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What is the logical difference between something being necessary in order for something else to be true; as opposed to something being sufficient to make something else true. i.e.
Fuel is sufficient to make an internal combustion engine run.
vs
Fuel is necessary to make an internal combustion engine run.
and what are some subtle examples of how the difference between these two things can greatly impact the meaning of a sentence, discussion, or conclusion.
The difference between "necessary" and "sufficient" is the direction of the logical arrow.
If you have A is sufficient for B it means that every time you have A you will have B, without exception:
A ⇒ B
If you have A is necessary for B it means that every time you have B you will have A, without exception
A ⇐ B
So as an example of A being sufficient for B, it is correct to say that every time you (successfully) kill someone, they will be dead, and the assertion that "Person X being killed is sufficient for Person X being dead" would be true. By contrast, it is not correct to say that every time someone is dead, it is because they have been killed. They could have died of natural causes, or there could have been some sort of accident. So the assertion "Person X being killed is necessary for Person X being dead" would be false.
This page has an excellent example of how the difference between these two concepts can change your conclusion. In his answer to the famous Seven Bridges of Königsberg problem, Euler demonstrated that in order to walk across each bridge exactly once it is necessary that the number of places with an odd number of bridges is either 0 or 2. Or put another way, it is necessary that a graph have either 0 or 2 nodes with an odd number of edges for you to be able to draw it without lifting your pen from the paper. However, this is not sufficient to ensure that such a walk or drawing is possible:
Either 0 or 2 places have an odd # of bridges ⇐ You can walk across each bridge once is true
but
Either 0 or 2 places have an odd # of bridges ⇒ You can walk across each bridge once is NOT!
So the difference between the two matters very much. Can you think why? Hint.
Just to add to the answers given, I would like to give a slightly different way to arrive at the same conclusion. My background is in mathematics, and so my answer might seem amateurish from a philosopher's or logician's perspective.
Sufficiency
I'll start with sufficiency, since it is easier. This answer is the same as those given. If p is sufficient for q, then, p is enough to have q. That is, if we have p, we have q, or
p ⇒ q.
Necessity
The statement "p is necessary for q," means, we must have p to get q. In other words, if we don't have p, then we can't have q. In symbols,
~p ⇒ ~q,
where I use ~ to denote negation. The logical equivalent, called the contrapositive, of this is
q ⇒ p.
Necessity and Sufficiency
Thus, p is necessary and sufficient for q means
p ⇔ q.
Another way this is commonly written is
p if and only if q.
It is sometimes abbreviated as
p iff q.
Example of confusing a claim with its converse
Consider the claim "All dogs go to heaven." This means
you are a dog ⇒ you go to heaven.
That is, being a dog is sufficient for entrance to heaven. The necessity statement, its converse, is this:
you go to heaven ⇒ you are a dog.
It's possible that you might be a cat in heaven, and so this is not true. The only way this could be true is if there is a law that requires all dogs and only dogs to enter heaven. That is, being a dog is necessary to enter heaven.
This is also sometimes called confusing cause and effect. There is more explanation and examples on this website:
http://www.nizkor.org/features/fallacies/confusing-cause-and-effect.html
Sufficient is an upper bound. Necessary is a lower bound.
Stopping the heart for X minutes (I don't know how long it needs to be to be legally dead) is necessary to make someone dead. (Whatever method you use to cause someone to be dead, it results in their heart stopping. If their heart does not stop as a result of the action the action has not made them dead.
Setting off a 100 megaton warhead next to someone is sufficient to make them dead. It does not require nearly this much to make someone dead, but if you do this they will certainly be dead.
There is room for error in both cases. you could decapitate someone but keep their heart beating through some artificial means, and you could theoretically devise some sort of clothing in some distant future that could protect against a nuclear blast but the situations in which these exceptions occur are bizarre and unlikely enough that we can effectively dismiss them. Otherwise we must accept that nothing is sufficient nor necessary to make someone dead.
Necessity excludes other methods or possibilities, sufficiency does not. To respond to your example, though I find your use of the phrase "make them dead" uncomfortable:
It is not necessary to kill someone to make them dead. You could incite someone else to kill them, and that would still be you starting a chain of events that "made" them dead. It is sufficient to kill someone to make them dead, as in killing them you've "made" them dead immediately.
Water is necessary to bake bread. Without water, you can't make bread. But water alone is not sufficient to make bread. You want to have some powder, too. And an oven, or at least a campfire.
Necessary means that something is required to achieve a goal. Sufficient means that is all you need to reach your objective.
In your case:
Killing someone is sufficient to make them dead.
Killing is sufficient to make someone dead.
Killing someone is necessary to make them dead.
Killing is not sufficient, you need killing plus something else. Your example is not a good one because, killing, by definition means that someone is going to be dead. You should use some other verb.
Maybe this help:
[Definition: A necessary condition for some state of affairs S is a condition that must be satisfied in order for S to obtain.]
[Definition: A sufficient condition for some state of affairs S is a condition that, if satisfied, guarantees that S obtains.]
From here. It has got examples and a tests. ☺
The conditional proposition can be other things besides necessary & sufficient for the record. Do not commit a fallacy of bifurcation thinking all conditionals must be one or the other.
Sufficient expresses that some element is able to satisfy the result. This means there are also OTHER means to also meet the same result. For instance a grade of 65 on an exam is sufficient to PASS the exam. Surely any score greater than 65 also passes.
Necessary expresses that the result would be impossible to occur in the absence of a essential element. A triangle necessarily MUST have three sides. Any more sides or less sides, then you absolutely do not have a triangle. Many people associate this conditional statement with cause & effect. As in science oxygen is necessary for combustion to occur or your standard chemistry balancing equation problems use the arrow to express this context.
To put conditionals another way we can use symbolic language to express these propositions in a clearer sense instead of the ambiguous arrow notation. To say a is sufficient for b is to say it is possible that you can obtain b without a. That is ~a V b. The tilde (~) represents a negation.
To say that you must have a to get b is to say it is impossible to get b without a. That is ~(b & ~a). This is read as it is impossible to have b and also not have a.
Some examples: if you are saved, then you have salvation. This expresses there is only ONE way to salvation. There are no others. This is necessary. You would use the latter notation above.
If you are a citizen then you have unemployment benefits. This expresses there are people who are eligible for unemployment benefits but all citizens do not automatically get benefits. Perhaps the law allows people on visas to collect unemployment benefits as well. It is a possibility that there are other conditions to grant the benefits. We are given that one of the many accepted conditions is being a citizen. This is not mandatory.
If I shoot you with my gun, you will die. This clearly is one way to die. So you can also die by drowning, by poison, getting a main blood vessel cut and bleeding out, shark attack, thrown out of a sky scraper, etc. This is not mandatory or necessary but sufficient to cause death.
If someone is a woman, then she is a human being. This is necessary by definition because it is impossible to have the traits of womanhood and not have the traits to be a human being. The fact that men are humans do not affect the truth if the conditional. As you know all humans are not women. This is impossible to be otherwise or else you are using the wrong term.
Let us start from material implication, then develop material equivalence, and explain what it means for something to be a sufficient condition, a necessary condition, and a necessary and sufficient condition.
The logical meaning of material implication is a sufficient condition. Therefore, (P -> Q) -> (P => Q).The material implication 'P -> Q' only fails to hold true (i.e. outputs 'false') in the case where P is true yet Q is false, because something true cannot imply something false. For all the other options, the implication holds (true). Something false can imply anything, because from falsity anything follows; that is, something false can imply something false as well as something true. For more information, please research the principle of explosion, which states "ex falso sequitur quodlibet" which means in Latin: "From falsity follows anything."
P -> Q: "P implies Q"
P => Q: "P is sufficient for Q".
The sufficiency of P for Q is equivalent to the necessity of Q for P; therefore...
Q <= P: "Q is necessary for P".
Here is an example to illustrate the difference between a necessary condition and a sufficient condition:
Necessity:
Rex is a chicken. A chicken is a bird. Being a bird is a necessary condition for being a chicken, because without being bird, a chicken cannot be, i.e., being bird is necessary (needed/required/pre-requisite), etc. in order to be a chicken.
Sufficiency:
Being chicken is sufficient for being a bird. Being a chicken implies being a bird. It is enough for something to be a chicken to be guaranteed that it be a bird. A chicken will always be a bird.
Moreover notice that:
Let:
The sufficiency of C for B is logically equivalent to the necessity of B for C. Therefore, since a chicken is a sufficient condition for being a bird, being a bird is a necessary condition for being a chicken.
Necessity and Sufficiency: What it means for something to be both a necessary and a sufficient condition for something else.
For the conditional P -> Q:
Note that:
The contrapositive of the original is logically equivalent the original implication:
The inverse of the original is logically non-equivalent to the original.
The converse of the original is logically non-equivalent to the original.
The inverse of the original is logically equivalent to the converse of the original.
P -> Q = 'P materially implies Q' = "If P, then Q" = Q if P. {Original}
Q -> P = 'Q materially implies P' = "If Q, then P" = Q only if P. {Converse}
When both [P -> Q] and [Q -> P] hold; that is, when 'P implies Q' and 'Q implies P': "P and Q imply one-another". Here, P and Q biconditionally materially imply each other. The symbol (<=>) denotes 'material biconditional'; that is the conjunction of the original conditional and its converse.
The original conditional "If P, then Q" is a material conditional which can be expressed as:
Original: P -> Q = "If P, then Q" = 'Q if P'
Converse: Q -> P = "If Q, then P" = 'P if Q' = 'Q only if P'
Therefore, the conjunction of the original conditional ('Q if P') and its converse conditional ('Q only if P) yields: 'Q if and only if P', which is logically equivalent to its converse 'P if and only if Q.
The term "if and only if" is abbreviated 'iff', where 'iff' stands for the logical biconditional, also known as "material equivalence", and is realized through an exclusive-nor, i.e., the "xnor" operator, the logical complement of the "xor" operator (exclusive disjunction). The "xnor" operator sets up a material biconditional in which a bidirectional (mutual) implication holds between P and Q: 'P <=> Q'.
If P and Q materially imply one another and then this bidirectional implication is called "material equivalence" is denoted by the symbol ("<=>"), and it sets up a logically biconditional if-then statement; where ("<=>") is implemented via "xnor", the logical complement of "xor".
The operator "xnor" is called the exclusive-nor. This operator ("xnor" = "iff" = "<=>") is called exclusive joint denial. The "xnor" operator only outputs a truth value of 'true' when P and Q are either both true together or both false together. The "xnor" operator logically excludes the remaining options in which exactly one of P and Q is true and the other false. The "xnor" operation between P and Q outputs only the truth value of 'true' when P and Q have the same truth values, and outputs 'false' when P and Q have different truth values.
The following three lines are logically equivalent:
