Arthur Pap

Logical Nonsense^*

Philosophy and Phenomenological Research, Vol. 9, No. 2 (Dec., 1948), 269-283.

 

This paper purports to be a study in logical analysis of language. Let it be made clear at the very outset, however, that the object-language analyzed is assumed to be non-systematized, e.e., it is the actual language of common sense and science, not an ideal language precisely characterized in terms of formation rules, designation rules, etc. With respect to an ideal language, the question whether a given sentence, formulated in that language, is significant, is logically decidable, since the formation rules of the language may be considered as a definition of "significant sentence" (in that language). Thus, one could imagine that the systematized language of quantum mechanics were to contain explicitly a formation rule from which it would follow that any sentence speaking about precise simultaneous values of position and momentum of an electron is meaningless. But, obviously, if a scientist insisted that such a sentence is meaningful, even though at the present time no method of "verifying it is available, you could not refute him by referring to the said formation rule: for he would simply vote against the adoption of such a formation rule. For a philosopher concerned with the analysis of natural (non-artificial) languages, then, the problem is rather to find formation rules that could serve as an adequate definition of the concept of a meaningful statement.

The phrase "logical nonsense" naturally suggests that the kind of nonsense referred to results from a violation of principles of logic, such as the principle of non-contradiction. Now, it is entirely possible that an analysis of statements intuitively recognized as nonsensical should lead to the conclusion that all nonsense which is not rooted in a violation of grammatical syntax is of this kind. However, we cannot take this for granted prior to an analysis of sample statements.

Consider the statement "Caesar's body was found dead to the left of space." Surely, the rules of grammatical syntax are not violated here. But, at first glance, this statement does not violate the principle of noncontradiction either, the way the statement "he stands—from my point of view—both to the left and to the right of me" does. Nevertheless, it may be presumed that nobody would hesitate to characterize it as nonsensical. What then makes it nonsensical? It is the fact that "space" is not a value that can be significantly^[1] substituted for any of the arguments of the sentential function "x is to the left of y." Being to the left of, in other words, is a spatial relation holding between bodies having position in space. Ascribing spatial relations to space is like ascribing temporal relations to time in saying "there was a time when God created time." One might argue, though, that, in spite of prima facie conformity to logic, there is a disguised contradiction here nonetheless. "Being to the left of" is defined (ostensively, at any rate) as a relation between bodies having spatial position ; hence it is analytic to say that only names or descriptions of bodies can be substituted for the arguments of the function "x is to the left of y." A similar example of such disguised contradictions might be this: suppose it is asserted that time is immovable. The range of significance of the function "x is immovable" is constituted by objects having position in space and time, "x is immovable," one might say, analytically entails "x has position in space and time." If this entailment is accepted, "time is immovable" violates the principle of non-contradiction. In every similar case, nonsense resulting from the failure to observe the range of significance^[2] of a propositional function can be reduced to nonsense resulting from a violation of logic, provided similar entailments are agreed upon at the outset. Thus, if it is stipulated that "x is colored" is to entail analytically "x is spatially extended," then the statement "thoughts are colored" is logically false. However, it might be replied that we have just been conditioned by experience to associate in our minds colors with surfaces and nothing else, and that therefore "x is colored" does not analytically involve "x is a surface" in the same sense in which, say, "x is a triangle" analytically involves "x has three sides."

Of course, one could hardly maintain that our inevitable associating of colors with surfaces is a result of experience in quite the same sense as, say, our associating the properties which define a crow with the property of being black is a result of experience: for we can easily conceive of a non-black crow, we know what our experience of seeing a non-black crow would be like. But we presumably do not know what it would be like to have a rectangular toothache or to hear a red chord.^[3] However, once the notion of inconceivability, as distinct from logical inconsistency, is brought on the scene, one is exposed to the argument from psychological relativity: how could you refute a man who claimed himself capable of having experiences in which the quality of b-flat is just as intimately integrated with the quality of blueness as the quality of spatial extension is integrated with the quality of coloredness in your experience? Or, to consider a hypothetical experience of a less fantastic kind : it is quite conceivable that we should live in a universe in which roundness and redness happen to be invariably concomitant properties. Obviously, these properties are logically independent in the sense that the analysis of neither property involves the other—as a matter of fact, ''round" and "red" designate simple qualities and are terms whose meanings can be communicated only through ostensive definitions. Is it too far fetched to suppose that, living in such a universe, we could not imagine what a red non-round object or a round non-red object would look like? Surely, if the terms "red" and "round" are only ostensively definable and hence designate unanalyzable qualities, we could not describe an χ fulfilling both functions "x is red" and "x is not round": such an χ could only be denoted, but by hypothesis there are no such x's in our hypothetical universe. Analogously, if redness is an unanalyzable quality, then "x is red and x is not extended" cannot possibly be a contradiction, hence a universe in which this sentential function is true for some value of x is a logically possible universe even though it is, for us, unimaginable.

It is customary among certain philosophers to say a statement is meaningless if it is not possible to describe the kind of situation which would verify it. This criterion of cognitive sense, however, is inadequate since it rules out as meaningless all predications of simple, unanalyzed (if not unanalyzable) qualities. How could anyone describe the kind of situation which would verify an assertion of the form "x is red"? Now, if color and extension are simple, unanalyzable^[4] determinables, how could I possibly describe the kind of situation which would verify a conjunction of the form "x is colored and x is not extended"? All I can do is to define "colored" and "extended" ostensively by pointing to instances of different determinates of the former determinable which are instances of the same determinate of the latter determinable, and to instances of different determinates of the latter determinable which are instances of the same determinate of the former determinable.^[5] Of course, I could not find any x which verified the function "x is colored and x is not extended" ; but this is merely to say that the statement "there are x's which are colored without being extended" is false, which does not at all imply that it is meaningless.

Analogous considerations seem to apply to all those statements which assert the co-inherence^[6] of different determinates of the same determinable. Take the classical example, discussed ad nauseam by philosophers interested in the criteria of meaningful language: "There is a surface which is at the same time both blue and red all over." The statements "this surface is both rectangular and blue" and "this surface is both red and blue" have precisely the same logical form. If the latter is argued to be contradictory on the ground that Red is an instance of Non-Blue, why not judge the former as likewise contradictory since rectangular is an instance of non-blue also? The proper reply, of course, is that blueness excludes redness with respect to co-inherence, whereas rectangularity does not. But is this exclusion a case of logical exclusion? By no means, it might be argued, for all we can say is that blueness and redness are determinates of the same determinable, and it is in a way a hard fact that determinates of the same determinable never co-inhere. Since "blue" cannot be verbally defined, it cannot be shown that the analysis of "x is blue" is such that "x is not red" follows from it. The principle that determinates of the same determinable never (or cannot) co-inhere would have to be introduced as a synthetic premiss in order to render such deduction possible.

For the reasons stated, an assertion of the co-inherence of different determinates of the same determinable cannot be argued to be nonsensical on the ground that such a situation could not be described. However, it might be argued to be meaningless on other grounds. To speak of the co-inherence of different pitches, for example, in the same simple tone, is nonsense because a tone thus qualified would either be said to be complex (a chord) or, if simple, to be two, not one: if you heard different pitches, you must have heard two tones. Similarly, if somebody claimed to have seen a surface which was both blue and red all over, we would either interpret him to say that the surface he saw was of a uniformly mixed color, composed of blue and red the way orange is composed of yellow and red, or we would say: you must have seen two surfaces, then, which were almost congruent, just as two notes an octave apart give rise to an almost simple tone sensation if they are played almost simultaneously. The general principle illustrated by these examples might be formulated as follows: different determinates of the same determinable cannot co-inhere, since exemplification of different determinates of the same determinable is taken as a criterion of the presence of different particulars.^[7] A less trivial application of this principle is the so-called principle of the "impenetrability of matter." Impenetrability is not a physical property of matter, the way mass and motion, say, may be said to be general physical properties of matter. For to say of a material particle that it is impenetrable is to say no more nor less than that it is impossible for two particles to occupy the same place at the same time. But identity of spatial position at a given time defines the particle's identity in the same way in which identity of pitch defines the identity of a tone. Just as the alleged plurality of pitch characterizing a simple tone would be regarded as evidence for the complex nature of the tone heard, thus any alleged penetration of a particle by another particle could be made intelligible only by assuming that the penetrated particle is composed of smaller particles separated by vacua into which the sub-particles of the penetrating particle could insert themselves. That is, it is analytic to say of an indivisible particle that it is impenetrable. A similarly analytic principle of "physics" is the impossibility of instantaneous motion. Instantaneous motion is motion with infinite speed. To assert, therefore, of a particle that it moves instantaneously is to assert the contradiction that it occupies different positions (relatively to the same reference frame) at the same time. It is logically possible that different places should be simultaneously occupied by exactly similar particles ; but it is logically impossible that they should be simultaneously occupied by one and the same particle. Or, if we preferred to speak of one particle thus "omnipresent" in space, we could hardly say that it moves. In spite of being in a sense^[8] self-contradictory, the concept of instantaneous motion played a prominent part in the history of physical theory. Thus Newton held that gravitational force was propagated instantaneously, and physicists tend to speak of the finite velocity of light as of a contingent, experimentally discovered fact. That the velocity of light has the specific value which it has is, of course, an empirical fact. But infinite speed of transmission is not an intelligible alternative to the finite speed of light: the alternative would be to say that the concept of motion cannot be significantly applied to light, just as it cannot be significantly applied to space or time. (Cf. Newton's statement that absolute time "flows equably"!). Once it is agreed to interpret optical phenomena in terms of either particle motion or wave motion, experiments could only warrant the assertion that the velocity of light is too great to admit of measurement, but not that it is infinite: the latter assertion would illustrate logical nonsense.

A counterpart, as it were, to the impossibility for determinates of the same determinable to co-inhere, is the necessity for certain determinables, such as shape and volume, or pitch and loudness, to co-inhere. Such co-inherent determinables are often called "dimensions" of the particulars in which they co-inhere: thus pitch, loudness and quality are said to be the dimensions of tones. Like the spatial coordinates of an event, these determinables are independent variables; this means that any value of one determinable is compatible with any values of the other determinables; however, the fact that one of them has a specific value entails that the other determinables have some value also. The question is, whether such entailment is analytic such that the denial of this kind of co-inherence in a given case would be logical nonsense in the sense of involving a contradiction. Consider the statement "I heard a tone of definite pitch and quality which, however, lacked any determinate degree of loudness." Now, if the class of tones is defined as a manifold whose elements are uniquely determined in terms of the three dimensions pitch, loudness and quality, such an event would, indeed, be logically impossible, and the proper reply would be "that tone really did have a determinate degree of loudness, only you did not perceive it."^[9] Yet, just as the word "point" could be theoretically dispensed with, and we could instead speak of ordered triplets of coordinates, thus we could describe our acoustic experiences without the use of nouns such as "tone," and instead describe just the values of the determinables simultaneously perceived. Instead of speaking of the co-inherence of properties in a particular, we could, dispensing with fictitious "substances," speak simply of the co-occurrence of certain properties. "This solid has a definite volume but it has no definite shape" is logical nonsense, because by the noun "solid" we mean something which, among other attributes, has both volume and shape. But if instead of speaking of co-inherence we spoke simply of co-occurrence of attributes, thus: "there occurs here-now a definite volume without there occurring a definite shape," the described fact would be inconceivable without nevertheless involving a contradiction: for certainly volume is not a complex characteristic which involves shape as an element, or vice versa.

To clarify this point, let us compare the following two implications: a) for every x, if χ is a triangle, then χ is a plane figure; b) for every x, if χ has volume, then x has shape, a) is analytic because the first attribute is a complex in which the second attribute is an element. Now, b) might be argued to be analytic in the following manner: every formal implication is simply an abbreviation for the logical product (finite or infinite) of implications that are its substitution instances. Thus b) may be expanded as follows: if a has volume, then a has shape, if b has volume, then b has shape, . . .if η has volume, then η has shape, where 'a', 'b', . . .'n' are the names of solids. But each conjunct in this logical product is analytic, since to say that 'a', 'b' . . .'n' are the names of solids is to say that they are the names of things which, among other attributes, possess both volume and shape. The fallacy in this argument can be revealed by showing that if the argument were valid, any formal implication could be made out as an analytic statement. Consider the implication "for every x, if x is white, then x is cold." Just like volume and shape, thus whiteness and coldness are logically independent attributes. But if the range of the quantifier be restricted to, say, instances of snow, then any substitution instance will, of course, be analytic, since "snow" is the name of a substance which is both white and cold. But tacit restrictions of the universal quantifier to a definite class of particulars should always be made explicit in the antecedent, thereby leaving the quantifier unrestricted. Now, while it is plainly true that "

is analytic, it is likewise plainly true that

is analytic. But from the latter the analyticity of the connection between having volume and having shape does not follow any more than it follows from the first statement that being white analytically entails being cold.

Let us, now, summarize the results which our discussion of logical nonsense has yielded so far. While conformity to the rules of grammatical syntax is a necessary condition for making logical (cognitive) sense, it plainly is not a sufficient condition. Is, then, logical nonsense avoided by additional conformity to the rules of "logical syntax," specifically the principle of non-contradiction? It has been shown that many statements which one is intuitively inclined to regard as meaningless in spite of their being prima fade self-consistent, turn out, upon analysis, to involve a violation of the logical requirement of consistency all the same. Often such contradictions may be revealed by specifying the range of significant application of the predicate (of whichever degree and level) involved. Thus, if we predicate penetrability of an indivisible particle, we forget that the predicate "penetrable" can be significantly applied only to divisible, "porous" bodies like sponges or gases. In other cases the intuited nonsense can be reduced to logical inconsistency only by appeal to what was called the "principle of the duality of discernibles" which functions as a criterion of numerical difference in a way similar to the way in which the principle of the identity of indiscernibles functions as a criterion of numerical identity. Finally, attention was called to a class of statements asserting the co-inherence of certain determinables, which we are inclined to regard as necessary and whose negations are nonetheless meaningful in the sense of involving no formally inconsistent use of language. This kind of necessity is, of course, reminiscent of the alleged non-analytic necessity of the synthetic a priori. The major issue, here, seems to be whether we can properly speak of two kinds of necessity of conditional statements, the one analytic, the other synthetic, or whether the word "necessary" is used ambiguously when it is applied to both analytic and synthetic statements. Maybe the necessity of alleged synthetic a priori judgments is just psychological, like the "necessary" connections between cause and effect? Now, if "p is necessary" is defined as "-p is logically false," evidently no room, so to speak, is left over for any non-analytic necessity of statements. · But if the generic property connoted by "necessary" (in its applications to statements) is non-inductiveness, in the sense that p is necessary if it can be known to be true without the use of the inductive methods by which scientists verify their hypotheses, then the synthetic a priori is not ruled out by definition. Everybody would agree, presumably, that it would be absurd to apply statistical methods in order to increase the probability of such statements as "all colored things are spatially extended" by observing instances. Hence the kind of necessity under discussion could hardly be said to have the sort of psychological origin which Hume claimed for the alleged necessity of causal laws. And if so, the fact that such statements have not yet been shown to be analytic hardly warrants the inference that they are not necessary.

It has been stated that inconsistencies of which many nonsensical statements are suspect may be revealed by specifying the range of significant application of a predicate—such specification involving more than just identification of the logical type of the predicate. But are we not begging the question if we argue to the meaninglessness of a statement from the premiss that a certain predicate cannot be significantly applied to entities not included in a certain class? Suppose a predicate 'P' is applied to an entity not included in a class L which is said to constitute the domain of significant application of 'P'.^[10] Then it cannot always be shown that such predications involve a contradiction. Sometimes, of course, this is the case. Thus, with respect to the predicate "triangle," L would be the class of plane figures. But to predicate triangularity of something that is not a plane figure is obviously self-contradictory, since 'P' here is defined in terms of the concept that determines L. However, if "x is blue" does not analytically entail "x is extended," where is the contradiction in applying the predicate "blue" to a mind, for example? Clearly, if "P" is verbally undefined and its meaning can only be discovered by observing its uses, it cannot be said that "x is P" analytically entails "x is a member of L." Living in the hypothetical universe described above, what should have prevented us from defining the class L, with respect to the predicate "red," as the class of round objects?^[11] Still we, living in a universe in which some non-round objects are red, would say that the inhabitants of the hypothetical universe would, in thus delimiting L, mistake the truth-range of "red" for its range of significance. They abstracted the concept of redness, we would say, in observing round objects, and had they never observed any round objects they would never have acquired that concept ; but they should not have accused that philosopher among them who speculated about red cubes, of misusing the word "red."

Similarly, a man might argue, for example, that it is no misuse of the phrase "being prior to," to say "essence is prior to existence," even though commonly we apply it to events and neither existence nor essence are events. He might say that we first apprehended the relation of priority by observing events; but some of us, gifted with metaphysical insight, see that it holds between different kinds of terms as well. Now, a little analysis will show that it is not the case that the self-same concept of priority, here, has been applied to different kinds of instances, but that actually the phrase "being prior to" is being used differently (whether such a use be- "proper" or not).

For, when we make the comparative judgment that event A preceded event B, it is always significant to ask the metrical question which was the time interval that separated the two events. Surely, it would be nonsense to ask how many time units separate essence and existence; hence, "priority," in this usage, must have a peculiar meaning. Indeed, what metaphysicians who use such language may have, more or less vaguely, in mind is the fact that it is impossible to talk about particulars without using predicates (and thus referring to universale), while it is possible to talk about universale without referring to particulars.^[12] Such an analysis, in destroying the emotive associations of the metaphysician's words, reveals the triviality of the metaphysical statement.

The above analysis illustrates how it is possible to discover the class L correlated with a predicate 'P', by examining the uses of 'P'. The search for L (or the concept determining L) is part of the search for the meaning (analysis) of "P". That is, the statement "if χ is P, then χ is a member of L" would express part of the meaning of "P". Thus, "for every χ and y, if χ is the cause of y, then χ and y are either events or states of continuants" would express part of the analysis of the concept of causality. And before a man who applied the term "cause" to, say, continuants (e.g., the self) or to universale (as Plato, when he speaks of the "chief good" as of the "author" of all forms) could be convicted of talking logical nonsense, it would have to be made explicit that "x is a cause" analytically entails "x is not a continuant and x is not a universal." Logical nonsense due to applications of "P" to non-members of L, does, then, seem to reduce to a kind of self-contradiction, provided P is an analyzable property.

Such logical nonsense, however, may in many cases be dimly recognized even though P is as yet unanalyzed. As a matter of fact, more or less intuitive recognitions of logical nonsense independently of an analysis of the relevant concept seem to be indispensable insofar as they serve as criteria of adequacy for a proposed analysis. Thus, G. E. Moore argued, in Principia Ethica, against any proposed definition of "good" in terms of  descriptive predicates, on the ground that it would always be meaningful to ask whether a good thing had in fact the natural property designated by the proposed definiens. If it could not be decided whether a statement is meaningful or not prior to knowing the analysis of its predicates, Moore's argument would be a petitio principii: for how, then, could he know that it is meaningful to deny goodness of a certain class of things, unless he knew already that being a member of that class formed no part of the meaning of "good"? Again, Russell, in the Principles of Mathematics, used the same sort of argument to refute Neumann's alternative to Newtonian absolute space as the reference frame with respect to which inertial motion should be defined. Neumann proposed to substitute for absolute space the "body alpha," such that "x moves uniformly" would always be elliptical for "x moves uniformly relatively to the body alpha." From such a definition it follows, of course, that it is meaningless to ask whether the body alpha itself is really in an inertial state or whether it accelerates; just as it would be meaningless to ask whether absolute space moves uniformly or not. But, Russell argued, with respect- to any body it is meaningful to ask whether its state is inertial or not; hence one cannot define inertial motion with respect to any material reference frame. One further example may be in order to bring out the important point that there may well be an absolute concept of contradiction which guides the analyst in his explications of concepts in use before definitional equivalences get stipulated in virtue of which sentences of the form "p is contradictory" become syntactically decidable. Suppose a man said "I have tossed this coin a 1000 times and obtained 498 heads and 502 tails; but I deny that the probability of throwing heads on the next trial is approximately 1/2." Independently of any analysis of the empirical probability concept that we may have, this would sound like a self-contradiction. Again, if he admitted that bread always nourished him in the past, but added that he thought it not at all improbable (to be distinguished, of course, from impossible) that bread would poison him the next time he ate it—we would feel, once more, that he said something absurd in the sense of self-contradictory. It is such intuitive recognitions of logical nonsense which lead to the explication of the empirical probability concept in statistical terms. If a statement involving a predicate 'P' seems meaningful even though it would be contradictory if a certain analysis of P were correct, it is of course possible to conclude that P has to be analyzed differently or is in fact unanalyzable. But it is likewise possible to conclude that the meaning of "P", in this usage, is different from the one which would make nonsense of the statement.^[13] The following illustration is intended to clarify this point. Is it meaningful to speak of the existence of unobservable entities?

In terms of the language of common sense, it seems that only observables could meaningfully be said to exist. If somebody asserted the existence of a physical object without being able to specify the kind of sensations which a hypothetical observer would have if that object existed, he would rightfully be accused of talking nonsense. Of course, a common sense object such as a tree could exist even if it had never been observed by anybody; but it would have to be logically possible to observe it, otherwise no possible facts could count as evidence against its existence. However, as we pass from the language of common sense to the language of science, we notice that scientists speak of the existence of entities such as electrons and atoms which might be argued to be in principle unobservable. Consider, for example, Newton's corpuscular theory of light, according to which light consists of particles which, when illumination occurs, are transmitted with great rapidity from the light source to the illuminated surface. Since these hypothetical particles are, as it were, the messengers that make surfaces visible by being reflected from them into the eye, it is obviously impossible to see them. Again, the early wave theory of light postulated the existence of an ether as the medium transmitting light waves; that is, vibrating ether particles were postulated to make light transmission intelligible. Surely, it is in principle impossible to see such ether particles.

Since there is no possible way of observing such scientific objects, some philosophers of science and scientists have characterized them as fictions that do not "really exist." This, however, is a misleading formulation. For, surely, atoms are not fictions in the same sense in which, for example, mermaids are fictions In asserting the fictitiousness of mermaids we assert no more than that there are not any, except in the imagination, but there would be no contradiction in the supposition that they are observable. But, if whatever is observable in the ordinary sense must have secondary qualities (such as degree of hardness, degree of hotness, color), then atoms, which by definition have only primary qualities (mass, volume, velocity, etc.) are not observable in the ordinary sense. Hence, in asserting that they exist the scientist could not mean the sort of thing one would mean if he asserted the existence of mermaids. The testability theory of meaning bids us to discover the meaning of such assertions of the existence of unobservable microcosmic things by asking what sort of facts the scientist regards as evidence for the truth of such assertions. What, for example, is the scientist's evidence for the truth of the assumption, involved in the kinetic theory of gases, that a gas consists of perfectly elastic molecules which obey the laws of Newtonian mechanics? Surely, it would not be possible to observe a pair of interacting molecules, to measure their kinetic energies before and after impact and thus to verify that the molecular impacts are perfectly elastic. The kinetic theory is verified indirectly by deducing from it empirical laws (such as Boyle's law) which have been or can be experimentally verified. What other cognitive content could the assertion of the existence of those gas molecules have but that the kinetic theory explains experimentally verifiable laws of the behavior of gases? The paradox, then, that statements asserting the existence of unobservable scientific objects seem meaningful while in everyday language statements of the form "x's exist" are logically equivalent to statements of the form "x's are observable," is easily resolvable by recognizing the ambiguity of the phrase "x's exist."

It has been seen that many grammatically correct statements which seem intuitively nonsensical turn out, upon analysis, to be implicitly contradictory. Now, it might be argyed that if a statement is contradictory it is false, and if it is false it is certainly meaningful. However, this argument against the thesis that much of what is intuitively regarded as nonsense in spite of its grammatical correctness js really disguised contradiction, does not seem to be conclusive. For, it is only factual falsity that entails significance, and contradictions, it is analytic to say, are not factually false. At least, if a significant sentence indicates a possible state of affairs, contradictory sentences, it is again analytic to say, are not significant. Moreover: if intuitively meaningless statements turn out to contain a contradiction, and one feels that contradictions are meaningful (may be on account of the rule that a conjunction is meaningful if each conjunct is meaningful), it is of course entirely possible to say: well, our intuitive feelings must have misled us. Yet, for the philosopher in search of an adequate analysis of the concept of non-syntactic nonsense, actual usage is presumably the criterion of adequacy. Hence, if there are many usages in which implicitly contradictory statements are characterized as meaningless, the least we have to admit is, that, in one sense of "nonsense," it is correct to say of a contradictory statement that it asserts nonsense; even though such an admission conflicts with the rule, which many a philosopher would not like to abandon, that if p is meaningless, then the denial of p is likewise meaningless.

It might be useful, before concluding, to examine a little more closely the two semantic rules which, as it seems, we have violated, viz. a) if p is significant, then — p is significant; b) if p is significant and q is significant, then p.q is significant, no matter whether q is logically independent of p or not. Let us assume to begin with that these rules are not arbitrary stipulations concerning the postulated usage of the word "significant"— the way the postulates of a non-Aristotelian sentential calculus may be said to amount to an arbitrary implicit definition of symbols functioning as logical constants—but are the outcome of an examination of actual uses of the word "significant." As to rule a), it is very likely the result of an examination of applications of "significant" to factual hypotheses. A factual hypothesis—or, more generally speaking, a factual statement—is often defined (or, at least, characterized) as a statement that could conceivably be false. But to say of a statement that its falsehood is conceivable, is in no way different from saying that its negation is not meaningless. Hence rule a). But why should the latter be extended to logically decidable statements? There is, of course, a sense of "meaningless" in which both tautologies and contradictions are meaningless: this is the sense of factual emptiness. However, while this is the only sense in which tautologies are meaningless, contradictions are not meaningless in just this sense. An implicit contradiction such as "time moves uniformly"^[14] is dismissed as nonsensical because it violates a principle of logic or "law of thought"; but tautologies do not violate the principles of logic; in fact they are substitution instances of the latter.

Essentially the same objections which we raised against rule a) may be brought against rule b). It is evidently derived from logically indeterminate conjunctions of factual statements. What right do we have to extend it to logically determinate conjunctions? Is it that we feel it is, in its general form, a substitution instance of the general principle "if p has a property P, and q has it, then p.q has it"? But surely, such a general principle is false; for example, if p is an atomic sentence, and q is an atomic sentence, it does not follow that "p.q" is an atomic sentence (in fact the opposite follows). Furthermore, it may even be questioned whether b) is valid if its applications are restricted to factual statements. For those, at least, who accept the testability theory of factual meaning, there seems to be agreement that within the language of quantum mechanics the conjunction "the particle P has position x₀ at to and P has momentum ρ ο at to" is meaningless, even though each conjunct, describing a case that is realisable with in principle unlimited approximation, is meaningful. If one sticks to rule b), one will, in the light of the quantum mechanical indeterminacy, have to abandon the testability criterion of meaning. But it is advisable to remember that rule b), being essentially a generalisation of descriptive semantics, certainly does not have any more prima facie plausibility than the much debated testability criterion of cognitive meaning.