Logic: a new beginning

2.1 Entailment, contrariety and contradiction:

the natural triangle

We start with the notions in terms of which the machinery of any logic is defined, namely contrariety, entailment, contradictoriness, and subcontrariety.

These are logical relations that may hold between two, or sometimes more, L-propositional types, depending on their logico-semantic properties, which, in their turn, are determined by the logical constants or operators occurring in them. (One remembers that an L-proposition is the type-level linguistic expression of a proposition in the logico-semantic language L_L.) Since these relations are not satisfied by world situations but by combinations of L-propositional types, we should really call them ‘metalogical relations’, as opposed to the object-language logical relations expressed by the truth-functional operators and quantifiers of LL. In practice, however, we use the term logical relations in all cases where no confusion is to be expected. In general, (meta)logical relations hold (are valid) in virtue of the meanings of the L-propositional arguments. Within a logical system, they hold in virtue of the meanings of the logical constants in the L-propositional arguments.

(Meta)logical relations should be clearly distinguished from the functors of the object language LL, also called operators or logical constants, mainly

ALL,^SOME,^NOT,^AND,^OR, and^IF. . . THEN, which produce truth or falsity depend-ing on token world situations. In standard logic, the functors of negation (^NOT) and of material implication (^IF P ^THEN Q) have a direct analog to a (meta)logical relation, as they correspond directly to contradictoriness and entailment, respectively: contradictoriness holds between any L-proposition and its negation; entailment holds when the material implication is true in all admissible situations.

One way of seeing how these (meta)logical relations get off the ground is the following. Let us take as our point of departure the requirement of consistency in linguistic interaction. Speakers who commit themselves to the truth of what they assert or who appeal to listeners to fulfil a request or an

order must maintain consistency, or else the social fabric will collapse. For that reason it is of the greatest importance, first to language users and secondly to language theorists, to spot inconsistencies or, in logical terms, to spot sets of propositions that cannot simultaneously be true. When such a set consists of just two propositions, we speak of contrary propositions. This is, in fact, how Aristotle set up his predicate logic in On Interpretation, which revolves around the notions of contrariety and contradictoriness. Aristotle called any two L-propositions which, for semantic reasons, cannot be simul-taneously true ‘contraries’, and any two L-propositions that can, again for semantic reasons, be neither simultaneously true nor simultaneously false

‘contradictories’. On the standard (yet oversimplified) assumption that all natural languages have a logical constant of negation—not in English—which inverts truth values (modulo key) under the axiom of bivalence, the contra-dictory of an L-proposition P is its negation^NOT(P), or¬P: the truth of the one entails the falsity of the other, either way.

To give a trivial example, the two sentences in (2.1a) are contraries, since anyone who asserts both is guilty of inconsistency (‘><’ stands for contrari-ety). This means that when (the L-proposition underlying) Joe has been murdered is true, Joe is not dead must be false and, therefore, Joe is dead must be true (assuming bivalence). In other words, Joe has been murdered entails Joe is dead, as stated in (2.1b). (One notes that we have silently passed from entailment in a general sense to logical entailment. This is legitimate given the definition of bivalent negation.)

(2.1) a. Joe has been murdered >< Joe is not dead b. Joe has been murdered  Joe is dead

Entailment can thus be defined in terms of contrariety and the contradiction-producing negation operator. It is also possible, of course, to define contrari-ety in terms of entailment and contradictoriness, but that seems less natural, given the basic requirement of consistency in the use of language. Nor is it how Aristotle proceeded.

Any relation of contrariety between two L-propositions P and ¬Q, thus brings along a relation of entailment from P to Q. Figure 2.1 shows the triangular relation arising from the assumed contrariety of P and ¬Q, which causes the entailment from P to Q, the contradictory of¬Q.

This triangle is arguably ‘natural’ in the sense that it may be taken to reflect natural, as opposed to constructed, set-theoretic structures and relations—a supposition that is further elaborated in Chapter3. It forms the natural basis of the logical system of propositional calculus with the operators^AND,^OR, and

NOT, and also of Aristotelian-Boethian predicate calculus (ABPC), which is based on the quantifiers^ALLand^SOMEand the propositional operator^NOT.

But one can go further, still under the bivalence axiom. Since P and¬Q in Figure2.1 are contraries, ¬Q entails the falsity of P, or: ¬Q ‘ ¬P. This is, of course, nothing but the well-known operation ofCONTRAPOSITION: when, for any L-propositions P and Q, P‘ Q, then also ¬Q ‘ ¬P. This is easily shown:

suppose Q is false, and therefore¬Q true, then P cannot be true, for if P were true, Q would also have to be true in virtue of the entailment P‘ Q. Since both P and Q have only two options, True and False, it follows that if P‘ Q, then in all cases where¬Q is true, ¬P is also true, that is, ¬Q ‘ ¬P. This can be added to the triangle of Figure2.1, leading to the square of Figure 2.2.

Now that the natural triangle has been extended to a square, a new logical relation has emerged, that of SUBCONTRARIETY. By definition, two L-proposi-tions form a pair of subcontraries when they cannot be simultaneously false, though they may be simultaneously true. In this case, Q and¬P cannot be simultaneously false, because, as the square shows, if Q is false, so is P, and when ¬P is false and thus P is true, Q is true. Clearly, the relation of

P

Q ¬Q

CD C

: entails CD: contradictories

C: contraries

FIGURE2.1 The natural logical triangle

Q ¬Q

CD C

CD

SC SC: subcontraries

¬P P

FIGURE2.2 The natural logical triangle extended with contraposition

contradictoriness is the combination of contrariety and subcontrariety—at least in a bivalent logic with the metalogical relations as defined.

Figure 2.3a shows a different arrangement of the vertices of the natural logical triangle, extended with contraposition to the square of Figure 2.3b.

This arrangement has a more familiar look as it reminds one of the famous Square of Opposition in traditional predicate logic, attributed to the Roman-Christian philosopher and statesman Boethius ( 480–524). For that reason, we call this the Boethian arrangement. It is shown, however, in Section2.2, that the Boethian arrangement is not a very good way of presenting metalo-gical relations and had better be given up in favour of the arrangement of Figure2.2.

Apart from this, it will be clear that the square of Figure2.2, or Figure 2.3b for that matter, cannot be the whole story, since, if it were, both the triangle and the square, in either form, would be dispensable, as they merely reiterate the entailment relation P‘ Q (or P  Q) in terms of contrariety, contradic-toriness and subcontrariety, given the negation as a truth-functional operator inverting binary truth values. The logic would also be pretty poor, consisting only of negation (important though that operator is) and a few new relations that can be defined in terms of each other—hardly sufficient to get a logic off the ground. What makes these new relations important for the machinery of logic is the establishing of logical relations between the external negations occurring in L-propositions of the form ¬P on the one hand, and corresponding L-propositions with internal negation on the other. What is meant by ‘internal negation’ is a question to which we turn now.

Q ¬P

CD C

CD

SC P ¬Q

P

Q

¬Q

CD C

a. b.

FIGURE2.3 The logical triangle and square in the Boethian arrangement

2.2 Internal negation and duality: the natural square and the Boethian square

Just about the most central operator in any logical system is negation, normally lexicalized as not in English. We concentrate first on ordinary external negation over L-propositions. This negation is typically a comple-ment selector. In the simplest case, it selects the complecomple-ment within a universe U of admissible situations.

To make this clear, we fall back again on the notion of valuation space, introduced in Van Fraassen 1971 and briefly discussed in Section 3.3.3 of Volume I. Leaving the formal definition of the notion of ‘admissible situation’

to Section2.3.3 below, we define ‘valuation space’ (VS) as follows:

V^ALUATIONS^PACE

For any L-proposition P expressing the proposition p, the ^VALUATION

SPACE(VS) of P, or /P/, is the set of admissible situations that make P true (modulo key).

Thus, the proposition p (or its L-propositional expression P) is true in all situations that are elements in the set of situations /P/. Given that the standard bivalent negation ¬ selects the complement of /P/

within the universe of all admissible situationsU, it follows that the proposi-tion expressed by ¬P is true in all situations that form the com-plement=P=ð¼ U =P=Þ of /P/ in U. This simplest case is what is found in standard logic. Since, by definition, an L-proposition P is true just in case the actual situation sit_act is an element of /P/ and false otherwise, it follows that¬P is true just in case sitact is an element of /¬P/—that is, =P= —and false otherwise. In the standard, strictly bivalent, system shown diagrammati-cally in Figure 2.4 no other possibilities than truth and falsity are left open:

U

/P/

/¬P/

FIGURE2.4 VS representation of /P/ and /¬P/ in U

‘not true’ equals ‘false’ and ‘not false’ equals ‘true’. Clearly, if P is necessarily false, as, for example, the L-proposition underlying the sentence He was dead for the rest of his life, /P/ ¼
; if P is necessarily true, as in He was alive for the rest of his life, /P/¼ U.

Thus defined, negation can be said to ‘toggle’ between truth and falsity. The property of the negation simply to invert truth values of propositions is expressed in the TRUTH TABLEof the standard negation shown in Figure2.5 (adapted to a valuation space interpretation), where ‘T’ stands for True and

‘F’ for False.

It is possible, however, to vary on this theme. For example, one may define a negation operator that selects the complement of /P/ within a subset of situa-tions inU defined on the basis of preceding context or the meaning of the main predicate in P, in which case room is made for more than one negation. In this book we argue that this is, in fact, the situation in natural language, where the main function of the negation^NOTis to toggle between the values ‘true’ and

‘minimally false’, minimal falsity being caused by those situations that are outside /P/ but within P’s subuniverse inU. Further comment on this issue is provided in Chapter3, but a full discussion has to wait till Chapter 10.

But what is meant by internal negation? So as not to complicate matters unduly at this stage, let us say that the internal negation is a negation not over an L-proposition but over a predicate. This is not an adequate definition, since, as is shown in Sections2.3.5.2 and 2.4.1, internal negation is better defined as a small scope negation over an embedded L-propositional structure, but it will do for the moment. If it is accepted that a predicate, for example human, expresses a property possessed by all objects in the world that are indeed human, then the ‘negation’ of this predicate, not-human, functioning as the internal negation of an L-proposition, expresses the lack of that property for all objects that are not human. Call the set of all human objects the^EXTENSIONof the predicate human, or [[Human]], in the set of all objectsOBJ. Then the set

P ¬P

T F

F T sit_act∈ /P/

sit_act∈ /¬P/ (= U –/P/)

FIGURE2.5 Truth table of the standard bivalent negation¬ in terms of VS-modelling

of objects that are not human is the extension of the predicate not-human, or [[^NOT-Human]], in OBJ. This makes [[^NOT-Human]] the complement of [[Human]]—that is,OBJ minus [[Human]], or ½½Human—as long as there are no vague boundaries between what is and what is not human.

The logical interest of internal negation, in the present context, lies in the relation of^DUALITY(Lo¨bner1990):

D^UALITY

Two logical constants X and Y are each other’s duals just in case there is logical equivalence between X preceded by the external negation and Y followed by the internal negation and, of course, vice versa.

In standard predicate calculus, for example, the quantifiers^ALLand^SOMEare each other’s duals, since in that system an L-proposition corresponding to the form ^{NOT ALL} F is G (where F and G are predicates) is equivalent with an L-proposition of the formSOME FIS NOT-G and, analogously, NOT SOME

F^ISG/^NO F is G is equivalent with^ALLF^{IS NOT}-G. These equivalences are standardly known as theCONVERSIONS. Henceforth, when dealing with external versus internal negation, we will use the standard symbol ‘¬’ for external negation and the symbol‘*’ for internal negation. That is, ¬P is the external negation of the L-proposition P and P* is the L-proposition P but with its main lexical predicate negated by the internal negation. Obviously, in a strictly bivalent system, double external negation and double internal nega-tion cancel out: for any L-proposinega-tion P,¬¬P
P and, since for any predicate C,^NOT-^NOT-C
C, P**
P.

It must be noted, at this point, that the logical properties of two L-propositions P and P* are identical for the simple reason that the choice of lexical predicates is irrelevant for the logic, which is defined by the logical constants only. We call this the ^MODULO-*-^PRINCIPLE. The relevance of the internal negation for predicate logic lies in any logical relation of duality or of a one-way entailment between two logical constants X and Y when one is preceded by the external negation and the other is followed by the internal negation. The Modulo-*-principle provides an extra check for the soundness of a predicate-logic system in that any system that violates the Modulo-*-principle is by definition unsound.

Logicians have observed that a similar relation of duality exists between the propositional operatorsANDandOR, a form of duality known as DEMORGAN’S LAWS. In standard propositional logic, the operators of negation, conjunction and disjunction are defined in such a way that for the L-propositions P and Q,

NOT(P^ANDQ) is equivalent with^NOT(P)^{OR NOT}(Q). Analogously,^NOT(P^ORQ)

is equivalent withNOT(P)AND NOT(Q), or in standard notation:¬(P ∧ Q)

¬P ∨ ¬Q and ¬(P ∨ Q)
¬P ∧ ¬Q.

This is, of course, an intriguing parallel with standard predicate calculus, but if we want to cash in on this parallelism it must first be made clear that here, too, we have to do with internal negation. To do that, the definition of internal negation as predicate negation must be generalized so as to cover the dual quantifiers and the dual propositional operators alike. This is done in Section 2.3, where all logical operators are reinterpreted as (abstract) predi-cates. The parallelism between the two calculi is then further elaborated in Chapter4. For the moment, we limit ourselves to the dual quantifiers.

Let us revert to the logical triangle of Figures2.1 and 2.3a, and the logical squares of Figures2.2 and 2.3b, respectively, assuming that P and Q are types of L-proposition defined by logical constants that are each other’s duals. That is, we assume that:

(2.2) P
¬Q* and consequently ¬P
Q*

Q
¬P* and consequently ¬Q
P*

Now the squares of Figures 2.2 and 2.3b are seen to be more than just restatements of the entailment relation and to have real logical power. For now the triangular Figures 2.1 and 2.3a can be expanded to quadrilateral figures that express the duality relation. This is done in Figure 2.6, where (a) corresponds with Figure2.2 and (b) (in either form) with Figure 2.3b.

It now becomes clear why the arrangement of Figure2.2 (expanded from the triangle in Figure2.1) is preferred to the Boethian arrangement of Figure 2.3b (expanded from Figure 2.3a). When the triangles are expanded to squares in such a way that the dual equivalences are expressed, Figure 2.1 results in Figure2.6a, which we call theNATURAL ISOMORPHIC SQUARE (two parallel lines stand for logical equivalence). But Figure2.3a turns into Figure 2.6b (in either form), which we call theIMPROVEDBOETHIAN SQUARES. The difference is that the natural isomorphic square consists of two triangles that areLOGICALLY ISOMOR-PHIC in that they display the same logical relations of entailment, (sub)-contrariety and contradiction, the only difference being that the right-hand side triangle has the internal negation added to P, Q, and¬Q, giving P*, Q*, and¬Q*, respectively—that is, with a negative instead of a positive predicate.

But since the choice of lexical predicates makes no difference to the logic, which is defined by the logical constants alone (the Modulo-*-principle), the two triangles making up the natural square are logically identical. It is their connection through the Conversions that makes a logic spring up.

The interest of their being coupled into the natural square lies in the dual equivalences. By contrast, the two improved Boethian squares in Figure2.6b

lack this regularity. There the triads <P,Q,Q*> and <P,Q,¬Q> of the left-hand side triangles correspond to the nonisomorphic triads <¬Q,¬P,¬Q*>

and <P*,Q*,¬P*>, respectively, of the right-hand side triangle. Clearly, the isomorphic representation displays more regularities in the logical system than the nonisomorphic representation.

Figure2.6c shows a second way of making a square out of two isomorphic triangles with the same logical power.¹ Here we have a triangle <P,¬P,Q>, linked up with the isomorphic triangle <P*,¬P*,Q*>. Now the relations involved are those of entailment, contradiction, and subcontrariety, with contrariety thrown in as a bonus due to the linking up through the Conver-sions. In Chapter3, however, it is argued that subcontrariety is not a basic-natural logical relation but a relation that requires a great deal of scholastic training to be grasped, whereas contrariety is just about maximally natural.

For that reason, the square of Figure2.6c is called ‘nonnatural isomorphic.’

From a strictly logical point of view, the difference is immaterial, but it is not when we are in search of natural logic. Since natural logic is what we are after, we consider Figure2.6a to be the preferred representation.

P

FIGURE 2.6 (a) the natural isomorphic square, (b) the two improved but noniso-morphic Boethian squares, and (c) the nonnatural isononiso-morphic square

1 I am indebted to Dany Jaspers (2005: 34–5) for calling this fact to my attention.

The isomorphy of a representation is important not only because it shows up more regularities in the system than nonisomorphic representations but also because it adds a criterion for the consistency of the logic in question.

The criterion can be formulated as follows:

T^HEI^SOMORPHYP^RINCIPLE

Given the Modulo-*-principle, the (meta)logical relations defined for the operators of predicate calculus (the quantifiers and negation) must be the same no matter whether the main or G-predicate is or is not characterized by internal negation.

In virtue of the Modulo-*-principle, the identity of the logical relations in the two triangles <P,Q,¬Q> and <P*,Q*,¬Q*> in Figure 2.6a, or in <P,¬P,Q>

and <P*,¬P*,Q* > in Figure 2.6c, is thus a condition for the soundness of the logic involved. Since the renderings shown in Figures2.6a and 2.6c actually show the isomorphy, whereas those of Figure2.6b do not, the former are at least didactically superior to the latter.

The fact that logical systems with X and Y as duals and an entailment

The fact that logical systems with X and Y as duals and an entailment