## 1. Philosophy and Conditions



An ambition of twentieth-century philosophy was to analyse and refine
the definitions of significant terms--and the concepts expressed
by them--in the hope of casting light on the tricky problems of,
for example, truth, morality, knowledge and existence that lay beyond
the reach of scientific resolution. Central to this goal was
specifying at least in part the conditions to be met for correct
application of terms, or under which certain phenomena could truly be
said to be present. Even now, philosophy's unique contribution
to interdisciplinary studies of consciousness, the evolution of
intelligence, the meaning of altruism, the nature of moral obligation,
the scope of justice, the concept of pain, the theory of perception
and so on still relies on its capacity to bring high degrees of
conceptual exactness and rigour to discussions in these areas.



If memory is a capacity for tracking our own past experiences and
witnessings then a *necessary* condition for Penelope
remembering giving a lecture is that it occurred in the past.
Contrariwise, that Penelope now remembers the lecture is
*sufficient* for inferring that it was given in the past. In a
well-known attempt to use the terminology of necessary and sufficient
conditions to illuminate what it is for one thing to be cause of
another thing, J. L. Mackie proposes that causes are at a minimum INUS
conditions, that is, "Insufficient but Necessary parts of a
condition which is itself Unnecessary but Sufficient" for their
effects (Mackie 1965). A schema like Mackie's became the
foundation for the "sufficient cause model" of disease in
epidemiology (see Rothman 1976) and continues to have influence on
medical thinking about disease causation, as well as on definitions of
causation in psychology and law (VanderWeele 2017).



What, then, is a necessary (or a sufficient) condition? This article
describes the difficulties in providing complete precision in
answering this question. Although the notion of sufficient condition
can be used in defining what a necessary condition is (and vice
versa), there appears to be no straightforward way to give a
comprehensive and unambiguous account of the meaning of the term
"necessary (or sufficient) condition" itself. 


## 2. The Standard Theory: Truth-functions and Reciprocity



The front door is locked. In order to open it in a normal way and get
into the house, I must first use my key. A necessary condition of
opening the door, without violence, then, is to use the key. So it
seems true that


1. If I opened the door, I used the key.



Can we use the truth-functional understanding of "if" to
propose that the consequent of any conditional (in (i), the consequent
is "I used the key") specifies a necessary condition for
the truth of the antecedent (in (i), "I opened the door")?
Many logic and critical thinking texts use just such an approach, and
for convenience it can be called "the standard theory"
(see Blumberg 1976, pp. 133-4, Hintikka and Bachman 1991, p.
328, Moore and Parker 2009, 310-11, and Southworth and Swoyer
2020, ch. 3.2 for examples of this approach).



The standard theory makes use of the fact that in classical logic, the
truth-function "*p* [?] *q*" ("If
*p*, *q*") is false only when *p* is true
and *q* is false. The relation between "*p*"
and "*q*" in this case is sometimes referred to as
*material implication* (see the entry on
 the logic of conditionals).
 On this account of "if *p*, *q*", if the
conditional "*p* [?] *q*" is true, and
*p* holds, then *q* also holds; likewise if *q*
is false, then *p* must also be false (if the conditional is
true). The standard theory claims that when the conditional
"*p* [?] *q*" is true the truth of the
consequent, "*q*", is *necessary* for the
truth of the antecedent, "*p*", and the truth of
the antecedent is in turn *sufficient* for the truth of the
consequent. This reciprocal relation between necessary and sufficient
conditions matches the formal equivalence between a conditional
formula and its contrapositive ("~*q* [?]
~*p*" is the *contrapositive* of "*p*
[?] *q*"). Descending from talk of truth of statements
to speaking about states of affairs, we can equally correctly say, on
the standard theory, that using the key was a necessary condition of
opening the door.



Given the standard theory, necessary and sufficient conditions are
converses of each other: *B*'s being a necessary
condition of *A* is equivalent to *A*'s being a
sufficient condition of *B* (and vice versa). So it seems that
any truth-functional conditional sentence states both a sufficient and
a necessary condition as well. Suppose that if Nellie is an elephant,
then she has a trunk. Being an elephant is a sufficient condition of
her having a trunk; having a trunk in turn is a necessary condition of
Nellie's being an elephant. Indeed, the claim about the
necessary condition appears to be another way of putting the claim
about the sufficient condition, just as the contrapositive of a
formula is logically equivalent to the original formula.



It is also possible--according the the standard theory--to
use "only if" to identify a necessary condition: we can
say that Jonah was swallowed by a whale only if he was swallowed by a
mammal, for if a creature is not a mammal, it is not a whale.
Equivalent to (i) above, on this account, is the sentence "I
opened the door only if I used the key"--a perfectly
natural way of indicating that use of the key was necessary for
opening the door.



This account of necessary and sufficient conditions is particularly
apposite in dealing with logical inferences. For example, from the
truth of a conjunction, it can be inferred that each component is true
(if "*p* and *q*" is true, then
"*p*" is true and "*q*" is
true). Suppose, then, that it is true that it is both raining and
sunny. This is a sufficient condition for "it is raining"
to be true. That it is raining is--contrariwise--a necessary
condition for it being true that it is both raining and sunny. A
similar account seems to work for conceptual and definitional
contexts. So if the concept of memory is analysed as the concept of a
faculty for tracking actual past events, the fact that an event is now
in the past is a necessary condition of my presently recollecting it.
If water is chemically defined as a liquid constituted mainly of
H2O, then if a glass contains water, it contains mainly
H2O. That the glass contains mostly H2O is a
necessary condition of its containing water.



Despite its initial appeal, objections to the standard theory have
been made by theorists from both philosophy and linguistics. In
summary, the objections build on the idea that "if" in
English does not always express a uniform kind of condition. If
different kinds of conditions are expressed by the word
"if", the objectors argue, then it would be wise to
uncover these before engaging in attempts to formalize and systematize
the concepts of *necessary* and *sufficient*. In trying
to show that there is an ambiguity infecting
"if"-sentences in English, critics have focused on two
doctrines they regard as mistaken: first, that there is a reciprocity
between necessary and sufficient conditions, and, second, that
"if *p*, *q*" and "*p* only if
*q*" are equivalent.


## 3. Problems for the Standard Theory



Given any two true sentences *A* and *B*, the
conditional "If *A*, then *B*" is true. For
example, provided it is true that the sun is made of gas and also true
that elephants have four legs, then the truth-functional conditional
"If elephants have four legs, then the sun is made of gas"
is also true. However, the gaseous nature of the sun would not
normally be regarded as either a conceptually, or even a contingently,
necessary condition of the quadripedality of elephants. Indeed,
according to the standard theory, any truth will be a necessary
condition for the truth of every statement whatsoever, and any
falsehood will be a sufficient condition for the truth of any
statement we care to consider.



These odd results do not arise in some non-classical logics where it
is required that premisses be relevant to the conclusions drawn from
them, and that the antecedents of true conditionals are likewise
relevant to the consequents. But even in those versions of relevance
logic which avoid some of these odd results, it is difficult to avoid
all of the so-called "paradoxes of implication"(see the
entry on
 the logic of conditionals
 and
 relevance logic).
 For example, a contradiction (a statement of the form
"*p* and not *p*") will be a sufficient
condition for the truth of any statement unless the semantics for the
logic in question allow the inclusion of inconsistent worlds. 



These oddities might be dismissed as mere anomalies were it not for
the fact that writers have identified a number of other problems
associated with the ideas of reciprocity and equivalence mentioned at
the end of the previous section. According to the standard theory,
there is a kind of reciprocity between necessary and sufficient
conditions, and "if *p*, *q*" sentences can
always be paraphrased by "*p* only if *q*"
ones. However, as writers in linguistics have observed, neither of
these claims matches either the most natural understanding of
necessary (or sufficient) conditions, or the behaviour of
"if" (and "only if") in English. Consider, for
example, the following case (drawn from McCawley 1993, p. 317):


2. If you touch me, I'll scream.



While in the case of (i) above, using the key was necessary for
opening the door, no parallel claim seems to work for (ii): in the
natural reading of this statement, my screaming is not a condition for
your touching me. James McCawley claims that the
"if"-clause in a standard English statement gives the
condition--whether epistemic, temporal or causal--for the
truth of the "then"-clause. The natural interpretation of
(ii) is that my screaming depends on your touching me. To take my
screaming as a necessary condition for your touching me seems to get
the dependencies back to front. A similar concern arises if it is
maintained that (ii) entails that you will touch me *only if* I
scream.



A similar failure of reciprocity or mirroring arises in the case of
the door example ((i) above). While opening the door depended,
temporally and causally, on using the key first, it would be wrong to
think that using the key depended, either temporally or causally, on
opening the door. So what kind of condition does the antecedent state?
It may be helpful to consider the following puzzling pair of
conditional sentences (a modification of Sanford 1989,
175-6):


3. If he learns to play, I'll buy Lambert a cello.
4. Lambert learns to play only if I buy him a cello.



Notice that these two statements are not equivalent in meaning, even
though the standard theory treats "if *p*,
*q*" as just another way of saying "*p* only
if *q*". While (iii) states a condition under which I buy
Lambert a cello (presumably he first learns by using a borrowed one,
or maybe he hires one), (iv) states a (necessary) condition of Lambert
learning to play the instrument in the first place (there may be
others too). Understood this way, the statements taken together leave
poor old Lambert with no prospect of ever getting the cello from me.
But if (iv) were just equivalent to (iii), combining the two
statements would not lead to an *impasse* like this.



But how else can we formulate (iii) in terms of "only if"?
A natural English equivalent is surprisingly hard to formulate.
Perhaps it would be something like:


5. Lambert has learned to play the cello only if I have bought him
one.



where the auxiliary ("has"/"have") has been
introduced to try to keep dependencies in order. Yet (v) is not quite
right, for it can be read as implying that Lambert's success is
dependent on my having first bought him a cello--something that
is certainly not implied in (iii). A still better (but not completely
satisfactory) version requires further adjustment of the auxiliary,
say:


6. Lambert will have learned to play the cello only if I have bought
him one.



This time, it is not so easy to read (vi) as implying that I bought
Lambert a cello before he learned to play. These changes in the
auxiliary (sometimes described as changes in "tense") have
led some writers to argue that conditionals in English involve
implicit quantification across times (see, for example, von Fintel
1998). 



An alternative view is that different kinds of dependency are
expressed by use of the conditional construction: (iv) is not
equivalent to (iii) because the consequent of (iii) provides what
might be called a *reason for thinking* that Lambert has
learned to play the cello. By contrast, the very same
condition--that I buy Lambert a cello--appears to fulfil a
different function in (iv) (namely that I first have to buy him a
cello before he learns to play). In the following section, the
possibility of distinguishing between different kinds of conditions is
discussed. The existence of such distinctions would seem to provide
some evidence for a systematic ambiguity about the meaning of
"if" and in the concepts of *necessary* (and
*sufficient*) *condition*.



The possibility of ambiguity in these concepts raises a further
problem for the standard theory. According to it--as Georg Henrik
von Wright points out (von Wright 1974, 7)--the notions of
necessary condition and sufficient condition are themselves
interdefinable:



*A* is a sufficient condition of *B*
=*df* the absence of *A* is a necessary
condition of the absence of *B*



*B* is a necessary condition of *A*
=*df* the absence of *B* is a sufficient
condition of the absence of *A*



Ambiguity would threaten this neat interdefinability. In the following
section, we will explore whether there is an issue of concern here.
The possibility of such ambiguity figures in work by Peter Downing on
so-called "subjunctive" conditionals (Downing1959) and is
explored subsequently as a more general problem in Wilson (1979),
Goldstein et al. (2005), ch. 6, and later writers. The general
argument, in brief, is that in some contexts there is a lack of
reciprocity between necessary and sufficient conditions understood in
a certain way, while in other situations the conditions do relate
reciprocally to each other in the way required by the standard theory.
If ambiguity is present, then there is no general conclusion that can
safely be drawn about reciprocity, or lack of it, between necessary
and sufficient conditions. Instead there will be a need to distinguish
the sense of condition that is being invoked in a particular context.
Without specification of meaning and context, it would also be wrong
to make the general claim that sentences like "if *p*,
*q*" are generally paraphrasable as "*p*
only if *q*". The philosophical literature contains
numerous ways to make sense of the lack of reciprocity between the two
kinds of conditions. Using a semi-formal argument, Carsten Held, for
example, has suggested a way of explaining why necessary and
sufficient conditions are not converses by making appeal to a version
of truthmaker theory (Held 2016 - see Other Internet Resources).
In what follows, the entry focuses on a rather different and possibly
more widely accepted approach: the attempt to make sense of the lack
of reciprocity in terms of the difference between inferential,
evidential and explanatory uses of conditionals.


## 4. Inferences, Reasons for Thinking, and Reasons Why



Are the following two statements equivalent? (Wertheimer 1968,
363-4):


7. The occurrence of a sea battle tomorrow is a necessary and
sufficient condition for the truth, today, of "There will be a
sea battle tomorrow."
8. The truth, today, of "There will be a sea battle
tomorrow" is a necessary and sufficient condition for the
occurrence of a sea battle tomorrow.



David Sanford argues that while (vii) is sensible, (viii) "has
things backward" (Sanford 1989, 176-7). He writes:
"the statement about the battle, if true, is true because of the
occurrence of the battle. The battle does not occur because of the
truth of the statement" (*ibid*.) What he may mean is
that the occurrence of the battle gives *a reason why* the
statement is true, but it is not conversely the case that the truth of
the statement provides any reason why the battle occurred. Of course,
people sometimes do undertake actions just to make true what they had
formerly said; so there will be unusual cases where the truth of a
statement is a reason why an event occurred. But this seems an
unlikely reading of the sea battle case.



Now let *S* be the sentence "There will be a sea battle
tomorrow". If *S* is true today, it is correct to infer
that a sea battle will occur tomorrow. That is, even though the truth
of the sentence does not explain the occurrence of the battle, the
fact that it is true licenses the inference to the occurrence of the
event. Ascending to the purely formal mode (see section 4 of the entry
on
 Rudolf Carnap),
 we can make the point by explicitly limiting inference relations to
ones that hold among sentences or other bearers of truth values. It is
perfectly proper to infer from the truth of *S* today that some
other sentence is true tomorrow, such as "there is a sea battle
today". Since "there is a sea battle today" is true
tomorrow if and only if there is a sea battle tomorrow, then we can
infer from the fact that *S* is true today that it is true that
a sea battle will occur tomorrow.



From this observation, it would appear that there is a gap between
what is true of inferences, and what is true of "reason
why" relations. There is an inferential sense in which the truth
of *S* is both a necessary and sufficient condition for the
occurrence of the sea battle. However, there is an explanatory sense
in which the occurrence of the sea battle is necessary and sufficient
for the truth of *S*, but not *vice versa*. It would
appear that in cases like (vii) and (viii) the inferences run in both
directions, while explanatory reasons run only one way. Whether we
read (vii) as equivalent to (viii) seems to depend on the sense in
which the notions of necessary and sufficient conditions are being
deployed.



Is it possible to generalize this finding? The door example ((i)
above) seems to be a case in point. The fact I used the key gives a
reason why I was able to open the door without force. That I opened
the door without force gives a ground for inferring that I used the
key. Here is a further example from McCawley:


9. If Audrey wins the race, we will celebrate.



Audrey's winning the race is a sufficient condition for us
having a celebration, and her winning the race is the reason why we
will be celebrating. Our celebration, however, is not likely to be the
reason why she wins the race. In what sense then is the celebration a
necessary condition of her winning the race? Again, there is a ground
for inferring: that we don't celebrate is a ground for inferring
that Audrey didn't win the race. English time reference appears
sensitive to the asymmetry uncovered here, in the way noted in the
previous section. The natural way of writing the contrapositive of
(ix) is not the literal "If we will not celebrate, then Audrey
does not win the race", but rather something like:


10. If we don't celebrate, Audrey didn't win the
race.



or


11. If we aren't celebrating, Audrey hasn't won the
race.



or even


12. If we don't celebrate, Audrey can't have won the
race.



Gilberto Gomes (Gomes 2019) discusses in detail a range of cases in
which sometimes elaborate paraphrasing of this, and other, kinds can
be used to preserve reciprocity between conditionals and their
contrapositives.



Ian Wilson (in Wilson 1979) proposes that there are five possible
relations symbolised by "if, ... then ..."
including the *reason why* reading. Subsequent authors have not
adopted this proposal, preferring to focus mainly on three of these
relations, each of which bears on questions of necessity and
sufficiency. First is the implication relation symbolised by the hook
operator, "[?]" or perhaps some relevant implication
operator. Such an operator captures some inferential relations as
already noted. For example, we saw that from the truth of a
conjunction, it can be inferred that each component is true (from
"*p* and *q*", we can infer that
"*p*" is true and that "*q*" is
true). Hook, or a relevant implication operator, seems to capture one
of the relations encountered in the sea battle case, a relation which
can be thought of as holding paradigmatically between bearers of truth
values, but can also be thought of in terms of states of affairs. For
this relation, we are able to maintain the standard theory's
reciprocity thesis with the limitations already noted.



Two further relations, however, are often implicated in reflections on
necessary and sufficient conditions. To identify these, consider the
different things that can be meant by saying


13. If Solange was present, it was a good seminar.



One scenario for (xiii) is the situation where Solange is invariably a
lively contributor to any seminar she attends. Moreover, the
contributions she makes are always insightful, hence guaranteeing an
interesting time for all who attend. In this case, Solange's
presence at least in part explains or is a *reason why* the
seminar was good. Note that some writers distinguish explanatory
reasons of this sort from full-bodied explanations (Nebel 2018). A
different scenario depicts Solange as someone who has an almost
unerring knack for spotting which seminars are going to be good, even
though she is not always active in the discussion. Her attendance at a
seminar, according to this story, provides a *reason for
thinking*--or is *evidence*--that the seminar is
good. We might say that according to the first story, the seminar is
good because Solange is at it. In the second case, Solange is at it
because it is good. Examples of this kind were first introduced in
Wilson (1979) following on from the attack on the validity of
contraposition in Downing (1975). Notice that the hook (as understood
in classical logic) does not capture the *reason for thinking*
relation, for it permits any true consequent to be inferred from any
other statement whatever. Where the conditional is a *reason for
thinking* relation, then the antecedent must provide some support
for the consequent--hence has to provide some supporting evidence
for accepting the consequent. A formal exploration of one *reason
for thinking* relation is given in Vincenzo Crupi and Andrea
Iacona's study of the *evidential conditional* (Crupi and
Iacona 2020, and see the fully developed logic of the evidential
conditional in Raidl, Iacona and Crupi 2021). These writers treat the
*reason for thinking* or evidentiary relation as one in which
the antecedent provides a "reason for accepting" the
consequent. See also Igor Douven on the "evidential support
thesis" (Douven 2008, 2016). 



The *reason why* and *reason for thinking that*
conditions may help to shed light on the peculiarities encountered
earlier. That I opened the door is a reason for thinking that I used
the key, not a reason why. In case (iii) above, that he learns to play
the instrument is the reason why I will buy Lambert a cello, and that
I buy him a cello is (in the same case) a reason for thinking
that--but not a reason why--he has learned to play the
instrument. Our celebrating is a reason for thinking that Audrey has
won the race in case (ix), but is unlikely to be a reason why.



Although there is sometimes a correlation between reasons why, on the
one hand, and evidentiary relations, on the other, few generalisations
about this can be safely made. If *A* is a reason why
*B* has occurred (and so perhaps also is evidence that
*B* has occurred), then the occurrence of *B* will
sometimes be a reason for thinking--but not a
guarantee--that *A* has occurred. If *A* is no more
than a reason for thinking that *B* has occurred, then
*B* will sometimes be a reason why--but not a guarantee
that--*A* has occurred. Going back to example (i) above,
my opening the door (without violence) was a reason for thinking, that
is to say evidence, that I had used the key. That I used the key,
however, was not just a reason for thinking that I had opened the
door, but one of the reasons why I was able to open the door. What is
important is that the "if" clause of a conditional may do
any of three things described in the present section. One of these is
well captured by classical truth-functional logic, namely (i)
introduce a sentence from which the consequent follows in the way
modelled by an operator such as hook. But there are two other jobs
that "if" may do as well, namely: (ii) state a reason why
what is stated in the consequent is the case; (iii) state a reason for
thinking that what is stated in the consequent is the case even when
this is not a reason why.



In general, if explanation is directional, it may not seem surprising
that when *A* explains *B*, it is not usually the case
that *B*, or its negation, is in turn an explanation of
*A* (or its negation). Audrey's winning the race explains
our celebration, but our failure to celebrate is not (normally) a
plausible explanation of her failure to win. Solange's presence
may explain why the seminar was such a great success, but a boring
seminar is not--in any normal set of circumstances--a reason
why Solange is not at it. This result seems to undermine the usual
understanding that if *A* is a sufficient condition of
*B*, it will typically be the case that *B* is a
necessary condition for *A*, and the falsity of *B* a
sufficient condition for the falsity of *A*.



In defence of contraposition, it might be argued that in the case of
causal claims there is at least a weak form of contraposition that is
valid. Gomes proposes (Gomes 2009) that where '*A*'
is claimed to be a causally sufficient condition for
'*B*', or '*B*' a causally
necessary condition of '*A*', then some form of
reciprocity between the two kinds of conditions holds, and so some
version of contraposition will be valid. Going back to example (ii),
suppose we read this as stating a causal condition--that your
touching me would cause me to scream. Gomes suggests that
'*A*' denotes a sufficient cause of *B*,
provided that (1) '*A*' specifies the occurrence of
an event that would cause another event '*B*', and
does this by (2) stating a condition the truth of which is sufficient
for inferring the truth of '*B*'. In such a case,
we could further maintain that '*B*', in turn,
denotes a necessary effect of '*A*', meaning that
the truth of *B* provides a necessary condition for the truth
of *A* (Gomes 2009, 377-9). It is notable that Crupi and
Iacona's evidential conditional also preserves contraposition,
which they claim as a distinctive feature of of their analysis, based
on their understanding that "provides evidence for" is
another way of saying "supports" (section 6 of Crupi and
Iacona 2020). For them, if Solange's presence provides evidence
for the seminar being good (see (xiii) above), then the
seminar's not being good is evidence of Solange's absence.
They--like Gomes (2019)--thus reject Downing's (1975)
general scepticism about contraposition. Wilson, by contrast, argues
that the seminar not being good might also be a reason why Solange
does not attend. But this, he claims, "saves only the appearance
of contraposition" (Wilson 1979, 274-5).



While it is possible to distinguish these different roles the
"if" clause may play (there may be others too), it is not
always easy to isolate them in every case. The appeal to
"reasons why" and "reasons for thinking"
enables us to identify what seem to be ambiguities both in the word
"if" and in the terminology of necessary and sufficient
conditions. Unfortunately, the concept of *what is explanatory*
itself may be too vague to be very helpful here, for we can explain a
phenomenon by citing a reason for thinking it is the case, or by
citing a reason why it is the case. A similar vagueness infests the
word "because", as we see below. Consider, for example,
cases where mathematical, physical or other laws are involved (one
*locus classicus* for this issue is Sellars 1948). The truth of
"that figure is a polygon" is sufficient for inferring
"the sum of that figure's exterior angles is 360
degrees". Likewise, from "the sum of the figure's
exterior angles is not 360 degrees" we can infer "the
figure is not a polygon". Such inferences are not trivial.
Rather they depend on geometrical definitions and mathematical
principles, and so this is a case of mathematically necessary and
sufficient conditions. But it appears quite plausible that
mathematical results also give us at least a reason for thinking that
because a figure is a polygon its exterior angles will sum to 360
degrees. We may even be able to think of contexts in which someone
claims that a figure's being a polygon is a reason why its
exterior angles sum to 360 degrees. And it might not be unnatural for
someone to remark that a certain figure is a polygon because its
exterior angles sum to 360 degrees.



A similar point holds for the theory of knowledge where it is
generally held that if I know that *p*, then *p* is true
(see the entry on
 the analysis of knowledge).
 The truth of *p* is a necessary condition of knowing that
*p*, according to such accounts. In saying this we do not rule
out claims stronger than simply saying that the truth of *p*
follows from the fact that we know that *p*. That a belief is
true--for example--may be (part of) a reason for thinking it
constitutes knowledge. Other cases involve inferences licensed by
physics, biology and the natural sciences--inferences that will
involve causal or nomic conditions. Again there is need for care in
determining whether *reason why* or *reason for
thinking* relations are being stated. The increase of mean kinetic
energy of its molecules does not just imply that the temperature of a
gas is rising but also provides a reason why the temperature is
increasing. However, if temperature is just one way of measuring mean
molecular kinetic energy, then a change in temperature will be a
reason for thinking that mean kinetic energy of molecules has changed,
not a reason why it has changed.



As mentioned at the start of the article, the specification of
necessary and sufficient conditions has traditionally been part of the
philosopher's business of analysis of terms, concepts and
phenomena. Philosophical investigations of knowledge, truth,
causality, consciousness, memory, justice, altruism and a host of
other matters do not aim at stating evidential or explanatory
relations, but rather at identifying and developing conceptual ones
(see Jackson 1998 for a detailed account of conceptual analysis, and
the supplementary entry on
 conceptions of analysis in analytic philosophy
 for an overview). But even here, the temptation to look for
*reasons why* or *reasons for thinking that* is not far
away. While conceptual analysis, like dictionary definition, eschews
evidential and explanatory conditions, evidential conditions seem to
be natural consequences of definition and analysis. That Nellie is an
elephant may not be a (or the) reason why she is an animal, any more
than that a figure is a square is a reason why it has four sides. But
some evidential claims seem to make sense even in such contexts: being
an elephant apparently gives a reason for thinking that Nellie is an
animal, and a certain figure may be said to have four sides because it
is a square, in an evidential sense of "because".



To specify the necessary conditions for the truth of the sentence
"that figure is a square" is to specify a number of
conditions including "that figure has four sides",
"that figure is on a plane", and "that figure is
closed". If any one of these latter conditions is false, then
the sentence "that figure is a square" is also false.
Conversely, the truth of "that figure is a square" is a
sufficient condition for the truth of "that figure is
closed". The inferential relations in this case are
modelled--even if inadequately--by an operator such as
hook.



Now consider a previous example--that of memory. That Penelope
remembers something--according to a standard account of
memory--means (among other things) that the thing remembered was
in the past, and that some previous episode involving Penelope plays
an appropriate causal role in her present recall of the thing in
question. It would be a mistake to infer from the causal role of some
past episode in Penelope's current remembering, that the
definition of memory itself involves conditions that are explanatory
in the *reason why* sense. That Penelope now remembers some
event is not a reason why it is in the past. Nor is it a reason for
thinking that it is in the past. Rather, philosophical treatments of
memory seek for conditions that are *a priori* constitutive of
the truth of such sentences as "Penelope remembers doing
*X*". The uncovering of such conditions simply provides
insight into whether, and how, "remember" is to be
defined. *Reason why* and *reason for thinking that*
conditions seem not to play a role in this part of the
philosopher's enterprise.



Finally, it should be noted that not all conditional sentences
primarily aim at giving either necessary or sufficient conditions. A
common case involves what might be called *jocular
conditionals*. A friend of Octavia's mistakenly refers to
"Plato's *Critique of Pure Reason*" and
Octavia says, "If Plato wrote the *Critique of Pure
Reason*, then I'm Aristotle". Rather than specifying
conditions, Octavia is engaging in a form of *reductio*
argument. Since it is obvious that she is not Aristotle, her joke
invites the listener to infer (by contraposition) that Plato did not
write the *Critique of Pure Reason*. Another case is the
so-called *concessive conditional,* where the antecedent does
not appear to be a condition on the consequent of even an inferential
kind. Suppose we plan on having picnic and hope it will be sunny. But
*even if* rain comes, we will still go. In such a case it does
not seem plausible to maintain that the threat of rain provides any
condition at all on accepting the consequent. Such concessive
conditionals do not admit of contraposition (Crupi and Iacona 2022,
and compare Gomes 2020). Others have argued that in cases where a
conditional construction does not appear to be putting forward any
genuine conditions, these are "nonconditional"
conditionals (Geis and Lycan 1993)--in other words they may just
be disguised affirmations.


## 5. Conclusion



Given the different roles for "if" just identified, it is
hardly surprising that generalisations about necessary and/or
sufficient conditions are hard to formulate. Suppose, for example,
someone tries to state a sufficient condition for a seminar being good
in a context where the speaker and all the listeners share the view
that Solange's presence is a reason why seminars would be good.
In this case, Solange's presence might be said to be a
sufficient condition of the seminar being good in the sense that her
presence is a reason why it is good. Now, is there a similar sense in
which the goodness of the seminar is a necessary condition of
Solange's presence? The negative answer to this question is
already evident from the earlier discussion. If we follow von
Wright's proposal, mentioned above, we get the following result:
that the seminar is not good is a sufficient condition of Solange not
being present. But this cannot plausibly be read as a sufficient
condition in anything like the sense of a reason why. At most, the
fact of the seminar not being a good one may be a reason for thinking
that Solange was not at it. So how can we tell, in general, what kind
of condition is being expressed in an "if" sentence? As
noted in the case of the sea battle, when rewriting in the formal mode
captures the sense of what is being said, and when the formulations
"if *p*, *q*" and "*p* only if
*q*" seem idiomatically equivalent, then an inferential
interpretation will be in order, von Wright's equivalences will
hold, and the material conditional gives a reasonable account of such
cases. As indicated above, there are limitations to such an approach
and it provides at best a partial account of the circumstances under
which conditional sentences express necessary or sufficient
conditions.



As already noted, even the inferential use of "if" is not
always associated primarily with the business of stating necessary and
sufficient conditions. This observation, together with the cases and
distinctions mentioned in the present article, shows the need for
caution when we move from natural language conditionals to analysis of
them in terms of necessary and sufficient conditions, and also the
need for caution in modelling the latter conditions by means of formal
operators. It appears that there are several kinds of conditionals,
and correspondingly several kinds of conditions. As the developments
outlined above have shown, there are as a result several formal
schemes for translating and making sense of the variety of
conditionals used in natural language and the conditions, if any, that
they express.