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 so-called quantification-theory the notions of ‘all’ ‘some’ and ‘none’.
Familiarity on the part of the reader with the techniques of modal logic and quantification-theory is however neither presupposed nor needed for understanding the arguments in this book.
Modal logic and quantification-theory may be said to rest on a more elementary branch of logical theory so-called 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 so-called truth-tables and the transformation of expressions into so-called normal forms. These techniques are described in any up-to-date text-book 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 that-clauses. A that-clause in English consists of the word ‘that’ followed by a sentence. For example ‘that London is the capital of England’ is a that-clause. That-clauses have the character of names of propositions. Propositions can be called the reference of that-clauses.
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 p-expressions. 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 p-expressions 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 p-calculus.
3. An important point of view from which so-called ‘classical’ propositional logic studies its objects propositions is the truth-functional point of view.
In classical propositional logic truth and falsehood are the two truth-values. It is assumed that every proposition has one and one only truth-value. If there are n logically independent propositions there are evidently 2n possible ways in which they can be true and/or false together. Any such distribution of truth-values over the n propositions will be called a truth-combination.
If the truth-value of one proposition is uniquely determined for every possible truth-combination in some n propositions then the first proposition is called a truth-function 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 truth-functions are of special interest to us:
The negation of a given proposition (is the truth-function 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 negation-sign.
The conjunction of two propositions (is the truth-function 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 conjunction-sign.
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 disjunction-sign.
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 truth-function of them which is true for all possible truth-combinations in those n propositions. The tautology has no special symbol.
The contradiction of n propositions is the truth-function of them which is false for all possible truth-combinations in those n propositions. Like the tautology the contradiction has no special symbol.
Truth-functionship is transitive. If a proposition is a truth-function of a set of propositions and if every member of the set is a truth-function of a second set of propositions then the first proposition too is a truth-function of the second set of propositions.
Thanks to the transitivity of truth-functionship every formula of propositional logic or p-expression expresses a truth-function of the propositions expressed by its atomic constituents. Which truth-function of its atomic constituents a given p-expression expresses can be calculated (decided) in a so-called truth-table. The technique of constructing truth-tables is assumed to be familiar to the reader.
Two formulae f1 and f2 are called tautologously equivalent if the formula f1 ↔ f2 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 ~(f1 & f2) and ~ f1 ∨ ~ f2 are tautologously equivalent and so are the formulae ~(f1 ∨ f2) and ~ f1 & f2. 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 truth-functions can be generalized so that one can speak of the conjunction and disjunction of any arbitrary number n of propositions.
The formulae f1 & (f2 ∨ f3) and f1 & f2 ∨ f1 & f3 are tautologously equivalent and so also the formulae f1 ∨ f2 & f3 and (f1 ∨ f2) & (f1 ∨ f3). These are called Laws of Distribution.
The formula f1 → f2 is tautologously equivalent to ~ f1 ∨ f2 and also to ~(f1 & f2). The formula f1 ↔ f2 again is tautologously equivalent to f1 & f2 ∨ ~f1 & f2. 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 truth-functionship every formula of propositional logic may be shown to possess certain so-called 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 2n different conjunction-formulae such that every one of the atomic formulae or its negation-formula is a constituent in the conjunction. (Conjunction-formulae 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 2n different conjunction-formulae may be said to ‘correspond’ to the 2n different truth-combinations in the propositions expressed by the atomic formulae. The conjunction-formulae are sometimes called state-descriptions. 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 state-descriptions 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 O-termed 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 truth-value? 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 to-day but not to-morrow; and it can be raining to-day in London but not in Madrid; and it can to-day 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 truth-value 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 truth-value; it is either true or false but not both. The generic proposition has by itself no truth-value. It has a truth-value 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 spatio-temporal 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 ‘thing-like’ 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 so-called 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 semi-generic or semi-individual. A further suggestion might be that semi-generic 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 spatio-temporal 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 well-known 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 truth-value 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 truth-function 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 well-known 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 writing-desk. 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 writing-desk. 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 writing-desk 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 semi-generic 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 substantive-clause 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 true-making 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 end-state.
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 end-state obtains or which contains the end-state. Such transformations will also be called state-transformations.
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 p-expressions. 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 end-state 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 p-world to a q-world. The states of affairs will also be called ‘features’ of the worlds.
We shall call expressions of the type T atomic T-expressions. We can form molecular compounds of them. By a T-expression we shall understand an atomic T-expression or a molecular compound of atomic T-expressions.
T-expressions may be handled in accordance with the rules of the p-calculus (propositional logic). As will be seen there also exist special rules for the handling of T-expressions. The rules for handling T-expressions we shall say define the T-calculus.
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 ‘not-event’ or a ‘not-transformation’.
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 state-transformations 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 T-expression we understand an atomic T-expression in which the letter to the left of T is either an atomic p-expression or an atomics p-expression preceded by the negation-sign and the letter to the right of T is this same atomic p-expression either with or without the negation-sign before itself.
8. We shall in this section briefly describe how every state-transformation—strictly speaking: proposition to the effect that a certain change or event takes place—may be regarded as a truth-function of elementary state-transformations.
Consider the meaning of pTq. A p-world changes to a q-world. p and q let us imagine describe logically independent features of the two worlds. The p-world either has or lacks the feature described by q. It is in other words either a p & q-world or a p & ~q-world. Similarly the q-world is either a p & q-world or a ~p & q-world. 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 p-world is a p & q-world and that the q-world is a p & q-world too. Then the transition from the initial state to the end-state 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 p-world is a p & q-world and that the q-world is a ~p & q-world. Then the transition from the initial state to the end-state 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 p-world is a p & ~q-world and the q-world a p & q-world. The world now changes from being a ~q-world to being a q-world but remains unchanged as p-world. 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 p-world is a p & ~q-world and the q-world a ~p & q-world. The world now changes from p-world to ~p-world and from ~q-world to q-world. 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 T-expression pTq is identical in meaning with the following disjunction-sentence of conjunction-sentences of elementary T-expressions:
(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 T-expression can become transformed into a molecular complex (disjunction-sentence of conjunction-sentences) of elementary T-expressions. Thus every atomic T-expression expresses a truth-function of elementary state-transformations. Since truth-functionship is transitive it follows that every molecular complex too of atomic T-expressions expresses a truth-function of elementary state-transformations.
Consider an arbitrary T-expression. We replace its (not-elementary) atomic constituents by disjunction-sentences of conjunction-sentences of elementary T-expressions. The original T-expression has thus become transformed into a molecular complex of elementary T-expressions. These last will be called the T-constituents of the original T-expression.
It follows from what has been said that every T-expression expresses a truth-function of (the propositions expressed by) its T-constituents. Which truth-function it expresses can be investigated and decided in a truth-table. This truth-table differs from an ‘ordinary’ truth-table of propositional logic only in the feature that certain combinations of truth-values are excluded from it. The excluded combinations are those and only those which would conflict with the principle that of the four elementary T-expressions which answer to a given atomic p-expression no two must be assigned the value ‘true’ and not all may be assigned the value ‘false’.
If a T-expression expresses the tautology of its T-constituents we shall call (the proposition expressed by) it a T-tautology. An example of a T-tautology is (pTp) ∨ (pT ~p) ∨ ( ~pTp) ∨ ( ~pT ~p).
The negation of a T-tautology is a T-contradiction. An example of a T-contradiction is (pTp) & (pT ~p). It follows that ~(pTp) ∨ ~(pT ~p) is a T-tautology.
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 T-tautology. Its normal form is (pTp) ∨ (pT ~p) ∨ ( ~pTp) ∨ ( ~pT ~p).
A special rule must be given for dealing with T-expressions in which contradictory p-expressions 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 O-termed disjunction. The rule which we need is simply this: An atomic T-expression in which the p-expression to the left or right of T expresses the contradiction of the propositions expressed by its atomic p-constituents expresses a T-contradiction. 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 T-expression. We replace the (not-elementary) atomic T-expressions of which it is a molecular complex by disjunction-sentences of conjunction-sentences of elementary T-expressions. Thereupon we transform the molecular complex thus obtained into its (perfect) disjunctive normal form. (See above Section 3.) This is a disjunction-sentence of conjunction-sentences of elementary T-expressions and/or their negation-sentences.
It may happen that some (or all) of the conjunction-sentences contain two (or more) elementary T-expressions of different type but of the same variable (atomic p-expression). For example: (pTp) & ( ~pT ~p). Since the four elementary types of state-transformations are mutually exclusive such conjunction-sentences are contradictory. We omit them from the normal form.
Consider next the negation-sentence of some elementary T-expression e.g. the formula ~(pTp). Since the four elementary types of state-transformations 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 disjunction-formula pT ~p ∨ ~pTp ∨ ~pT ~p.
Because of the joint exhaustiveness of the four elementary types of state-transformations we can replace each negated elementary T-expression by a three-termed disjunction-sentence of (unnegated) elementary T-expressions. 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 conjunction-sentences which contain disjunction-sentences as their members into disjunction-sentences of conjunction-sentences of elementary T-expressions. The formula thus obtained we call the positive normal form of the original arbitrary T-expression. It is a disjunction-sentence of conjunction-sentences of elementary T-expressions. No negated T-expressions occur in it.
10. p-expressions we have said (Section 5) may be regarded as (schematic) descriptions of (generic) states of affairs. T-expressions again are schematic descriptions of generic changes. Thus in a general sense p-expressions could be called ‘state-descriptions’ and T-expressions ‘change-descriptions’. Following an established terminology however we here make a restricted use of the term state-description to mean a conjunction-sentence of n atomic p-expressions and/or their negation-sentences (cf. Section 3). By analogy we shall make a restricted use of the term change-description to mean a conjunction-sentence of some n elementary T-expressions of n different atomic variables (p-expressions). Thus for example (pTp) & (qT ~q) is a change-description.
n atomic p-expressions (variables p q etc.) determine 2n different possible state-descriptions. To each state-description of n atomic p-expressions there correspond 2n possible change-descriptions n atomic p-expressions therefore determine in all 2n×2n or 22n different possible change-descriptions. Thus for example to the state-description p & ~q there correspond the four change-descriptions (pTp) & (~qT ~q) and (pTp) & (~qTq) and (pT ~p) & (~qT ~q) and (pT ~p) & (~qTq).
Given n atomic p-expressions we can list in a table the 2n state-descriptions and the 22n change-descriptions which answer to the atomic variables. This is a list for the case of two atomic variables p and q:
|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 normal-form of a T-expression which contains n variables for states of affairs is a disjunction-sentence of (none or) one or two… or 22n conjunction-sentences of n elementary T-expressions. If the disjunction has no terms the T-expression expresses a T-contradiction. If it has 22n terms the T-expression expresses a T-tautology.
- 1. f, g, and f1, f2, etc., are here used as so-called meta-variables. They represent arbitrary formulae or p-expressions. The constant-signs of propositional logic are used ‘autonymously’ for the purpose of building up molecular compounds of meta-variables. Such compounds represent arbitrary p-expressions 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.