Randall Wray on Necessary & Sufficient Conditions in Economics

In a recent post about debt-free money, Randy Wray makes some remarks about the relationship between the role of taxation and the movement of the monetary circuit. It is my purpose here to explore the character of this relationship from a logical perspective. Wray contends that taxes are a sufficient but not necessary condition for driving the monetary circuit. As his comments were not entirely clear to me, I asked him what he meant. Among other comments he made, which we will get to, he made these remarks.

If I said “taxes are a necessary and sufficient condition” then that means you must have taxes and nothing else will do. I don’t say that.

If I said taxes are a necessary condition, then that means you must have taxes, but maybe that alone is not enough. I don’t say that.

If I say taxes are a sufficient condition then that means if you have taxes you will drive a currency. However something else might drive it, too.

I think these statements are relatively clear even if not entirely logically so. In order to show what I mean by this, let me specify, how, logically, necessary and sufficient conditions are specified. First, we have a variable, x, that ranges over taxes. Now, we are in a position to specify the exact character of necessary and sufficient conditions relevant to Wray’s concerns.

Sufficient condition (for tax to drive monetary circuit): if x is a tax, then x drives the monetary circuit.

Necessary condition (for tax to drive monetary circuit): is x drives the monetary circuit, then x is a tax.

Necessary & Sufficient condition (for same): x is a tax iff x drives the monetary circuit.

It is probably apparent that my formulations do not quite conform to Wray’s formulations.  We can ignore the stipulation about tax being a necessary and sufficient condition for driving the monetary circuit, as it is obviously false. Therefore, we need concentrate only on the other two. First, we must note that Wray is engaged in a complexly compacted argument. Let us take sufficient condition first.

The first sentence is nothing more than a statement of what a sufficient condition is. When he mentions that other factors may drive the circuit, he is merely pointing out that for something to be a sufficient condition for something else, this does not rule out the possibility of other sufficient conditions being operative. He is right about this; it is a logically elementary feature of the context.

When we come to Wray’s discussion of necessary conditions, we encounter a little prolixity. When Wray says that were taxes a necessary condition, this would mean that one must have taxes but that this might not be enough, it is not entirely clear, logically, what he is trying to say. For tax to be a necessary condition,this means that if you don’t have taxes (via an application of modus tollens), the circuit will stop unless some other factor can drive the circuit (like the situation with the sufficient condition). While Wray may be clear what he means, the logical structure of what he is trying to get across certainly isn’t.

However, in denying that taxes are a necessary condition for the movement of the monetary circuit,he seems to be saying more than that if x drives the monetary circuit, then x is a tax, from which follows, if x is not a tax, then x does not drive the monetary circuit. This situation does not allow one to contend, as Wray does, that if taxes were a necessary condition of monetary motion, one must have taxes. Additional assumptions are needed in order to derive this conclusion. Hence, I can do no other than to conclude that Wray’s comments in this regard obfuscate the point he is trying to make. Which I think is a straightforward one. it is that the stipulation that taxes are a necessary condition for the driving of the monetary circuit are false because he denies the implication that if something isn’t a tax, then it doesn’t drive the monetary circuit. From which one can conclude that only taxes can do this driving. Which is false. There are other factors that can drive the monetary circuit besides taxes, such as fines, tithes, and the like.

Taxes as a sufficient condition allow for these additional factors to be drivers of the monetary circuit along with taxes. Hence, Wray’s contention that taxes are only sufficient but not necessary.

Upshot: while the logical structure of Wray’s argument isn’t entirely clear, he is consistent. And the role he assigns taxes vis-a-vis the monetary circuit seems to be more than reasonable.