# What logical principle allows you to conclude from “A or B; B;”, “not A”?

Is the following example a valid argument:

``````Either Your hair is short or long.
It is long.
Therefore it is not short.
``````

This is the form of the argument: Either P or Q. P. So not Q.

• No, it is not. In logic "or" is inclusive, i.e. it is true when at least one of the disjuncts is true. Commented Oct 24, 2016 at 9:49
• This is the form of the argument: Either P or Q. P. So not Q. Commented Oct 24, 2016 at 9:57
• Are you sure that " Either P or Q. P. So not Q" is not valid? Commented Oct 24, 2016 at 10:04
• "Either Napoleon is French or it is a male" is true and also "Napoleon is French" is true. Then, we have to conclude that "Napoleon is not a male" ???? Commented Oct 24, 2016 at 10:17
• @MauroALLEGRANZA: "Either / or" is usually read as "one is true and the other is not", different from plain "or". Commented Oct 24, 2016 at 15:09

I disagree with the answers above.

Either / or corresponds unequivocally, in English, to an exclusive or (XOR). The XOR function returns true only if P and Q have opposite values (T/F, F/T).

Hence the statement "You hair is not short" returns true.

The problem was merely in the translation into formula, becaused it used "OR"; but it is obvious from the question that it should have been formulated as XOR.

• Agree. In plain English, the word "Either" means only one of the two possibilities can be true. As opposed to "your hair is short or long" which would be true if my hair was short and also if it were long. Commented Oct 25, 2016 at 16:06
• A question about language : what is in "plain English" the negation of "Either ... or ..." ? Commented Nov 2, 2016 at 15:16
• @MauroALLEGRANZA Sorry for the late answer: "Neither A nor B", i.e. not A and not B. Putting an "n" in front of "either" and "or" kind of makes sense. Commented Oct 18, 2020 at 18:01

The original example: Either Your hair is short or long. It is long. Therefore it is not short.

Assumptions. (1) Short is defined as 'not long' (S= ~L) and vice versa. One term is the negation of the other. (2) 'Either/or' means either one or the other, but not both.

On those assumptions the example is valid reasoning.

It is difficult to apply exclusive 'or' to quantities due to the sorites paradox (en.wikipedia.org/wiki/Sorites_paradox) -- 'Either a collection of grains is a heap, or it is not' is simply not true. Likewise 'Either a hair is long or short' is also simply not true.

Further, there is a problem with collective application of properties -- not all of the individual hairs are necessarily of comparable length, so it might be meaningless to declare 'your hair' long or short. If you have 'bangs', you still have long hair, but some of your hair is short.

There is nothing wrong with the logic, but the premise oversimplifies the semantics of measurement and collective reference.

So this could be considered valid but not sound, or it could be considered invalid because some of the statements involved have ambiguous truth values, and valid arguments only work on binary truth, depending upon how much semantics you consider part of the form.

English has some clarity problems as a language in that we do not have separate terms for "inclusive or" and "exclusive or" in our everyday language.

The closest we have is something along the lines of the phrases "A or B, or both" and "A or B, but NOT both."

I'm going to take a slightly different take on this than the other answers: the first premise is ambiguous, so there's not enough information to determine if it's valid or not.

As others have indicated, the main issue here is "translating" this sentence into its equivalent logic statements.

Logically,

``````A \/ B
A
Therefore, ~B
``````

is very clearly invalid because "or" ( / ) is inclusive (i.e. "at least one of these statements is true) in formal logic.

In spoken language, however, people will often use the word "or" to denote "exclusive or" (i.e. exactly one of these statements is true). In fact, it would probably seem rather odd (and possibly misleading) if someone used it in some other sense. If we translate this sentence in that way, this would become

``````A xor B
A
Therefore, ~B
``````

which is perfectly valid.

Thus, whether this is "valid" or not really depends on the context of the original argument. If it's from some kind of "ordinary" spoken language, they probably meant "xor," in which case it's valid. However, if it's from some kind of more rigorous text or context it probably means "inclusive or," in which case it's clearly not valid.

• It is ambiguous, but not because of the 'or'. Either X or Y is rather unambiguous in English.
– user9166
Commented Oct 25, 2016 at 21:17

Logically, in the usual, contemporary sense of the word where we do not take into account semantics and judgement - but merely it's form - it's a valid argument - as pointed out by Mark Andrews answer

However, taking one step away, we can judge that it has produced no new information, so though valid it's not useful.

Judgement, is a part of philosophical logic, if not formal logic; and is key to reflection as pointed out by Arendt when discussing Kants political philosophy.

• By that standard, valid arguments are almost always not useful. All strict logic is a tautology at some point.
– user9166
Commented Oct 25, 2016 at 21:16
• @jobermark: that wasn't what I was driving at; I'm merely pointing out - though that's probably not clear from my brief answer - that the conclusion is simply one the premises:one of the premises is, 'it is long', and the conclusion is, 'it is long'. Commented Oct 26, 2016 at 0:34
• Whereas in a syllogism such as 'Socrates is himan, humans are mortal; therefore Socrates is mortal', produces a conclusion that isn't one of the premises - it's a new sentence; it's that sense of new that I'm talking about. Commented Oct 26, 2016 at 0:37
• The conclusion, as I see it, is likely to often be false, due to the sorites paradox. There is new information here, because it is likely, under realistic circumstances to disagree with reality in a way in which the given information would not. Information-theoretically, these are not equivalent, one is iffy, the other is not.
– user9166
Commented Oct 26, 2016 at 16:32