A handy tool in the search for precise definitions is thespecification of necessary and/or sufficient conditions for theapplication of a term, the use of a concept, or the occurrence of somephenomenon or event. For example, without water and oxygen, therewould be no human life; hence these things are necessary conditionsfor the existence of human beings. Cockneys, according to thetraditional definition, are all and only those born within the soundof the Bow Bells. Hence birth within the specified area is both anecessary and a sufficient condition for being a Cockney.

Like other fundamental concepts, the concepts of necessary andsufficient conditions cannot be readily specified in other terms. Thisarticle shows how elusive the quest is for a definition of the terms“necessary” and “sufficient”, indicating theexistence of systematic ambiguity in the concepts of necessary andsufficient conditions. It also describes the connection betweenpuzzles over this issue and troublesome issues surrounding the word“if” and its use in conditional sentences.

• 1. Philosophy and Conditions
• 2. The Standard Theory: Truth-functions and Reciprocity
• 3. Problems for the Standard Theory
• 4. Inferences, Reasons for Thinking, and Reasons Why
• 5. Conclusion
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Philosophy and Conditions

An ambition of twentieth-century philosophy was to analyse and refinethe definitions of significant terms—and the concepts expressedby them—in the hope of casting light on the tricky problems of,for example, truth, morality, knowledge and existence that lay beyondthe reach of scientific resolution. Central to this goal wasspecifying at least in part the conditions to be met for correctapplication of terms, or under which certain phenomena could truly besaid to be present. Even now, philosophy’s unique contributionto interdisciplinary studies of consciousness, the evolution ofintelligence, the meaning of altruism, the nature of moral obligation,the scope of justice, the concept of pain, the theory of perceptionand so on still relies on its capacity to bring high degrees ofconceptual exactness and rigour to discussions in these areas.

If memory is a capacity for tracking our own past experiences andwitnessings then anecessary condition for Peneloperemembering giving a lecture is that it occurred in the past.Contrariwise, that Penelope now remembers the lecture issufficient for inferring that it was given in the past. In awell-known attempt to use the terminology of necessary and sufficientconditions to illuminate what it is for one thing to be cause ofanother thing, J. L. Mackie proposes that causes are at a minimum INUSconditions, that is, “Insufficient but Necessary parts of acondition which is itself Unnecessary but Sufficient” for theireffects (Mackie 1965). A schema like Mackie’s became thefoundation for the “sufficient cause model” of disease inepidemiology (see Rothman 1976) and continues to have influence onmedical thinking about disease causation, as well as on definitions ofcausation in psychology and law (VanderWeele 2017).

What, then, is a necessary (or a sufficient) condition? This articledescribes the difficulties in providing complete precision inanswering this question. Although the notion of sufficient conditioncan be used in defining what a necessary condition is (and viceversa), there appears to be no straightforward way to give acomprehensive 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 getinto the house, I must first use my key. A necessary condition ofopening the door, without violence, then, is to use the key. So itseems true that

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

Can we use the truth-functional understanding of “if” topropose that the consequent of any conditional (in (i), the consequentis “I used the key”) specifies a necessary condition forthe truth of the antecedent (in (i), “I opened the door”)?Many logic and critical thinking texts use just such an approach, andfor 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 Swoyer2020, ch. 3.2 for examples of this approach).

The standard theory makes use of the fact that in classical logic, thetruth-function “p ⊃q” (“Ifp,q”) is false only whenp is trueandq is false. The relation between “p”and “q” in this case is sometimes referred to asmaterial implication (see the entry onthe logic of conditionals). On this account of “ifp,q”, if theconditional “p ⊃q” is true, andp holds, thenq also holds; likewise ifqis false, thenp must also be false (if the conditional istrue). The standard theory claims that when the conditional“p ⊃q” is true the truth of theconsequent, “q”, isnecessary for thetruth of the antecedent, “p”, and the truth ofthe antecedent is in turnsufficient for the truth of theconsequent. This reciprocal relation between necessary and sufficientconditions matches the formal equivalence between a conditionalformula and its contrapositive (“~q ⊃~p” is thecontrapositive of “p⊃q”). Descending from talk of truth of statementsto speaking about states of affairs, we can equally correctly say, onthe standard theory, that using the key was a necessary condition ofopening the door.

Given the standard theory, necessary and sufficient conditions areconverses of each other:B’s being a necessarycondition ofA is equivalent toA’s being asufficient condition ofB (and vice versa). So it seems thatany truth-functional conditional sentence states both a sufficient anda necessary condition as well. Suppose that if Nellie is an elephant,then she has a trunk. Being an elephant is a sufficient condition ofher having a trunk; having a trunk in turn is a necessary condition ofNellie’s being an elephant. Indeed, the claim about thenecessary condition appears to be another way of putting the claimabout the sufficient condition, just as the contrapositive of aformula is logically equivalent to the original formula.

It is also possible—according the standard theory—touse “only if” to identify a necessary condition: we cansay that Jonah was swallowed by a whale only if he was swallowed by amammal, for if a creature is not a mammal, it is not a whale.Equivalent to (i) above, on this account, is the sentence “Iopened the door only if I used the key”—a perfectlynatural way of indicating that use of the key was necessary foropening the door.

This account of necessary and sufficient conditions is particularlyapposite in dealing with logical inferences. For example, from thetruth of a conjunction, it can be inferred that each component is true(if “p andq” is true, then“p” is true and “q” istrue). Suppose, then, that it is true that it is both raining andsunny. This is a sufficient condition for “it is raining”to be true. That it is raining is—contrariwise—a necessarycondition for it being true that it is both raining and sunny. Asimilar account seems to work for conceptual and definitionalcontexts. So if the concept of memory is analysed as the concept of afaculty for tracking actual past events, the fact that an event is nowin the past is a necessary condition of my presently recollecting it.If water is chemically defined as a liquid constituted mainly ofH₂O, then if a glass contains water, it contains mainlyH₂O. That the glass contains mostly H₂O is anecessary condition of its containing water.

Despite its initial appeal, objections to the standard theory havebeen made by theorists from both philosophy and linguistics. Insummary, the objections build on the idea that “if” inEnglish does not always express a uniform kind of condition. Ifdifferent kinds of conditions are expressed by the word“if”, the objectors argue, then it would be wise touncover these before engaging in attempts to formalize and systematizethe concepts ofnecessary andsufficient. In tryingto show that there is an ambiguity infecting“if”-sentences in English, critics have focused on twodoctrines they regard as mistaken: first, that there is a reciprocitybetween necessary and sufficient conditions, and, second, that“ifp,q” and “p only ifq” are equivalent.

3. Problems for the Standard Theory

Given any two true sentencesA andB, theconditional “IfA, thenB” is true. Forexample, provided it is true that the sun is made of gas and also truethat 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 notnormally 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 necessarycondition for the truth of every statement whatsoever, and anyfalsehood will be a sufficient condition for the truth of anystatement we care to consider.

These odd results do not arise in some non-classical logics where itis required that premisses be relevant to the conclusions drawn fromthem, and that the antecedents of true conditionals are likewiserelevant to the consequents. But even in those versions of relevancelogic which avoid some of these odd results, it is difficult to avoidall of the so-called “paradoxes of implication”(see theentry onthe logic of conditionals andrelevance logic). For example, a contradiction (a statement of the form“p and notp”) will be a sufficientcondition for the truth of any statement unless the semantics for thelogic in question allow the inclusion of inconsistent worlds.

These oddities might be dismissed as mere anomalies were it not forthe fact that writers have identified a number of other problemsassociated with the ideas of reciprocity and equivalence mentioned atthe end of the previous section. According to the standard theory,there is a kind of reciprocity between necessary and sufficientconditions, and “ifp,q” sentences canalways be paraphrased by “p only ifq”ones. However, as writers in linguistics have observed, neither ofthese claims matches either the most natural understanding ofnecessary (or sufficient) conditions, or the behaviour of“if” (and “only if”) in English. Consider, forexample, 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 foropening the door, no parallel claim seems to work for (ii): in thenatural reading of this statement, my screaming is not a condition foryour touching me. James McCawley claims that the“if”-clause in a standard English statement gives thecondition—whether epistemic, temporal or causal—for thetruth of the “then”-clause. The natural interpretation of(ii) is that my screaming depends on your touching me. To take myscreaming as a necessary condition for your touching me seems to getthe dependencies back to front. A similar concern arises if it ismaintained that (ii) entails that you will touch meonly if Iscream.

A similar failure of reciprocity or mirroring arises in the case ofthe door example ((i) above). While opening the door depended,temporally and causally, on using the key first, it would be wrong tothink that using the key depended, either temporally or causally, onopening the door. So what kind of condition does the antecedent state?It may be helpful to consider the following puzzling pair ofconditional 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, eventhough the standard theory treats “ifp,q” as just another way of saying “p onlyifq”. While (iii) states a condition under which I buyLambert a cello (presumably he first learns by using a borrowed one,or maybe he hires one), (iv) states a (necessary) condition of Lambertlearning to play the instrument in the first place (there may beothers too). Understood this way, the statements taken together leavepoor old Lambert with no prospect of ever getting the cello from me.But if (iv) were just equivalent to (iii), combining the twostatements would not lead to animpasse 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 himone.

where the auxiliary (“has”/“have”) has beenintroduced to try to keep dependencies in order. Yet (v) is not quiteright, for it can be read as implying that Lambert’s success isdependent on my having first bought him a cello—something thatis certainly not implied in (iii). A still better (but not completelysatisfactory) version requires further adjustment of the auxiliary,say:

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

This time, it is not so easy to read (vi) as implying that I boughtLambert a cello before he learned to play. These changes in theauxiliary (sometimes described as changes in “tense”) haveled some writers to argue that conditionals in English involveimplicit quantification across times (see, for example, von Fintel1998).

An alternative view is that different kinds of dependency areexpressed by use of the conditional construction: (iv) is notequivalent to (iii) because the consequent of (iii) provides whatmight be called areason for thinking that Lambert haslearned to play the cello. By contrast, the very samecondition—that I buy Lambert a cello—appears to fulfil adifferent function in (iv) (namely that I first have to buy him acello before he learns to play). In the following section, thepossibility of distinguishing between different kinds of conditions isdiscussed. The existence of such distinctions would seem to providesome evidence for a systematic ambiguity about the meaning of“if” and in the concepts ofnecessary (andsufficient)condition.

The possibility of ambiguity in these concepts raises a furtherproblem for the standard theory. According to it—as Georg Henrikvon Wright points out (von Wright 1974, 7)—the notions ofnecessary condition and sufficient condition are themselvesinterdefinable:

A is a sufficient condition ofB=_df the absence ofA is a necessarycondition of the absence ofB

B is a necessary condition ofA=_df the absence ofB is a sufficientcondition of the absence ofA

Ambiguity would threaten this neat interdefinability. In the followingsection, we will explore whether there is an issue of concern here.The possibility of such ambiguity figures in work by Peter Downing onso-called “subjunctive” conditionals (Downing1959) and isexplored subsequently as a more general problem in Wilson (1979),Goldstein et al. (2005), ch. 6, and later writers. The generalargument, in brief, is that in some contexts there is a lack ofreciprocity between necessary and sufficient conditions understood ina certain way, while in other situations the conditions do relatereciprocally to each other in the way required by the standard theory.If ambiguity is present, then there is no general conclusion that cansafely be drawn about reciprocity, or lack of it, between necessaryand sufficient conditions. Instead there will be a need to distinguishthe sense of condition that is being invoked in a particular context.Without specification of meaning and context, it would also be wrongto make the general claim that sentences like “ifp,q” are generally paraphrasable as “ponly ifq”. The philosophical literature containsnumerous ways to make sense of the lack of reciprocity between the twokinds of conditions. Using a semi-formal argument, Carsten Held, forexample, has suggested a way of explaining why necessary andsufficient conditions are not converses by making appeal to a versionof truthmaker theory (Held 2016 – see Other Internet Resources).In what follows, the entry focuses on a rather different and possiblymore widely accepted approach: the attempt to make sense of the lackof 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 andsufficient condition for the truth, today, of “There will be asea battle tomorrow.”
8. The truth, today, of “There will be a sea battletomorrow” is a necessary and sufficient condition for theoccurrence of a sea battle tomorrow.

David Sanford argues that while (vii) is sensible, (viii) “hasthings backward” (Sanford 1989, 176–7). He writes:“the statement about the battle, if true, is true because of theoccurrence of the battle. The battle does not occur because of thetruth of the statement” (ibid.) What he may mean isthat the occurrence of the battle givesa reason why thestatement is true, but it is not conversely the case that the truth ofthe statement provides any reason why the battle occurred. Of course,people sometimes do undertake actions just to make true what they hadformerly said; so there will be unusual cases where the truth of astatement is a reason why an event occurred. But this seems anunlikely reading of the sea battle case.

Now letS be the sentence “There will be a sea battletomorrow”. IfS is true today, it is correct to inferthat a sea battle will occur tomorrow. That is, even though the truthof the sentence does not explain the occurrence of the battle, thefact that it is true licenses the inference to the occurrence of theevent. Ascending to the purely formal mode (see section 4 of the entryonRudolf Carnap), we can make the point by explicitly limiting inference relations toones that hold among sentences or other bearers of truth values. It isperfectly proper to infer from the truth ofS today that someother sentence is true tomorrow, such as “there is a sea battletoday”. Since “there is a sea battle today” is truetomorrow if and only if there is a sea battle tomorrow, then we caninfer from the fact thatS is true today that it is true thata sea battle will occur tomorrow.

From this observation, it would appear that there is a gap betweenwhat is true of inferences, and what is true of “reasonwhy” relations. There is an inferential sense in which the truthofS is both a necessary and sufficient condition for theoccurrence of the sea battle. However, there is an explanatory sensein which the occurrence of the sea battle is necessary and sufficientfor the truth ofS, but notvice versa. It wouldappear that in cases like (vii) and (viii) the inferences run in bothdirections, while explanatory reasons run only one way. Whether weread (vii) as equivalent to (viii) seems to depend on the sense inwhich the notions of necessary and sufficient conditions are beingdeployed.

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 areason why I was able to open the door without force. That I openedthe door without force gives a ground for inferring that I used thekey. 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 ushaving a celebration, and her winning the race is the reason why wewill be celebrating. Our celebration, however, is not likely to be thereason why she wins the race. In what sense then is the celebration anecessary condition of her winning the race? Again, there is a groundfor inferring: that we don’t celebrate is a ground for inferringthat Audrey didn’t win the race. English time reference appearssensitive to the asymmetry uncovered here, in the way noted in theprevious section. The natural way of writing the contrapositive of(ix) is not the literal “If we will not celebrate, then Audreydoes not win the race”, but rather something like:

10. If we don’t celebrate, Audrey didn’t win therace.

or

11. If we aren’t celebrating, Audrey hasn’t won therace.

or even

12. If we don’t celebrate, Audrey can’t have won therace.

Gilberto Gomes (Gomes 2019) discusses in detail a range of cases inwhich sometimes elaborate paraphrasing of this, and other, kinds canbe used to preserve reciprocity between conditionals and theircontrapositives.

Ian Wilson (in Wilson 1979) proposes that there are five possiblerelations symbolised by “if, … then …”including thereason why reading. Subsequent authors have notadopted this proposal, preferring to focus mainly on three of theserelations, each of which bears on questions of necessity andsufficiency. First is the implication relation symbolised by the hookoperator, “⊃” or perhaps some relevant implicationoperator. Such an operator captures some inferential relations asalready noted. For example, we saw that from the truth of aconjunction, it can be inferred that each component is true (from“p andq”, we can infer that“p” is true and that “q” istrue). Hook, or a relevant implication operator, seems to capture oneof the relations encountered in the sea battle case, a relation whichcan be thought of as holding paradigmatically between bearers of truthvalues, but can also be thought of in terms of states of affairs. Forthis relation, we are able to maintain the standard theory’sreciprocity thesis with the limitations already noted.

Two further relations, however, are often implicated in reflections onnecessary and sufficient conditions. To identify these, consider thedifferent 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 alively contributor to any seminar she attends. Moreover, thecontributions she makes are always insightful, hence guaranteeing aninteresting time for all who attend. In this case, Solange’spresence at least in part explains or is areason why theseminar was good. Note that some writers distinguish explanatoryreasons of this sort from full-bodied explanations (Nebel 2018). Adifferent scenario depicts Solange as someone who has an almostunerring knack for spotting which seminars are going to be good, eventhough she is not always active in the discussion. Her attendance at aseminar, according to this story, provides areason forthinking—or isevidence—that the seminar isgood. We might say that according to the first story, the seminar isgood because Solange is at it. In the second case, Solange is at itbecause it is good. Examples of this kind were first introduced inWilson (1979) following on from the attack on the validity ofcontraposition in Downing (1975). Notice that the hook (as understoodin classical logic) does not capture thereason for thinkingrelation, for it permits any true consequent to be inferred from anyother statement whatever. Where the conditional is areason forthinking relation, then the antecedent must provide some supportfor the consequent—hence has to provide some supporting evidencefor accepting the consequent. A formal exploration of onereasonfor thinking relation is given in Vincenzo Crupi and AndreaIacona’s study of theevidential conditional (Crupi andIacona 2020, and see the fully developed logic of the evidentialconditional in Raidl, Iacona and Crupi 2021). These writers treat thereason for thinking or evidentiary relation as one in whichthe antecedent provides a “reason for accepting” theconsequent. See also Igor Douven on the “evidential supportthesis” (Douven 2008, 2016).

Thereason why andreason for thinking thatconditions may help to shed light on the peculiarities encounteredearlier. That I opened the door is a reason for thinking that I usedthe key, not a reason why. In case (iii) above, that he learns to playthe instrument is the reason why I will buy Lambert a cello, and thatI buy him a cello is (in the same case) a reason for thinkingthat—but not a reason why—he has learned to play theinstrument. Our celebrating is a reason for thinking that Audrey haswon the race in case (ix), but is unlikely to be a reason why.

Although there is sometimes a correlation between reasons why, on theone hand, and evidentiary relations, on the other, few generalisationsabout this can be safely made. IfA is a reason whyB has occurred (and so perhaps also is evidence thatB has occurred), then the occurrence ofB willsometimes be a reason for thinking—but not aguarantee—thatA has occurred. IfA is no morethan a reason for thinking thatB has occurred, thenB will sometimes be a reason why—but not a guaranteethat—A has occurred. Going back to example (i) above,my opening the door (without violence) was a reason for thinking, thatis 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 thedoor, but one of the reasons why I was able to open the door. What isimportant is that the “if” clause of a conditional may doany of three things described in the present section. One of these iswell captured by classical truth-functional logic, namely (i)introduce a sentence from which the consequent follows in the waymodelled by an operator such as hook. But there are two other jobsthat “if” may do as well, namely: (ii) state a reason whywhat is stated in the consequent is the case; (iii) state a reason forthinking that what is stated in the consequent is the case even whenthis is not a reason why.

In general, if explanation is directional, it may not seem surprisingthat whenA explainsB, it is not usually the casethatB, or its negation, is in turn an explanation ofA (or its negation). Audrey’s winning the race explainsour celebration, but our failure to celebrate is not (normally) aplausible explanation of her failure to win. Solange’s presencemay explain why the seminar was such a great success, but a boringseminar is not—in any normal set of circumstances—a reasonwhy Solange is not at it. This result seems to undermine the usualunderstanding that ifA is a sufficient condition ofB, it will typically be the case thatB is anecessary condition forA, and the falsity ofB asufficient condition for the falsity ofA.

In defence of contraposition, it might be argued that in the case ofcausal claims there is at least a weak form of contraposition that isvalid. Gomes proposes (Gomes 2009) that where ‘A’is claimed to be a causally sufficient condition for‘B’, or ‘B’ a causallynecessary condition of ‘A’, then some form ofreciprocity between the two kinds of conditions holds, and so someversion of contraposition will be valid. Going back to example (ii),suppose we read this as stating a causal condition—that yourtouching me would cause me to scream. Gomes suggests that‘A’ denotes a sufficient cause ofB,provided that (1) ‘A’ specifies the occurrence ofan event that would cause another event ‘B’, anddoes this by (2) stating a condition the truth of which is sufficientfor inferring the truth of ‘B’. In such a case,we could further maintain that ‘B’, in turn,denotes a necessary effect of ‘A’, meaning thatthe truth ofB provides a necessary condition for the truthofA (Gomes 2009, 377–9). It is notable that Crupi andIacona’s evidential conditional also preserves contraposition,which they claim as a distinctive feature of of their analysis, basedon their understanding that “provides evidence for” isanother way of saying “supports” (section 6 of Crupi andIacona 2020). For them, if Solange’s presence provides evidencefor the seminar being good (see (xiii) above), then theseminar’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, arguesthat the seminar not being good might also be a reason why Solangedoes not attend. But this, he claims, “saves only the appearanceof 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 notalways 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 sufficientconditions. Unfortunately, the concept ofwhat is explanatoryitself may be too vague to be very helpful here, for we can explain aphenomenon by citing a reason for thinking it is the case, or byciting a reason why it is the case. A similar vagueness infests theword “because”, as we see below. Consider, for example,cases where mathematical, physical or other laws are involved (onelocus 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 360degrees”. Likewise, from “the sum of the figure’sexterior angles is not 360 degrees” we can infer “thefigure is not a polygon”. Such inferences are not trivial.Rather they depend on geometrical definitions and mathematicalprinciples, and so this is a case of mathematically necessary andsufficient conditions. But it appears quite plausible thatmathematical results also give us at least a reason for thinking thatbecause a figure is a polygon its exterior angles will sum to 360degrees. We may even be able to think of contexts in which someoneclaims that a figure’s being a polygon is a reason why itsexterior angles sum to 360 degrees. And it might not be unnatural forsomeone to remark that a certain figure is a polygon because itsexterior angles sum to 360 degrees.

A similar point holds for the theory of knowledge where it isgenerally held that if I know thatp, thenp is true(see the entry onthe analysis of knowledge). The truth ofp is a necessary condition of knowing thatp, according to such accounts. In saying this we do not ruleout claims stronger than simply saying that the truth ofpfollows from the fact that we know thatp. That a belief istrue—for example—may be (part of) a reason for thinking itconstitutes knowledge. Other cases involve inferences licensed byphysics, biology and the natural sciences—inferences that willinvolve causal or nomic conditions. Again there is need for care indetermining whetherreason why orreason forthinking relations are being stated. The increase of mean kineticenergy of its molecules does not just imply that the temperature of agas is rising but also provides a reason why the temperature isincreasing. However, if temperature is just one way of measuring meanmolecular kinetic energy, then a change in temperature will be areason 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 ofnecessary and sufficient conditions has traditionally been part of thephilosopher’s business of analysis of terms, concepts andphenomena. Philosophical investigations of knowledge, truth,causality, consciousness, memory, justice, altruism and a host ofother matters do not aim at stating evidential or explanatoryrelations, but rather at identifying and developing conceptual ones(see Jackson 1998 for a detailed account of conceptual analysis, andthe supplementary entry onconceptions of analysis in analytic philosophy for an overview). But even here, the temptation to look forreasons why orreasons for thinking that is not faraway. While conceptual analysis, like dictionary definition, eschewsevidential and explanatory conditions, evidential conditions seem tobe natural consequences of definition and analysis. That Nellie is anelephant may not be a (or the) reason why she is an animal, any morethan that a figure is a square is a reason why it has four sides. Butsome evidential claims seem to make sense even in such contexts: beingan elephant apparently gives a reason for thinking that Nellie is ananimal, and a certain figure may be said to have four sides because itis 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 ofconditions including “that figure has four sides”,“that figure is on a plane”, and “that figure isclosed”. If any one of these latter conditions is false, thenthe sentence “that figure is a square” is also false.Conversely, the truth of “that figure is a square” is asufficient condition for the truth of “that figure isclosed”. The inferential relations in this case aremodelled—even if inadequately—by an operator such ashook.

Now consider a previous example—that of memory. That Peneloperemembers something—according to a standard account ofmemory—means (among other things) that the thing remembered wasin the past, and that some previous episode involving Penelope playsan appropriate causal role in her present recall of the thing inquestion. It would be a mistake to infer from the causal role of somepast episode in Penelope’s current remembering, that thedefinition of memory itself involves conditions that are explanatoryin thereason why sense. That Penelope now remembers someevent is not a reason why it is in the past. Nor is it a reason forthinking that it is in the past. Rather, philosophical treatments ofmemory seek for conditions that area priori constitutive ofthe truth of such sentences as “Penelope remembers doingX”. The uncovering of such conditions simply providesinsight into whether, and how, “remember” is to bedefined.Reason why andreason for thinking thatconditions seem not to play a role in this part of thephilosopher’s enterprise.

Finally, it should be noted that not all conditional sentencesprimarily aim at giving either necessary or sufficient conditions. Acommon case involves what might be calledjocularconditionals. A friend of Octavia’s mistakenly refers to“Plato’sCritique of Pure Reason” andOctavia says, “If Plato wrote theCritique of PureReason, then I’m Aristotle”. Rather than specifyingconditions, Octavia is engaging in a form ofreductioargument. Since it is obvious that she is not Aristotle, her jokeinvites the listener to infer (by contraposition) that Plato did notwrite theCritique of Pure Reason. Another case is theso-calledconcessive conditional, where the antecedent doesnot appear to be a condition on the consequent of even an inferentialkind. Suppose we plan on having picnic and hope it will be sunny. Buteven if rain comes, we will still go. In such a case it doesnot seem plausible to maintain that the threat of rain provides anycondition at all on accepting the consequent. Such concessiveconditionals do not admit of contraposition (Crupi and Iacona 2022,and compare Gomes 2020). Others have argued that in cases where aconditional construction does not appear to be putting forward anygenuine conditions, these are “nonconditional”conditionals (Geis and Lycan 1993)—in other words they may justbe disguised affirmations.

5. Conclusion

Given the different roles for “if” just identified, it ishardly surprising that generalisations about necessary and/orsufficient conditions are hard to formulate. Suppose, for example,someone tries to state a sufficient condition for a seminar being goodin a context where the speaker and all the listeners share the viewthat Solange’s presence is a reason why seminars would be good.In this case, Solange’s presence might be said to be asufficient condition of the seminar being good in the sense that herpresence is a reason why it is good. Now, is there a similar sense inwhich the goodness of the seminar is a necessary condition ofSolange’s presence? The negative answer to this question isalready evident from the earlier discussion. If we follow vonWright’s proposal, mentioned above, we get the following result:that the seminar is not good is a sufficient condition of Solange notbeing present. But this cannot plausibly be read as a sufficientcondition in anything like the sense of a reason why. At most, thefact of the seminar not being a good one may be a reason for thinkingthat Solange was not at it. So how can we tell, in general, what kindof condition is being expressed in an “if” sentence? Asnoted in the case of the sea battle, when rewriting in the formal modecaptures the sense of what is being said, and when the formulations“ifp,q” and “p only ifq” seem idiomatically equivalent, then an inferentialinterpretation will be in order, von Wright’s equivalences willhold, and the material conditional gives a reasonable account of suchcases. As indicated above, there are limitations to such an approachand it provides at best a partial account of the circumstances underwhich conditional sentences express necessary or sufficientconditions.

As already noted, even the inferential use of “if” is notalways associated primarily with the business of stating necessary andsufficient conditions. This observation, together with the cases anddistinctions mentioned in the present article, shows the need forcaution when we move from natural language conditionals to analysis ofthem in terms of necessary and sufficient conditions, and also theneed for caution in modelling the latter conditions by means of formaloperators. It appears that there are several kinds of conditionals,and correspondingly several kinds of conditions. As the developmentsoutlined above have shown, there are as a result several formalschemes for translating and making sense of the variety ofconditionals used in natural language and the conditions, if any, thatthey express.

Bibliography

• Blumberg, Albert E., 1976.Logic: A First Course, NewYork: Alfred E. Knopf.
• Crupi, Vincenzo and Iacona, Andrea 2020. “The EvidentialConditional”,Erkenntnis 2020.doi:10.1007/s10670-020-00332-2
• –––, 2022. “On the Logical Form ofConcessive Conditionals”,Journal of PhilosophicalLogic. doi:10.1007/s10992-021-09645-1/li>
• Douven, Igor, 2008. “The Evidential Support Theory ofConditionals”,Synthese, 164(1): 19–44.doi:10.1007/s11229-007-9214-5
• –––, 2016,The Epistemology of IndicativeConditionals: Formal and Empirical Approaches, Cambridge:Cambridge University Press. doi:10.1017/CBO9781316275962
• Downing, Peter, 1959. “Subjunctive Conditionals, Time Orderand Causation”,Proceedings of the AristotelianSociety, 59: 126–40.
• –––, 1975. “Conditionals, Impossibilitiesand Material Implications”,Analysis, 35:84–91.
• Geis, Michael L., and Lycan, William G., 1993.“Nonconditional Conditionals”,PhilosophicalTopics, 21: 35–56.
• Gomes, Gilberto, 2009. “Are Necessary and SufficientConditions Converse Relations?”,Australasian Journal ofPhilosophy, 87: 375–87.
• –––, 2019. “Meaning PreservingContraposition of Natural Language Conditionals”,Journal ofPragmatics 152: 46–60.
• –––, 2020. “Concessive conditionalswithoutEven ifand non-concessive conditionals withEvenif”,Acta Analytica, 35: 1–2.doi:10.1007/s12136-019-00396-y
• Goldstein, Laurence, Brennan, Andrew, Deutsch, Max and Lau, JoeY.F., 2005.Logic: Key Concepts in Philosophy, London:Continuum.
• Hintikka, Jaakko and Bachman, James, 1991.What If …?Toward Excellence in Reasoning, London: Mayfield.
• Jackson, Frank, 1998.From Metaphysics to Ethics: A Defence ofConceptual Analysis, Oxford: Oxford University Press.
• Mackie, J. L., 1965. “Causes and Conditions”,American Philosophical Quarterly, 12: 245–65.
• McCawley, James, 1993.Everything that Linguists have AlwaysWanted to Know about Logic* (Subtitle: *But Were Ashamed toAsk), Chicago: Chicago University Press.
• Moore, Brooke N., and Parker, Richard 2009.CriticalThinking, ninth edition, New York: McGraw Hill.
• Nebel, Jacob 2019. “Normative Reasons as Reasons Why WeOught”, Mind 128(510): 459–484.doi:10.1093/mind/fzy013
• Raidl, Eric, Iacona, Andrea, and Crupi, Vincenzo 2021. “TheLogic of the Evidential Conditional”,The Review of SymbolicLogic, First View: 1–13. doi:10.1017/S1755020321000071
• Rothman, K.J. 1976. “Causes”,American Journal ofEpidemiology, 104:587–92
• Sanford, David H., 1989.If P, then Q: Conditionals and theFoundations of Reasoning, London: Routledge.
• Sellars, Wilfrid, 1948. “Concepts as involving laws andinconceivable without them”,Philosophy of Science, 15:289–315.
• Southworth, Jason and Swoyer, Chris, 2020. “CriticalReasoning: A User's Manual, v.4.0”.Philosophy OpenEducational Resources, 2. [Southworth and Swoyer 2020 available online]
• VanderWeele, Tyler J., 2017. “Invited Commentary: TheContinuing Need for the Sufficient Cause Model Today”,American Journal of Epidemiology, 185, 11: 1041–1043.doi:10.1093/aje/kwx083
• Von Fintel, Kai, 1997. “Bare Plurals, Bare Conditionals andOnly”,Journal of Semantics, 14: 1–56.
• Von Wright, Georg Henrik, 1974.Causality andDeterminism, New York: Columbia University Press.
• Wertheimer, Roger, 1968. “Conditions”,Journal ofPhilosophy, 65: 355–64.
• Wilson, Ian R., 1979. “Explanatory and InferentialConditionals”,Philosophical Studies, 35:269–78.
• Woods, M., Wiggins, D. and Edgington D. (eds.), 1997.Conditionals, Oxford: Clarendon Press.

Academic Tools

	How to cite this entry.
	Preview the PDF version of this entry at theFriends of the SEP Society.
	Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO).
	Enhanced bibliography for this entryatPhilPapers, with links to its database.

Other Internet Resources

• Held, Carsten, 2016.Conditions, download from philsci-archive.pitt.edu/12164.
• The Concepts of Necessary and Sufficient Conditions, maintained by Norman Swartz, Philosophy, Simon FraserUniversity.

Related Entries

conditionals |conditionals: counterfactual |definitions |logic: classical |logic: conditionals |logic: modal |logic: relevance

Acknowledgments

I am grateful to Richard Borthwick, Jake Chandler, Laurence Goldstein,Fred Kroon, Y.S. Lo, Jesse Alama, Edward Zalta and Uri Nodelman fortheir generous help and advice relating to this entry.