that vs. if and whether

I don't find your examples to be following your own rules.
"I wonder if that is a fact?" THAT IS A QUESTION. A question displays uncertainty or doubt.
"That is a fact." is a statement lacking uncertainty or doubt, which is why THAT is used.
"You know that that's going to fall." This is an example of what you are describing. The speaker is telling the listener that "THAT" is going to fall.
"You know if that's going to fall?" This is a question asking if "THAT" is going to fall.

"From my experience, about 95% of english speaking people (even educated people) employ this grammar"

Ninety-five percent, eh?

Then perhaps it's correct.

I do not think I have ever heard (much less read) people using THAT in the way that you say.

“I wonder THAT this is correct” does not mean the same thing as “I wonder IF (or WHETHER) this is correct”, it is a rather awkward way of saying something like "I am (or will be) amazed to find that this is correct."

Banjo

Reply to EPI:

"I don't find your examples to be following your own rules.":

"You know if that's going to fall?" This is a question asking IF "THAT" is going to fall.
Your sample sentence is, in my humble opinion, grammatically correct, but seems to point to something grammatically inappropriate. "that" in this sentence refers to some item which is, or is not going to fall, rather than the grammatical "that" used incorrectly to mean WHETHER or NOT, or, IF.

To clarify for you:
"Do you know IF that's (item or object) going to fall ?" Question, compare with:
"Do you know THAT that's (item or object) going to fall ? - implies a certainty.

Repeat mention of radio personality:
“I don’t know THAT it was cleaned much…” (from a radio personality this very evening)
(He was was referring to "it" being a shirt, and was wondering whether of not the shirt in question had been cleaned much, if at all, over the last 3 weeks while it had been in use)

95% certainly suggests common usage.
However, as can be seen from the majority of posts in the forum, common usage is not always correct usage.
I would agree that some of the examples quoted certainly do sound strange and stilted and I certainly would not use such phraseology.
Some people do have an unfortunate tendency toward using peculiar phraseology in the misguided belief that it makes them sound a bit more erudite.

I marvel that this is correct.

Dear Rachel, I can assure you that I am of most sound mind, as well as educated sufficiently to perceive many things, but I simply do not get what you are referring to with your comment.
"Please explain"

Plus: here's another that gets me:

The frequently used words "ecology" and economy or "economic", pronounced by many as if the words were spelled with double c, e.g. ecconomic etc.

Why this pain to our great language ?
Is it ignorance or do some people introduce these things just to be different ?

@Banjo

At the risk of going off topic, how does the pronunciation of a double "c" differ from that of a single "c"?

Sounds such as "echo", eccentric, etc. employ either an extra "c", or sometimes an "h" to indicate to the user how to pronounce the word in question.
I thought that this is the way upon which language is usually built, spelling to indicate how to pronounce, and grammar rules to indicate how to make sense to each other what we mean with a minimum of confusion.

Hence, the single "c" is meant to pronounce words like say, "economics" with an "ee" sound, rather than an "e" sound such as in "echo". We don't pronounce that word as "eecho" do we ?

@Banjo
Thanks for the explanation.
Just shows that one is never too old to learn.
Do you have an example that covers words like schedule, mandatory, premier, buoy, route, debut etc?
There are a lot of people out there who need enlightenment.
;)

One thing is sure. Your need to add "OR NOT" to the WHETHER usage is a redundancy. The inclusion of "whether" already stipulates a request for an understanding between the pro and con side of the stated argument. Adding "OR NOT" only doubles this, like saying "I wonder WHETHER OR NOT this is a fact or not". It should simply and correctly read "I wonder WHETHER this is a fact" because it either is or it isn't.

The notion that the single "c" requires an "ee-" pronunciation is not a standard English pronunciation rule. In the variants of econo... ecolo... etc, the first e may often be pronounced with many options, including "ih-", "eh-", "uh-" (well, really "shwa"), and, yes, "ee-". Actually, in all of the dictionaries I've checked, the "ee-" pronunciation is not even listed in mostof the variants, and then, when it is, always the last pronunciation. They are pronounced the same as the double c, not out of ignorance, but out of correctness.

I would say 'if that' because 'that' is relative pronoun that introduces a noun, adjective, or an adverb clause. When it's an adverb clause, it means a purpose or result, while 'if' is used for condition, i.e. "If I read now, I will be done with home work sooner."

I wonder where this figure of 95% comes from; it certainly doesn't seem to apply to British English and like njtt I don't think I've ever heard "I wonder that" used instead of "I wonder if/whether". Perhaps it's a regional thing. These are all the instances of the three variations at the British National Corpus:

"I wonder if" - 840
"I wonder whether" - 126
"I wonder that" - 7 - all of which have the meaning "I'm surprised that" pointed out by njtt

I make that precisely 0%

At Netspeak, based on a web corpus (ie mainly American), out of all "I wonder *" possibilities:
"I wonder if" gets 40.5%,
"I wonder whether" - 0.7%
"I wonder that" - doesn't even register, meaning less than 0.0%

Google Books show over 63 million for "I wonder if", 8 million for "I wonder whether" and just under 2 million for "I wonder that",with most on the first page looking to be valid uses of "that"

Facebook - "I wonder if" - 28.4 million, "I wonder whether" - 199,000. Admittedly "I wonder that" has 915,000, but a glance at the first page suggests that many of these are non-native speakers, and in any case, that would still be only about 3% max.

Twitter - "I wonder if" - 74 million, "I wonder whether" - 1 million, "I wonder that" 473,000

(All Google Search figures are front page figures, which aren't particularly accurate, but should do for comparison purposes).

And remember that all these searches for "I wonder that" includes things like "I wonder that, too", as well as the "surprised" meaning. Substitutions for "if/whether" certainly exist, but appear to be relatively few.

So I don't think we need to bother too much arguing about common usage on this one, as common usage it obviously ain't.

Economy, if memory serves, was originally spelled 'oeconomy' and would therefore have had (still does) a long O.
Americans tend to pronounce Oedipus, which they spell Edipus, with a short O, which sounds odd to me.

Economy comes from Greek for houselhold management cf "oikos" meaning a house which survives as wick, wic, wich in placenames like Chiswick, Norwich, Highwic.

Thanks, jayles. Economy, ecology and ecumenical are all derived from it. The English spellings (oeconomy, oecology and oecumenical) have finally given way to the simpler form, just as the 'ae' in, for example, mediaeval, has been shortened to plain 'e'.
In Britain we keep the spelling 'aesthete', pronounced with a long 'e'. Americans spell it 'esthete' and give it a short 'e'. As with 'Edipus', that sounds odd to me.

Whether and if can be used interchangeably when reporting yes/no questions.
eg.
The policeman asked if / whether I had seen the accident.

Use Whether only
eg.
He asked whether I wanted to go by air or by sea.
He asked whether or not I wanted to insure my luggage.
Note: Using whether is far more common. It is certainly more formal.

I think you better check it in http://www.grammar-monster.com/easily_confused/if_and_whether.htm

Whether and if can be used interchangeably when reporting yes/no questions.
eg.
The policeman asked if / whether I had seen the accident.

Use Whether only
eg.
He asked whether I wanted to go by air or by sea.
He asked whether or not I wanted to insure my luggage.
Note: Using whether is far more common. It is certainly more formal.

I think you better check it in http://www.grammar-monster.com/easily_confused/if_and_whether.htm

@Olivia Queen - "Using whether is far more common" - Here, Grammar-Monster was only referring to "whether/if ... or ..." constructions, as in "She didn't know whether to stay or go".

In normal indirect questions, "if" is "far more common", even in written English:

Netspeak - "He asked me if" - 78,000, "He asked me whether" - 5000
Ngram - "He asked me if" - 0.0000190%, "He asked me whether" - 0.0000020%
British National Corpus - "He asked me if" - 37, "He asked me whether" - 5

Incidentally, this question wasn't really about using "if" or "whether", but the use of "that" instead of either.

Re: Banjo's spelling idea -

In most words starting "ec", a single c is followed by a consonant, "eclaire, eclectic, eclipse, ecstatic, ectoderm, ectoplasms, eczema" - all with a short e as in "bed", and "ech" words simply follow the same pattern, as do "ecc" words. (For a list of all words starting "ec" see: http://www.morewords.com/starts-with/ec/)

With a single consonant + single vowel we would normally also have a short e sound. And apart from "ecarte"(from French), the only vowel that follows initial "ec" is "o" - and this is where the only variation I can see comes: in words starting "eco", which all belong to two word families based on "economy" and "ecology" (and related words with the prefix "eco" as in "ecowarrior").

So, as porsche has pointed out, the long "ee" pronunciation many of us use for these words is not down to a standard spelling rule (you can't really make a rule from what are basically two roots).

In dictionaries, the waters are rather muddy. Whereas Oxford Concise lists only /ɪ:/ ( the long ee sound) for "economy, economize , econometrics", it allows both long and short e for "economic, economical". For "ecology, ecologist, ecological" it allows both long and short e. But the /ɪ:/ (long ee) is always listed first. For those with the "eco" prefix "ecofreak, ecosystem, ecoterrorist", only the long /ɪ/ is given.

The American picture is rather different. The Free Dictionary is equally schizophrenic, giving /i/ (more like "ik"than "eek") only for "economy, economize, economist", and both long and short e for "economic" and "economical". They give /i/ for "ecology, ecologist" and long and short e for "ecological". But unlike Oxford, here the short e is listed first. With "ecosystems, ecoterrorism" both long and short e are given, but here long e is listed first.

So, although I agree with porsche that there are several variants - in standard dictionaries alone we can find /i:c/ (eek), /ic/ (ik), /ek/ (as in heck), I don't quite agree that "ee" doesn't get listed very much, although in American English it may not be so long - /i/ rather than /i:/. Where both are possible, recordings sometimes give one, sometimes the other.

One conclusion seems to be that these two dictionaries, one British, one American, list the nouns and verbs - "economy, economist, economize, ecology, ecologist" with "eek" or "ik" (AmE), and the adjectives and adverbs (and nouns based on them) are listed with both long and short e. Somebody who pronounces "economics" with a short e might well pronounce "economy" with a longer one.

And so what about Skeeter Lewis's suggestion about origins - ecology certainly entered the English language as oecology (1873 - from German) and you can find a few examples of oeconomy in eighteenth century books, although I'm not sure how "oe" was pronounced. But the economy family seemed to come to us via French, where they are definitely pronounced with a short 'e'.