CONDITIONAL STATEMENTS AND DIRECTIVES
DAVID MAKINSON
OUTLINE
Conditional assertions are used in many different ways in ordinary language, with great
versatility. Corresponding to their different jobs, they have different logical behaviour. For this
reason, there is not just one logic of conditionals, but many. In this brief overview, we describe
some of the main kinds of conditional to be found in common discourse, and how logicians have
sought to model them.
In the first part, we recall the pure and simple notion of the truth-functional (alias material)
conditional and some of its more complex elaborations. We also explain the subtle distinction
between conditional probability and the probability of a conditional.
In the second part, we explain the difference between a conditional proposition and a conditional
directive. The logical analysis of directives has lagged behind that of the propositions, and we
outline the recently developed concept of an input/output operation for that purpose.
1. CONDITIONAL PROPOSITIONS
1.1. What is a Conditional Proposition?
Suppose that in your house the telephone and the internet are both accessed by the same line, but
the telephone is in the living room on the ground floor, and the computer is upstairs. You explain
to a guest:
If the telephone is in use then the computer cannot access internet.
This is an example of a conditional proposition. Verbally, it is of the form if…then…, which is
perhaps the most basic and common form for conditionals in English – although, as we will see,
there are many others. We make such conditional propositions in daily life, as well as in
mathematics and the sciences.
1.2. What is the Truth-functional Conditional?
The simplest of all models for stand-alone conditionals is the truth-functional one, also often
known as material implication. Its very simplicity, so much less subtle than ordinary language,
hid it from view for a long time. The idea of a truth-table dates from the beginning of the
twentieth century - two thousand years after Aristotle first began codifying the discipline called
logic. After the event, it was realized that in antiquity, certain Stoic philosopher/logicians had
grasped the concept of material implication although, as far as anybody knows, they did not used
the graphic device of a table. But their ideas never became part of mainstream thinking in logic,
which until the late nineteenth century was dominated by the Aristotelian tradition.
To make a truth-table, whether for conditionals or other particles such as and, or, and not, we
need a simplifying assumption. We need to assume that the assertions that we are dealing with
are always either true or false, but never both; as it is usually put, that they are two-valued. It is
also assumed that if a proposition appears more than once in the context under study, its truthvalue is the same in each instance. The truth-values are written as T,F or more commonly, 1,0.
Certain ways of forming complex propositions out of simpler ones are called truth-functional, in
the sense that the truth-value of the complex item is a function (in the usual mathematical sense)
of the truth-values of its components. In the case of the material conditional, this may be
expressed by the following table, where the right-hand column is understood as listing the values
of a function of two variables, whose values are listed in the left columns.
p
q
pq
1
1
1
1
0
0
0
1
1
0
0
1
In other words: when p is true and q is false, the material conditional pq is false, but in all the
other three cases pq is true. As simple as that.
1.3. Some Odd Properties of the Material Conditional
2
Most of the properties of the truth-functional conditional are very natural. For example, it is
reflexive (the conditional proposition pp is always true, for any proposition p) and also
transitive (pr is true whenever pq and qr are). But there are others, reflecting the simple
definition, that are less natural. Among them are the following, often known as the paradoxes of
material implication:

Any conditional pq with a false antecedent p is true, no matter what the consequent q
is, and no matter whether there is any kind of link between the two. For example, the
proposition ‘If Sydney is the capital of Australia then Shakespeare wrote Hamlet’ is true,
simply because of the falsehood of its antecedent (the capital is in fact Canberra). We
could replace the consequent by any other proposition, even its own negation, and the
material conditional would remain true.

Any conditional pq with a true consequent q is true, no matter what the antecedent p is,
whether or not there is any kind of link between the two. For example, the proposition ‘If
the average temperature of the earth’s atmosphere is rising then wombats are marsupials’
is true, simply because of the truth of its consequent. We could replace the antecedent by
any other proposition, even its own negation, without affecting the truth of the entire
material conditional.

Given any two propositions p and q whatsoever, either the conditional pq or its
converse qp is true. For example, either it is true that if my car is in the garage then
your computer is turned on, or conversely it is true that if your computer is turned on then
my car is in the garage – when these propositions are understood truth-functionally.
Of the entries on the right hand side of the truth-table for , the one in the second row appears to
be incontestable. But the first may be a little suspicious, while the entries in the third and fourth
rows can appear quite arbitrary. Now as we will see in the following sections, it cannot be
pretended that the truth-functional conditional captures all the subtleties of content of
conditionals of everyday discourse. Nevertheless the table has its rationale and a certain
inevitability. A story told by Dov Gabbay illustrates this.
A shop on high street is selling electronic goods, and to promote sales offers a free printer to
anyone who buys more than £200 worth in a single purchase. The manager puts a sign in the
3
window: ‘If you buy more than £200 in electronic goods here in a single purchase, we give you a
free printer’. You are on the fraud squad, and you suspect this shop manager of making false
claims. You try to nail him by sending inspectors disguised as little old ladies, teenage punks etc.,
making purchases and asking for the free printer. The first inspector buys for £250, and is given
the printer. So far, no grounds for charging the manager. The second inspector buys for only
£150, asks for the free printer, and is refused. Still no grounds for a charge. The third inspector
buys for £190, asks for a free printer, and because the manger likes the colour of her eyes, gives
one. Still no grounds for a charge of fraud. There is only one way of showing that the shopwindow conditional is false: getting an instance where the customer buys for £200 or more, but is
not offered the printer.
What is the moral of this story? If we want our connective to be truth-functional, in other words
to be determined by some truth-table, then there is only one table that can do the job acceptably –
the one that we have chosen.
This said, it would be very misleading to say that the truth-table gives us a full analysis of
conditional propositions of everyday language, for they are normally used to convey much more
information than is given in the table. In fact, it is fair to say that the truth-functional conditional
almost never occurs in daily language in its pure form. We look at some of the ways in which this
can happen.
1.4. Implicit Generalization
You tell a student: ‘if a relation is acyclic then it is irreflexive’. What kind of conditional is this?
In effect, you are implicitly making a universal quantification. You are saying that: for every
relation r, if r is acyclic, then it is irreflexive. There is an implicit claim of generality.
For those who have already seen the notation of quantifiers in logic, the statement says that
r((A(r)I(r)), where  is material implication,  is the universal quantifier, and the letters A,I
stand for the corresponding predicates, with A(r) for ‘r is acyclic’ and I(r) for ‘r is irreflexive’.
The truth-functional connective is present, but it is not working alone.
To show that this proposition is false, you would have to find at least one relation that is acyclic
but not irreflexive, i.e. that satisfies the antecedent A(r) but falsifies the consequent I(r), briefly
that gives the combination (1,0) for antecedent and consequent. In fact, in this example, the
4
combinations (1,1), (0,1), (0,0) all arise for suitable relations r, but that does not affect the truth
of the conditional. The combination (1,0) does not exist for any relation r, and that is enough to
count the conditional as true.
We can generalize on the example. In pure mathematics, the if…then… construction is typically
used as a universally quantified material conditional, with the quantification often left implicit.
The same happens in daily language. In the example from the electronics shop, the conditional in
the window is implicitly generalizing, or as we say quantifying, over all customers and sales. In
the telephone/internet example, we are quantifying over times or occasions, saying something
like ‘whenever the telephone is in use, the computer cannot access internet’, i.e. ‘at any time t, if
the house telephone is in use at time t then the computer cannot access internet at t’.
Again in the notation of logic, this may be written as t(T(t)A(t)), where the letters T,A serve
as predicates, i.e. T(t) means ‘the telephone is in use at time t’, and A(t) means ‘computer cannot
access internet at time t’.
But for ordinary discourse, even this level of simplicity is exceptional. Suppose you are working
on your computer without a virus protection. Your friend advises: ‘if you open an attachment
with a virus, it will damage your computer’. In the spirit of the previous example, we can see this
as a universal quantification over times or occasions, with an embedded material implication.
Thus as first approximation we have: for every time t (within a certain range left implicit), if you
open an attachment with a virus at time t, then your computer will be damaged, i.e.
t(A(t)D(t)).
But this representation sins by omission, for it leaves unmentioned two aspects of the advice:

Futurity. Your hard disk will not necessarily be damaged immediately, even if it is
immediately infected. There may be a time lapse.

Causality. The introduction of the virus is causally responsible for the damage.
The representation also sins by commission, for it says more than we probably mean. It says
always, when we may mean something a bit less. Ordinary language conditionals often have the
property of:
5

Defeasibility. Your friend may not wish to say that such an ill-advised action will always
lead to damage, but that it will do so usually, probably, under natural assumptions, or
barring exceptional circumstances.
All three dimensions are pervasive in the conditionals of everyday life. Logicians have attempted
to tackle all of them. We will say a few words on each.
1.5. Futurity
Success is most apparent in the case of temporal futurity and other forms of temporal crossreference. Since the 1950s, logicians have developed a range of what are known as temporal
logics. Roughly speaking, there are two main kinds of approach.
One is to remain within the framework of material implication and quantification over moments
of time, but recognize additional layers of quantificational complexity. In the example, the
representation becomes something like: t(A(t)t(ttD(t))), although this still omits any
indication of the vague upper temporal bound on the range of the second, existential, quantifier.
Another approach is to introduce non-truth-functional connectives on propositions to do the same
job. These are called temporal operators, and belong to a broad class of non-truth-functional
connectives called modal operators. Writing x for ‘it will always be the case that x’, the
representation becomes (ad), where the predicates A(t), D(t) are replaced by propositions
a,d, and appropriate principles are devised to govern the temporal propositional operator . The
study of such temporal logics is now a recognized and relatively stable affair.
1.6. Causality
The treatment of causality as an element of conditionals has not met with the same success,
despite some attempts that also date back to the middle of the twentieth century. The reason for
this is that, to be honest, we do not have a satisfying idea of what causality is. Part of its meaning
lies in the idea of regular or probable association, and for this reason, can be considered as a form
of implicit quantification. But it is difficult to accept, as the eighteenth-century philosopher David
Hume did, that this is all there is to the concept, and even more difficult to specify clearly what is
missing. There is, as yet, no generally accepted way of representing causality within a
proposition, although there are many suggestions.
6
1.7. Defeasibility and Probability
The analysis of defeasibility has reached a state somewhere between those of futurity and
causality – not as settled as the former, but much more advanced than the latter. It is currently a
very lively area of investigation. Two general lines of attack emerge: a quantitative one using
probability, and a non-quantitative one using ideas that are not so familiar to the general scientific
public. The non-quantitative approach is reviewed at length in a companion chapter in this
volume (‘Bridges between classical and nonmonotonic logic’) and so we will leave it aside here,
making only some remarks on the quantitative approach.
The use of probability to represent uncertainty is several centuries old. For a while, in the
nineteenth century, some logicians were actively involved in the enterprise – for example Boole
himself wrote extensively on logic and probability. But in the twentieth century, the two
communities tended to drift apart, with little contact.
From the point of view of probability theory, the standard way of representing an uncertainty in
conditional contexts is via the notion of conditional probability. Given any probability
distribution P on a language, and any proposition a of the language such that P(a)  0, one
defines the function Pa by the equality Pa(x) = P(ax)/P(a) where the slash is ordinary division.
In the limiting case that P(a) = 0, Pa is left undefined. When defined, Pa is itself a probability
distribution on the language, called the conditionalization of P on a.
This concept may then be used to give probabilistic truth-conditions for defeasible conditionals.
In particular, the notion of a threshold probabilistic conditional may be defined as follows. We
begin by recalling the definition of the latter. Let P be any probability distribution on the
language under consideration, and let t be a fixed real in the interval [0,1]. Suppose a,x are
propositions of the language on which the probability distribution is defined, and P(a)  0. We
say that a probabilistically implies x (under the distribution P, modulo the threshold t), and we
write a |~P,t x, iff Pa(x)  t, i.e. iff P(ax)/P(a)  t
Thus there is not one probabilistic conditional relation |~p,t but a family of them, one for each
choice of a probability distribution P and threshold value t. Each such relation is defined only for
premises a such that P(a)  0, for it is only then that the ratio P(ax)/P(a) is well-defined in
conventional arithmetic. There is some discussion whether it is useful to extend the definition
7
with a special clause to cover the limiting case that P(a)  0, but that is a question which we need
not consider.
From a logician’s point of view, this is a kind of relation that is rather badly behaved. It is in
general nonmonotonic, in the sense that we may have a |~P,t x but not ab |~P,t x. This is only to
be expected, given that we are trying to represent a notion of uncertainty, but worse is the fact
that it is badly behaved with respect to conjunction of conclusions. That is, we may have a |~P,t x
and a |~P,t y but not a |~P,t xy. Essentially for this reason, the relation also fails a number of other
properties, notably one known as cumulative transitivity alias cut. But from a practical point of
view, it is a relation that has been used to represent uncertain conditionality in a number of
practical domains.
1.8. Conditional Probability versus Probability of a Conditional
Note that the conditional assertion a |~P,t x tells us that the conditional probability Pa(x) is
suitably high, but it does not tell us that the probability of any conditional proposition is high.
Underlying this is the fact that while Pa(x) is the conditional probability of x given a, it is not the
probability of any conditional proposition. This is a subtle conceptual distinction, but an
important one; we will attempt to explain it in this section.
To begin with, Pa(x) cannot be identified with P(ax), where  is material implication. For
ax is classically equivalent to ax, and so is highly probable whenever a is highly
improbable, even when x is less probable in the presence of a than in its absence.
This suggests the question whether there is any kind of conditional connective, call it , such
that Pa(x) = P(ax) whenever the left-hand side is defined. For a long time it was vaguely
presumed that there must some such ‘probability conditional’ somewhere; but in a celebrated
paper of 1976 David Lewis showed that this cannot be the case. The theorem is a little technical,
but is worth stating explicitly (without proof), even in a general review like the present one.
Consider any propositional language L with at least the usual truth-functional connectives (or
equivalently, consider its quotient structure under logical equivalence, which is a Boolean
algebra). Take any class P of probability distributions on L. Suppose that P is closed under
conditionalization, that is, whenever P  P and a  L then Pa  P. Suppose finally that there are
a,b,c  L and P  P such that a,b,c are pairwise inconsistent under classical logic, while each
8
separately has non-zero probability under P. This is a condition that is satisfied in all but the most
trivial of examples. Then, the theorem tells us, there is no function  from L2 into L such that
Px(y) = P(xy) for all x,y  L with P(x)  0, i.e. whenever the left-hand side is defined.
An impressive feature this result is that it is does not depend on making any apparently innocuous
but ultimately questionable assumptions about properties of the conditional . The only
hypothesis made on the connective is that it is a function from L2 into L. Indeed, an analysis of
the proof shows that even this hypothesis can be weakened. Let L0 be the purely Boolean part of
L, i.e. the part built up with just the truth-functional connectives from elementary letters. Then
there is no function  even from L02 into L that satisfies the property that for all x,y  L0 with
P(x)  0, Px(y) = P(xy). In other words, the result does not depend on iteration of the
conditional connective  in the language. Interestingly, however, it does depend on iteration of
the operation of the operation of conditionalization in the sense of probability theory – the proof
requires consideration of a conditionalization (Px)y of certain conditionalizations Px of a
probability distribution P.
Lewis’ impossibility theorem does not show us that there is anything wrong with the notion of
conditional probability. Nor does it show us that there is anything incorrect about with classical
logic. It shows that there is a difference between conditional probability on the one hand and the
probability of a conditional proposition on the other, no matter what kind of conditional
proposition we have in mind. There is no way in which the former concept may be reduced to the
latter.
Some logicians have argued that if we are prepared to abandon the classical basis of logic,
conditional probability can be identified with the probability of a suitable conditional.
Specifically, if we abandon the two-valued interpretation of the Boolean connectives between
propositions, pass to a suitable three-valued interpretation, and at the same time reconstruct
probability theory so that probability distributions have real intervals rather than real points as
values, then we can arrange matters so that the two coincide. There does not appear to be a
consensus on this question – whether it can be made to work, what conceptual costs it entails, and
whether it gives added value for the additional complexity. For an entry into the literature, see the
“Guide to further reading”.
9
1.9. Counterfactual conditionals
Returning again to daily language, we cannot resist mentioning a very strange kind of
conditional. We have all heard statements like ‘If I were you, I would not sign that contract’, ‘If
you had been on the road only a minute earlier, you would have been involved in the accident’, or
‘If the Argentine army had succeeded in its invasion of the Falklands, the military dictatorship
would had lasted much longer’.
These are called counterfactual conditionals, because it is presumed by the speaker that, as a
matter of common knowledge, the antecedent condition is in fact false (I am not you, you did not
in fact cross the road a minute earlier, the invasion did not succeed). Grammatically, in English at
least, they are often signalled by the subjunctive mood in the antecedent and the auxiliary
‘would’ in the consequent. Notoriously, they cannot be represented as material conditionals, for if
they were, the falsehood of the antecedent would make them all true (which, by the way is
another counterfactual). They may sometimes be intended as assertions without exceptions, more
often as defeasible ones. They are occasionally used in informal mathematical discourse,
especially on the oral level, but never in formal pure mathematics.
Logicians have been working on the representation of counterfactual conditionals for several
decades, and have developed some fascinating mathematical constructions to model them. To
describe these constructions would take us too far from our main thread; once again we refer to
the “Guide” at the end of this paper.
1.10. So Why Work with the Truth-Functional Conditional?
Given that the conditional statements of ordinary language usually say much more than is
contained in the truth-functional analysis, why should we bother with them? There are several
reasons. From a pragmatic point of view:

If you are doing computer science, then the truth-functional conditional will give you a
great deal of mileage. It forms the basis of any language that communicates with a
machine.

If you are doing pure mathematics, the truth-functional conditional will also serve your
purposes perfectly well, provided that you recognize that it is usually used with an
implicit universal quantification.
10
From a more theoretical point of view, there are even more important reasons:

The truth-functional conditional is the simplest possible kind of conditional. And in
general, it is a good strategy to begin any formal modelling in the simplest way,
elaborating it later if needed.

Experience has shown that when we do begin analysing any of the other kinds of
conditional mentioned above (including even such a maverick one as the counterfactual),
the truth-functional one turns up in one form or another, hidden inside. It is impossible
even to begin an analysis of any of these more complex kinds of conditional unless you
have a clear understanding of the truth-functional one.
Thus despite its eccentricities and limitations, the material conditional should not be thrown
away. It gives us the kernel of conditionality, even if not the whole fruit.
2. CONDITIONAL DIRECTIVES
2.1. The problem
So far, we have been looking at conditional propositions, where a proposition is something that
can be regarded as true or false. But when we come to consider conditional directives, we are
faced with a new problem.
By a conditional directive we mean a statement that tells us what to do in a given situation. It
may be expressed in the imperative mood as in “if you take the rubbish out for collection, put it
in a plastic bag”, or in the indicative mood as in “if you take the rubbish out for collection, you
should put it in a plastic bag”. The former has a purely directive function, whereas the latter can
be used to either issue a directive, report the fact that such a directive has been made (or is
implicit in one made), express acceptance of the directive, or (most commonly) a combination of
the three. In what follows, we focus on the purely directive function, ignoring the elements of
report and acquiescence. We also abstract from the grammatical mood in which the directive is
expressed, indicative or imperative.
Philosophically, it is widely accepted that directives and propositions differ in a fundamental
respect. Propositions may bear truth-values, in other words may be true or false; but directives are
11
items of another kind. They may be respected (or not), and may also be assessed from the
standpoint of other directive as, for example, when a legal requirement is judged from a moral
point of view (or vice versa). But it makes no sense to describe directives as true or as false.
If this is the case, how is it possible to construct a logic of directives, and in particular, of
conditional directives? The whole of classical logic revolves around the distinction between truth
and falsehood. In this section, we will sketch one approach to this problem, recently developed
by the present author with Leendert van der Torre. Called input/output logic, it provides a means
of representing conditional directives and determining their consequences, without treating them
as bearing truth-values.
2.2. Simple-minded output
For simplicity, we write a conditional directive ‘in condition a, do x’ as ax. To break old
habits, we emphasise again that we are not trying to formulate conditions under which ax is
true, for as we have noted, directives are never true, nor false. Our task is a different one. Given a
set G of conditional directives, which we call a code, we wish to formulate criteria under which a
conditional directive ax is implicit in G.
Another way of putting this is that we would like to define an operation out(.), such that out(G)
consists of all the conditional directives ax that are implicit in those already in G. Equivalently,
an operation out(.,.) such that out(G,a) is the set of all propositions x such that ax is implicit in
G. These are equivalent because given either one we can define the other by the rule x  out(G,a)
iff ax  out(G).
The simplest kind of input/output operation is depicted in Figure 1. It has two arguments G and a,
where G is a code and a is an input proposition. The operation has three phases. First, the input a
is expanded to its classical closure Cn(a), i.e. the set of all propositions y that are consequences of
a under classical (truth-functional) logic. Next, this set Cn(a) is ‘passed through’ G, which
delivers the corresponding immediate output G(Cn(a)). Here G(X) is defined in the standard settheoretic manner as the closure of a set under a relation, so that G(Cn(a)) = {x: for some b 
Cn(a), (b,x)  G}. Finally, this is expanded by classical closure again to out1(G,a) =
Cn(G(Cn(a))). We call this simple-minded output.
12
Cn(G(Cn(a)))
G
a
G(Cn(a))
Cn(a)
Figure 1: Simple-Minded Output
out1(G,a) = Cn(G(Cn(a)))
Despite its simplicity, this is already an interesting operation. It gives us the implicit content of an
explicitly given code of conditional directives, without treating the directives themselves as
propositions: only the items serving as input and as output are so treated. It should be noted that
the operation out1(G,a) does not satisfy the principle of identity, which in this context is called
throughput. That is, in general we do not have that a  out1(G,a). In the parallel notation, we do
not have that ax  out1(G). It also fails contraposition. That is, in general x  out1(G,a) does
not imply a out1(G,x). In the parallel notation, we can have ax  out1(G) without having
x a  out1(G). Reflection on how we think of conditional directives in real life indicates
that this is how it should be.
As an example, let the code G consist of just three conditional directives: (b,x), (c,y), (d,z). We
call b,c,d the bodies of these directives, and x,y,z their respective heads. Let the input a be the
conjunction b(cd)ed. Then the only bodies of elements of G that are consequences of a
are b,c, so that G(Cn(a)) = {x,y} and thus out1(G,A) = Cn(G(Cn(a))) = Cn(x,y). In the other
notation, we can say that for any proposition z, we have az out1(G) iff z  Cn(x,y).
13
It can easily be shown that simple-minded output is fully characterized by just three rules. When
formulating these rules it is convenient to use the notation ax  out1(G); indeed, as G is held
constant in all the rules, we drop the unvarying part “ out1(G)” from each of them. These are
notational conventions to make the formulations easier to read. The rules characterizing simpleminded output are three:
Strengthening Input (SI):
From ax to bx whenever a  Cn(b)
Conjoining Output (AND):
From ax, ay to axy
Weakening Output (WO):
From ax to ay whenever y  Cn(x).
It can be shown that these three rules suffice to provide passage from a code G to any element
ax of out1(G), by means of a derivation tree with leaves in G{tt} where t is any classical
tautology, and with root the desired element.
2.3. Stronger output operations
Simple-minded output lacks certain features that may be appropriate for some kinds of directive.
In the first place, the treatment of disjunctive inputs is not very sophisticated. Consider two inputs
a and b. By classical logic, we know that if x  Cn(a) and x  Cn(b) then x  Cn(ab). But there
is nothing to tell us that if x  out1(G,a) = Cn(G(Cn(a))) and x  out1(G,b) = Cn(G(Cn(b))) then x
 out1(G,ab) = Cn(G(Cn(ab))), essentially because G is an arbitrary set of ordered pairs of
propositions. In the second place, even when we do not want inputs to be automatically carried
through as outputs, we may still want outputs to be reusable as inputs – which is quite a different
matter.
Operations satisfying each of these two features can be provided with explicit definitions,
pictured by diagrams in the same spirit as that for simple-minded output. They too can be
characterized by straightforward rules. We thus have four very natural systems of input/output,
which are labelled as follows: simple-minded alias out1 (as above), basic (simple-minded plus
input disjunction: out2), reusable (simple-minded plus reusability: out3), and reusable basic (all
together: out4).
For example, reusable basic output out4 may be given a diagram and definition as in Figure 2.
The definition tells us that: out4(G,a) consists of just those propositions that are in every set of the
form Cn(G(V)), where V ranges over the complete sets of propositions that both contain a and are
14
closed under G. Here, a complete set is one that is either maximally consistent or equal to the set
of all formulae.

Cn(G(V1))
G(V1)
G
G
V1
out4(G,a)
a
G(V2)
V2

Cn(G(V2))
Figure 2: Reusable Basic Output:
out4(G,a) = {Cn(G(V)): a  V  G(V), V complete}
It can be shown that these three stronger systems may be characterized by adding one or both of
the following rules to those for simple-minded output:
Disjoining input (OR):
From ax, bx to abx
Cumulative transitivity (CT):
From ax, axy to ay.
There is a great deal more to input/output logics than we have sketched here. In particular, there
is the problem, which must be faced by any approach to the logic of directives, of dealing
adequately with what are called contrary-to-duty conditional directives. In general terms, the
problem may be put as follows: given a set of norms, how should we determine which obligations
are operative in a situation that already violates some among them. In the context of input/output
logics, one way of approaching this question is via the imposition of consistency constraints on
the application of input/output operations. Another question of interest is that of understanding
15
conditional permissions. Again, the input/output approach provides a convenient platform for
clarifying the well-known contrast between positive and negative permission, as well as for
distinguishing between different kinds of positive permission. However, in this brief review we
leave these questions aside, directing the reader to the guide below.
SUMMARY
There are many kinds of conditional in human discourse. They can be used to assert, and they can
be used to direct. On the level of assertion, the simplest kind of conditional is the truth-functional,
alias material, conditional. It almost never occurs pure in daily language, but provides the kernel
for a range of more complex kinds of conditional assertion, involving such features as universal
quantification, temporal cross-reference, causal attribution, and defeasibility. Unlike conditional
assertions, conditional directives cannot be described as true or false, and their logic has to be
approached in a more circumspect manner. Input/output logic does this by examining the notion
of one conditional directive being implicit in a code of such directives, bringing the force of
classical logic to play in the analysis without ever assuming that the directives themselves carry
truth-values.
GUIDE TO FURTHER READING
The truth-functional conditional
All elementary textbooks of modern logic present and discuss the truth-functional conditional. A
well-known text that carries the discussion further than most is W.V.O. Quine Methods of Logic,
fourth edition 1982. Harvard University Press.
Temporal conditionals
The pioneering work on temporal logics was A.N. Prior Time and Modality, Greenwood
Publishing 1979. For an introductory review, see e.g. Johan van Benthem “Temporal Logic”, in
Handbook of Logic in Artificial Intelligence and Logic Programming. vol 4, Gabbay, Hogger and
Robinson eds Oxford University Press 1995, pp 241-350, or Yde Venema “Temporal logic” in
Lou Goble ed., The Blackwell Guide to Philosophical Logic (Oxford:Blackwell, 2001).
16
Defeasible conditionals: quantitative approaches
All elementary textbooks of probability discuss conditional probability. There is a short but clear
introduction, with pointers to the literature, in chapter 14 of Stuart J. Russell and Peter Norvig
Artificial Intelligence: A Modern Approach, Prentice Hall: Upper Saddle River, NJ. 1995. David
Lewis’ impossibility result was established in his paper “Probabilities of conditionals and
conditional probabilities” The Philosophical Review 85: 297-315, 1976, reprinted with a
postscript in his Philosophical Papers, Oxford University Press 1987, pp133-156. On the attempt
to bypass this result by falling back onto a three-valued logic and a modified probability theory,
see e.g. Didier Dubois and Henry Prade “Possibility theory, probability theory and multiplevalued logics: a clarification” Annals of Mathematics and Artificial Intelligence 32: 2001, 35-66.
Defeasible conditionals: non-quantitative approaches
The pioneering classics, in chronological order, are: R. Reiter “A logic for default reasoning”
Artificial Intelligence 13: 81-132, 1980, reprinted in M.Ginsberg ed, Readings in Nonmomotonic
Reasoning Morgan Kaufmann, Los Altos CA, 1987 pp 68-93; David Poole “A logical framework
for default reasoning” Artificial Intelligence 36: 27-47, 1988; Yoav Shoham Reasoning About
Change, MIT Press, Cambridge USA 1988. For a more recent overview of the literature: David
Makinson “General Patterns in Nonmonotonic Reasoning”, in Handbook of Logic in Artificial
Intelligence and Logic Programming, vol. 3, ed. Gabbay, Hogger and Robinson, Oxford
University Press, 1994, pages 35-110. The chapter “Bridges between classical and nonmonotonic
logic” in the present volume shows how these nonmonotonic relations emerge naturally from
classical consequence.
Counterfactual conditionals
The best place to begin is probably still the classic presentation: David K. Lewis Counterfactuals,
Blackwells, Oxford 1973. For a comparative review of the different uses of minimalization in the
semantics of counterfactuals, preferential conditionals, belief revision, update and deontic logic,
see David Makinson "Five faces of minimality", Studia Logica 52: 339-379, 1993.
Conditional directives
17
For an introduction, see Makinson, David and Leendert van der Torre “What is input/output
logic?” in Foundations of the Formal Sciences II: Applications of Mathematical Logic in
Philosophy and Linguistics. Dordrecht: Kluwer, Trends in Logic Series (to appear, 2002). For
details see the following three papers by the same authors: “Input/output logics” Journal of
Philosophical Logic (2000) 29: 383-408, “Constraints for input/output logics” Journal of
Philosophical Logic 30 (2001) 155-185, “Permission from an input/output perspective” (to
appear).
Last revised 03.10.02
18