1. The author became interested in the logic of norms and normative concepts (also called ‘deontic logic’) through the observation that the notions of ‘ought to’ ‘may’ and ‘must not’ exhibit a striking analogy to the modal notions of necessity possibility and impossibility. His interest in modal logic again had been awakened by the observation that its basic concepts show an analogy to the basic concepts of socalled quantificationtheory the notions of ‘all’ ‘some’ and ‘none’.
Familiarity on the part of the reader with the techniques of modal logic and quantificationtheory is however neither presupposed nor needed for understanding the arguments in this book.
Modal logic and quantificationtheory may be said to rest on a more elementary branch of logical theory socalled propositional logic. The orthodox logical techniques used in this work nearly all belong to this elementary theory. We shall in the next two sections briefly recapitulate its fundamentals. This recapitulation however is too summary to give anyone who is not already familiar with the subject a working knowledge of its techniques.
By the ‘techniques’ of propositional logic I mean principally the construction of socalled truthtables and the transformation of expressions into socalled normal forms. These techniques are described in any uptodate textbook on (mathematical or symbolic) logic.
2. The objects which propositional logic studies are usually called by logicians and philosophers propositions.
Propositions may be said to have two ‘counterparts’ in language. One of these is (indicative) sentences. An example would be the sentence ‘London is the capital of England’. Sentences express propositions. Propositions can be called the meaning or sense of sentences.
The second linguistic counterpart of propositions is thatclauses. A thatclause in English consists of the word ‘that’ followed by a sentence. For example ‘that London is the capital of England’ is a thatclause. Thatclauses have the character of names of propositions. Propositions can be called the reference of thatclauses.
Names of propositions must not be confused with names of sentences. A conventional way of naming a sentence is to enclose (a token of) this sentence within quotes. This method we used above when we gave an example of a sentence.
When we speak about sentences and propositions we have to refer to them by means of their names. Thus for example when we say that the German sentence ‘London ist die Hauptstadt Englands’ expresses the proposition that London is the capital of England. Instead of the phrase ‘expresses the proposition’ we could also have used the word ‘means’.
By expressions or formulae of propositional logic we understand certain (linguistic) structures which are built up of two kinds of signs called variables and constants. As variables we shall use lowercase letters p q r etc. The constants which we use are the signs ~ & v → and ↔. The formulae we also call pexpressions. They are defined recursively as follows:

(i) Any variable is a formula.

(ii) Any formula preceded by ~ is a formula; any two formulae joined by & ∨ → or ↔ is a formula.
The variables themselves we also call atomic formulae. A formula which is not atomic is called molecular or is said to be a molecular complex or compound of atomic formulae.
For the building up of molecular formulae as we do it here brackets are needed. For our use of brackets we adopt the convention that the sign & has a stronger binding force than v → and ↔; the sign ∨ than → and ↔; and the sign ↔ than ↔. Thus for example we can instead of: (((p & q) ∨ r) → s) ↔ t write simply: p & q ∨ r → s ↔ t.
(Brackets are a third kind of signs of propositional logic and should be mentioned in a full recursive definition of formulae. They are however signs of a ‘subsidiary’ nature. Under a different way from ours of defining the formulae one can dispense with the use of brackets altogether.)
We shall have to think of the letters p q r etc. in expressions of propositional logic as standing for or representing (arbitrary) sentences which express propositions. The pexpressions could be called sentence—schemas. What the techniques of propositional logic literally ‘handle’ are thus schemas for arbitrary sentences and their compounds. This is perhaps a reason why some logicians prefer to call propositional logic by the name ‘sentential logic’ or ‘sentential calculus’. We shall sometimes call it by the name pcalculus.
3. An important point of view from which socalled ‘classical’ propositional logic studies its objects propositions is the truthfunctional point of view.
In classical propositional logic truth and falsehood are the two truthvalues. It is assumed that every proposition has one and one only truthvalue. If there are n logically independent propositions there are evidently 2^{n} possible ways in which they can be true and/or false together. Any such distribution of truthvalues over the n propositions will be called a truthcombination.
If the truthvalue of one proposition is uniquely determined for every possible truthcombination in some n propositions then the first proposition is called a truthfunction of the n propositions. It is not difficult to calculate that there exist in all 2^{(2n)} different truth functions of n logically independent propositions.
The following truthfunctions are of special interest to us:
The negation of a given proposition (is the truthfunction of it which) is true if and only if the given proposition is false. If p expresses a proposition then ~p will by convention express the negation of this proposition. ~ is called the negationsign.
The conjunction of two propositions (is the truthfunction of them which) is true if and only if both propositions are true. If p and q express propositions p & q expresses their conjunction. & is called the conjunctionsign.
The disjunction of two propositions is true if and only if at least one of the propositions is true. If p and q express propositions p ∨ q expresses their disjunction ∨ is called the disjunctionsign.
The (material) implication of a first proposition called the antecedent and a second proposition called the consequent is true if and only if it is not the case that the first is true and the second false. If p and q express propositions p → q expresses their implication.
The (material) equivalence of two propositions is true if and only if both propositions are true or both false. If p and q express propositions p ↔ q expresses their equivalence.
The tautology of n propositions is the truthfunction of them which is true for all possible truthcombinations in those n propositions. The tautology has no special symbol.
The contradiction of n propositions is the truthfunction of them which is false for all possible truthcombinations in those n propositions. Like the tautology the contradiction has no special symbol.
Truthfunctionship is transitive. If a proposition is a truthfunction of a set of propositions and if every member of the set is a truthfunction of a second set of propositions then the first proposition too is a truthfunction of the second set of propositions.
Thanks to the transitivity of truthfunctionship every formula of propositional logic or pexpression expresses a truthfunction of the propositions expressed by its atomic constituents. Which truthfunction of its atomic constituents a given pexpression expresses can be calculated (decided) in a socalled truthtable. The technique of constructing truthtables is assumed to be familiar to the reader.
Two formulae f_{1} and f_{2} are called tautologously equivalent if the formula f_{1} ↔ f_{2} expresses the tautology of its atomic constituents.1
The formulae f and ~ ~f are tautologously equivalent. That this is the case is called the Law of Double Negation. ‘Double negation cancels itself.’
The formulae ~(f_{1} & f_{2}) and ~ f_{1} ∨ ~ f_{2} are tautologously equivalent and so are the formulae ~(f_{1} ∨ f_{2}) and ~ f_{1} & f_{2}. These are the Laws of de Morgan. The first says that the negation of a conjunction of propositions is tautologously equivalent to the disjunction of the negations of the propositions. The second says that the negation of a disjunction of propositions is tautologously equivalent to the conjunction of the negations of the propositions.
Conjunction and disjunction are associative and commutative. Thanks to their associative character the truthfunctions can be generalized so that one can speak of the conjunction and disjunction of any arbitrary number n of propositions.
The formulae f_{1} & (f_{2} ∨ f_{3}) and f_{1} & f_{2} ∨ f_{1} & f_{3} are tautologously equivalent and so also the formulae f_{1} ∨ f_{2} & f_{3} and (f_{1} ∨ f_{2}) & (f_{1} ∨ f_{3}). These are called Laws of Distribution.
The formula f_{1} → f_{2} is tautologously equivalent to ~ f_{1} ∨ f_{2} and also to ~(f_{1} & f_{2}). The formula f_{1} ↔ f_{2} again is tautologously equivalent to f_{1} & f_{2} ∨ ~f_{1} & f_{2}. These equivalences may be said to show that implication and equivalence is definable in terms of negation conjunction and disjunction.
Formulae may become ‘expanded’ or ‘contracted’ in accordance with the laws that a formula f is tautologously equivalent to the formulae f & f and f ∨ f and f & (g ∨ ~g) and f ∨ g & ~g.
Thanks to these equivalences and the transitivity of truthfunctionship every formula of propositional logic may be shown to possess certain socalled normal forms. A normal form of a given formula is another formula which is tautologously equivalent to the first and which satisfies certain ‘structural’ conditions. Of particular importance are the (perfect) disjunctive and the (perfect) conjunctive normal forms of formulae. The techniques of finding the normal forms of given formulae are assumed to be familiar to the reader.
Given n atomic formulae one can form 2^{n} different conjunctionformulae such that every one of the atomic formulae or its negationformula is a constituent in the conjunction. (Conjunctionformulae which differ only in the order of their constituents e.g. p & ~q and ~q & p are here regarded as the same formula.)
It is easily understood in which sense these 2^{n} different conjunctionformulae may be said to ‘correspond’ to the 2^{n} different truthcombinations in the propositions expressed by the atomic formulae. The conjunctionformulae are sometimes called statedescriptions. The conjunctions themselves can be called possible worlds (in the ‘field’ or ‘space’ of the propositions expressed by the atomic formulae).
The (perfect) disjunctive normal form of a formula is a disjunction of (none or) some or all of the statedescriptions formed of its atomic constituents. If it is the disjunction of them all the formula expresses the tautology of the propositions expressed by its atomic constituents. This illustrates a sense in which a tautology can be said to be true in all possible worlds. If again the disjunctive normal form is Otermed the formula expresses the contradiction of the propositions expressed by its atomic constituents. A con tradiction is true in no possible world. Propositions which are true in some possible world(s) but not in all are called contingent.
Sentences which express contingent propositions we shall call descriptive or declarative sentences.2
4. What is a proposition?—An attempt to answer this question in a satisfactory way would take us out on deep waters in philosophy. Therefore we shall confine ourselves to a few scattered observations only. In the first place I should like to show that the term ‘proposition’ as commonly used by logicians and philosophers covers a number of different entities which for the specific purposes of the present study we have reason to distinguish.
Someone may wish to instance that it is raining as an example of a proposition. Or that Chicago has more inhabitants than Los Angeles. Or that Brutus killed Caesar.
Is it not the case that the proposition that it is raining has one and one only truthvalue? Surely someone may say it must be either raining or not raining and cannot be both. But of course it can be raining in London today but not tomorrow; and it can be raining today in London but not in Madrid; and it can today be raining and not raining in London viz. raining in the morning but not in the afternoon. So in a sense it is quite untrue to say that the proposition that it is raining has one and one only truthvalue or to say that it cannot be both raining and not raining.
When we insist that it cannot be both raining and not raining we mean: raining and not raining at the same place and time. Or as I shall prefer to express myself: on one and the same occasion. But a proposition may be true on one occasion and false on another.
These observations give us a reason for making a distinction between generic and individual propositions. The individual proposition has a uniquely determined truthvalue; it is either true or false but not both. The generic proposition has by itself no truthvalue. It has a truthvalue only when coupled with an occasion for its truth or falsehood; that is when it becomes ‘instantiated’ in an individual proposition.
We cannot here discuss in detail the important notion of an occasion. It is related to the notions of space and time. It would not be right however to identify occasions with ‘instants’ or ‘points’ in space and time. They should rather be called spatiotemporal locations. Two occasions will be said to be successive (in time) if and only if the first occasion comes to an end (in time) at the very point (in time) where the second begins.
Occasions are the ‘individualizes’ of generic propositions. Their logical role in this regard is related to old philosophic ideas of space and time as the principia individuationis.
Occasions must not be confused with (logical) individuals. Individuals could be called ‘thinglike’ logical entities. Not all logical individuals however are called ‘things’ in ordinary parlance. ‘London’ and ‘the author of Waverley’ refer to individuals; but neither a city nor a person is it natural to call a thing. The counterparts of individuals in language are proper names and socalled definite descriptions (uniquely descriptive phrases).
When a sentence which expresses a proposition contains proper names and/or definite descriptions the corresponding logical individuals we shall say are constituents of the expressed proposition. But the occasion for a proposition's truth or falsehood we shall not call a constituent of the proposition.
It should be observed that it is not the occurrence of individuals among its constituents which decides whether a proposition is generic or individual. That Brutus killed Caesar is an individual proposition. But this is not so because of the fact that the proposition is about the individuals Brutus and Caesar; it is due to the logical nature of the concept (universal) of being killed. A person can be killed only once on one occasion. That Brutus kissed Caesar is not an individual proposition. This is so because a person can be kissed by another on more than one occasion.
It may be suggested that only generic propositions among the constituents of which there are no logical individuals are eminently or fully generic. Generic propositions among the constituents of which there are individuals might then be called semigeneric or semiindividual. A further suggestion might be that semigeneric propositions ‘originate’ from fully generic propositions by a process of substituting for some universal in the generic proposition some individual which falls under that universal. But we need not discuss these questions here.
The relation of universal to logical individual must be distinguished from the relation of generic proposition to individual proposition. But the two relations though distinct are also related.
Sometimes there are intrinsic connexions between a logical individual and the spatiotemporal features which constitute an occasion for a proposition's truth or falsehood. The individuals to which geographical names refer have a fixed location on the surface of the earth. The proposition that Paris is bigger than New York is false now but was true two hundred years ago. The occasion on which the proposition is true or false has only the temporal dimension. This is so because the individuals which are constituents of the proposition have intrinsically a fixed spatial location. If individually the same town could move from one country to another it might be true to say that Paris was bigger than New York at the time when the former was situated in China. As things are logically to say this does not even make sense.
The distinction which we are here making between individual and generic propositions must not be confused with the wellknown distinction between singular or particular propositions on the one hand and universal or general propositions on the other hand. As far as I can see the division of propositions into individual and generic applies only to particular propositions. General propositions such as e.g. that all ravens are black or that water has its maximum density at 4° C have a determined truthvalue but are not instantiations in the sense here considered of some generic propositions. There are no ‘occasions’ for the truth or falsehood of general propositions. Such propositions are therefore also as has often been noted in a characteristic way independent of time and space.
To propositional logic in the traditional sense it is not an urgent problem whether we should conceive of its objects of study propositions as generic or individual. It is perhaps true to say that primarily propositional logic is a formal study of individual (particular) propositions. If we conceive of its objects as generic propositions we must supplement such statements as that no proposition is both true and false by a (explicit or tacit) reference to one and the same occasion. And we must bear in mind that it is only via the notion of an occasion that the notion of truth and of truthfunction reaches generic propositions.
For the formal investigations which we are going to conduct in the present work the distinction between individual and generic propositions is of relevance. We shall here have to understand the variables p q etc. of propositional logic as schematic representations of sentences which express generic propositions. Thus for example we could think of p as the sentence ‘The window is open’ but not as the sentence ‘Brutus killed Caesar’. A further restriction on the interpretation of the variables will be introduced in the next section.
5. When a (contingent) proposition is true there corresponds to it a fact in the world. It is a wellknown view that truth ‘consists’ in a correspondence between proposition and fact.
There are several types of fact. Here we shall distinguish three types:
Consider the propositions (true at the time when this was written) that the population of England is bigger than that of France and that my typewriter is standing on my writingdesk. The facts which answer to these propositions and make them true we commonly also call states of affairs.
Consider the proposition that it is raining at a certain place and time. Is the fact which would make this proposition true rainfall or the falling of rain also a state of affairs? We sometimes call it by that name. But the falling of rain is a rather different sort of state of affairs from my typewriter's standing on my writingdesk. One could hint at the difference with the words ‘dynamic’ and ‘static’. Rainfall is something which ‘goes on’ ‘happens’ over a certain period of time. Rainfall is a process; but my typewriter's being or standing on my writingdesk we would not in ordinary speech call a process.
Consider the proposition that Brutus killed Caesar. The corresponding fact nobody—with the possible exception of some philosophers—would call by the name ‘state of affairs’. Nor would we call it ‘process’ although processes certainly were involved in the fact e.g. Brutus's movements when he stabbed Caesar and Caesar's falling to the ground and his uttering of the famous words. The type of fact which Caesar's death exemplifies is ordinarily called an event. Like processes events are facts which happen. But unlike the happening of processes the happening of events is a taking place and not a going on.
The three types of fact which we have distinguished are thus: states of affairs processes and events. It is not maintained that the three types which we have distinguished are exhaustive of the category of facts. The truth of general propositions raises special problems which we shall not discuss here at all.
Just as we can distinguish between generic and individual propositions so we can distinguish between generic and individual states of affairs processes and events. Whether we should also distinguish between generic and individual facts is a question which I shall not discuss. Someone may wish to defend the view that facts are necessarily individual states of affairs processes and events.
Rainfall is a generic process of which the falling of rain at a certain place and time is an instantiation. Dying is a generic event of which e.g. Caesar's death is an instantiation. The superiority with regard to population of one country over another is a generic state of affairs of which the present superiority with regard to population of England over France is an instantiation. But in the past the relative size of the populations of the two countries was the reverse. Thus there is also a generic or semigeneric state of affairs viz. the superiority with regard to population of England over France which is instantiated in the present situation.
A sentence which expresses a contingently true proposition will be said to describe the fact which makes this proposition true. (Cf. above p. 22 on the term ‘descriptive sentence’.) Thus e.g. the sentence ‘Caesar was murdered by Brutus’ describes a fact.
Facts can also be named. The name of a fact is a substantiveclause such as e.g. ‘Caesar's death’ or ‘the present superiority with regard to population of England over France’. One also speaks of the fact that e.g. Caesar was murdered by Brutus. This may be regarded as an abbreviated way of saying that the proposition that Caesar was murdered by Brutus is true (‘true to fact’). The phrase ‘that Caesar was murdered by Brutus’ names a proposition. (Cf. above p. 18.)
Even if we do not want to distinguish between individual and generic facts it seems appropriate and natural to say that sentences which express contingent generic propositions describe generic states of affairs or processes or events. Thus e.g. the sentence ‘It is raining’ can be said to describe a generic process the name of which is ‘rainfall’.
To propositional logic as such it makes no difference whether we think of the truemaking facts of propositions as states of affairs or processes or events. But to the study of deontic logic these distinctions are relevant. This is so because of the paramount position which the concept of an act holds in this logic.
We have already stipulated that the variables p q etc. should be understood as schematic representations of sentences which express generic propositions. We now add to this the stipulation that the sentences thus represented should describe generic states of affairs.
6. The three types of fact (and correspondingly of proposition) which we have distinguished are not logically independent of one another.
We shall not here discuss the question how processes are related to events and to states of affairs. Be it only observed that the beginning and the end (stopping) of a process may be regarded as events.
There is a main type of event which can be regarded as an ordered pair of two states of affairs. The ordering relation is a relation between two occasions which are successive in time. We shall not here discuss the nature of this relation in further detail. Simplifying we shall speak of the two occasions as the earlier and the later occasion. The event ‘itself’ is the change or transition from the state of affairs which obtains on the earlier occasion to the state which obtains on the later occasion. We shall call the first the initial state and the second the endstate.
The event for example which we call the opening of a window consists in a change or transition from a state of affairs when this window is closed to a state when it is open. We can also speak of the event as a transformation of the first state to the second. Alternatively we can speak of it as a transformation of a world in which the initial state obtains or which contains the initial state into a world in which the endstate obtains or which contains the endstate. Such transformations will also be called statetransformations.
Sometimes an event is a transition not from one state to another state but from a state to a process (which begins) or from a process (which ceases) to a state. Sometimes an event is a transition from one process to another process. Sometimes finally it is a transition from one ‘state’ of a process to another ‘state’ of the same process—e.g. from quicker to slower or from louder to weaker.
Events of these more complicated types we shall in general not be considering in this inquiry. ‘Event’ will unless otherwise expressly stated always mean the transition from a state of affairs on a certain occasion to a state of affairs (not necessarily a different one) on the next occasion. If the occasion is specified the event is an individual event; if the occasion is unspecified the event is generic.
7. We introduce a symbol of the general form T where the blanks to the left and to the right of the letter T are filled by pexpressions. The symbol is a schematic representation of sentences which describe (generic) events. The event described by pTq is a transformation of or transition from a certain initial state to an endstate viz. from the (generic) state of affairs described by p to the (generic) state of affairs described by q. Or as we could also put it: pTq describes the transformation of or transition from a pworld to a qworld. The states of affairs will also be called ‘features’ of the worlds.
We shall call expressions of the type T atomic Texpressions. We can form molecular compounds of them. By a Texpression we shall understand an atomic Texpression or a molecular compound of atomic Texpressions.
Texpressions may be handled in accordance with the rules of the pcalculus (propositional logic). As will be seen there also exist special rules for the handling of Texpressions. The rules for handling Texpressions we shall say define the Tcalculus.
Let p mean that a certain window is open. ~p then means that this same window is closed (=not open). ~pTp again means that the window is being opened strictly speaking: that a world in which this window is closed changes or is transformed into a world in which this window is open. Similarly pT ~p means that the window is being closed (is closing). We could also say that ~pTp describes the event called ‘the opening of the window’ and that pT ~p describes the event named ‘the closing of the window’.
Consider the meaning of pTp. The letter to the left and that to the right of T describe the same generic state of affairs. The occasions on which this generic state is thought to obtain are successive in time. Hence pTp expresses that the state of affairs described by p obtains on both occasions irrespective of how the world may have otherwise changed from the one occasion to the other. In other words: pTp means that the world remains unchanged in the feature described by p on both occasions. It is a useful generalization to call this too an ‘event’ or a ‘transformation’ although it strictly speaking is a ‘notevent’ or a ‘nottransformation’.
In a similar manner ~pT ~p means that the world remains unchanged in the generic feature described by ~p on two successive occasions.
Again let p mean that a certain window is open. pTp then means that this window remains open and ~pT ~p that it remains closed on two successive occasions.
We shall call the events or statetransformations described by pTp pT ~p ~pTp and ~pT ~p the four elementary (state) transformations which are possible with regard to a given (generic) state of affairs or feature of the world. The four transformations be it observed are mutually exclusive; no two of them can happen on the same pair of successive occasions. The four transformations moreover are jointly exhaustive. On a given occasion the world either has the feature described by p or it lacks it; if it has this feature it will on the next occasion either have retained or lost it; if again it lacks this feature it will on the next occasion either have acquired it or still lack it.
By an elementary Texpression we understand an atomic Texpression in which the letter to the left of T is either an atomic pexpression or an atomics pexpression preceded by the negationsign and the letter to the right of T is this same atomic pexpression either with or without the negationsign before itself.
8. We shall in this section briefly describe how every statetransformation—strictly speaking: proposition to the effect that a certain change or event takes place—may be regarded as a truthfunction of elementary statetransformations.
Consider the meaning of pTq. A pworld changes to a qworld. p and q let us imagine describe logically independent features of the two worlds. The pworld either has or lacks the feature described by q. It is in other words either a p & qworld or a p & ~qworld. Similarly the qworld is either a p & qworld or a ~p & qworld. The event or transformation described by pTq is thus obviously the same as the one described by (p & q ∨ p & ~q) T(p & q ∨ ~p & q).
Assume that the pworld is a p & qworld and that the qworld is a p & qworld too. Then the transition from the initial state to the endstate involves no change at all of the world in the two features described by p and q respectively. The schematic description of this transformation is (p & q) T(p & q) and the transformation thus described is obviously the same as the conjunction of the two elementary transformations described by pTp and qTq.
Assume that the pworld is a p & qworld and that the qworld is a ~p & qworld. Then the transition from the initial state to the endstate involves a change from ‘positive’ to ‘privative’ in the feature described by p. The transformation described by (p & q) T( ~p & q) is obviously the same as the conjunction of the elementary transformations described by pT ~p and qTq.
Assume that the pworld is a p & ~qworld and the qworld a p & qworld. The world now changes from being a ~qworld to being a qworld but remains unchanged as pworld. The transformation described by (p & ~q) T(p & q) is the conjunction of the elementary transformations described by pTp and ~qTq.
Assume finally that the pworld is a p & ~qworld and the qworld a ~p & qworld. The world now changes from pworld to ~pworld and from ~qworld to qworld. The transformation described by (p & ~q) T( ~p & q) is the conjunction of the elementary transformations described by pT ~p and ~qTq.
Thus the atomic Texpression pTq is identical in meaning with the following disjunctionsentence of conjunctionsentences of elementary Texpressions:
(pTp) & (qTq) ∨ (pT ~p) & (qTq) ∨ (pTp) & ( ~qTq) ∨ (pT ~p) & ( ~qTq).
From the example which we have been discussing it should be plain that every atomic Texpression can become transformed into a molecular complex (disjunctionsentence of conjunctionsentences) of elementary Texpressions. Thus every atomic Texpression expresses a truthfunction of elementary statetransformations. Since truthfunctionship is transitive it follows that every molecular complex too of atomic Texpressions expresses a truthfunction of elementary statetransformations.
Consider an arbitrary Texpression. We replace its (notelementary) atomic constituents by disjunctionsentences of conjunctionsentences of elementary Texpressions. The original Texpression has thus become transformed into a molecular complex of elementary Texpressions. These last will be called the Tconstituents of the original Texpression.
It follows from what has been said that every Texpression expresses a truthfunction of (the propositions expressed by) its Tconstituents. Which truthfunction it expresses can be investigated and decided in a truthtable. This truthtable differs from an ‘ordinary’ truthtable of propositional logic only in the feature that certain combinations of truthvalues are excluded from it. The excluded combinations are those and only those which would conflict with the principle that of the four elementary Texpressions which answer to a given atomic pexpression no two must be assigned the value ‘true’ and not all may be assigned the value ‘false’.
If a Texpression expresses the tautology of its Tconstituents we shall call (the proposition expressed by) it a Ttautology. An example of a Ttautology is (pTp) ∨ (pT ~p) ∨ ( ~pTp) ∨ ( ~pT ~p).
The negation of a Ttautology is a Tcontradiction. An example of a Tcontradiction is (pTp) & (pT ~p). It follows that ~(pTp) ∨ ~(pT ~p) is a Ttautology.
We consider finally some special formulae. The first is (p ∨ ~p) Tp. Its normal form is (pTp) ∨ (~pTp). The formula in other words expresses a true proposition if and only if on the later of two successive occasions the world has the feature described by p independently of whether it had this feature or lacked it on the earlier of the two occasions.
The second is (p ∨ ~p) T(p ∨ ~p). It is a Ttautology. Its normal form is (pTp) ∨ (pT ~p) ∨ ( ~pTp) ∨ ( ~pT ~p).
A special rule must be given for dealing with Texpressions in which contradictory pexpressions occur. This is necessary because of the fact that a contradictory formula has no perfect disjunctive normal form. Or as one could also put it: its normal form ‘vanishes’ is a Otermed disjunction. The rule which we need is simply this: An atomic Texpression in which the pexpression to the left or right of T expresses the contradiction of the propositions expressed by its atomic pconstituents expresses a Tcontradiction. The intuitive meaning of this is obvious: since a contradictory state of affairs cannot obtain it cannot change or remain unchanged either. Nor can it come into existence as a result of change.
9. Consider an arbitrary Texpression. We replace the (notelementary) atomic Texpressions of which it is a molecular complex by disjunctionsentences of conjunctionsentences of elementary Texpressions. Thereupon we transform the molecular complex thus obtained into its (perfect) disjunctive normal form. (See above Section 3.) This is a disjunctionsentence of conjunctionsentences of elementary Texpressions and/or their negationsentences.
It may happen that some (or all) of the conjunctionsentences contain two (or more) elementary Texpressions of different type but of the same variable (atomic pexpression). For example: (pTp) & ( ~pT ~p). Since the four elementary types of statetransformations are mutually exclusive such conjunctionsentences are contradictory. We omit them from the normal form.
Consider next the negationsentence of some elementary Texpression e.g. the formula ~(pTp). Since the four elementary types of statetransformations are jointly exhaustive the negation of the formula for one of the types will be tautologously equivalent to the disjunction of the unnegated formulae for the three other types. Thus e.g. the formula ~(pTp) is tautologously equivalent to the disjunctionformula pT ~p ∨ ~pTp ∨ ~pT ~p.
Because of the joint exhaustiveness of the four elementary types of statetransformations we can replace each negated elementary Texpression by a threetermed disjunctionsentence of (unnegated) elementary Texpressions. We make these replacements throughout in the above perfect disjunctive normal form of the molecular complex—having omitted from the normal form the contradictory conjunctions if any which occur in it. Thereupon we distribute the conjunctionsentences which contain disjunctionsentences as their members into disjunctionsentences of conjunctionsentences of elementary Texpressions. The formula thus obtained we call the positive normal form of the original arbitrary Texpression. It is a disjunctionsentence of conjunctionsentences of elementary Texpressions. No negated Texpressions occur in it.
10. pexpressions we have said (Section 5) may be regarded as (schematic) descriptions of (generic) states of affairs. Texpressions again are schematic descriptions of generic changes. Thus in a general sense pexpressions could be called ‘statedescriptions’ and Texpressions ‘changedescriptions’. Following an established terminology however we here make a restricted use of the term statedescription to mean a conjunctionsentence of n atomic pexpressions and/or their negationsentences (cf. Section 3). By analogy we shall make a restricted use of the term changedescription to mean a conjunctionsentence of some n elementary Texpressions of n different atomic variables (pexpressions). Thus for example (pTp) & (qT ~q) is a changedescription.
n atomic pexpressions (variables p q etc.) determine 2^{n} different possible statedescriptions. To each statedescription of n atomic pexpressions there correspond 2^{n} possible changedescriptions n atomic pexpressions therefore determine in all 2^{n}×2^{n} or 2^{2n} different possible changedescriptions. Thus for example to the statedescription p & ~q there correspond the four changedescriptions (pTp) & (~qT ~q) and (pTp) & (~qTq) and (pT ~p) & (~qT ~q) and (pT ~p) & (~qTq).
Given n atomic pexpressions we can list in a table the 2^{n} statedescriptions and the 2^{2n} changedescriptions which answer to the atomic variables. This is a list for the case of two atomic variables p and q:
Statedescriptions  Changedescriptions 

p & q  (pTp) & (qTq) (pTp) & (qT ~q) (pT ~p) & (qTq) (pT ~p) & (qT ~q) 
p & ~q  (pTp) & ( ~qT ~q) (pTp) & ( ~qTq) (pT ~p) & ( ~qT ~q) (pT ~p) & ( ~qTq) 
~p & q  ( ~pT ~p) & (qTq) ( ~pT ~p) & (qT ~q) ( ~pTp) & (qTq) ( ~pTp) & (qT ~q) 
~p & ~q  ( ~pT ~p) & ( ~qT ~q) ( ~pT ~p) & ( ~qTq) ( ~pTp) & ( ~qT ~q) ( ~pTp) & ( ~qTq) 
The positive normalform of a Texpression which contains n variables for states of affairs is a disjunctionsentence of (none or) one or two… or 2^{2n} conjunctionsentences of n elementary Texpressions. If the disjunction has no terms the Texpression expresses a Tcontradiction. If it has 2^{2n} terms the Texpression expresses a Ttautology.
 1. f, g, and f_{1}, f_{2}, etc., are here used as socalled metavariables. They represent arbitrary formulae or pexpressions. The constantsigns of propositional logic are used ‘autonymously’ for the purpose of building up molecular compounds of metavariables. Such compounds represent arbitrary pexpressions of the corresponding molecular structure.
 2. We shall, for the sake of typographical convenience, throughout avoid the use of quotes round symbolic expressions such as p, ~ p, p & q, etc. When mentioning the expressions, we use the expressions themselves ‘autonymously’. When speaking of the meanings of the expressions, we shall use locutions of the type ‘the proposition expressed by p’, ‘the state of affairs described by p & q’, etc.