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I know the ideas of inclusive or and exclusive or. Usually, p or q is true if any one or both of p and q are true. Either p or q (meaning it exclusively) is true only when exactly any one of p and q are true.

But what confuses me are statements like this:

'All integers are either odd or even'.

It seems that the component statements are, 'All integers are odd' and 'All integers are even'.

None of them are true and thus 'All integers are either odd or even' must be false.

But it is true that all integers are either even, i.e. divisible by 2, or odd, i.e. not divisible by 2. It seems that the statement 'All integers are either odd or even' has two meanings. Is it so?

If not, what exactly are the truth functional value conditions of 'p or q' and 'either p or q'?

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    "or" in natural language is sometimes exclusive and sometimes inclusive... "All integers are either odd or even" is $\forall x (\text {Even}(x) \lor \text {Odd}(x))$, that is true. What is not true is : $(\forall x \text {Even}(x)) \lor (\forall x \text {Odd}(x))$, that reads : 'Either all integers are odd or all integers are even". – 2017-02-03
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    This is the benefit of symbolic writing... – 2017-02-03
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    @MauroALLEGRANZA I am somewhat confused by the notation here. In that last expression; shouldn't it read: "Either all integers are odd or all integers are even or both"? Also; will not the first expression allow for the case of the second? – 2017-02-04
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    Ok for "... or both". – 2017-02-04
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    The tqo are clearly **not** equivalent. The first says "**every** (natural) number is (either even or odd)", and this is true. The second one says "Either (**every** number is even) or (**every** number is odd)", that is not. See the different position of the universal quantifiers with respect to "either ... or ...". – 2017-02-04

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