: King’s College Publications. Texts in Computing, vol 5 2005.

For a thorough exploration of how nonomonotonic logic relates to scientific

and practical reasoning see

• Rott, H. Change, choice and inference, Oxford Logic Guides, OUP,

2001

For a cognitive-science perspective see

• Stenning, K., and van Lambalgen, M. Human reasoning and cogni-

tive science. MIT, 2008.

CHAPTER 4

Further projects

In addition to the projects recommended in Section 2.1.2 and in Section

1.3, here is a list of further projects that will allow you to further advance

your knowledge of non-classical logics and their applications to practical

reasoning.

1. Abduction is non-monotonic

The following quotation is taken from

• C.S. Peirce Collected papers of C. S. Peirce, Harvard University

Press, 1932-1963. § 7.202.

Accepting the conclusion that an explanation is needed

when facts contrary to what we should expect emerge, it

follows that the explanation must be such a proposition as

would lead to the prediction of the observed facts, either as

necessary consequences or at least as very probable under

the circumstances. A hypothesis then, has to be adopted

which is likely in itself, and renders the facts likely. This

step of adopting a hypothesis as suggested by the facts is

what I call abduction.

The project involves delving in the nonmonotonic aspect of abduction. This

can be related to diagnostic problems as in the paper

• D. Poole, Representing diagnosis knowledge , Ann Math Artif Intell

(1994) 11: 33

or it can be related to the issue of providing causal explanations, as in the

paper

• A. Bochman, A causal approach to nonmonotonic reasoning Arti-

ficial Intelligence Volume 160, Issues 1–2, December 2004, Pages

105-143

49

50 4. FURTHER PROJECTS

2. 2,3,4 . . . infinity

This project relies on the fact that the compositionality of classical logic can

survive the extension of the set of truth values, but non-contradiction does

not. To recap:

Principle 2.1 (Two-valuedness). For all v ∈ V and for all θ ∈ SL

v(θ) ∈ 2.

A logic (more precisely its representation) is two-valued when the valuation

function on its language has a codomain of size 2.

Principle 2.2 (Non-contradiction). For all v ∈ V and for all θ ∈ EL the

image of θ under v is exactly one between 0 and 1.

The principle of non-contradiction expresses the rather obvious intuition to

the effect that we cannot map a propositional variable into both 0 and 1.

This is because valuations are functions, of course.1

Note however that this principle immediately loses cogency if the codomain

of valuation functions extends 2.

C.S. Peirce put it like this to William James in a letter of 26th February

1909:

I have long felt that it is a serious defect in existing logic

that it takes no heed of the limit between two realms. I do

not say that the Principle of Excluded Middle is downright

false; but I do say that in every field of thought whatsoever

there is an intermediate ground between positive assertion

and positive negation which is just as real as they. Math-

ematicians always recognize this, and seek for that limit

as the presumable lair of powerful concepts; while meta-

physicians and oldfashioned logicians –the sheep and goat

separators– never recognize this. The recognition does not

involve any denial of existing logic but it involves a great

addition to it.

From a formal point of view, the generalisation to richer codomains for

the valuation function which underlies a logic’s representation is extremely

natural. However this raises a clear interpretational problem: what is the

meaning of intermediate truth values?

1Things get more complicated if we allow set-valued functions for valuations.

2. 2,3,4 . . . INFINITY 51

The case of three values is apparently simple. We can interpret i as inde-

terminate and consider two distinct senses in which a truth value may not

be determinate.

The first goes back to Bochvar2 and takes the value i as meaningless. This

interpretation is partly motivated by the desire to set up a logic for reasoning

about self-reference and the related paradoxes. Building on this Kleene

suggested an even more cogent interpretation of i as the output of a Turing

Machine which has an indeterminate process as its own component. This

intended interpretation leads to the truth-table 1 of what is today known as

weak Kleene logic K3w .

p q ¬p p ∧ q p ∨ q p→ q

1 1 0 1 1 1

1 i i i i

1 0 0 1 0

i 1 i i i i

i i i i i

i 0 i i i

0 1 1 0 1 1

0 i i i i

0 0 0 0 1

Table 1. K3w . Note that this generalises classical proposi-

tional connectives

The second interpretation of the value i is again due to Kleene and amounts

to supposing that a computation may be determined even if it includes

a potentially indeterminate sub-computation. The result is strong Kleene

logic, K3s , whose connectives are described by Table 2

Moving away from two values leads naturally to ending up with infinitely

many values. Let us call W∞ the set

w : L → [0, 1].

We can now defineF∧, F∨, F→ : [0, 1]2 → [0, 1]

F¬ : [0, 1]→ [0, 1]

compositionally in such a way that they generalise classical propositional

connectives. The following are popular proposals in the literature.

2His original 1937 paper is translated in Bochvar, D. A., & Bergmann, M. (1981). On

a Three-valued Logical Calculus and Its Application to the Analysis of the Paradoxes of

the Classical Extended Functional Calculus. History and Philosophy of Logic, 2, 87–112.

52 4. FURTHER PROJECTS

p q ¬p p ∧ q p ∨ q p→ q

1 1 0 1 1 1

1 i i 1 i

1 0 0 1 0

i 1 i i 1 1

i i i i i

i 0 0 i 1

0 1 1 0 1 1

0 i 1 i i

0 0 0 0 1

Table 2. K3s

Definition 2.1 (Fuzzy logic).

F¬(x) = 1− xF∧(x, y) = min{x, y}F∨(x, y) = max{x, y}F→(x, y) = min{1, 1− x+ y}

Definition 2.2 (Product logic).

F¬(x) = 1− xF∧(x, y) = xy

F∨(x, y) = x+ y − xy

F→(x, y) =

{1 (x ≤ y)yx (y < x)

Definition 2.3 ( Lukasiewicz logic).

F¬(x) = 1− xF∧(x, y) = max{0, x+ y − 1}F∨(x, y) = min{1, x+ y}F→(x, y) = min{1, 1− x+ y}

For general presentations of many-valued logics you look at

• Gottwald, S. 2001. A Treatise on Many-Valued Logics. Oxford

University Press.

• Hajek, P. 1998. Metamathematics of Fuzzy Logic. Kluwer Aca-

demic Publishers. [mathematically more advanced than the previ-

ous one]

2. 2,3,4 . . . INFINITY 53

Here are two projects on many-valued logics:

(1) Lukasiewicz logic is among the most well-known many-valued log-

ics, partly because it is rich in mathematical properties, and partly

because it has long been considered a serious candidate for the log-

ical representation of reasoning under vagueness. This however is

faced with a number of epistemological difficulties related to the

actual meaning of intermediate truth values. References for this

project:

• Hajek, P. 2009. “On Vagueness, Truth Values and Fuzzy Log-

ics.” Studia Logica 91 (3): 367–82.

• Smith, N. J. 2008. Vagueness and Degrees of Truth. Oxford

University Press.

(2) What if we really did not need all the intermediate values? Refer-

ences for this project:

• Suszko, R. 1977. “The Fregean Axiom and Polish Mathemat-

ical Logic in the 1920s.” Studia Logica 36 (4): 377–80.

• Wansing, H., and Y. Shramko. 2008. “Suszko’s Thesis, Infer-

ential Many-Valuedness, and the Notion of a Logical System.”

Studia Logica 88 (3): 405–29.