There is a theory that there is no actual money in modern state systems of currency. Instead what is commonly referred to as "fiat" money is actually a money substitute (e.g. a legally-enforceable claim for money). A money substitute is an obligation to redeem the substitute for the borrowed money that it represents, so even definitionally this presents a problem - the basis of the term "loop". The theory relies on the observation that the state both issues the currency and accepts it, implying an obligation to do so, such as in the cancellation of debt to the state (e.g. taxes). As such at issuance the claim is a credit against future tax settlement, etc. (i.e. the actual money).

Yet money substitutes are claims to a definite amount of money, as otherwise they are not fungible. The amount of tax liability represented by a $100 note, in payment of $100 of tax, is defined in terms of itself (i.e. the logical fallacy of circular reasoning). The amount it offsets is whatever the state is willing to trade for it. This would be the case for any money, including 100 ounces of gold or 100 units of fiat. Money does not represent any amount of another good, it represents whatever it can be traded for.

The state incurs no debt in declaring that it will accept a money, whether it be gold or fiat. Similarly a business that declares that it will take a particular money incurs no debt by doing so. The debt of representative money (a form of money substitute) such as a gold certificate, is expressed in the trade of the gold for the certificate-holder's claim against it. Issuance of the money does not change this fact. The state or a business can certainly issue gold in trade without the gold being considered a debt. State fiat enjoys monopoly protection on issuance, guaranteeing the state a profit from doing so. But this is not relevant to the question of whether the fiat is money or debt.

No money has intrinsic value. Fiat is distinguished from commodity money, such as gold, only by the presumption of no use value. But given that value is subjective, this is not a material distinction. Nor is it an actual one, as paper money can be burned for heat. If the state mined, minted and accepted gold or bitcoin, the theory would have to consider units of gold and bitcoin debt under the same criteria it applies to fiat.

The theory represents a misunderstanding of the nature of money substitutes. A claim cannot be a claim for itself. In such a scenario, the claim would settle itself. In other words, if $100 was a claim for $100 worth of anything, holding the claim is satisfaction of the claim. It would not be a claim at all, it would be money. As such the theory is invalid.

The transition from claim to fiat happens when representative money is abrogated by its issuer. The U.S. Dollar was monetized in 1934 when its redeemability was canceled. People were compelled to exchange redeemable dollars for irredeemable dollars. To the extent that formerly-redeemable dollars remain in circulation, as many still do, they are converted when encountered by the Federal Reserve. The retention of the phrase "Federal Reserve Note" on the irredeemable Dollar is anachronistic.

All money implies money substitutes, as a consequence of lending. We can classify four hypothetical scenarios for money substitutes in terms of debt regression, where each step in the regression is a promissory note.

• no regression (money)
• single regression (representative money)
• finite regression (money substitute)
• infinite regression (impossible money)

A note may be a claim for another type of claim, but not for itself (i.e. whatever it can be traded for). Otherwise there is no actual regression and the supposed claim is money. This holds in the case where the claim is directly or indirectly entirely circular, as implied by the term "loop", as the note settles itself. So the term "debt loop" is simply another description for "money". Examples include Gold, Bitcoin, and the irredeemable (modern) U.S. Dollar.

A direct claim (single regression) for money is a representative money, though this term is generally reserved for a tangible note that represents a commodity money. The note directly represents money. The redeemable U.S. Dollar was a representative money.

An indirect claim represents any finite progression of claims against others. When all claims are settled, the money is held by its rightful owner with all claims closed out, and any circular claims fully netted. Note that if the claims are fully circular there is nothing to settle (i.e. the claim is money).

An infinite regression of claims cannot exist. Consider a hypothetical note issued by the state treasury with redeemability in terms of offsetting state tax liability.

• $1 settles the tax liability on $10 of income.
• $10 settles the tax liability on $100 of income.
• $100 settles the tax liability on $1000 of income.
• and so on...

While the note does not represent itself, its regression is infinite. A claim can only be made against a finite number of other claims. In this case any such instrument is not actually a note and could only trade as money.