# Is it jointly sufficient without being individually necessary?

I am going through online article on necessary condition and sufficient condition. The explanation given is very clean and concise, but I am stuck on this example.

Example 5.1 - A set of conditions that are (jointly) sufficient without being individually necessary.

A sufficient condition for travelling from Calgary to Vancouver would be your taking an uneventful trip as a passenger on a regularly scheduled air flight. But while that method of getting from the one city to the other would suffice, it is by no means necessary. There are all sorts of other conditions that would also suffice for your getting from Calgary to Vancouver.

You could take VIA rail; or
You could travel by car; or
You could hike; or
You could ride a bicycle; or
You could travel on horseback;

I don't understand, why above conditions are jointly sufficient? We can either take only one of them at the given time. In my understanding, a single condition out of given all options should be necessary and sufficient condition at the same time.

"A is sufficient for B" is "If A then B".

This means that "it is not the case that we have $$A$$ without $$B$$".

And also "B is necessary for A" is "If A then B".

Thus, the sentence

"A sufficient condition for travelling from Calgary to Vancouver would be your taking an uneventful trip as a passenger on a regularly scheduled air flight",

will be :

"if you take an uneventful trip as a passenger on a regularly scheduled air flight, then you will travel from Calgary to Vancouver".

And the author says :

"But while that method of getting from the one city to the other would suffice, it is by no means necessary",

that means : "getting from the one city to the other [I presume : with a regularly air flight] is not necessary for travelling from Calgary to Vancouver".

Thus, the author is denying that :

"if you travel from Calgary to Vancouver, then you are getting from one city to the other [by plane]".

And this is consistent with the final sentences :

all sorts of other conditions that would also suffice for your getting from Calgary to Vancouver,

that means : "if you take VIA rail, then you will travel from Calgary to Vancouver"; "if you go by car, then you will travel from Calgary to Vancouver", etc.

This amounts to expressing the necessary condition as a disjunction : the list of all possible cases :

"if you travel from Calgary to Vancouver, then (either You could take VIA rail, or You could travel by car, or You could hike, or You could ride a bicycle, or You could travel on horseback, or You could take a flight)."

And you are right : "jointly" must be read "and".

IMO, the "trick" is that the list of possibilities are expressed as sentences ending with "or". Thus, the disjunction is obtained simply by juxtaposing the individual sentences.

Conclusion : the convoluted example must be read : each "travel method" is individually sufficient for going from Calgary to Vancouver, because if it is the case that you have taken a car, we can conclude that you have travelled.

But each "travel method" is not individually necessary, because the fact that you have travelled is not enough to conclude about which way you have chosen. The only thing we can cocnlude is that you have used one of the possibility.

Thus, each individual method is not a necessary and sufficient condition.

If we agree to consider the "joint" condition as the juxtaposition of the possible cases listed above, we have seen that the "joint" sentence is a necessary condition.

Obviously it is also a sufficient one, because "if (A or C) then B" implies "if A then B", and thus it is true that "if (either You could take VIA rail, or You could travel by car, or You could hike, or You could ride a bicycle, or You could travel on horseback, or You could take a flight), then you travel from Calgary to Vancouver".

Thus, the conditions (plural) are (jointly) necessary and sufficient.

• On same page "jointly" is read as "and" while giving definition of $$sufficient$$ while going through definition of Walkman: sfu.ca/~swartz/conditions1.htm#section3 – Ubi hatt Feb 6 '19 at 7:30
• My confusion-- necessary conditions are joint to make one sufficient condition while sufficient conditions are given in the fashion of options. – Ubi hatt Feb 6 '19 at 7:32
• Also, thanks for investing your time for my confusion! – Ubi hatt Feb 6 '19 at 7:33
• @EVG - you are welcome :-) And a second Conclusion (or advice) : try to avoid necessary and suffient" and use "if..., then...". – Mauro ALLEGRANZA Feb 6 '19 at 7:53
• Sure! thanks for nice advice. – Ubi hatt Feb 6 '19 at 8:04