II.1 Introduction

There is high demand for certainty about the tax position of tax payers. First of all, the tax payers themselves like to know their tax position. Are their activities subject to taxation? If so, how much tax do they have to pay? If these questions cannot be answered with absolute certainty, then the tax payers at least like to know the probability of the possible outcomes. This is especially true if they have to make a decision and the tax consequences play a (possibly decisive) role in that decision.

A tax payer stands in different kind of relations to others. For example, a company has a (business) relation with bankers, debtors, shareholders, suppliers of goods and services, etc. In order to assess their risks, these bankers, debtors, shareholders and suppliers want a true and fair view of the assets and liabilities of the company. Part of these assets and liabilities are deferred tax debits and deferred tax credits. The Dutch and international financial reporting guidelines contain specific rules for the valuation of deferred tax debits and credits. The general principle is that all possible outcomes of an uncertain tax position are to be identified and that the probability of each of the possible outcomes is assessed. However, in reality this turns out be a very difficult task.

The key question of my research is to what extent it is possible to connect a probability value to each of the possible outcomes of a hard case tax problem. A hard case tax problem is an uncertain tax problem with more than one potential outcome (tax consequence).

An important question for tax advisers and their clients, a question that is closely related to the key question, is what position a tax payer may take in a hard case without risking a penalty.

In hard cases, different outcomes are possible. At first sight, this provides tax payers with the possibility to choose any outcome, normally the most favorable one. Uncertainty thus gives freedom of choice. However, the question is how ‘justifiable’ the chosen outcome is.

The tax authorities can impose a penalty if a tax payer files an incorrect tax return. However, a penalty will not be imposed if the tax payer takes a position that might at the end be wrong, but is still justifiable.

What makes a tax position justifiable? A quantitative analysis of the concept ‘justifiable position’ seems to be impossible. Maybe we could make a qualitative analysis, but that is difficult as well. Although there should not be any gradations in ‘justifiable’ (a tax position is justifiable or not), it is not possible to exactly pinpoint how much argumentation is sufficient to make a position justifiable. When assessing if a position is justifiable or not, one might think that all arguments need to be considered. If different steps are necessary to justify a position, this justification is only as strong as the weakest link. But it seems that, in principle, only arguments ‘for’ have to be taken into account. However, strong arguments ‘against’ could potentially still throw a spanner in the works.

The uncertainty related to a specific hard case will show up when trying to solve the case. Thus, when trying to answer the key question, we need to understand how tax cases are solved.

I differentiated between two methods of legal problem solving: a forward approach and a backward approach. In the forward approach, the tax consequences are deduced step by step from the relevant facts and tax rules. The legal discovery process then yields one or more possible tax consequences, as well as the underlying argumentation. In the backward approach, the problem solving starts with one or more possible tax consequences. Via backward chaining, the relevant facts and tax rules are then added.

For every tax case, we can imagine a set of all possible propositions (such as facts and tax rules) that could be relevant for solving the case. I call this the ‘problem space’. The case is solved within this problem space. At the start of the problem solving process, a tax specialist already has a set of initial propositions. The rest of the problem space will be uncovered during the problem solving process (with regard to tax rules) or added to the problem space (with regard to facts). The ultimate goal is to find one or more tax consequences (heuristics) and to find the relevant arguments (justification).

The ‘outcome space’ is the set of all possible outcomes (tax consequences) of the specific case. In order to find the possible tax consequences of a specific case and to find the relevant arguments for these tax consequences, we also need meta rules that tell us how we can combine facts and tax rules and deduce new propositions (and eventually the possible tax consequences). I differentiated between logical inference rules and legal inference rules. Together they form the set of all inference rules. The inference rules allow us to combine facts, tax rules and previously deduced propositions, and deduce new propositions (including, at the end, the possible tax consequences).

All facts and tax rules that, in one or more steps, lead to a tax consequence, constitute the evidence of that tax consequence. I call the set of all these facts and tax rules the ‘evidence set’. An evidence set does not necessarily lead to the truth of a tax consequence; it is also possible that an evidence set leads to a tax consequence that is more or less probable. The evidence set contains all elements of the problem space that either function as an argument ‘for’ or an argument ‘against’ the tax consequence.

Each of the facts and tax rules and the inference rules may hold a certain degree of uncertainty. This uncertainty is expressed in the probability of the propositions. Every proposition in the problem space has a probability value. Where every proposition plays a role when deducing the tax consequences, the probability of that proposition plays a role when assessing the probability of the deduced tax consequence.

In easy cases, the probability of the single tax consequence always equals 1. In hard cases, the probability of the different possible tax consequences is lower than 1.

The problem space of a specific case consists of all relevant facts, tax rules all other propositions deduces during the problem solving process. These elements of the problem space, as well as the inference rules, may all hold a certain amount of uncertainty.

Regarding the facts, the first question is if all relevant facts are known and thus part of the problem space. And of the facts that are known, the question is how they have to be interpreted because facts are not always unambiguously clear. I have shown that facts are relations between subjects (people or companies) and/or objects. And the interpretation of these relations can be a cause of uncertainty. Uncertainty can also show up due to the autonomous nature of tax law where the interpretation of facts has in some cases diverged from civil law. Another relevant principle is that of economic reality (substance over form), where of course the question is: what is ‘real’ in a specific case?

Regarding tax rules, the first question again is if all relevant tax rules are known. (They can be found in the Tax Code and case law, but there is always a risk of missing some during the problem solving process.) This is especially tricky because of the interaction between facts and tax rules.

I differentiated between different kinds of tax rules and focussed on definition rules and ‘if-then’ rules. In case of definition rules, uncertainty can rise from the interpretation of the definition in the rule. In case of ‘if-then’ rules, uncertainty can rise from the interpretation of the antecedent, the consequent or the relation between the antecedent and the consequent. When the tax legislator or the Supreme Court uses vague terms, the interpretation of a tax rule is always uncertain. There are several methods of interpretation that could help to explain tax rules, but there is no fixed order for the application of these methods of interpretation. So the application of the methods of interpretation also brings about a certain degree of uncertainty which affects the probability of a specific tax rule. The application of legal principles also leads to uncertainty because they are not applicable in an allor- nothing fashion. Instead, principles have a dimension of weight. The same could probably be said for some rules as well. Finally, tax rules may conflict with each other, thereby bringing uncertainty to the problem solving process. There are some conflict rules, but there is no overarching rule that tells us how conflict rules should be applied.

Finally, there are inference rules that tell us how we can combine facts, tax rules and propositions deduced earlier during the problem solving process. Here, I differentiated between logical inference rules and legal inference rules. It is not always immediately clear which inference rules can be applied in a certain case. And there is no overarching rule that can help us decide which inference rule to apply in a certain case. This means that inference rules also add to the uncertainty of the possible tax consequences of a specific case.

Every uncertainty of facts, tax rules and inference rules has an impact on propositions that are deduced from these facts, tax rules and inference rules and eventually on the possible tax consequences thus influencing the probability of the outcomes of a hard case.

I introduced probability as a measurement for the uncertainty of the outcomes of a hard tax case. But what do we mean by ‘probability’? Can we quantitatively determine the probability of the outcomes of a specific case?

In hard cases, there is more than one possible outcome. The outcome space contains all possible outcomes (tax consequences) of a specific case.

The probability of a possible outcome is 0, 1 or in between. Probability is mathematically defined by the three axioms of probability theory. From these axioms other theorems of probability theory can be derived. Using the axioms and the subsequent theorems, it is possible to calculate with probabilities. This is very interesting if we want to calculate the probability of the possible tax consequences of a hard tax case.

There are different interpretations of probability, such as the classical, frequency, subjective and inductive-logical interpretations of probability. The question then is: what kind of probability are we interested in if we want to calculate the probability of the possible tax consequences of a hard tax case?

The basis of the classical interpretation of probability is symmetry: all possible outcomes of an event have the same probability. Here, the so called principle of indifference plays an important role. Every tax specialist will understand that a symmetrical probability distribution does not alwaysmatch real life. An asymmetric probability distribution is also possible within the classical interpretation, but that leads to a somewhat artificial expansion of the number of possible outcomes. Moreover,we like to see an objective analysis of the probability distribution and the classical interpretation offers us no solution.

The premise of the frequency interpretation is a repeated experiment. As the number of experiments increases, the approximation of the probability distribution of the possible outcomes becomes more accurate. However, a hard tax case is a ‘single case’ that cannot be repeated in order to obtain a reliable probability distribution of the possible outcomes. This means that the frequency interpretation of probability is not much help either.

The subjective interpretation is based on the degree of believe of a certain person in one or more outcomes. The disadvantage of this interpretation is that it is very hard to obtain an objective approximation of the probability of the possible outcomes. The only tool we have to objectify the subjective probability is to offer a bet. The cost/profit ratio at which a person agrees to accept a bet for a certain outcome is the measure of believe of that person in the outcome and thus the objectified subjective probability of that outcome. In real life, this bet is not a viable instrument. Furthermore, we are generally not interested in the subjective assessment of individuals (such as a tax adviser or a tax inspector). Instead, we are more interested in an objective analysis of the probability distribution.

The inductive-logical interpretation goes beyond the subjective interpretation. The premise of inductive-logical interpretation is the degree of belief of a rational reasoning person given the available evidence. The relation between the probability of the possible outcomes and the available evidence takes away the subjective part. But the introduction of objectivity does not mean that the probability of the possible outcomes of a hard tax can be determined quantitatively. The different interpretations of probability theory do not offer much help whenwewant to quantify the probability of the possible outcomes of a hard tax case. However, the inductive-logical interpretation does offer an interesting base for further investigation.

The Bayes formula seems a powerful tool for calculating the effect of evidence on the probability of a particular outcome of a hard tax case. If it is possible to find a quantitative relation between the evidence and the probability of a particular outcome, then the Bayes formula seems to be a good starting point.

We are interested in the posterior probability of a possible outcome, given certain evidence. However, when applying the Bayes formula, we encounter a number of practical problems. First of all, we have to determine the priori probability of the outcome before applying the evidence. The principle of indifference seems a likely candidate to do so. With the lack of any information (the evidence comes later) all possible outcomes have equal probability. However, the application of the principle of indifference is nothing more than a ‘best guess’. Other than that we reason from a lack of knowledge, I see no justification for applying the principle of indifference.

Secondly, we have to determine the probability of the evidence and the conditional probability of the evidence, given that outcome we are investigating is true. Unlike other sciences and disciplines (such as criminal law and medicine research), we do not have statistical evidence to effectively calculate these probabilities. Therefore, I do not think it is possible to use Bayes formula for a precise calculation of the probability of the possible outcomes of a hard tax case. This means that the answer to the key question is that it is very hard, if not impossible to connect a probability value to each of the possible outcomes of a hard tax case.

Nevertheless, Bayes formula does show us how our believe in one outcome increases and our believe in another outcome decreases, given a certain piece of evidence. The likelihood ratio (which can be deduced from Bayes formula) shows us the effect of a single piece of evidence on the probability ratio between two competing, mutually exclusive outcomes. Using the likelihood ratio, we are able to asses to what extend the evidence increases our believe in one outcome and decreases our believe in the other outcome. This believe is not a measure of a subjective probability, but a measure of an objective probability because we do not rely on our (gut) feeling but on evidence to determine the probability. The Bayes formula and all theorems of probability theory deduced from Bayes formula are tools for an objective analysis of the effect of evidence on the probability of the possible outcomes of a hard case. But again: I see no room for an exact quantitative analysis of the probability of the possible outcomes of a hard tax case.

Evidence plays an important role in the inductive-logical interpretation of probability theory. The evidence of a given outcome of a hard tax case is a measurement for the believe of a rational person in that outcome and thus the probability of that outcome (from the perspective of that person).

This means that a thorough understanding of the concept of evidence is necessary in order to determine the probability of the outcomes of a specific hard case. In my definition, the evidence of a specific outcome is all information available to support or refute that outcome. This is epistemic evidence and should not be confused with legal evidence. If in a specific situation a tax position needs to be justified (for example to convince a tax inspector or a judge), then legal evidence is brought forward. A tax payer (or his tax adviser) will only put such propositions on the table that will support his position, not propositions that will refute his position. This means that only a part of the epistemic evidence will be put on the table.

When confronted with a hard tax case that has to be solved, the tax adviser, tax inspector and judge will probably make an assessment of the probability of the possible outcomes of that case. They might unconsciously think that the outcome with the highest probability is the final outcome, the final solution. So when for example a tax adviser and a tax inspector are facing each other in a dispute, they will be inclined only to share the supporting elements of their respective evidence sets. This way, they will try to influence the other party’s perception of the probability of the outcomes in dispute, hoping that the other will eventually swing around.

Although it is not possible to exactly determine the probability of the outcomes of a hard tax case, we can make an estimate using Bayes formula; even if we do not make detailed calculations but use Bayes ‘intuitively’. This way, it is possible to influence the other party’s perception by selectively sharing pieces of evidence.

The key question was to what extent it is possible to connect a probability value to each of the possible outcomes of a hard case tax problem. The answer is that a quantitative analysis of the probability of the possible outcomes of a hard case tax problem is not possible.

Probability theory in mathematics allows us to calculate probabilities. Bayes theorem shows how evidence influences the probability of a certain outcome. It allows us to understand the impact of evidence on the probability of the possible outcomes of a hard case tax problem. However, Bayes theorem is not suited as a quantitative tool when it comes to hard case tax problems.

What does that mean in real-life? First of all, it is not possible to inform tax payers about the exact probability of possible tax consequences of (proposed) action. This makes it difficult for tax payers to make a calculated decision about a proposed action (for example about a business succession or about how investments should by financed) if taxes play a significant role when a decision has to be made.

The fact that it is not possible to calculate the exact probability of the possible outcomes of a hard case tax problem also affects the valuation of uncertain tax positions. For this valuation, the probability of all possible outcomes must be assessed. However, a quantitative assessment is not possible. It also teaches us that tax advisers who want to issue a tax opinion about an uncertain tax position, cannot provide an exact probability value (the chance of success) for that tax position. Since a quantitative analysis is not possible, the only thing left to do for the tax adviser is to give a qualitative analysis of the chances of success for a tax position, using words like ‘might’ or ‘should’.

My final conclusion is not necessarily the end-of-the-line. The starting point of my model is that the possible outcomes of a hard tax case are deduced from facts and tax rules using logical and legal inference rules. This yields the possible outcomes of the tax case, as well as the arguments to justify these outcomes.

But we can also ignore the arguments and only focus on the outcomes. We can also ignore the logical and legal relations between the facts and tax rules on one hand and the possible outcomes on the other hand. This may sound strange, but this is what we do when we feed all data necessary to predict the outcome of hard tax cases to a computer and allow the computer to find correlations and patterns in that data. From these correlations and patters, it might be possible to construct a heuristic system that allows the computer to predict the possible outcomes of hard tax cases, including the probability of these outcomes. Will this be the future? That’s hard to say. Further research will be necessary.