Abbreviate Necessary Conditions to NC and Sufficient Conditions to SC.

Source: p 545, A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley

Sometimes the context provides an immediate clue to the sense in which “cause” is being used. If we are trying to prevent a certain phenomenon from happening, we usually search for a cause that is a NC, and if we are trying to produce a certain phenomenon, we usually search for a cause that is a sufficient condition. For example, in attempting to prevent the occurrence of smog around cities, scientists try to isolate a necessary condition or group of necessary conditions that, if removed, will eliminate the smog. [...]
Another important point is that whenever an event occurs, at least one sufficient condition is present and all the necessary conditions are present. The conjunction of the necessary conditions is the sufficient condition that actually produces the event.

  1. To prevent something from happening, why do we 'usually search for a cause that is a NC'? Why not SC? NCs do not reveal the entire picture for the production of something?

I explain my confusion by detailing the above example on smog. Suppose that smog production requires Chemical 1, ..., Chemical 10 as NCs (ie: The 10 NCs for smog = Chemical 1 ∧ ... ∧ Chemical 9 ∧ Chemical 10); but Chemical 10 is fatal though hardly detectable.
I deliberately omit describing the SCs for smog (eg: specific meteorological conditions).

  1. The quote above implies no need to care about all 10 Chemicals, but this appears wrong?
  • The necessary conditions for smog involve 1) one or more of a list of chemicals, together with 2) a given meteorological effect (stagnation, inversion, cyclical winds...) that keeps them from blowing away fast enough. Each of the list of chemicals then does not become a necessary condition. You can forget most of them and the remainder may still be enough to provide the chemical half of the cause. So no given one of them is ever truly necessary.
    – user9166
    Jan 8, 2016 at 22:24

6 Answers 6


Regarding your question 1 :

  1. To prevent something from happening, why do we usually search for a cause that is a NC? Both prevention and production of something need the SC, because NCs do not reveal the entire 'picture'.

This appears to be backwards to me. A SC captures one reason for a consequent, but not necessarily the only reason for a consequent. Suppose you wish to know if a number is even. A sufficient condition for evenness is divisibility by 4, but obviously this is not a necessary condition for evenness. An NC for evenness would be divisibility by 2, and this does "capture the whole picture".

In the quoted text, Hurley states : "scientists try to isolate a necessary condition or group of necessary conditions that, if removed, will eliminate the smog". Let's suppose: their efforts have succeeded and they have identified the, say, 10 chemicals that cause smog. (For now, let's ignore your point about the difficulty in detecting chemical10.) Then, quasi-formally we have :

  1. smog = (chem1 ∨ chem2 ∨ ... ∨ chem10)

and therefore

  1. ¬smog = (¬chem1 ∧ ¬chem2 ∧ ... ∧ ¬chem10)

Some of your confusion may arise from the fact that the disjunctive statement for smog (ie 3) is both a NC and a SC for smog since we are assuming that scientists have identified all of the chemical components of smog. Suppose that 'smog' were not defined by 3, but instead by the statement

  1. An SC for smog = chem1 ∨ chem7.

Then 5 is not a NC for smog since we can still have smog if there is no chem1 or chem7 present - e.g., just chem3 is present in the air. In other words, a SC is sufficient to cause a consequent, but it is not the only way to cause a consequent.

If we wish to eliminate smog, then 4 (the conjunctive statement for ¬smog) tells us that we must necessarily eliminate each of the 10 chemicals. Again, this is both a NC and a SC for no smog since we are assuming that scientists have succeeded in identifying all of the chemical constituents of smog. Suppose that ¬smog were not defined by 4, but instead by the statement

  1. ¬smog = ¬chem4 ∧ ¬chem8.

Then 6 is a NC for not having any smog, but 6 is not a SC since other problem chemicals may be present. The Argument Form of 6 is the 'mirror opposite' of the Argument Form of 5 (the subgroup of the disjunctive statement illustrated in the previous paragraph). Beware that I use 'mirror opposite' to describe the relation between NC and SC in the two examples, not to the formal content of 5 and 6 (since 5 and 6 are not the negations of each other): ie, 5 is a SC but not a NC, while 6 is a NC but not a SC, hence 'mirror opposite'. The use of 'mirror opposite' is probably not the best choice of words since it is perhaps too strong in this context.

The reason I have bolded try in the text quoted from Hurley is that this may be why you question the case "but Chemical 10 is fatal but hardly detectable". If scientists have not succeeded in identifying all of the constituents of smog, or are unable to detect all of the constitutents, then establishing NCs and SCs are more selective. Assuming other unknown or undetectable chemicals are required and not listed, the disjunctive statement for smog would become SC but not NC, while the conjunctive statement for ¬smog would become NC but not SC.

  • +1. Thanks as always. I failed to perceive possible ambiguity in my question; when you wrote your answer above, how did you interpret uppose some NCs for smog as Chem(ical) 1, ..., Chemical 99, Chemical 100? That all 100 Chemicals are needed for smog, or only any one of these 100?
    – user8572
    Jan 20, 2016 at 5:03
  • @LePressentiment Hi! I didn't mean to imply any ambiguity on your behalf. I was highlighting Hurley's use of the word "try" as a possible source of a lack of precision by the author. Is this what you refer to?
    – nwr
    Jan 22, 2016 at 22:20
  • Hello again! Sorry for any confusion; I did not mean that you impl[ied] any ambiguity on your behalf. Rather, I am diagnosing and exposing my own ambiguity! No, I meant to ask: How did you interpret my writing that Suppose some NCs for smog as Chem(ical) 1, ..., Chemical 99, Chemical 100. Did you interpret this as a conjunction of 100 chemicals, or a disjunction? I meant it as a conjunction but neglected to specify so; so I am unsure how others interpreted my writing.
    – user8572
    Jan 23, 2016 at 3:10
  • 1
    @LePressentiment You're more than welcome. As I said before, your experience learning logic is very similar to my own. It can be very frustrating at times, but rewarding when things finally click. Like many things, one can read something dozens of times without making any sense of it, but for some reason it does make sense on the umpteenth reading.
    – nwr
    Jan 29, 2016 at 20:45
  • 1
    +1. Thanks again. At least my experience will be less painful with thanks to your continued benevolent aid.
    – user8572
    Jan 29, 2016 at 21:14

For preventing something, why do we usually search for the Necessary and not the Sufficient Conditions?

See this post for "necessary and sufficent conditions".

We have that "B is a necessary condition for A" means A → B. This is equivalent to : ¬B → ¬A (see : contraposition).

This means that "car traffic is a necessary condition for pollution", i.e. "if pollution, then car traffic", is equivalent to "if not car traffic, then not pollution".

In other terms :

in order to prevent pollution from happening, we have to "remove" car traffic (the necessary condition).


Usually you use both approaches, depending on how you are dealing with the problem.

One major reason for the focus on NC rather than SC, is that a focus on SC can easily lead to pulling the trigger on excessive solutions. For example, it's famously said that Nuclear War can end Hunger. However, nobody would consider Nuclear War to be an acceptable solution to Hunger because the costs are so drastic.

NC are useful when you are trying to surgically isolate the least undesirable solution to a problem. Consider the case of a bacterial infection. Sufficient cause is "Bacteria are present and the humors of the patient are out of alignment." Modern Western medicine would rather treat the NC: "Bacteria are present," and leave the humors out of it entirely.

Of course, a focus on one or the other will always lead to easy-to-construct synthetic situations where the chosen approach fails miserably. In some cases, like your multiple chemicals situation, it may be terribly difficult to isolate the NC, but easier to identify a SC, so we may act on the SC instead. Part of this stems from NC and SC being hamstringed by being tied to Boolean logic. In many cases, a more nuanced viewpoint does a better job of making decisions. Consider WWII and the dropping of the atomic bombs on Japan. There was no guarantee the bomb was necessary to end WWII, for we had some infantry attacks planned against Japan which could do it instead. There was also no guarantee that the bomb sufficed to end WWII, for we may have underestimated the Japanese resolve (in fact, one might argue we worried that this was the case, because we dropped a second one). In this real life situation, neither "sufficient" nor "necessary" were effective for decision making purposes.


Going for the short answer here since it's easiest to explain:

I'll work with the old Clark Kent (CK) & Superman logic. Assume throughout that we're living in a world where everyone knows CK and Superman are the same person - not the dumbass comic book world where people somehow haven't caught on.

Being Clark Kent is necessary for being Superman. Seeing a man wearing a superhero suit emblazoned with a large "S" (etc.) is sufficient for seeing Superman (if you see Superman out and about, he'll be wearing the suit).

So to relate this to your question: if we want to show that some imposter isn't Superman, we go find Clark Kent and say "Look! This is Superman, not this imposter!" Being Clark Kent is a necessary condition for being Superman, after all.

By contrast, if we only looked as far as the suit with the "S" on it, we'd be up the wrong tree. While seeing Superman entails seeing a man with a cape and suit, the converse is not true - any cosplayer can dress up like Superman without having his totally awesome powers, after all.

Hopefully that's a quick and dirty answer to your problem.


A sufficient condition S for event E has the form S → E.
A necessary condition N takes the form E → N.

If you want to make sure E happens, make sure one of its sufficient conditions is true, because then S → E ensures E.

Using modus tollens it also follows that ¬N → ¬E. So if you don't want E to happen, make sure one of its necessary conditions is false.

The quote above implies no need to care about all 10 Chemicals, but this appears wrong?

It is in the name that a necessary condition has to be true for the consequent (smog) to be true. So, if you have ten necessary conditions, they all have to be true. Therefore, if one is not true, the consequent is not true.

And in that case, you can rewrite n necessary conditions E → N1, E → N2, ..., E → Nn to one necessary condition E → (N1N2 ∧ ... ∧ Nn). It then becomes clear that if one Ni is false, E has to be false as well.

So, your error was to write "Chemical 1 ∨ Chemical 2 ∨ ...", i.e. to use ∨ instead of ∧.


There is a junction near my home that had a few accidents.

A sufficient condition for preventing these accidents is to completely remove the junction and all roads around it in a 10 mile radius.

I hope you agree that this sufficient condition is not a good solution to the problem.

The approach of trying to prevent a phenomenon is usually suboptimal. We are probably guided by the fact that then phenomenon has a cost that we want to avoid, but preventing the phenomenon also has a cost. Instead we should use an approach where we reduce the cost of the phenomenon with a low cost for achieving that reduction. It may even happen if we already spend money on prevention, that spending a lot less on prevention will cause only a slight increase in the cost of the phenomenon, so we should do less to prevent it.

Trying to completely prevent a phenomenon is usually excessively expensive. "Murder" is a phenomenon that we would like to prevent. Try imagining what measures you would have to take to absolutely 100% prevent any murder in your country. Then consider that according to Christian mythology, person #3 murdered person #4 and you'll figure out that your measures are probably not sufficient. (Persons in that order: Adam, Eve, Cain, Abel).

(This also should give you some thoughts for "necessary conditions". What necessary conditions would there be for murder? You might say "two living people" are a necessary condition for murder, but not even that is true because a person can set events in motion that will kill another person, and then take their own life).

  • Sorry, but I do not understand Then consider that according to Christian mythology, person #3 murdered person #4 and you'll figure out that your measures are probably not sufficient. Would you please clarify?
    – user8572
    Jan 24, 2016 at 3:36
  • Would you please respond in your answer, which is easier to read than comments?
    – user8572
    Jan 24, 2016 at 3:37

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