The advice of Bill Brown, Mike Chappie, Dave Craig, Wilpen Gorr, Doug Keenan, Dan Rosenblum, and Wilfred Sieg is gratefully acknowledged. The author is responsible for all shortcomings. This research was supported by the Center for Public Financial Management, Carnegie-Mellon University.

Contents

1. Introduction

1.1. Overview

Two independent benefits of a good notation should be distinguished.

1. Effective law: A clear notation exposes the underlying conceptual structure of a proposed provision and makes its implications more apparent. It also tends to reveal any ambiguity that may exist. This facilitates the drafter’s task of designing provisions that will carry out their intended objectives.

2. Comprehensible law: Clarity reduces the costs of interpreting the law, of arriving at an understanding of its provisions. It reduces uncertainty and improves the ability of economic agents to foresee the consequences of alternative courses of action. In this sense, it reduces “information friction” in society and promotes efficiency.

While maintaining that symbolic logic notation can serve the goals of effectiveness and comprehensibility, this paper proposes that these same goals can be further promoted by a richer, “mathematical” notation. The notation proposed here includes the AND, OR, NOT of symbolic logic, but is enhanced by additional formal elements, sets and functions in particular, that amplify its expressive power.

I emphasize that the sole purpose of the proposed notation is to enhance human insight and communication. Therefore, it should be used as a convenient supplement to English, where appropriate. As in mathematics, economics, political science, and other disciplines in which math notation is considered indispensable in dealing with many topics, excessive formality should be avoided in favor of convenience and comprehensibility.

The organization of this paper is as follows. Section 2 discusses prior proposals for introducing symbolic notation into the language of the law. Section 3 is a complete, self-contained introduction to the notation proposed by this paper (to the extent that it is used here) and related concepts. Section 4 illustrates the use of the notation by presenting a restatement of the provisions of section 333 of the Internal Revenue Code (which pertains to certain corporate liquidations), as well as a discussion of the logical weaknesses of the statute, some of which are exposed in the process of restatement. Section 5 concludes. §333 is reproduced in Appendix A.

1.2. Traditional Language and the Importance of Notation

One look at any digest of cases, a study of the litigation that has turned repeatedly (and in many directions) on the interpretation not of layman’s words but of law words (Sections 120-123, 126), brings a conviction of imprecision or incompetency, or of both.[4]

Belying extravagant claims he cites of the law’s precision (“rank[ing] in the exactitude of their language with the classic studies in physics and natural science”[5]), Mellinkoff’s impressively researched work provides ample documentation of the imprecision of the language of the law. Mellinkoff points to numerous ill-defined words and phrases of the law as one source of trouble. On “aforesaid,” for example:

Coke ... warned [in the year 1628] that aforesaid (Latin, praedictus) was no good when precision was called for, because it was not clear that aforesaid always referred to what went immediately before.[6] Despite Coke, lawyers have permitted aforesaid to give the same ambiguous directions ever since. It may refer to what is next-before[7], to the next-to-the-next-before[8] or to everything that has gone before.[9] ( [Mellinkoff 63], p.305. Endnotes are Mellinkoff’s.)

Mellinkoff provides similar documentation of many more imprecise words and terms.[10] For others, Mellinkoff suggests:

Visit for a moment that “... comprehensive depositary ...,” that “... vast storehouse of judicial definitions ...” known as Words and Phrases. (The seventy volumes -- in Spring, 1962 -- of this useful boneyard of words must be distinguished from the smaller and more selective English publication, Words and Phrases: Judicially Defined ....) To the extent that the publishers have done a good job, Words and Phrases is an impressive demonstration of lack of precision in the language of the law.[11]

Mellinkoff also describes a historical prejudice against punctuation[12] and documents various types of ambiguity -- including one he dubs “the Wandering Afterthought” which was recognized in the 16th century -- that continue to produce confusion today.[13]

One might expect a work so vividly repudiating the attitude that the language of the law is precise to draw objections from speakers of the language. The Index of Legal Periodicals provides references to book reviews appearing in about 300 law journals. Of the fifteen reviews of Mellinkoff’s The Language of the Law that it cites, not one seriously challenges Mellinkoff’s charges of imprecision.[14]

The opacity of the law has been castigated for centuries. Mellinkoff quotes attacks by Jeremy Bentham[15], Thomas Jefferson[16], John Adams[17] and others, and documents how pleading another man’s cause in exchange for payment was outlawed in some early American colonies because of popular antipathy for the iniquity and unintelligibility of the law.[18]

[Allen 80a] has presented empirical evidence that law can be represented in a more comprehensible form without alteration of its provisions. Allen used symbolic logic notation to express a particular New York statute (his apt choice was the 1978 “Plain Language” consumer protection statute) and showed experimentally that the logical form leads to faster and more accurate comprehension of the law.[19]

While the above considerations suggest that a more comprehensible medium of expression is attainable, the goal of effective law deserves equal emphasis. A notation can have an enormous impact on how deeply and thoroughly a complex structure, like many tax law statutes, can be grasped. This is important for the drafter seeking to design a structure of provisions that is free of loopholes and that deals correctly with the problems it addresses. The mathematician, Alfred North Whitehead, wrote of the importance of notation in the field of mathematics, where the inadequacy of natural language as a medium of expression is generally appreciated:

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers .... By the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain. It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.[20]

Symbolic notation offers precisely the same benefits in law as in mathematics. The same exhortations made hundreds of years ago, before symbolic notation became widespread in mathematics, seem appropriate today:

....Which Treatise being not written in the usuall synthetical manner, nor with verbous expressions, but in the inventive way of Analitice, and with symboles or notes of things instead of words, seemed unto many very hard; though indeed it was but their owne diffidence, being scared by the newness of the delivery; and not any difficulty in the thing it selfe. For this specious and symbolicall manner, neither racketh the memory with multiplicity of words, nor chargeth the phantasie with comparing and laying things together; but plainly presenteth to the eye the whole course and processe of every operation and argumentation.[21]

2. Prior Proposals for Symbolic Notation

It has been observed for some time that the appropriate use of symbolic notation is a useful and even a necessary tool in drafting better-quality legislation and understanding existing legislation, particularly when the logical structure is relatively elaborate.[22] [Allen 57a] proposed the use of the formal connectives AND, OR, NOT of symbolic logic, and an indenting convention to portray logical structure. One of Allen’s illustrations of his proposal involves section 354 of the Internal Revenue Code, which has the form shown in Figure 2-1. The T’s in the figure represent the elementary propositions[23] of the statute. Figure 2-2 displays the “normalized” form of the same statute.[24]

(a) General Rule.--
   (1) In General.- Tl if T2.
   (2) Limitation.- Paragraph (1) shall not apply if-
      (A) T3, or
      (B) T4.

(b) Exception.-
   (1) In General.  Subsection (a) shall not apply to T5, unless-
      (A) T6; and
      (B) T7.

IF
   1. S2
   AND
   2. a) 1. S3
         2. S4
         AND
         3. a) S5 
            OR
            b) 1. S6 
               AND
               2. S7
      OR
      b) S9
THEN
   3. S1

[Allen 78] argues compellingly and cites experimental evidence showing that the “normalized” representation of 354 of the Internal Revenue Code is clearer, more readily grasped, and more precise than the statute in its actual form.

The symbolic logic approach is to extract the elementary propositions from a semantic body and exhibit how they are logically interrelated. The enhancement in comprehensibility can be seen by comparing Figure 2-2 with Figure 2-1. The limitation of this approach, however, is that it cannot go beyond the level of the elementary proposition. The symbolic logic approach is to defer to natural language in the expression of an elementary proposition, even when the proposition is complicated and has a structure of its own.

As an illustration of the limitations of symbolic logic, consider how subsection 333(c) of the Internal Revenue Code, defining “qualified electing shareholder,” might be represented. First, the elementary propositions would be isolated and given names, as follows:

Given these definitions, the subsection could be expressed as shown in Figure 2-3.

IF
   1. T1
   2. T2
   AND
   3. a) 1. NOT T4 
         AND
         2. T6 
      OR
      b) 1. T4
         2. NOT T5
         AND
         3. T7
THEN
   X a "qualified electing shareholder."

Although there has been a clarification of structure, one senses an unfulfilled need. The “elementary” propositions are complex and wordy, not very elementary. As long as symbolic logic is being applied, one wonders why the notation stops short of the open use of arithmetic functions like division in expressing T6 and T7 themselves more clearly. Also, T6 and T7 are so closely parallel it seems the concepts they have in common should be extracted and given names for concise reference. These names could then be used in terse definitions of T6 and T7 in which their differences and their similarities would be readily apparent. One may also rightly suspect that the surviving complexity continues to harbor a latent ambiguity: in T6, if we define A and B so that the condition is

Symbolic logic offers no means of significantly improving the representation given in Figure 2-3. Symbolic logic is inadequate for expressing law for the same reason it would be inadequate in mathematics. The subject matter demands a more elaborate notation. This is not to suggest that there is anything that cannot be expressed under the symbolic logic approach illustrated above. If a difficult proposition cannot be broken down into a structure of elementary propositions, the proposition can simply itself be treated as an elementary proposition. However, the example of §333(c) suggests that symbolic logic offers less communicative power than we might expect in a symbolic notation.

What we have been referring to as “symbolic logic” is the notation proposed by Allen, which is a mix of English and formal language. The most simple axiomatic system of conventional symbolic logic is called “propositional calculus” (PC).[25] Allen’s notation is drawn almost exclusively from PC. (The exception is the notation’s use of propositional constants, i.e. symbolic names, like S3, denoting specific propositions.)

The adequacy of PC for expressing law has already been questioned, in [Maggs 72] and [Finan 82](pp. 702-706). [Maggs 72], reporting on efforts to translate statutes into symbolic logic form (as a first step in making statutes interpretable by computer), state that “careful examination of the text of a number of statutory materials shows that it is extraordinarily difficult, because of the internal logical structure of the individual sentences, to use a totally simple sentence-based logical structure for processing.”[26] The authors offer section 2-104(1) of the Uniform Commercial Code as an example:

Attempts to translate this section into sentence logic encounter great difficulties, because the logical connectives -- the “or’s” -- are not found between complete clauses, but are embedded within the syntactic structure of the sentence. Two alternatives are available: to move toward one of the more complex and sophisticated forms of mathematical logic -- some form of the predicate calculus -- or to attempt to transform this complicated section into a number of sentences in the propositional logic. In the long run, probably the former approach is the real solution, because the more powerful system of the predicate calculus could lead to solutions of not only this, but of a number of other problems. In the short run, however, the simplest approach. and the one we have adopted, is that of translation to sentence form. To take a very simple example the sentence “One who employs an agent or broker is a merchant,” could be transformed into the two sentences “One who employs an agent is a merchant,” and “One who employs a broker is a merchant.” With somewhat more complex application of the same rule even something as complicated as section 2-104 could be translated into a series of sentences which retain the exact meaning of the original.[27]

3. A Mathematical Notation

The mathematical notation proposed here, like Allen’s, is an amalgam of English and formal notation. This paper does not seek to provide general prescriptions for determining the proper mix, or for devising the ideal representation for any law provisions. There is a realm apart from law, however, in which good organization in the expression of complex structures of exact rules has commercial significance and is a continual concern. Certain principles that have gained wide recognition there can provide useful guidance in drafting legislation mathematically.

The purpose of certain areas of legislative law (such as the Internal Revenue Code) is closely analogous to that of computer software. It aims to articulate rules and procedures that will properly regulate events that have not yet occurred. The passage of time can bring many disparate events under the governance of a statute. The variety and unanticipated circumstances of these events are a severe test for a statute, and ambiguity, inconsistency, or clear but unintended provisions are typically exposed. (Most flaws of this sort in a preliminary version of a computer program will manifest themselves immediately in the “debugging” stage of software development.)

While “interpretation” can sustain the intent of a feeble statute, the consequences of incorrect software are generally immediate and unmistakable. A certain approach to the intellectual problem of designing a structure of exact rules has come to be generally accepted in the discipline of computer science. Some of the terms used to describe aspects of the approach are top-down problem-solving; modularity; consistency; and generality. There is also operative, as in mathematics, a strong element of aesthetic judgement. A discussion of the approach is beyond the scope of this paper, but has been written about in the computer science context. It is striking how directly the principles of good programming apply to drafting law. The reader is referred to [Kernighan 74], [Kernighan 76], and [Ledgard 75].

This section of the paper introduces: abstract objects; variables, which are short names by which objects are referred to[28]; and the essential concept of function. These are the basic types of tools that a legal drafter should freely have at his disposition. They are described informally in this section. In addition, certain formal terms that facilitate precise expression (IF, THEN, ELSE) are given here.

It is impossible to foresee and define all the specific objects or functions a drafter may require. The drafter must define these as the need arises, as computer programmers routinely do. There are certain functions, however, that are basic and generic[29], for example, AND, OR, MAX, MIN, +, ×. These are presented here rather than in section 4. Similarly, sets are such an important kind of object that they, along with certain symbols and functions traditionally associated with them, are presented in this section.

The following discussion avoids unnecessary formality and seeks only to make the pragmatic use of the notation plain. The objective is to minimize the background in mathematics or formal logic that is required of readers.[30]

3.1. Variable

A variable is a name used to represent an object. The object may be abstract or concrete -- anything that can be conceptualized. A variable is shown in italicized capital letters and is usually chosen to be a mnemonic for the object it represents. For example. FMV represents the fair market value of the property received in exchange for a particular share of a corporation (a more exact definition is given below).

A variable’s (or a function’s) definition may make appeal to human discretion[31]. This is an important point to note, since it refutes the frequent objection to codification, that explicit codification excludes necessary human judgement. The definition of FMV, for example, implicitly calls on human judgement to determine the value of property. Similarly, the value of TADP (the time of adoption of the corporate liquidation plan) will hinge, sometimes crucially, on the determination of a judge. (See discussion in section 4.)

3.2. Set

A set is a particular kind of object It is a collection of other objects, which are referred to as elements of the set. An element can be another set, a shareholder, a moment in time, or any other object.

Two methods are used here to specify a set The first is the “roster” approach. under which the set containing the 3 elements X, Y, Z, is represented as:

{X, Y, Z}

The other method would be used as follows to specify, for example, the set of all men whose fathers were born before 1910:

{X: X is a man and X’s father was born before 1910.}

The symbol X is chosen arbitrarily.

{Y + 3 : Y is a number between 6 and 8, inclusive}

is the set of numbers between 9 and 11, inclusive.

3.3. Conventional mathematical symbols

This paper uses the following symbols:

as well as:

× (multiplication),
/ (division),
{} (explained in the discussion of sets above).

Some of the less familiar symbols are defined here:

A - B, if both A and B are sets, means the set containing all elements that are in A and that are not in B.

means A is an element of the set B. Thus,

is true.

means A is a subset of the set B. Thus,

is true.

, if both A and B are sets, means the “union” of A and B; that is, the set containing all elements that are either in A or B (or both).

A > B means the number A is greater than the number B. Thus, 5>5 is false.

means A is greater than or equal to B. Thus,

is true.

3.4. Function

The term function is used in the conventional mathematical sense. + is an example of a function. Given two arguments, 5 and 3, it yields the result 8. The arguments of a function in the notation used here normally appear as a list within parentheses following the function name, with semi-colons used to separate arguments. (“+”, “×”, “AND”, and others are exceptions to this rule.) The name of a function (when it is not a symbol, like “+”) is given in italicized capital letters, like variable names.

In the cases of AND and OR, the function name appears between the last and second-last argument, e.g.,

A; B; C; AND D

MIN is defined here as the function whose result is the lowest of the numbers among its arguments.

MIN (10; 2; 15; 2; 20; 6) yields 2.

The usefulness of MIN is immediately apparent when considering the awkwardness of language such as:

so much of the gain as is not in excess of his ratable share of the earnings and profits[32]

This number can be more simply represented by the expression:

MIN (GAIN; SHARE)

(Appropriate definitions for GAIN and SHARE must, of course, also be provided.)

MAX is defined correspondingly as yielding the maximum of its arguments, rather than the minimum.

The expressive power of functions is enhanced by the option of representing an argument of one function as an expression involving other functions. Thus,

MAX ( MIN (100 ; 50 ; 80); 7 + MIN (200 ; 40 ; 5) )

is equivalent to

MAX (50 ; 7 + 5)

which in turn is equivalent to

MAX (50 ; 12)

which in turn is equivalent to 50.

This “nesting” of functions can make an expression appear quite unapproachable, but evaluating it is a simple matter of mechanically applying the function definitions in the appropriate order.

The concept of function is completely general and allows either the argument, or the result, or both, to be any kind of object, numeric or otherwise. A number of non-numeric functions are useful in expressing the provisions of §333. For example, the function E takes, as its single argument, some set of shareholders in the corporation under consideration, and yields the subset containing those shareholders who have elected the provisions of §333.[33]

MVP (“Maximum Voting Power”) might be another useful function for conveying the intended provisions of §333.[34] Consider first the concept of voting power implied by the statute in the following language:

stock possessing 50 percent or more of the total combined voting power of all classes of stock entitled to vote on the adoption of [the plan of liquidation][35]

Using this concept, we can define the “voting power” of any group (set) of shareholders, at a given point in time, as a proportion of total combined voting power. This voting power, since it represents a proportion, may be expressed as a percentage.

The expression,

MVP (Point in time; set of shareholders) ,

denotes the percentage representing the voting power of the shareholders in the set, at the indicated point in time.

To illustrate the definition of MVP, suppose that shareholders X and Y together held 20% of voting power on October 3, 1985. Then

MVP ( October 3, 1985 ; {X, Y}) = 20%.

Subsection (c), involving concepts related to the functions E and MVP, illustrates the convenience of representing arguments as expressions involving other functions (as described above). Consider the following passage from the statute:

if written elections have been filed by shareholders (other than corporations) who at the time of the adoption of the plan of liquidation are owners of stock possessing at least 80 percent of the total combined voting power (exclusive of voting power possessed by stock owned by corporations) of all classes of stock entitled to vote on the adoption of such plan of liquidation ...[36]

If we define TADP as the time of the adoption of the plan of liquidation, and NS as the set of shareholders of the corporation that are not themselves corporations, the intended condition might be expressed as:

Since subsection (b) makes use of a more general concept of voting power, it is useful to also permit MVP’s first argument to represent a period of time, rather than a single moment This makes it possible to exploit MVP in expressing provisions like those of subsection (b) as well as (c). Accordingly,

MVP (Period of time; set of shareholders)

is defined as the maximum voting power held by the shareholders in the set, over the period of time indicated.

To illustrate the extended definition of MVP, suppose that shareholders X and Y together held 20% of voting power in January, 42% in February, and 38% in March. Then

MVP(January to March inclusive; {X, Y} ) = 42%.

The extended MVP can be used to express the definition implied in the following passage from the statute:

...the term “excluded corporation” means a corporation which at any time between January 1, 1954, and the date of the adoption of the plan of liquidation, both dates inclusive, was the owner of stock possessing 50 percent or more of the total combined voting power of all classes of stock entitled to vote on the adoption of such plan[37]

Using our mathematical notation, the definition can be expressed as follows:

A corporation X is excluded iff[38]

MVP(1/1/54 to date of plan’s adoption ; {X}) 50%

The extended MVP illustrates the desirability of defining a function to be as powerful and general as possible. MIN and MAX are similarly extended by permitting each to take a set as its sole argument. Thus,

MIN S

represents the least number in the set of numbers, S. (Similarly for MAX.) This permits the use of MAX in the formal definition of the set XCS in subsection 4.1.

Sometimes the definition of a function cannot be conveniently expressed in English, and a more formal mechanism is desirable. This is the case with CAP and DIV, defined in subsection 4.1. To illustrate the notation used to define those functions, the following example defines the function PREM (not used beyond this example), which yields the annual premium for a standard automobile insurance policy, according to the sex of the object specified in its argument

PREM(X):
IF X is male
   THEN PREM = 400 
   ELSE PREM = 250

Thus, PREM(John) represents the number, 400, and PREM(Sue) represents the number, 250.

In other cases, the ELSE clause may be omitted. Note also that the expression between IF and THEN is always one which is either true or false.

3.5. Special terms

Certain words are sometimes used in a formal sense. Following [Allen 80b] these words are fully capitalized (and not italicized) when they are being used in their formal sense. The words so used in this paper are:

• IF, THEN, ELSE (defined by illustration above)[39]

• DISTRIBUTION (see subsection 4.1)

4. Section 333 of the Internal Revenue Code, Translated

In June, 1937, President Roosevelt sent a letter to Congress asking for legislation making the “present tax structure evasion-proof.”[40] A 1933 Congressional investigation into tax avoidance had led to enactment of legislation in 1934 that purported to prevent abuses, but the legislation was flawed and ineffective. The investigation had pinpointed the so-called “incorporated pocket book,” or the personal holding company (PHC), as perhaps the most prevalent tax-avoidance device of all.[41]

The appeal of PHCs for wealthy individuals was the great disparity in individual and corporate income tax rates. (The top individual rate was 79% in 1939, as opposed to a top rate of 19% for corporations.) By transferring income-earning assets to their corporations, such individuals could realize substantial tax savings.

Although tax law has contained provisions since 1913 to discourage excess accumulations of profits, these provisions did not function effectively to prevent the PHC method of tax avoidance.[42] The legislation of 1934 created a special category and separate penalty provisions for “personal holding companies,” but it is only the “Loophole Law” of 1937 amending the 1934 provisions that made PHC’s very unappealing and produced a strong incentive to liquidate them.

The treatment of shareholders of corporations undergoing liquidation was the same at the time as that specified today by section 331 of the Internal Revenue Code: a capital gain or loss is recognized, in the amount of the difference between the shareholder’s basis in his stock (that is, the amount he paid for his stock when he acquired it) and the fair market value of the money and other property he receives in the liquidation.

The House Ways and Means Committee determined in 1938[43] that PHC owners seeking to escape the punitive PHC tax by liquidating, in accordance with Congressional intent, faced an excessively onerous capital gains tax. In cases in which non-liquid assets were received in liquidation, the shareholder could be forced to sell the assets on unfavorable terms.

This is the problem that led to the enactment of §112(b)(7) in 1938, the forerunner of today’s section 333.[44]

Generally speaking, §112(b)(7) of the 1938 Code, which is similar to §333 of today’s Code, permits deferral of capital gains tax on the appreciation of assets transferred by a liquidating corporation to the time that the shareholder sells the assets. However, each shareholder must pay tax at ordinary income rates on his share of the corporation’s “earnings and profits” account, which reflects the corporation’s accumulated (and undistributed) income. The reasoning was that earnings and profits represent income from which shareholders have escaped taxation at individual rates. PHCs offer no corresponding advantage in the case of appreciated assets, however, since taxation at the time that the individual disposes of the asset would not be affected by the fact that the asset was held for a time by a PHC.[45]

§333 is an elective provision, offering shareholders an alternative, in some situations, to normal capital gains taxation of their gain. §333 is generally of interest when earnings and profits are low. A complex maze of rules must be satisfied if a liquidation is to qualify under §333. Also, an ill-advised election (as when earnings and profits are under-estimated) may be irreversible and result in a much higher tax burden than would have been faced if no election had been made.

In this section I present §333 in the proposed mathematical notation. References to sub-parts of §333 are given in endnotes.

The translation is given in two parts. The first part (subsection 4.1) is an alphabetically ordered list of definitions[46] of variables, functions and special terms (e.g., DISTRIBUTION). The second part (subsection 4.2) is a single IF ... THEN proposition stating the provisions of section 333.

A third subsection provides a few independent elements of the translation that can be quickly grasped and usefully applied in §333 situations by practicing lawyers.

4.1. Definitions

4.2. Restatement of Section 333 of the Internal Revenue Code, as of Aug. 15, 1984

Let OWNER be the owner of a share, STK, of a corporation, at TADP.[63]

IF

1. the corporation is domestic[64];

2. the corporation distributes property in a complete liquidation[65] in pursuance of LIQP[66];

3. the corporation is not a collapsible corporation to which §341(a) applies[67];

4. FMV > BAS[68]

5. RS(DIST) contains “all the stock”[69];

6. TIMTR(DIST) M, where M is the set containing all the moments in some one calendar month. (This month is referred to as the “liquidation month”)[70];

7. OWNER receives some part of PR(DIST) in exchange for STK[71]; AND

8. [Links to endnotes within the following expression: [72], [73]]

   

THEN

• DIV(OWNER) will be recognized and treated as a dividend on STK to OWNER[74]

• CAP(OWNER) will be recognized and treated as short or long term capital gains on STK to OWNER, as the case may be.[75]

4.3. Numerical Computation of Dividend and Capital Gains Amounts

This subsection focusses exclusively on the problem of calculating the amount that will be recognized and treated as a dividend, and the amount that will be recognized and treated as a (long- or short-term} capital gain, to a shareholder who is a “qualified electing shareholder” under §333.

The computation depends on amounts that are denoted, for convenient reference, by the variables BAS, FMV, MSEC, SHR, and EP. These variables are given definitions below similar to those in subsection 4.1:

The results of the computation are:

The representation of the computational procedure involves the functions MIN and MAX, explained in section 3.

The amount recognized and treated as a dividend to a shareholder X is:

IF X is a corporation
   THEN DIV = 0
   ELSE DIV = MIN (FMV - BAS; SHR × EP)

The amount recognized and treated as a dividend to shareholder X is denoted DIV(X) in the similar representation that is given of the computation procedure for defining CAP:

IF X is a corporation
   THEN CAP = MAX (MSEC; SHR × EP)
   ELSE CAP = MIN (FMV - BAS - DIV(X); MAX (0 ; MSEC - SHR) × EP))

                                         corp   non-corp
                                         ----   ---------
      EP    SHR   BAS   FMV  MSEC  gain   CAP   CAP   DIV
     500   0.05   200   400     0   200    25     0    25
     500   0.05   200   400    10   200    25     0    25
     500   0.05   200   400    20   200    25     0    25
     500   0.05   200   400    30   200    30     5    25
     500   0.05   200   400    40   200    40    15    25
     500   0.05   200   400   100   200   100    75    25
     500   0.05   200   400   200   200   200   175    25
     500   0.05   200   400   250   200   250   175    25
     500   0.05   200   400   300   200   300   175    25
     500   0.05   200   400   350   200   350   175    25
     500   0.05   200   400   400   200   400   175    25
     500   0.50   200   400     0   200   250     0   200
     500   0.50   200   400   200   200   250     0   200
     500   0.50   200   400   400   200   400     0   200
   10000   0.01   200   400     0   200   100     0   100
   10000   0.01   200   400   200   200   200   100   100
   10000   0.01   200   400   400   200   400   100   100
   50000   0.01   200   400     0   200   500     0   200
   50000   0.01   200   400   200   200   500     0   200
   50000   0.01   200   400   400   200   500     0   200

The reader can test these representations in the hypothetical situations given by the rows of Figure 4-1. For the first row, for example, the expression giving the amount taxed as a dividend to a non-corporate shareholder is

DIV = MIN (FMV - BAS; SHR × EP)

i.e., DIV = MIN (400 - 200 ; .05 × 500)

i.e., DIV = MIN (200; 25)

i.e., DIV = 25.

4.4. Criticism of Section 333

The foregoing is an attempt to exactly replicate the provisions of §333 in mathematical notation with its ambiguities preserved. In the remainder of this section the ambiguities are discussed, with reference to the mathematical statute representation. [McGaffey 64]’s analysis of §333 is drawn on for many of the following points. The conditions referred to are the numbered propositions between IF and THEN in the mathematical representation of §333 given in subsection 4.2 above.

4.4.1. TADP (Time of Adoption of Liquidation Plan)

TADP is not operationally defined, despite its potentially crucial role in determining satisfaction of the requirement that OWNER own STK at TADP, and condition 8. Can it be the moment an oral agreement was reached? --even if not all parties required for the vote had yet been consulted? Or is it the moment a written plan was formally adopted? What if the written plan does not unequivocally mandate a liquidation?

The definition of TADP has been the central issue in several cases, for example, Shull v. Commissioner[76], which was heard in the U.S. Tax Court three times and by the U.S. Court of Appeals twice. In this case, Shull filed an election for § 112(b)(7) tax treatment[77] on 4/29/52, not realizing that his corporation’s earnings and profits would as a result be taxed to him as a dividend. Accompanying the election document was a certification that a liquidation plan had been adopted in a meeting on 3/31/52. After an unfavorable 1955 audit by the IRS, he sought to void his election by claiming no meeting had truly taken place on 3/31/52, and that the liquidation plan had in fact been adopted months earlier.[78] (The election would be voided if it could be established that it was made more than 30 days after the adoption of the liquidation plan.) The issue ultimately addressed by the Court of Appeals was whether a Certificate of Dissolution issued by the State Corporation Commission on 3/27/52, and the fact that the corporation was actually in a state of liquidation on that date, evidenced the existence of a plan of liquidation on that date.

TADP may be the type of variable that relies for its value on judicial discretion in questionable cases. But the judge is offered no statement of what in principle he must seek to ascertain. Should he attempt to estimate at what moment liquidation became probable? (This could be years earlier.) Or the moment it was a firm intention for parties formally controlling sufficient voting power? --Or controlling sufficient voting power in pragmatic terms?

The unpleasant attention this variable calls to itself highlights the puzzle of why it is allowed to exist. The requirement of filing elections within 30 days of TADP, possibly before the shareholder knows the values of MSEC and EP, seems arbitrary. A more appropriate moment for the purpose of condition 8 would seem to be some moment more closely corresponding to the liquidating distribution.[79] McGaffey questions what policy is served by Condition 8[80], and the requirement that OWNER hold STK at TADP also seems superfluous. TADP appears to create legally consequential issues for no purpose.

4.4.2. Conditions 2 and 8

Another difficulty in deriving an operational interpretation of the statute is the omission of a definition of DIST, the DISTRIBUTION referred to by the statute in (a)(2). McGaffey points out[81] the incentive to pay dividends to corporate shareholders during the liquidation month. Are these part of PR(DIST)? The same incentive exists to pay dividends before the liquidation month, or before the nominal adoption of the plan. Can these be deemed to be associated with DIST (entailing the consequence, possibly ruinous for the shareholders, that condition 6 is violated)? McGaffey states:

If the corporation to be liquidated has mainly corporate shareholders, it would be deemed to be a distribution pursuant to the plan of liquidation made outside the one month, which would void the section 333 election.[82]

Thus an incentive exists in certain situations, yet the statute’s effect is unclear in cases where the incentive is acted upon. Some will take a risk; others will play it safe. The product is arbitrary taxation.

4.4.3. EP (Earnings and Profits)

McGaffey, searching for a rationale for section 333’s taxation of earnings and profits as ordinary income, states:

Some tax should be imposed upon the earnings and profits of a corporation upon liquidation because of the double-tax character of our corporate tax system. If the tax were not imposed upon earnings and profits upon liquidation. such earnings would be subject to only one tax, a treatment that is not permitted operating profits. However. in other situations where there is a bulk distribution of assets, the tax on the distribution is imposed at capital gain rates. Therefore, some other reason is necessary for imposing the tax at ordinary income rates.

The tax on earnings and profits at ordinary rates may derive from the historical purpose of section 333 and its predecessors which were intended to apply to personal holding companies. Since earnings of such companies are normally distributed, they are normally taxed at both the corporate and individual level at ordinary income rates .... if the reason for taxing earnings and profits at ordinary rates is peculiar to personal holding company corporations, only those corporations should be required to recognize earnings and profits at ordinary rates, and the earnings and profits of other corporations should be taxed at capital gain rates.[83]

According to (e)(1), EP is determined at the close of the liquidation month. However, the statute also states that EP “includ[es] ... all amounts accrued up to the date on which the transfer of all the property under the liquidation is completed ....” McGaffey points out: “This provision raises the question whether an item of income or expense accruable at some later time -- perhaps because the amount is not ascertainable -- may be accrued at a date after the liquidation, or if because of the liquidation, it becomes accruable at that particular date.”[84]

McGaffey points out the distinction between the dates of last transfer and close of month is of no consequence, since “Presumably, all income, no matter to whom paid, which is earned on assets after they have been distributed to the stockholders, is the stockholder’s property and would not be considered earnings and profits of the corporation.”[85]

McGaffey also points out (p. 331) that the distributions referred to by the statute’s “... without diminution by reason of distributions made during such month ...”[86] are probably intended to be only the distributions made pursuant to the plan of liquidation. This calls into question the definition of the subvariable D in the definition of EP (see subsection 4.1). Although the statute’s language corresponds to the definition, deemed Congressional intent could be invoked in certain cases to arrive at a different construction.

4.4.4. MSEC (Money, Stock or Securities)

The references in (e) and (f) to “money, or ... stock or securities” may signal a legislative intent indicating a broader category of property than is apparent. Since a judicial decision may attempt to serve a view of Congressional intent different from the statute’s literal language, MSEC is in effect ill-defined.

The reasoning underlying the inclusion of stock or securities in MSEC was apparently as follows:[87] If MSEC included only money, the unfavorable treatment of MSEC could be circumvented by having the corporation convert money to liquid securities and distribute the securities, then having the distributee convert back to money.

If this is accepted as legislative intent, then a court could also consider other liquid assets, such as receivables, to be included in “money, or ... stock or securities.”[88]

Stepping back further for a broader view, legislative intent might be used to argue that only money, and not even stock or securities, should be included in MSEC. If legislative intent in including stock and securities is to prevent the abuse outlined above, then as McGaffey points out, the inclusion is superfluous because the scheme simply is not sound. Congress “closed a loophole which in fact did not exist,”[89] since if the distributee indirectly obtained money in this way, his gain would be recognized the moment he converted the stocks to money.

McGaffey points out that the unnecessary inclusion of stock or securities in MSEC

make(s) this section less and less available as the years pass, penalizing those who have acquired post-1953 securities for reasons other than avoidance of the tax on money. It may be in accord with the intent of Congress to limit the application of this section to the years shortly after 1954.... It is submitted that trying to limit the applicability of section 333 in this manner is extremely arbitrary, as it is questionable whether there are such differences between a corporation which made substantial investments and changes in its portfolio since 1953 and one which did not as to justify different tax treatment.[90]

The unnecessary inclusion of stock or securities in MSEC also produces an incentive to convert this kind of asset to another kind of liquid asset prior to distribution, in order to obtain the benefits of tax deferral.

Further ambiguity in the definition of MSEC arises from the unqualified use of the word “acquires” in “... stock or securities acquired by the corporation after December 31, 1953 ....”[91] If a shareholder transfers stock to a corporation in a §351 transaction, the time of transfer might be considered to be the moment of the corporation’s acquisition, for purposes of §333. However, as pointed out by McGaffey, under §1223 the corporation’s holding period for the stock begins at the time the transferor-shareholder acquired the stock. It would seem consistent to deem the moment of the corporation’s acquisition referred to in §333 to be the beginning of its holding period.

The statute does not resolve this question. The IRS has ruled[92] that in the case of an individual transferring to a corporation created after December 31, 1953, in a section 351 transaction, securities he has acquired prior to December 31, 1953, the transfer date is the date of acquisition for purposes of §333. However, there is authority for carrying over the “acquisition date” in the case of stock dividends, mergers, replacement of lost or destroyed certificates, and the change in denomination of certificates.[93]

4.4.5. Condition 8 (80% Election)

The inordinate and unnatural complexity of this condition is perhaps the clearest indication here of the dividends a mathematical approach in drafting could deliver.

The first problem is that the concept of “voting power” appealed to by the statute can be meaningless for certain corporations. Suppose there are two classes of corporate shares, A and B, and 60% of each is required for a plan of liquidation to be adopted. Under such circumstances, the “voting power” of the shares in class A is meaningless.

Let us consider a corporation in which this problem does not arise. That is, we assume that at any particular moment

• A certain number of votes is necessary and sufficient for the adoption of a liquidation plan;

• Each share is entitled to cast a designated number of votes (0 or more) on a plan of liquidation (the number of votes a particular share is entitled to cast may depend on which class of shares it belongs to); and

• There is no other source of votes (for example, directors, etc.)

Given this simplified liquidation approval mechanism (apparently taken for granted by the current law’s drafters), it is likely that a mathematical approach would have led to the articulation of the intended condition in the following relatively natural manner:

where LVOTES (T ; S) denotes the number of votes that shares owned by shareholders in the set S at time T, can cast at time T, on the issue of adopting LIQP.

The following additional ambiguities also arise:

• What is the meaning of VP? The proportional voting power of a set of shares may depend on the issue being voted upon. A share that normally has voting rights may be barred from voting on questions of liquidation. This ambiguity is not resolved by the statute’s restriction of VOTSTK(TADP) to shares that are entitled to vote on a liquidation plan.

• What is the definition of VOTSTK(T)? Could it include treasury shares, or lost shares? This question would not arise if the statute’s language corresponded to the relatively natural

  VP (TADP; OWN (TADP; E(NXCS)); OWN (TADP; NXCS))

  instead of

  VP (TADP; OWN (TADP; E (NXCS)) ; VOTSTK (TADP) - OWN (TADP; XCS NCS))

4.4.6. Condition 5 (Redemption of All Stock)

This condition is not clear because “all stock” is ambiguous. “All” stock outstanding at what point in time? What if new shares are sold by the corporation after this point?

4.4.7. Non-Specification of Gain Transaction

The statute fails to mandate a firm link between the gain on corporate shares that it addresses, and the corporation’s liquidating distribution.

Subsection (a) states: “In the case of [property in PR(DIST)] ... gain on the shares owned by [the shareholder] at [TADP] ...” Thus the statute does not appear to require that OWNER own STK at the time of distribution, redemption, or transfer. §333 should be applicable to OWNER’s gain in the following situation:

Suppose the distribution occurs over a period of time, and Z, another shareholder, is the first to have his shares redeemed. If Z uses the property distributed to him to buy STK from OWNER, then OWNER should be able to claim §333 treatment of his resultant gain. This introduces uncertainty when the property referred to is money. One can determine if a palpable asset is received, directly or indirectly, by OWNER in exchange for STK. But is it meaningful to ask whether money received by OWNER from a third party was at some point part of the corporation’s assets? This directly arises in §333(e), which states: “... that portion of the [implicitly, corporation’s] assets received by [OWNER] which consists of money, or of ....”

Note also that OWNER can invoke §333 repeatedly if he sells STK repeatedly (with intervening repurchases).

This problem could be avoided by replacing condition 7 by the requirement that OWNER own STK continuously from the beginning of the liquidation month to the time the corporation redeems STK.

Further uncertainty in interpreting the statute arises if only part of what OWNER receives for STK belongs to PR(DIST).

5. Conclusions

This paper concentrates on the use of math notation at an advanced stage in the drafting process, when the provisions are already determined. However, the discussion in section 4 suggests that the use of a math notation could have a desirable influence not only on the clarity and precision with which a provision is expressed, but also on the determination of what the provision should be. The full potential benefits of this notation can only be realized by applying the notation during earlier stages, when the provisions are being designed. This is where the notation can enhance the effectiveness of legislation.

Just as math notation is useful in expressing legislation, so can it serve in articulating the public policy purpose underlying legislation. An exact statement of the intent of §333, for example -- something expressing the tax avoidance problem arising from PHCs and the objective of providing liquidation incentives -- would enable a methodical analysis which could quite conceivably lead to very different provisions than those found in §333.

The qualities of clarity and precision this paper seeks to illustrate in connection with tax legislation could indicate a broad role for math notation in the public policy debate and formulation process. If indeed the current, relatively formal process of moving from policy objectives to tax legislation can be improved by this mechanism, we can expect similar gains in many areas of government decision-making.

Every writer has experienced how the discipline of articulating ideas in explicit language can reshape and transform those ideas. Often one’s initial inclination is reversed in the exercise of articulation. If one grants the view that math notation demands a greater degree of articulation than English and leads to more explicit expression, one can accept that math notation has a place in virtually all areas of pragmatic communication -- in nearly every realm other than those of art and entertainment.

In tax law, it is foreseeable that the use of math notation could significantly reduce the incidence of tax loopholes. Proposed provisions rendered in a mathematical form would lend themselves more readily to conceptual manipulation. Loopholes or other unexpected outcomes would be more prone to detection in the drafting stage. Conceptual manipulability would also permit closer analyses of the cumulative effect of many interacting statutes. Given these considerations, math notation could be an important tool in the drafting stage of tax law and of many other complex areas of law, such as banking regulation, administrative law, securities law, and corporate law.[94]

The institutionalization of math notation is feasible, notwithstanding the possibility that few legislators would evidence enthusiasm for working directly with such a notation. The case of modern corporations that rely on computer software is exactly parallel: management is not involved in the technical aspects of computer operations, yet it routinely communicates its complex business requirements to computer professionals who then formalize these requirements and represent them in the precise languages of computer programming. The expression of a body of law in a precise notation is no greater a challenge.

Appendix A: Text of IRC Section 333

SEC. 333. ELECTION AS TO RECOGNITION OF GAIN IN CERTAIN LIQUIDATIONS.

   (a)  GENERAL RULE.--In the case of property distributed in complete liquidation of a domestic corporation (other than a collapsible corporation to which section 341 (a) applies), if--

   (b)  EXCLUDED CORPORATION.--For purposes of this section, the term “excluded corporation” means a corporation which at any time between January 1, 1954 and the date of the adoption of the plan of liquidation, both dates inclusive, was the owner of stock possessing 50 percent or more of the total combined voting power of all classes of stock entitled to vote on the adoption of such plan.

   (c)  QUALIFIED ELECTING SHAREHOLDERS.--For purposes of this section the term “qualified electing shareholder” means a shareholder (other than an excluded corporation) of any class of stock (whether or not entitled to vote on the adoption of the plan of liquidation) who is a shareholder at the time of the adoption of such plan, and whose written election to have the benefits of subsection (a) has been made and filed in accordance with subsection (d), but--

   (d)  MAKING AND FILING OF ELECTIONS.--The written elections referred to in subsection (c) must be made and filed in such manner as to be not in contravention of regulations prescribed by the Secretary. The filing must be within 30 days after the date of the adoption of the plan of liquidation.

   (e)  NONCORPORATE SHAREHOLDERS.--In the case of a qualified electing shareholder other than a corporation--

   (f)  CORPORATE SHAREHOLDERS.--In the case of a qualified electing shareholder which is a corporation, the gain shall be recognized only to the extent of the greater of the two following--

Endnotes

(Use your browser’s “Back” function to return to the endnote reference in the text -- see Websurfing Tips.)

[1]	With the support of consumer groups, the Plain English movement has been successful since the 1970s in getting legislation enacted at the federal and state levels that mandates clearer laws in certain areas. See [Hathaway 83].
[2]	[Allan 57a], [Allen 78], [Allan 80a], [Allen 80b].
[3]	[Allen 78], [Allen 80a].
[4]	[Mellinkoff 63], p. 387.
[5]	pp. 290-295
[6]	Sir Edward Coke, The First Part of the Institutes of the Laws of England, or a Commentary upon Littleton, 10th ed. London: William Rawlins and Samuel Roycroft, 1703. [1st published. 1628.] f. 20b and compare f. 46b (10th ed. 1703).
[7]	Black’s Law Dictionary (4th ed. 1951); 1 Words and Phrases: Judicially Defined (1946); Allentown v. Pennsylvania Public Ulility Commission, 173 Pa. Super. 219, 222 (1953); State v. Youngblood, 199 Tenn. 519, 523 (1955); contra: State v. Fields, 70 Kan. 391, 393 (1904).
[8]	Sanborn v. Camberlin, 101 Mass. 409, 418 (1869).
[9]	In re Pearsons, 98 Cal. 603, 608 (1893); Central National Bank v. Pratt, 115 Mass. 539, 545 (1874); Doe dem Gibson v. Gell, 107 Eng. Rep. 535 (K.B. 1824).
[10]	cf. pp. 304-366, 374-383.
[11]	[Mellinkoff 63], p. 375. Footnotes omitted.
[12]	pp. 247-252
[13]	p. 366-368, 371-374.
[14]	[ILP 64] and the subsequent volume cite the following reviews: [Grad 64], [Gordon 64], [Probert 64], [Cohen 64], [Littleton 64], [Stolz 64], [Turack 64], [Currie 64], [Casad 64], [Baker 65], [Goldfarb 64], [Lewis 64], [Mayda 65], [Munro 65], [Rossman 64]. Only [Baker 65], while conceding “much force in Mr. Mellinkoff’s criticisms,” protests that Mellinkoff’s claims of imprecision are “occasionally ... less than fair.” At least one of the rejoinders he gives to three specific criticisms he selects is incorrect. The reviews are generally laudatory. One encounters recurring tributes to the work’s scholarship, research and organization. Samples: “With the bulk of the author’s targets there can be no serious dispute” ([Gordon 64]); “[The chapter on precision (13)] is by far the longest in the book. It is detailed; it is documented; it is reasonable; it is devastating” ([Currie 64]); “Directed against a profession which is so innately involved in the business of communication, this is a critical and serious indictment. The charge is supported by devastating scholarship and thoroughness.” ([Goldfarb 64])
[15]	pp. 261-266
[16]	pp. 252-256
[17]	pp. 253-254
[18]	pp. 201-204, 208-212.
[19]	Allen reports on the preliminary results of an experiment in which law students, lawyers, and laymen were tested on their ability to determine the implications of the statute in ten problems. Half the members of each of the three groups were permitted to refer only to the statute itself; the others worked only with Allen’s “normalized” version (after a brief introduction to this style of logical notation). Partial data from the 89 subjects indicates that in all 3 groups, the time spent on the problems was about the same, but the responses of those working with Allen’s normalized version were 80% more accurate.
[20]	A.N. Whitehead, An Introduction to Mathematics (New York and London, 1911), p. 59. Quoted in [Cajori 29], pp. 332-333. Similar views are expressed by J.W.L. Glaisher, quoted p. 328.
[21]	William Oughtred, The Key of the Mathematicks (London, 1647), Preface. Quoted in [Cajori 28], p. 199. For counter-arguments, see Thomas Hobbes, John Keill and others. Quoted pp.426-431.
[22]	[Allen 57a], [Allen 57b], [Allen 78], [Allen 80a], [Allen 80b], [Bauer-Bernet 80], [Finan 82] bolster the argument with illustrations. Also [Lowry 79], [Tammelo 55]. The latter gives further references. A translation to essentially normalized form is also a preliminary to implementing certain computer systems that mechanically apply statutory rules, as in [Welch 82], [Hellawell 82], [Maggs 72]. [Montrose 53] presents his own distinctive symbolic notation to overcome the awkwardness of using natural language. [Mellinkoff 63] makes several fleeting references to the symbolic logic approach (pp. 394-395; also 388, 294) but is not among its supporters. He cites the necessity of being intelligible to many people; the general notion of absolutes running against the grain of human experience, and the process of fundamental language changes requiring “eons, not lifetimes.”
[23]	A “proposition” is a sentence or portion thereof which asserts something and is therefore true or false. A proposition may be elementary, e.g., “It is raining,” or it may be compound, e.g., “It is raining AND God is dead.”
[24]	This illustration is a minor modification of that given in [Allen 78]. Note that parentheses can be used in the place of indentation, as in, ( S2 AND ( S3; S4; AND (S5 OR ( S6 AND S7 ))) OR S9 ) S1 Parentheses are used to specify priority of operations, just as in arithmetic: 4 + (3 × 2) is 10, whereas (4 + 3) × 2 is 14.
[25]	[Copi 73] is a standard textbook treatment of PC and lower predicate calculus.
[26]	p. 162
[27]	[Maggs 72], p. 163.
[28]	The term “constant” may appear more appropriate to logicians than “variable.” FMV, for example, seems to be defined below as a specific number. Formally, however, FMV is indeed a variable, having specified relationships to other variables. It can be assigned to any object, although under certain assignments the major antecedent in subsection 4.2 will be false and therefore the consequent will not be implied. As a simple example of this general idea, consider the law: “IF X commits perjury THEN X is liable for a fine.” This is a standard logic conditional expressing an empirical (not tautological) truth about what the law provides: the variable X may be assigned to any object. When the assignment makes the antecedent true, then the conditional delivers the empirical information that X is liable for a fine.
[29]	The analogue in a computer language is its primitive functions.
[30]	[Suppes 57] provides formal definitions of certain mathematical concepts, such as function and set.
[31]	Such variables are analogous to a special kind of variable in certain computer languages, exemplified by in APL. When a reference to this variable is encountered in the course of processing, execution is interrupted and a person must provide the value.
[32]	333(e)
[33]	333(d)
[34]	MVP is not used beyond this example. The statute translations suggested in this example may well represent the intended provisions. However, in order to exactly adhere to the technical provisions entailed by the actual language of the statute, a more complex representation is required.
[35]	333(b). This poses problems of interpretation, but we postpone discussion of these until section 4.
[36]	333(c)
[37]	333(b)
[38]	“iff” is used in this paper as an abbreviation for “if and only if”.
[39]	As a consequence of the informality of this presentation, the use of IF ... THEN in defining functions may appear inconsistent with the use of these terms in expressing a conditional such as: “IF X commits perjury THEN X is liable for a fine.” FormaIly, however, there is no difference. “IF X is male THEN PREM = 400” is a conditional linking propositions about PREM’s argument and PREM’s result.
[40]	Quoted in [Paul 37], p. 48.
[41]	[Libin 65]. pp. 422-423.
[42]	[Rudick 39], p. 173-176.
[43]	1 H.R. Rep. No. 1860, 75th Cong., 3d Sess.30 (1938), cited in [Eaton 52], p. 10.
[44]	See [Eaton 52], pp. 9-12, for a more thorough discussion.
[45]	This assumption overlooks certain subtleties involving the effect that temporary ownership by the PHC may have had on the individual basis in the asset.
[46]	In a formal framework these can be thought of as additional propositions supplementing the conjunction constituting the major antecedent in Subsection 4.2.
[47]	a
[48]	f
[49]	e2
[50]	(b), (d)
[51]	(a)
[52]	f
[53]	e1
[54]	d
[55]	(e)(1). The Statute permits both interpretations. C will normally be negative, because distributions reduce earnings and profits.
[56]	e1, f2
[57]	a
[58]	e2, f1
[59]	e, f2
[60]	a, c
[61]	(b) and (c)
[62]	b
[63]	a; a1
[64]	a
[65]	a
[66]	a1
[67]	a
[68]	a: “...gain on the shares...” indicates the statute only concerns cases of positive gain.
[69]	a2
[70]	a2
[71]	a: “In the case of property distributed...”
[72]	(c)(1)
[73]	(c)(2)
[74]	e, f
[75]	e, f
[76]	291 F.2d 680, 8 AFTR 2d 5010 (1961).
[77]	112(b)(7) is the predecessor of the 1954 Internal Revenue Code’s section 333.
[78]	30 T.C. 827-8.
[79]	cf. McGaffey, p.342.
[80]	McGaffey, pp. 341-342.
[81]	p. 331
[82]	p. 331
[83]	pp. 329-330
[84]	p. 331
[85]	p. 331
[86]	(e)(1) and (f)(2)
[87]	See [Eaton 52], p. 52. The legislative intent here is also that attributed by Bittker, Federal Income Taxation of Corporations and Shareholders, p. 266 (1959); cited by McGaffey, p. 332.
[88]	It has been held that receivables are not part of “money, or ... stock or securities” in the §333 context. See 4 T.C. 919-920 (1945).
[89]	McGaffey, p. 333. McGaffey observes that a revision proposal by the American Bar Association Section on Taxation also evidences apparent concern over such a loophole.
[90]	p.333
[91]	(e)(2); similar language in (f)(1)
[92]	Rev. Ruling 58-92, 1958-1 Cum. Bull. 174
[93]	McGaffey, p. 336.
[94]	It should be noted here that the subject must have some inherent complexity in order for the apparatus under discussion to offer any advantages.

References

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