1. By an elementary d-expression we shall understand an expression which is formed of the letter d followed by an elementary T-expression (within brackets). The letter f followed by an elementary T-expression will be said to form an elementary f-expression.
By an atomic d-expression we shall understand an expression which is formed of the letter d followed by a (atomic or molecular) T-expression. The letter f followed by a T-expression will be said to form an atomic f-expression.
By df-expressions finally we shall understand atomic d- and atomic f-expressions and molecular complexes of atomic d- and/or f-expressions.
Examples: d(qT ~q) is an elementary d-expression. f((p & ~q) T (r ∨ s) ∨ ~pTp) is an atomic f-expression. d(pTp) & f( ~qTq) is a df-expression.
Elementary d-expressions describe elementary acts and elementary d-expressions elementary forbearances. Generally speaking a df-expression describes a certain (mode of) action which is performed by one and the same unspecified agent on one and the same unspecified occasion.
The logic of df-expressions or the df-calculus is a fragment of a (general) Logic of Action.
2. We shall in this section briefly discuss the logical relations between the eight elementary acts and forbearances which answer to one given state of affairs.
Firstly we note that corresponding elementary acts and forbearances are mutually exclusive. One and the same agent cannot both do and forbear the same thing on the same occasion. But one and the same agent can do something on some occasion and forbear doing the (generically) same thing on a different occasion.
Secondly we note that any two of the four types of elementary act (relative to a given state of affairs) are mutually exclusive. Consider for example the acts described by d(pTp) and by d( ~pTp). They cannot be both done by the same agent on the same occasion. This is a consequence of the fact that no occasion constitutes an opportunity for doing both acts. This again is so because a given state and its contradictory state cannot both obtain on the same occasion. Or consider the acts described by d(pTp) and by d(pT ~p). They too cannot both be done by the same agent on the same occasion. For a given state of affairs either changes or remains unchanged. If independently of action it would remain unchanged the agent may destroy it but cannot preserve it. That is: there is then an opportunity for destroying it but not for preserving it. If again independently of action the world would change in the feature under consideration the agent may preserve but cannot destroy this feature.
Thirdly we note that any two of the four types of elementary forbearances are mutually exclusive. Since no state and its contradictory state can both obtain on the same occasion no agent can for example both forbear to preserve and forbear to suppress a given state on a given occasion. And since a state which obtains either changes or remains unchanged independently of action no agent can for example both forbear to preserve and forbear to destroy it on one and the same occasion.
From the above observations we may conclude that all the eight elementary acts and forbearances which answer to one given state of affairs are mutually exclusive.
The question may be raised: Are the eight elementary acts and forbearances jointly exhaustive? Let for example the state of affairs described by p be that a certain window is closed. Is it necessarily true given an agent and an occasion that this agent will on that occasion either close the window or leave it open open the window or leave it closed keep the window close or let it (become) open or keep the window open or let it close?
I think we must when answering this question take into account considerations of human ability. Assume that the state of affairs is one which the agent can neither produce nor suppress if it does not exist nor destroy or preserve if it exists. Then he can of course not be said truly to produce or suppress or destroy or preserve it. But neither can he be rightly said to forbear to produce or suppress or destroy or preserve it. For forbearing as we understand it here makes sense only when the act can be done. (See Ch. III Sect. 8.)
There are many states towards whose production or suppression or destruction or preservation human beings can do nothing. Most states of the weather are of this kind and states in remote parts of the universe. And there are states with which some agents cannot interfere in any way whatsoever but with which other more ‘powerful’ agents can interfere in some if not in every way. A child may have learnt to open a window but not to close it.
The correct answer to the above question concerning the jointly exhaustive character of the eight elementary acts and forbearances answering to a given state of affairs therefore is as follows:
Only on condition that the agent can produce and suppress and destroy and preserve a given state of affairs is it the case that he necessarily will on any given occasion either produce or forbear producing suppress or forbear suppressing destroy or forbear destroying or preserve or forbear preserving this state of affairs.
In the subsequent discussion it will be assumed that this requirement as regards ability is satisfied and that consequently the eight types of elementary acts and forbearances may be treated as not only mutually exclusive but also jointly exhaustive.
3. Every df-expression expresses a truth-function of elementary d- and/or f-expressions. This is so because of the fact that the operators d and f have certain distributive properties. These properties are ‘axiomatic’ to the df-calculus: that is they cannot be proved in the calculus. Their intuitive plausibility however can be made obvious from examples.
Consider an atomic d-expression. Let the T-expression in it be in the positive normal form (Ch. II Sect. 9). It is then normally a disjunction of conjunctions of elementary T-expressions. The conjunctions describe mutually exclusive ways in which the world changes and/or remains unchanged. Obviously the proposition that some of these ways is effected through the action of some unspecified agent on some unspecified occasion is equivalent to the proposition that the first of these ways is effected through the action of that agent on that occasion or… or the last of these ways is effected through the action of that agent on that occasion.
For example: d( ~pTp ∨ pT ~p) says that some agent on some occasion either produces the state described by p or destroys it. The same thing is also expressed by d( ~pTp) ∨ d(pT ~p).
Thanks to the disjunctive distributivity of the d-operator every atomic d-expression may become replaced by a disjunction of atomic d-expressions in which the d-operator stands in front of a change-description (Ch. II Sect. 10).
Now consider for example the meaning of d((pT ~p) & (qT ~q)). An agent on some occasion through his action makes both of two states vanish. Does this not mean that he makes the one and makes the other state vanish i.e. does the above expression not mean the same as d(pT ~p) & d(qT ~q)?
I shall answer in the affirmative and accept the identity of the expressions. I also think this answer accords best with ordinary usage. Be it observed however that ordinary usage is not perfectly unambiguous in cases of this type. To say that somebody through his action has become ‘responsible’ for two changes in the world could be taken to mean that he effected one of the two changes whereas the other took place independently of his action. But to say that he effected or produced the two changes would not seem quite accurate unless he actually produced the one and also produced the other. We must not however be pedantic about actual usage. But we must make the intended meaning of our symbolic expressions quite clear. Therefore we rule that the d-operator is conjunctively distributive in front of change-descriptions.
Consider an atomic f-expression. Is the f-operator too disjunctively distributive in front of a disjunction which describes some mutually exclusive alternative changes in the world? What does it mean to say that an agent forbears this or that? Since the changes (the ‘this’ and the ‘that’) are mutually exclusive the occasion in question cannot afford an opportunity for forbearing to produce more than one of the changes. The agent therefore on the occasion in question either forbears to produce the first or… or forbears to produce the last of the mutually exclusive alternative changes in the world.
It is essential to this argument that the changes are mutually exclusive. To forbear this or that when both things can be done on the same occasion would I think ordinarily be understood to mean that the agent forbears both things i.e. does neither the one nor the other.
Thus the f-operator too is disjunctively distributive in front of a T-expression in the perfect normal form. For example: f(( ~pTp) ∨ (pT ~p)) means the same as f( ~pTp) ∨ f(pT ~p).
Remains the case when the f-operator stands in front of a change-description. For example: f((pT ~p) & (qT ~q)). What does the agent do who on some occasion forbears to destroy two existing states? The question can be answered in more than one way. If however we stick to the view that forbearing is not-doing on an occasion for doing and accept the above interpretation of d((pT ~p) & (qT ~q)) then we must answer the question as follows: To forbear to destroy two existing states is to forbear the destruction of at least one of them. f((pT ~p) & (qT ~q)) thus equals f(pT ~p) ∨ f(qT ~q). Generally speaking: the f-operator is disjunctively distributive in front of change-descriptions.
These four rules for the distributivity of the d- and f-operators secure that every atomic d- or f-expression expresses a truth-function of elementary d- or f-expressions. Since truth-function-ship is transitive it follows a fortiori that every df-expression expresses a truth-function of elementary d- and/or f-expressions.
The elementary d- and/or f-expressions of which a given df-expression expresses a truth-function will be called the df-constituents of the df-expression. Which truth-function of its df-constituents a given df-expression expresses can be investigated and decided in a truth-table. The distribution of truth-values over the df-constituents in the table is subject to the limitations imposed by the mutually exclusive and jointly exhaustive nature of the eight types of elementary acts and forbearances (relative to the same state of affairs).
If a df-expression expresses the tautology of its df-constituents we shall call it a df-tautology. If it expresses their contradiction we call it a df-contradiction.
d(pTp) & f(pTp) is an example of a df-contradiction. Hence ~d(pTp) ∨ ~f(pTp) is a df-tautology.
Assume that the T-expression in an atomic d- or f-expression is a T-contradiction. Then the positive normal form of the T-expression is a O-termed disjunction. We cannot use the distributivity of the d- and f-operators for transforming the atomic d- or f-expression into a molecular complex of elementary d- and/or f-expressions. A special rule has to be introduced for the case. The rule is simple: the atomic d- or f-expression in question is a df-contradiction. The intuitive meaning of this rule is obvious: If it is logically impossible that a certain change should happen then it is also logically impossible to effect or leave uneffected this change through one's action.
4. On the assumption which we are here making that the eight types of elementary acts and forbearances are jointly exhaustive of ‘logical space’ every df-expression has what I propose to call a positive normal form. (Cf. Ch. II Sect. 9.) It is a disjunction-sentence of conjunction-sentences of elementary d- and/or f-expressions. It is called ‘positive’ because it does not contain negation-sentences of elementary d- and/or f-expressions.
The positive normal form of a given df-expression is found as follows: The df-expression is first transformed into a molecular complex of elementary d- and/or f-expressions according to the procedure described in Section 3. The new df-expression thus obtained is thereupon transformed into its perfect disjunctive normal form. This is a disjunction-sentence of conjunction-sentences of elementary d- and/or f-expressions and/or negation-sentences of elementary expressions. We replace each negation-sentence of an elementary d- or f-expression by a 7-termed disjunction-sentence of elementary expressions. The new df-expression thus obtained is transformed into its perfect disjunctive normal form. From the normal form we omit those conjunction-sentences if there are any which contain two or more different elementary d- or df-expressions of the same variable (p q etc.). What remains after these omissions is the perfect normal form of the original df-expression.
We give a simple example to illustrate this procedure:
Let the df-expression be d(pTp) ∨ d(qTq). Its perfect disjunctive normal form is d(pTp) & d(qTq) ∨ d(pTp) & ~d(qTq) ∨ ~d(pTp) & d(qTq). We replace ~d(pTp) by the 7-termed disjunction-sentence d(pT ~p) ∨ d( ~pTp) ∨ d( ~pT ~p) ∨ f(pTp) ∨ f(pT ~p) ∨ f(∪pTp) ∨ f( ~pT~p) and ~d(qTq) by the 7-termed disjunction-sentence d(qT ~q) ∨ d( ~qTq) ∨ d( ~qT ~q) ∨ f(qTq) ∨ f(qT ~q) ∨ f( ~qTq) ∨ f( ~qT ~q). After distribution we get the 15-termed disjunction-sentence of 2-termed conjunction-sentences d(pTp) & d(qTq) ∨ d(pTp) & d(qT ~q) ∨ d(pTp) & d( ~qTq) ∨ d(pTp) & d(∪qT ∪q) ∨ d(pTp) & f(qTq) ∨ d(pTp & f(qT ~q) ∨ d(pTp & f( ~qTq) ∨ d(pTp) & f(∪qT ∪q) ∨ d(pT ~p) & d(qTq) ∨ d( ~pTp) & d(qTq) ∨ d( ~pT ~p) & d(qTq) ∨ f(pTp) & d(qTq) ∨ f(pT ~p) & d(qTq) ∨ f( ~pTp) & d(qTq) ∨ f( ~pT ~p) & d(qTq). This is the positive normal form of the original df-expression. It is a complete enumeration of the 15 mutually exclusive generic modes of action which are covered by the description d(pTp) ∨ d(qTq).
5. We have previously (Ch. II Sects. 3 and 10) introduced the notions of a state-description and a change-description. By analogy we now introduce the notion of an act-description. An act-description is a conjunction-sentence of some n elementary d- and/or f-expressions of n different atomic variables. Thus for example d(pTp) & f(qT ~q) is an act-description.
As we know n atomic variables determine 2^{n} different possible state-descriptions and 2^{2n} different possible change-descriptions (cf. Ch. II Sect. 10). An act-description is obtained from a given change-description through the insertion of the letter d or the letter f in front of each of the n T-expressions in the change-description. The insertion can take place in 2^{n} different ways. Consequently the total number of act-descriptions which are determined by n atomic variables is 2^{n}×2^{2n} or 2^{3n}.
(pTp) & (qT ~q) is a change-description. To it answer four act-descriptions viz. d(pTp) & d(qT ~q) and d(pTp) & f(qT ~q) and f(pTp) & d(qT ~q) and f(pTp) & f(qT ~q).
Given n atomic variables we can list in a table the 2^{n} state-descriptions the 2^{2n} change-descriptions and the 2^{3n} act-descriptions which these variables determine. On the next page there is a fragment of such a list for the case of two variables p and q.
The positive normal form of a df-expression which contains n variables for states of affairs is a disjunction-sentence of (none or) one or two or… or 2^{3n} conjunction-sentences of n elementary d- and/or f-expressions. If the disjunction-sentence has no members the df-expression expresses a df-contradiction. If it has 2^{3n} members the df-expression expresses a df-tautology.
It is often convenient to regard the positive normal form of a df-expression as consisting of ‘bits’ or segments answering to the various conditions (change-descriptions) which constitute opportunities for doing the act in question. Thus for example the 15-termed disjunction-sentence which is the positive normal form of the expression d(pTp) ∨ d(qTq) (Section 4) may become divided into the following seven ‘bits’:
State-descriptions | Change-descriptions | Act-descriptions |
p & q | (pTp) & (qTq) |
d(pTp) & d(qTq) d(pTp) & f(qTq) f(pTp) & d(qTq) f(pTp) & f(qTq) |
p & q | (pTp) & (qT ~q) | |
p & q | (pT ~p) & (qTq) | |
p & q | (pT ~p) & (qT ~q) | |
4. ~p & ~q | ( ~pT ~p) & ( ~qT ~q) | |
4. ~p & ~q | ( ~pT ~p) & ( ~qTq) | |
4. ~p & ~q | ( ~pTp) & ( ~qT ~q) | |
4. ~p & ~q | 16. ( ~pTp) & ( ~qTq) |
d( ~pTp) & d( ~qTq) d( ~pTp) & f( ~qTq) f( ~pTp) & d( ~qTq) 64. f( ~pTp) & f( ~qTq) |
d(pT ~p) & d(qTq) ∨ f(pT ~p) & d(qTq) answering to (pTp) & (qT ~q); d(pTp) & d(qT ~q) ∨ d(pTp) & f(qT ~q) answering to (pT ~p) & (qTq); d(pTp) & d(qTq) ∨ d(pTp) & f(qTq) & f(pTp) & d(qTq) answering to (pT ~p) & (qT ~q); d(pTp) & d( ~qTq) ∨ d(pTp) & f( ~qTq) answering to (pT ~p) & ( ~qT ~q); d(pTp) & d( ~qT ~q) ∨ d(pTp) & f( ~qT ~q) answering to (pT ~p) & ( ~qTq); d( ~pTp) & d(qTq) ∨ f( ~pTp) & d(qTq) answering to ( ~pT ~p) & (qT ~q); and d( ~pT ~p) & d(qTq) ∨ f( ~pT ~p) & d(qTq) answering to ( ~pTp) & (qT ~q).
6. We shall distinguish between the external and the internal negation of a df-expression.
External negation is negation in the ‘ordinary’ sense. Its symbol is ~. If the positive normal form of a given df-expression has m members (conjunction-sentences) then the positive normal form of the external negation of this df-expression has 2^{3n}—m members n being the number of atomic variables of the expression. Thus for example the positive normal form of ~(d(pTp) ∨ d(qTq)) is a disjunction-sentence of 49 i.e. of 64–15 conjunction-sentences of two elementary df-expressions. It is readily seen that this normal form has 16 ‘bits’ of which the shortest is f(pTp) & f(qTq). The other segments are either 2-termed or 4-termed disjunction-sentences (of conjunction-sentences of two elementary df-expressions).
The internal negation of a given df-expression is obtained as follows: The expression is transformed into its positive normal form and the normal form is divided up into segments. We form the disjunction-sentence of all those conjunction-sentences (of elementary df-expressions of the same atomic variables) which do not occur in the segments but answer to the same conditions for acting (change-descriptions) as the conjunction-sentences in the segments. The expression thus formed is the (positive normal form of the) internal negation of the given df-expression.
For example: The internal negation of d(pTp) ∨ d(qTq) is the 13-termed disjunction-sentence d(pT ~p) & f(qTq) f(pT ~p) & f(qTq) ∨ f(pTp) & d(qT ~q) ∨ f(pTp) & f(qT ~q) ∨ f(pTp) & f(qTq) ∨ f(pTp) & d( ~qTq) & f(pTp) & f( ~qTq) ∨ f(pTp) & d( ~qT ~q) ∨ f(pTp) & f( ~qT ~q) ∨ d( ~pTp) & f(qTq) ∨ f( ~pTp) & f(qTq) ∨ d( ~pT ~p) & f(qTq) & f( ~pT ~p) & f(qTq).
The internal negation of d(pTp) & d(qTq) is the 3-termed disjunction-sentence d(pTp) & f(qTq) & f(pTp) & d(qTq) ∨ f(pTp) & f(qTq). Its external negation is (in the normal form) a 63-termed disjunction-sentence.
The internal negation of d(pTp) is f(pTp). Generally speaking: the internal negation of doing is forbearing.
The external negation of d(pTp) is in the normal form the 7-termed disjunction-sentence d(pT ~p) ∨ d( ~pTp) ∨ d( ~pT ~p) ∨ f(pTp) ∨ f(pT ~p) ∨ f( ~pTp) ∨ f( ~pT ~p).
The external negation says that the action described by the expression in question is not done (by the agent in question on the occasion in question). The internal negation says that under the same conditions of action the ‘opposite’ of the action described by the expression in question is done (by the agent in question on the occasion in question).
An action and its external negation are incompatible (modes of action). This means: they cannot both be performed by the same agent on the same occasion. An action and its internal negation are also incompatible.
We can distinguish between external and internal incompatibility of actions (and of expressions for action). Two actions will be called externally incompatible when the proposition that the one has been performed (by some agent on some occasion) entails the proposition that the external negation of the other has been performed (by the same agent on the same occasion). Two actions will be called internally incompatible when the proposition that the one has been performed entails the proposition that the internal negation of the other has been performed.
For example: The actions described by d(pTp) & d(qTq) and by d(pT ~p) & d(qT ~q) are externally incompatible. The actions described by d(pTp) & d(qTq) and d(pTp) & f(qTq) are internally incompatible. Also: the actions described by d(pTp) and f(pT ~p) are externally the actions described by d(pTp) and f(pTp) internally incompatible.
It is readily seen that internal incompatibility entails external incompatibility but not vice versa.
The notions of external and internal incompatibility can be generalized so as to become applicable to any number n of actions (and of descriptions of actions).
n actions are externally incompatible when they cannot be all performed by the same agent on the same occasion n actions are internally incompatible when they are externally incompatible and the conditions under which each of them can be performed are the same.
Speaking of descriptions of action we can say that n df-expressions are externally incompatible when their conjunction is a df-contradiction. They are internally incompatible when they are externally incompatible and answer to the same change-descriptions.
Three or more actions can be (externally or internally) incompatible even though no two of them are incompatible. An example would be the three actions described by d(pTp) & d(qTq) ∨ d(pTp) & f(qTq) and d(pTp) & f(qTq) ∨ f(pTp) & d(qTq) and d(pTp) & d(qTq) ∨ f(pTp) & d(qTq). Their incompatibility moreover is internal since the condition under which each of them can be performed is the same viz. (pT ~p) & (qT ~q).
7. We shall also distinguish between the external and the internal consequences of (the proposition expressed by) a given df-expression.
A df-expression entails (in the Logic of Action) another df-expression if and only if the implication-sentence whose antecedent is the first and whose consequent is the second df-expression is a df-tautology. When a df-expression entails another the second is called an external consequence of the first.
For example: d(pTp) & d(qTq) entails d(pTp) & d(qTq) ∨ d(pT ~p) & d(qT ∨q). ‘If a person on some occasion continues both of two states then trivially he either continues them both or destroys them both.’ This entailment is valid already by virtue of the laws of the Logic of Propositions.
A df-expression is an internal consequence of another df-expression if and only if the first is a (external) consequence of the second and the two expressions answer to the same change-description (conditions of action).
For example: d(pTp) & d(qTq) & f(pTp) & f(qTq) is an internal consequence of d(pTp) & d(qTq). ‘If an agent on some occasion continues both of two states then trivially he either continues both or lets both vanish.’
8. Two or more df-expressions which contain exactly the same variables for states of affairs will be called uniform with regard to the variables. Expressions which are not uniform can be made uniform by a vacuous introduction of new variables into them.
If e.g. the variable p does not occur in a given df-expression we can introduce it into the expression by forming the conjunction-sentence of the given df-expression and e.g. the df-expression d(pTp) ∨ ~d(pTp). In a similar manner the variable p can be introduced into a given T-expression by conjoining the expression with (pTp) ∨ ~(pTp) and into a given p-expression by conjoining it with p ∨ ~p.
Consider the T-expression pTp. If we want to introduce the variable q into it we can form the conjunction-sentence (pTp) & (qTq ∨ ~(qTq)) or the conjunction-sentence (pTp) & (qTq ∨ qT ~q & ~qTq & ~qT ~q). But we can achieve the same by replacing p in the original expression by the conjunction-sentence p & (q ∨ ~q). The reader can easily satisfy himself that the two operations lead to the same result i.e. that after the appropriate transformations we reach in the end the same T-expression. Because of this fact we say that T-expressions are extensional with regard to p-expressions. This means generally speaking that if for some p-expression which occurs in a T-expression we substitute a (in the p-calculus) tautologously equivalent p-expression the new T-expression which we get through the substitution is (in the T-calculus) tautologously equivalent to the original T-expression.
df-expressions be it observed are not extensional with regard to p-expressions nor with regard to T-expressions. If for some p-expression which occurs in a df-expression we substitute a (in the p-calculus) tautologously equivalent p-expression the new df-expression is not necessarily (in the df-calculus) tautologously equivalent to the first. And similarly if for some T-expression which occurs in a df-expression we substitute a (in the T-calculus) tautologously equivalent T-expression. In the said respect df-expressions may be said to be intensional and the df-calculus may be called an intensional calculus.
Consider some elementary df-expression e.g. d(pTp). As known from the Logic of Propositions p is tautologously equivalent to p & q ∨ p & ~q. Consider now the atomic d-expression d((p & q ∨ p & ~q) T (p & q ∨ p & ~q)). According to the laws of the Logic of Change (p ∨ q ∨ p & ~q) T (p & q ∨ p & ~q) is tautologously equivalent to pTp & (qTq ∨ qT ~q ∨ ~qTq ∨ ~qT ~q). Consider next the atomic d-expression d(pTp & (qTq ∨ qT ~q ∨ ~qTq ∨ ~qT ~q)). According to the Logic of Action this is tautologously equivalent to the molecular d-expression d(pTp) & d(qTq ∨ qT ~q ∨ ~qTq ∨ ~qT ~q) which in its turn is equivalent to d(pTp) & (d(qTq) ∨ d(qT ~q) ∨ d( ~qTq) ∨ d( ~qT ~q)).
Let us compare the first and the last of our above d-expressions. Do the two mean the same? The first says that a certain agent on a certain occasion through his action preserves a certain state of affairs e.g. keeps a certain door open. The second says that a certain agent on a certain occasion does this same thing and also another thing in addition to it. This additional thing is that he through his action either preserves or destroys or produces or suppresses a certain state of affairs e.g. the state of affairs that a car is parked in the front of his house. It is plain that even if it were (which it need not be) possible for the agent to do the first thing and one of the mutually exclusive four other things on one and the same occasion it is not necessary that he should do any of the four other things on an occasion when he does the first. Hence the meaning of d(pTp) is not the same as the meaning of d(pTp) & (d(qTq) ∨ d(qT ~q) ∨ d( ~qTq) ∨ d( ~qT ~q)).
That the two meanings must be different is not at all difficult to understand. The disjunction of changes described by qTq ∨ qT ~q ∨ ~qTq ∨ ~qT ~q is a tautology something which necessarily happens on any occasion. But neither the disjunctive act described by d(qTq ∨ qT ~q ∨ ~qTq ∨ ~qT ~q) nor the equivalent disjunction of acts described by d(qTq) ∨ d(qT ~q) ∨ d( ~qTq) ∨ d( ~qT ~q) is a tautology i.e. something which will necessarily be done on every occasion. If for example an agent forbears to do one of the four acts then he does not do any of them. And if for some reason or other he cannot do any of them then he neither does nor forbears any of them on a given occasion.
Though it is easy to see that the two expressions have different meanings it may yet appear as something of a paradox that there should be this difference—considering how the two expressions are related to each other ‘formally’. We reached the last from the first through a series of substitutions of tautologously equivalent expressions and of a series of transformations of expressions into tautologously equivalent forms. We have no reason to deny or to doubt any of these equivalences. What we have to do then is to reject some of the substitutions (as not leading from one expression to another which is tautologously equivalent to the first). The substitution which we reject is the first. The act described by d(pTp) is not the same as the act described by d((p & q ∨ p & ~q) T(p & q & p & ~q))—although the change described by (p & q ∨ p & ~q) T (p & q ∨ p & ~q) is the same as the change described by pTp and the state described by p & q ∨ p & ~q is the same as the state described by p.
When df-expressions are uniform with regard to the variables and in the positive normal form it can instantly be seen from the ‘look’ of the expressions whether they are compatible or not. They are compatible if and only if the normal forms have at least one disjunct in common.
When df-expressions are uniform with regard to the variables and in the positive normal form it can also instantly be seen from the ‘look’ of the expressions whether the one entails (or is a consequence of) the other. The one entails the other if and only if the normal form of the first is a part of the normal form of the other.