Sometimes in the course of political conversations I make valid arguments whose premises are either obviously true or easily verified empirically. Those who disagree with the conclusions of these arguments often point out to me that the vast majority of economists disagree. I note that this, if true, carries the unfortunate implication that the vast majority of economists are wrong. I am then typically told that it is more likely that I am wrong than that the economists are wrong.

This is known as the *argumentum ad verecundiam*, often classified unjustly as a fallacy. The Latin name is superior to the English “appeal to authority”, since the word “authority” in some of its senses begs the most important question. But as Susan Stebbing pointed out, such an argument:

is sometimes fallacious, namely, when a point in dispute is supposed to be settled by showing that some respectable person has held the disputed view. If, however, the authority in question is an expert in the subject and the opponent is ignorant, the appeal to authority is justifiable. (A Modern Elementary Logic, 161)

I do not believe myself to be entirely ignorant of economics. And as for the authorities in question being experts, that is a point to be proven.

How do we know that the economists are experts in the relevant sense? The most common answer makes reference to a significant consensus among economists. By itself this also is fallacious, and the fact that it gets any traction shows only that the teaching of intellectual history requires more funding. It was not all that long ago that G.K. Chesterton made himself the laughingstock of the ‘scientific community’ by proposing that there might be something wrong with eugenics.

The inference from consensus to likely correctness only becomes valid given a further assumption. If there were any easy refutation of the conclusions of mainstream economics, it is said, this would already have been discovered by an economist, who would then have gone on to enjoy extraordinary fame for having made the discovery. Since nobody has won a Nobel Prize for shattering the foundations of mainstream economics, we can safely assume that those foundations are sound, or at least that nobody knows they are unsound.

This argument should not be too readily dismissed. It is, however, based on an imperfect analogy with the natural sciences. A chemist or physicist who found a sufficiently compelling case against a deeply established theory would publish her results in a prominent journal and become famous. Science journalists, who love a controversy, would report the story in the mainstream media. There might be some attempts to silence the dissenter along the way, but truth would out.

Why do we believe that? If the scientific community typically behaved any other way – if it typically formed conspiracies to silence dissenters from the accepted body of theory – then the success of science would be very hard to explain. If aerodynamic theory was fundamentally flawed, a conspiracy of theorists might silence those who durst offer the flaws to the press. But they couldn’t keep the aeroplanes from falling out of the sky. The predictive successes of natural science and the extraordinary technology that has emerged from it testify to the existence of mechanisms within the scientific community for finding, reporting, and making good on any obvious mistakes.

Lee Smolin, the physicist, wrote a book claiming among other things that String Theory holds an undeserved monopoly over research in theoretical physics. I have no idea whether his claim is true, but it does seem *possible*. There is, after all, no equivalent to aeroplanes falling out of the sky with theoretical physics – not at its current level of abstraction. And so the sort of illegitimate forced consensus Smolin describes could, for all I know, exist without anyone outside the field knowing about it for a fair while.

Whatever we think of theoretical physics, this is certainly the condition of economics. It makes very few successful predictions. Look at some of the R-squared values (from this paper) assessing the predictive power of a cutting-edge DSGE model:

And as for technological improvements, we have only economic policy to judge. There might have been some slight improvements in policy since 1929, though it is naïve to ignore the regress that has come with the progress. Anyway, what should we use as the measure of success? Certainly nothing from economics has improved economic policy in the way that the integrated circuit has improved computing. The problem is that until the 2007-8 crash it *looked* as though it had, and the public and politicians are slow to bin the conviction although it has passed its expiry date.

It is the success of the natural sciences that persuades us that legitimate dissent is not suppressed. Economics has no such success, and so there is no reason to infer that suppression is not widespread. There are in fact many reports suggesting that it is. And so the fact that no economist has become famous for pointing out obvious flaws in the foundations of mainstream economics does not justify any inference to the conclusion that such flaws do not exist or that they are not readily visible to anybody willing to look for them.

Furthermore, the criticisms that I make of mainstream economics do not concern topics that economists are expertly trained to consider. If I were criticising their linear algebra then it would be fair to assume that I am a crank who does not understand. But generally it is their logic that I find objectionable. Economists are not trained in logic at all, and they do not formalise their logic, which makes it difficult for others to detect their mistakes.

Economists typically reply to this that their use of mathematics, and particularly accounting identities, makes the logic of their arguments perfectly transparent. This is a point in my favour, since it shows that they do not know what logic is. Let me give some examples to illustrate this point.

Logical Mistakes in Macroeconomics

Willem Buiter’s “Joys and Pains of Public Debt” explores the question of fiscal sustainability within the framework of standard economics. He begins by stating that:

The fiscal-financial-monetary programme of the state is sustainable if the implementation of the programme does not threaten the solvency of the state, now or in the future. (3)

The implicit point is that if a state does not maintain a sustainable programme, it will at some point be unable to borrow. Now Buiter explores two different definitions of sustainability, one involving the assumption that “the world is known to come to an end at some fixed future date” (6) and one for which that assumption is suspended.

In the first case, sustainability is “easy to define. When the last trumpet sounds, the state cannot leave any net debt.”

This strikes me as a simple mistake in logic. On Buiter’s assumptions about how borrowing works (and I do not want to discuss fiscal operations here), in order for a sovereign to be able to borrow all throughout its lifetime there must be no point during its lifetime at which a payment falls due – an interest payment or repayment of principal – that it cannot make.

Buiter supposes that if the world ends, and there is net debt outstanding, then this condition will not be met all throughout the sovereign’s lifetime. This is quite false. Buiter is interested not in the possibility of governments perishing in the Apocalypse but rather in the possibility of their becoming insolvent. If the world ends, there is no time for the government to pay its outstanding debts. But there is also no time for those payments to fall due.

In logical terms, I suspect that the end-of-the-world conceit conceals an implicit operator-shift fallacy. If it is the case that a government is solvent (at some time), then there is a government that is solvent (at that time). But if it is *not* the case that a government is solvent then there are two possibilities: (a) there is a government and it is insolvent and (b) there is no government at all. A government is solvent if and only if the following holds:

That is, roughly, there exists some government, g, and g is solvent (the meaning of “S(g)”). More precisely, there is some value of g for which the value of the function S(g) is truth (the quantifier is untensed, but we could easily add in a time variable to express the claim that there is a solvent government at a time). The negation of this, however, is not:

This is an illegitimate operator-shift: the negation operator that should have been outside the quantifier has somehow wormed its way inside it. This is equivalent to reasoning that if there are no tall men in the room then there are men in the room who are not tall (which does not follow because there might be no men in the room at all).

Likewise, Buiter has become confused between a situation in which there is no solvent government and a situation in which there is an insolvent government. The government cannot be expected to remain solvent beyond the duration of its existence; it can only be expected to remain solvent for the entire duration of its existence, and this latter condition is *consistent* with there being *net debt* (whose payments have not yet fallen due) at the time of the government’s perishing or, *a fortiori*, “when the last trumpet sounds”.

It could be argued that if the end date is known, then sometime before that date the creditors will clamour at the government’s door, and no new lenders will be forthcoming to refinance the debt; in that case if the government’s budget is not near balance it will risk defaulting. But the antecedent is false, and anyway this is not how Buiter argues.

I am not disparaging Buiter in pointing out this fallacy. Peter Geach gave a lecture entitled “History of a Fallacy” in which he found operator-shift fallacies in the work of Plato, Aristotle, Spinoza, Berkeley, Bertrand Russell, and others. It is not an undergraduate’s blunder; it is a monster of the intellect that vigilance can only keep at bay. But economists don’t have the weapons to fight it. An instance of the same fallacy occurs in the popular belief that if the government would just stop spending so much money there would be more money for the rest of us.

Vaguely aware, perhaps, that something has gone wrong in his imagining of the finite case, Buiter moves on to discussing the case “[w]hen there is no finite terminal date” (7). As he puts it:

The infinite-horizon concept of solvency requires that the present discounted value of the terminal debt be non-positive in the limit as the terminal date retreats infinitely far into the future.

Buiter here inserts a reference to an article co-written with Kenneth Kletzer, in which they point out that a certain sort of tax regime is required for this concept to properly apply in the infinite-horizon case. But in any case there is a logical mistake in identifying the infinite-horizon case with a case in which there is no finite terminal date. The limit of a function as a certain argument approaches infinity is not the same as the value of the function if the argument (*per impossibile*, as it might be) *reaches* infinity.

Here is a function for determining the present discounted value (at t) of government debt (at t+n):

It is obvious that if interest rates, i, exceed GDP growth, g, then this debt will be unsustainable in Buiter’s terms. As n approaches infinity, the present discounted value of the debt at t approaches infinity. Does this show that an infinitely long-lived government – a government with no finite terminal date – will be insolvent under those conditions? Not at all.

Suppose that i is 2% and g is 1%. Debt will double roughly every 35 years, while GDP will double roughly every 70 years. Thus, starting from 100, debt will be 400 after the first 70 years, 1600 after the next 70 years, and so on. Meanwhile, again starting from 100, GDP will be 200 after the first 70 years, 400 after the next 70 years, and so on. It appears that the government gets further and further away from being able to pay its debt.

But the situation is identical to that of Bertrand Russell’s Paradox of Tristram Shandy. Because it takes Shandy a year to write one day of his life, if Tristram Shandy lives only a finite timespan then he will be unable to complete his autobiography. But, writes Russell:

I maintain that, if he had lived for ever, and had not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten. For consider: the hundredth day will be described in the hundredth year, the thousandth in the thousandth year, and so on. Whatever day we may choose as so far on that he cannot hope to reach it, that day will be described in the corresponding year. Thus any day that may be mentioned will be written up sooner or later, and therefore no part of the biography will remain permanently unwritten. This paradoxical but perfectly true proposition depends upon the fact that the number of days in all time is no greater than the number of years. (Mysticism and Logic, 89)

Our situation with the ‘unsustainable’ government debt is formally equivalent. After the first 140 years, the government will have earned enough to service the portion of the debt accrued in the first 70 years. After 280 years, the government will have earned enough to service the portion of the debt accrued in the first 140 years. In general, after 2n years, the government will have earned enough to service the portion of the debt accrued in the first n years. It follows that* there is no portion of the debt that the government is not able to service.* As Russell would say: this paradoxical but perfectly true proposition depends upon the fact that the number of 35-year periods in all time is no greater than the number of 70-year periods. In fact the government earns *the same amount* as it borrows, even though its debt doubles faster than its earnings. Infinity is strange.

Buiter’s mistake here is not an operator-shift fallacy, but still it is a mistake in mathematical logic insofar as he ignores a domain restriction. In a formula defining the limit of a function as time approaches infinity, the continuum of real numbers is taken as a model for time. Infinity is not a real number. And so it is clear that the formula cannot give us a value where time reaches infinity. If Buiter had expressed the limit statement using a quantifier, the mistake might have come out clearly. Thus:

To use a quantifier is to imply a domain, and when the domain is specified the inapplicability of the argument to the case described becomes quite explicit. The mathematics of limits provide a good device for knowing what happens to a function when various real numbers are assigned to an argument. But if we want to talk about cases where infinite values are assigned to arguments, then it is simply the wrong instrument. What we need is logic and set theory. And economists are of course not trained in these.

You might think that this hardly matters, since no government will exist for an infinite timespan. But if we are in fact discussing the case of a finite sovereign lifespan, then we are returned to Buiter’s reasoning concerning the finite-horizon case, and we have seen that this is fallacious for a different reason.

One final example. Thomas Palley’s article, that I discussed previously, includes the accounting identity: G ≡ T + θ + β. I mentioned that this identity can be read “right-handed” or “left-handed”, but that there is nothing in the formal expression of the identity to show this. Economists who claim that accounting ‘lays out the logic of their argument’ seem to be unaware that accounting notation is deficient in this regard.

Suppose, for simplicity, that we are dealing with a government that doesn’t ‘print money’ – θ is zero (what this really means is that the government does not let the money it creates by spending stay as excess cash balances; instead it drains all excess away by selling bonds). The standard policy question is whether the government should set G and T to equal each other over some given period, i.e. balance the budget. But the logically prior question is whether this is even possible. It might not be. If people insist on using the proceeds from government spending to buy government bonds, and the government tries to prevent this by ceasing to issue bonds, then people might simply cut back on their spending and thus reduce T (the paradox of thrift). What we want to know is whether the fact that the government can set G implies that the government can set the ratio T/β. *Prima facie* the answer would appear to be no, since T/β seems like it should be a function of non-government net savings desires, which the government does not set.

Economists like to talk about endogenous and exogenous variables. The question here is whether the ratio T/β is exogenous, and controlled by the government, or endogenous – determined by ‘market forces’ within the economy. But economists, restricting themselves to equations and inequalities, have no way of formalising endogeneity and exogeneity. In physics this is not a serious problem, since experiments can readily determine which variables are independent and which are dependent. But in economics the reasoning occurs in abstraction from experiment, and a proper formalisation is called for.

Quantifiers fill the gap nicely. What is unquestionable to anyone who confronts the identity is that for any value of G there is some *combination* *of values* (T, β) that preserves the identity. A government can always (let’s say ‘in normal times’) drain the surplus of its spending over its tax revenue by selling bonds. Or as a mainstream economist will insist on putting it, a government can always borrow to fill the gap between its tax revenue and its spending. But can it bring about a situation in which there is no excess to drain / gap to fill? What we want to know is whether for any *combination of values *(G, T) or (G, β) there is *some* value of β or T (respectively) that can preserve the identity. The identity itself does not tell us that this is or is not possible. Nor can any valid inference from the identity on its own, since the following is not a valid inference schema:

What economists who advocate for balanced budgets need to show is that it is even possible to balance the budget in the case in question. The accounting identity does not show that it is possible. Nor does simple inference from established facts. Many economists (such as Thomas Piketty) seem to think that the government *chooses* how much of its spending to ‘pay for’ with tax and how much to ‘borrow’. Where is the argument for this? There are (controversial) models that could show it, but it is most often accepted without argument, perhaps because a glance at the accounting identity can make it appear obvious. What would help a great deal would be to attach to the identity the quantifiers that show what we *do* know to follow from it, so that we can be clear on where further argument is needed to derive more substantial conclusions.

I could multiply examples, but I hope I have made my point. Economists are trained in applied mathematics and other important skills. But they still make mistakes in their logic, as we all do, and these mistakes often invalidate their conclusions.

Why don’t economists always notice each other’s mistakes? First, they don’t formalise their arguments with logical operators so that logical mistakes are hard to spot. This is due to their misconception that mathematical notation on its own suffices to make explicit the logic of their arguments. Secondly, they are not trained in the sort of formal logic or pure mathematics that would allow them to notice mistakes of this sort.

Experts they might be, but they aren’t the right sort of experts for my criticisms to be ignored.