It's a perfectly normal, actual word, even though it's uncommon.
Logicalness is defined by Princeton WordNet as:
correct and valid reasoning
It also provides the synonym logicality.
According to the American Heritage Dictionary, logicalness and logicality are the corresponding nouns to the adjective logical.
By the way, your fourth link does not list logicalness, but does include logicality under logical.
Apart from the fact that logicalness is listed in some dictionaries (and the synonym logicality provided), the notion that a word has to appear in a dictionary before it can be considered a valid English word is just wrong. -ness is a productive English suffix, i.e. you are allowed to attach it to any suitable word in order to form a new word, regardless of whether anyone has ever done it with that particular word before. What is more, in the case of -ness, basically every English word is suitable, even though the suffix is primarily applied to adjectives. But some philosophers study the relation between thisness and thatness, and you don't need a philosophing license before you are allowed to coin such words. (The reason nobody refers to the qualities distinguishing definite articles and indefinite articles as one between theness and aness is probably just phonetics. PS: I should have known better than to make such claims without checking. These silly words have been used before - if you count E.E. Cummings.)
Unless you are really asking in a Scrabble context, the puzzlingness of your question is as if your inquireness were about the permittedness of attaching -ly to a rare adjective that you have never seen used adverbially before, or whether it's OK to form the regular plural of a noun or the regular past tense of a verb when there is no indication of irregularity.
Whether it is good style to use logicalness is, of course, an entirely different matter. For certain types of English words, a noun can be formed with -ity instead of -ness, and for these it's usually preferred and is better style - especially if they already appear in dictionaries. But even then you may still choose to use both and make a more or less subtle distinction. E.g., provided you are explicit about your definitions, you might choose to refer to the "logicalness but lack of logicality" of the following wrong (hence illogical) but in a sense purely logical statement: "If every x is a y, then every y is an x."
Among your links, (1) is for logical, (2) and (3) are for logicalness. However, in (3), it says it is just a synonym of "logicality".
[Oxford Online Dictionary] in your link (4) has a noun form "logicality" under "logical" which means:
Of or according to the rules of logic or formal argument: ‘a logical impossibility’
If you compare the above definition with your link (2), you will see no difference.
Most noun forms of adjectives ending with "al" have "ality" instead of "alness".