# Can a Condition be neither Necessary nor Sufficient?

I've consulted the four introductory logic textbooks below, and none moot the case of an un-necessary and in-sufficient condition. Do such conditions exist?

I don't quote from Peter Smith's An Introduction to Formal Logic (Cambridge Introductions to Philosophy) for my library doesn't carry the 2 edn (Aug. 6 2020).

Nicholas J.J. Smith, Logic: The Laws of Truth (2012). p 181.

“P is a sufficient condition for Q” means that having the property P is enough for something to have the property Q; that is, if something is P, then it is Q. So we regard this statement as meaning the same thing as “all Ps are Qs,” and we translate it as $$∀x(Px →Qx)$$. For example, “weighing more than a ton is sufficient for being heavy” says the same as “anything that weighs more than a ton is heavy.” “P is a necessary condition forQ” means that something cannot possess the property Q if it does not possess the property P—in other words, something possesses the property Q only if it possesses the property P. So we regard this statement as meaning the same thing as “all Qs are Ps,” and we translate it as $$∀x(Qx →Px)$$. For example, “weighing more than a pound is necessary for being heavy” says the same as “only things that weigh more than a pound are heavy” and as “anything that is heavy weighs more than a pound.” Thus, “P is a necessary and sufficient condition for Q” says the same thing as “all Ps and only Ps are Qs,” and translates as $$∀x(Px →Qx) ∧ ∀x(Qx →Px)$$, or equivalently $$∀x(Px ↔Qx)$$.

Copi, Cohen, Rodych. Introduction to Logic (2019 15 ed). p 282.

The notions of necessary and suffi cient conditions provide other formulations of conditional statements. For any specifi ed event, many circumstances are necessary for it to occur. Thus, for a normal car to run, it is necessary that there be fuel in its tank, that its spark plugs be properly adjusted, that its oil pump be working, and so on. So if the event occurs, every one of the conditions necessary for its occurrence must have been fulfilled. Hence to say

That there is fuel in its tank is a necessary condition for the car to run.

p 283.

can equally well be stated as

The car runs only if there is fuel in its tank.

which is another way of saying that

If the car runs then there is fuel in its tank.

Any of these is symbolized as $$R \supset F$$ . Usually “ q is a necessary condition for p” is symbolized as $$p \supset q$$. Likewise, “ p only if q ” is also symbolized as $$p \supset q$$.
For a specified situation there may be many alternative circumstances, any one of which is sufficient to produce that situation. For a purse to contain more than a dollar, for example, it is sufficient for it to contain five quarters, or eleven dimes, or twenty-one nickels, and so on. If any one of these circumstances obtains, the specified situation will be realized. Hence, to say “That the purse contains five quarters is a sufficient condition for it to contain more than a dollar” is to say “If the purse contains five quarters then it contains more than a dollar.” In general, “ p is a sufficient condition for q ” is symbolized as $$p \supset q$$.

Lepore, Cumming. Meaning and Argument: An Introduction to Logic Through Language (2012 2nd rev. edn.) p 83

If $$\alpha$$ is a sufficient condition for $$\beta$$, then if $$\alpha$$ obtains, $$\beta$$ obtains as well.

p 84

If $$\alpha$$ is a sufficient condition for $$\beta$$, then if $$\beta$$ obtains, $$\alpha$$ obtains as well.

I couldn't find anything relevant in Harry Gensler's Introduction to Logic (2017 3 ed).

• Could whitepace be considered a condition? Or perhaps syntactical validity, though that might be necessary...
– Emil
Aug 23, 2020 at 5:13
• I think whitespace could do the trick, semantically it could be something that adds no information.
– Emil
Aug 23, 2020 at 5:33
• If whitespace would be considered necessary and sufficient, you could write a math book that was empty (except for connectives) ;)
– Emil
Aug 23, 2020 at 5:52
• (I suggest whitespace as connective)
– Emil
Aug 23, 2020 at 5:59
• $5\le n\le10$ is neither necessary nor sufficient for $n$ being a prime. Aug 24, 2020 at 1:56

Given that $$(P\rightarrow Q)\lor (Q\rightarrow P)$$ is a tautology (it's the same as (¬𝑃∨𝑄)∨(¬𝑄∨𝑃)), it is impossible for a statement to be both not necessary and not sufficient under the definitions given by Copi, Cohen and Rodych.

However, if we use the definition given by Nicholas J.J. Smith which involves quantification, it is possible for a statement to both be not necessary and not sufficient.

Consider $$\forall x(Px\rightarrow Qx)\lor\forall x(Qx\rightarrow Px)$$ in a model with a domain consisting of two objects $$\{a,b\}$$, and interpretation $$Pa=1,Qa=0,Qb=1,Pb=0$$. This means that neither of $$\forall x(Px\rightarrow Qx)$$ or $$\forall x(Qx\rightarrow Px)$$ are true.

A worded example of this would be the following: An odd number is neither a necessary nor sufficient condition for that same number to be even.

• Nice answer! =) Aug 23, 2020 at 11:03
• Is it anywhere given that $(P\rightarrow Q)\lor (Q\rightarrow P)$ is a tautology? Aug 23, 2020 at 12:36
• I think md2perpe has a relevant question. Does ((P→Q)∨(Q→P)) hold in relevance logic? If not, I don't find it too difficult to imagine someone devising some system where it doesn't hold, since it's had objections before as I recall. Aug 23, 2020 at 21:49
• damn i miss logic
– JKEG
Aug 24, 2020 at 0:38
• @DougSpoonwood md2perpe also has an intuitive question; $(P\to Q)\lor(Q\to P)$ is not intuitionistically valid. I think it \'s valid in linearly ordered Kripke frames but not in general partially ordered ones. Aug 24, 2020 at 3:31