Necessities of Nature, Formal Truths and the Blur in Between
I. Contrasting Natural Necessities with Formal Necessities 1
1. Natural necessities are true of physical objects. Formal necessities are true of ideal objects. 2
2. Necessities of nature earn their truth. 7
3. Many kinds off necessities besides. 10
4. Necessities of nature exhaust the relevant content. 11
5. Transcendent determinacy. 14
6. Conceptual nesting. 16
7. Web of belief. 18
8. Immunity to observation. 19
9. Packed virtuality. 20
10. Forms of thought that make their objects, make their truths. 22
11. Outcome. 23
Necessities of Nature, Formal Truths and the Blur in Between
I. Contrasting Natural Necessities with Formal Necessities
The necessities of nature are notably unlike formal necessities. I talk first about statements not about the realities stated. Contrast: "Magnets attract iron," "E=MC2," "F=MA," "copper corrodes," "water conducts electricity," with "[(pv - p)> (p> pv - p)]," "A2 + B2 = C2," and "not: xn + yn = zn, where x, y, and z are non-zero integers and n is an integer greater than 2"; and "the bisector of the angle at the apex of an isosceles triangle perpendicularly bisects the base." The first group, though diverse in itself, contrasts with the second, which is also internally diverse.
The two kinds of statements are true and necessary in different senses because they have (i) different sorts of truth-making, (ii) different "compliant-realities" (one spatio-temporal, the other immaterial), (iii) vastly different "objects" (referents of common and proper names), (iv) opposed existential commitments, (v) divergent conditions of certification, and (vi) truth is earned by necessities of nature, and made by inflation for formal truths. I will examine each of those points.
Still, "true" applies to "2+2=4" and "falling objects accelerate," under common umbrella notions, protean, plastic notions, like "the truth is what to believe," "the truth is what is so," and "it is what is right to say and right to think." Similarly, "necessary" applies to both under umbrella notions, like "what has to be," "what couldn't fail to be," but means different things and has different explanations. These are what Gilbert Ryle called polymorphous concepts. Such general conceptions are the outcome of our understanding how to use "true" and "necessary" in varied contexts where the words are semantically captured and differentiated into meanings that are prior to the pliant notion. As a result, our acquisition of such notions is dependent on our knowing how to use the words in differentiating contexts, and not the reverse (as philosophers have usually supposed). So, we have to explain the meanings in particular contexts from the peculiarities of that kind of discourse (say about numbers, or being, or opinions, or experience), and not, chiefly, from the pliant common notions. Analogue notions are not prior to experience (either in discourse or observation) but the result of it.
1. Natural necessities are true of physical objects. Formal necessities are true of ideal objects.
Necessities of nature, though otherwise various, have certain traits of common opposition to formal necessities (which are not all of a kind, either).
I include among natural necessities (a) observations available to common sense ("iron rusts," "cement cracks," "stars twinkle," "water expands with freezing," "animals die"), and (b) elementary science ["only mammals have hair," "all and only birds have feathers," "insects are cold blooded," "mammals suckle their young," "birds are warm-blooded but are not mammals," "rusting is oxidation," "heated gasses expand"), and (c) mathematized scientific idealizations, ["E=MC2," "F=MA"], and equations that relate physical idealizations to one another (like temperature to the volume and pressure of gasses, and the mass, and momentum of real objects), for instance "g=at2," or the physics of rocket propulsion.
Many will say that none of these statements are necessary. I won't stop to argue that point now, only remarking that in each case "the opposite cannot happen" any more than there can be a perpetual motion machine, and that is not to be explained by how we think or talk but the reverse.
Necessities of nature are all "existentially embedded" in contingent things, and known (or found out) through experience, even those "pure" laws, like the laws of pure mechanics, that apply directly to idealizations and indirectly to real objects, accelerations, velocities and momentum.
The general words in natural necessities, like "expands," "distributes," and "com-presses," apply to observable particulars: to iron, to the freon in my refrigerator, to air pressures from a compressor, to the resistance in a given wire, to acceleration of a given engine, to the 1-inch self-compression of a 1000-foot steel column in a building.
The abstractions, the idealization, are fashioned to fit real things. Generalities are thought to be "true of" things they are not, strictly, about: for they are recognized by the scientists -- I don't know about the philosophers -- to be, strictly or really, about idealizations. So, I don't find deviation between theoretically predicted values and actually observed values as hard to reconcile as N. Cartwright does . The objects of pure theory are all obtuse abstractions, abstractions with formally neat features that have been substituted for rough and complex features of real things. Physical abstractions arrange the phenomena into background and foreground features so that idealized values apply to the foreground (which may have augmented, or simplified, or replaced elements), while the divergent real values apply to the unarranged phenomena under all their forces and constraints. Thus, when calculating amps by dividing watts by volts, we ignore the age of the wires, the "dirty" volts, the variation in currents as background. And in most cases of adding a fifteen hundred watt toaster to a fifteen amp circuit with several other lamps on it, we'll get a prediction we can rely upon because our tolerances are so generous. So the objects of empirical scientific description and prediction are material objects, though the theory strictly is about idealized objects.
In contrast, the objects of formal truths (say, truths of propositional calculus, arithmetic, set-theory, Q.M.L., category theory, group theory, topology, the rules of chess and games of cards) are never material objects, not even the real but idealized aspects of material objects. Formal objects are not obtuse abstractions (with made-up features replacing real features) but are pure abstractions made entirely by abstraction for theoretical purposes. Formal truth does not depend on there being any particular material objects. (You can play cards without any physical cards and do mental arithmetic.) Nor can a formal truth be falsified or refuted by any physical condition. Real conditions and shapes do not satisfy, but only approximate, the formal definitions. There are no exact rectangles or pure doughnut shapes, but formulas ("rectangles have equal diagonals") do apply as precisely as we can measure. And no arrangement of physical conditions is sufficient to make any such statement true, nor can such formal truth conditions be stated as physical conditions.
Individual material objects exist contingently, while formal objects "exist" necessarily but intentionally. Some people doubt both parts of this. I say they are confused. There are no numbers or sets in the same sense that there are humans, or even shelters for humans because humans occupy places, numbers and sets do not; humans happen to exist; it cannot just happen that there are numbers, too [see Ross, 1981]. Even though Quine [1987(b) and 1984] and Field  valiantly insist that "exists" is univocal (or reducible to univocal statements about singulars), I can only repeat that speaker intentions and convictions have nothing to do with it: words are captured by their semantic contexts. Quine and Field cannot defend their desert ontologies by re-interpreting what we mean. Just as I don't mean by "Socrates is human" that Socrates participates in the Platonic Form Human, so I don't mean by "there are natural numbers" anything like that singular things arrange themselves numerically or that there are numbers just as there are Quine and Field.
Additionally, material things cannot be exact (that is, entirely comprehended in conceptions or definitions) as formal objects have to be. Material things overflow our conceptions; their hidden, overflow necessities are the basis for solving a number of problems as we shall see. There is nothing more to a hypercube than is logically contained in its geometric definition and system. There has to be more to a sugar cube than is contained in our conception of it, no matter how comprehensive it becomes. For one thing, our knowledge of the basic necessities of nature is not encompassed within the meaning of words for such things, and our knowledge of the necessities of nature, for instance of the behavior of crystals will expand for a long time, eventually to include things not presently within our grasp.
Finally, what is true of formal objects is true because the form of thinking compels it, whereas the necessities of nature -- the ones not about our own thinking -- are true on account of realities independent of our thinking. Truth, with formal necessities, is caused by the form of thinking -- as form, not as a particular act of a particular thinker by which we are justified in affirming them. That means anyone who does the thinking in the same way will get the same results. No one doubts that about arithmetic, or speaking French, or following the rules of chess, as against playing a game of chess.
Not so, with natural necessities except for some self-referring thoughts. Although, thinking that P where P is so is what makes the thought true, it is not because of the form of the thinking but because reality is compliant. The same holds for contingent truths, too. The formal objects have to be as much made by thinking as the assertions are authorized by a form of thinking; otherwise we would not have entia rationis, beings consequent upon thought. Why say formal objects exist necessarily when, on this account, there aren't actually any until they are thought of, and presumably wouldn't be any if all mathematical thought ceased and all records of it were destroyed? Formal objects seem to be especially contingent because they are entirely dependant on ways of thinking; yes, but they are necessitated, like a sixty-four square game board, by the judgments that are true of them.
With formal truths, the thinking that justifies our asserting what we do, also assures the truth of what we think by causing compliant intentional objects. Not so with either necessities of nature or contingent truths. The thinking that justifies our affirming "iron rusts" is not what assures that such thinking is true. The thinking that justifies our saying "iron rusts" or saying, "food now is cheaper than a century ago," does not cause or compel what we say to be true. But the thinking that justifies our saying "2+2=4" does assure and compel the truth of what we say. (At least when we are not reciting arithmetic by rote, but seeing the truth that lies within our assertions.) In both cases our thinking aright causes a truth, but thinking that way causes what we think to be true only for formal truth.
2. Necessities of nature earn their truth.
Necessities of nature earn their truth from the real conditions of things (that iron oxidizes, is magnetized; that some things cost more now than a century ago), whereas formal truth arises by inflation, by belonging (with various logical constraints) to blocks of authorized assertions. I do not think logic, mathematics, or other forms are innate, or are the structure of rational thought. Rather, we make the forms of thought that are expressed as mathematics, just as we make the forms of thought that are expressed as musical compositions. Aristotle was probably right to suppose that mathematical thinking begins with our natural abstraction from various quantitative aspects of things, which aspects are made into the subject matter of families of judgments and reasoning. But the levels of abstraction have over the centuries been so refined that for many purposes the original basis for abstraction is no longer relevant to the pure mathematical practice.
You might think "an object in uniform motion continues forever unless acted on by another force" would be certified or decertified directly in experience, but it is too general for that. Nothing has ever moved forever. And nothing has ever been in motion under a single force. It tells us what an object in motion would do, not what anything actually does. So, its truth has to be by inheritance (or outright coinage), and not earned from some object's moving forever under a single force. Here is a truth supposedly true of everything in nature that has never had an exact case. That should alert us that although truth may be earned from physical realities, and although we say that what makes "snow is white" true is that snow is white, we should not say the compliant reality is what productively causes the statement to be true. It is what judgment conforms to. Thinking is always the cause of truth (and of falsity, too), but diversely for diverse things.
Continuing the contrast, formal truths have (a) no real objects, (b) no relevant observations, (c) no pragmatic decertifiability, (d) no "working" that fails to determine truth, and (e) no explanatory force without truth. Instead formal truths have truth by inflation, by an internal condition within our thinking that provides complete certification by the "objects of discourse" but excludes their having independent objects (or natures), because their designata cannot have any attributes not constructed (or otherwise "contained") by the system. That's an enormous difference, enough to differentiate "necessary" and "true" [see Ross, 1981, pp. 33-34, 47-51]. "Necessary," as "true under all consistent conditions," is satisfied so differently by the two groups (NN and FT) that the predicates "is true" and "is necessary" apply analogically, though with common focal notions of "being so," "being completely certified," "having to be so," "being the way things are," etc.
Formal truths require formal objects that cannot be real. Yet such objects have to "exist" under all consistent conditions, and belong to replete domains, like numbers and sets, that are "the same in all possible worlds," and "form a fixed universal domain" (where that is coherent). Such objects are entia rationis. That is exactly the opposite from empirical generalizations for which there is no "fixed domain for all worlds" (neither of objects nor predicates nor kinds nor relations nor metapredicates nor kinds of kinds), because it is contingent that there are any material things at all, or things of any given kinds.
So, natural necessity cannot be "certification by the designated objects under all consistent conditions," but only "certification actually, and whenever there is a suitable subject, without de-certification under any consistent conditions." And even that is not a defining condition because singular judgments of existence with proper names, "Socrates existed," satisfy the conditions stated but are not naturally necessary. Nevertheless, those are obviously different truth-conditions, and they are meaning-relevant [see Ross, 1981, Chapter 8]. So, "is true" and "is necessary" apply to necessities of nature and to formal truths in different senses.
3. Many kinds of necessities besides.
The contrasts between formal truth and natural necessities have intermediates. For example, metric theory for diatonic music has formal and arithmetic features, natural and conventional features (e.g., it applies both to sounds and to notes). Theories of "card counting" in gambling have pure mathematical and natural and conventional features, too. The rules of poker, gin rummy, and the like, are like formal truths because they are about made-up objects, objects all of whose features are caused by the truths; whereas the rules of baseball are more like laws of nature because they apply to real events of hitting, running, etc., but under a made-up classification by which the real events fall under the rules, so that "crossing the plate" is more the product of an umpire's judgment than a physical intersection. That means there are other kinds of necessities, say, ones whose designata are not completely grounded physically, but are not merely intentional objects either; e.g., games of cards and tricks and plays in a card game.
Other features of things, though fully grounded physically are also partly made up: like, the rotational laterality of a building (how much it tends to move on its base in a wind velocity up to the 150-year maximal recorded wind), or its vertical stability (how much it tends to tip over like a cereal box, in a 150-year maximal recorded wind). As defined, there were no such conceptual properties before we had wind-velocity records; but there was always a complete physical basis for the tendency of a building to tip and to turn, and it was determinate how much a certain sort of building would turn or tip, though we did not know it. What about "lamination," the property of very long steel beams more than two or three inches thick, "to slice into thin layers" like pastry because of "locked in stresses," a property that caused the West Gate Bridge in Australia to collapse [see Salvador, 1980]? That seems to have been one of the hidden necessities unknown to materials science until it dramatically displayed itself. The "space" between purely formal truths and simple necessities of nature is crammed with intermediates, sharing the features of the extremes in various arrangements. So, if "so" and "has to be so" are plastic, as are their opposites, the fact that the classification we use is made-up is irrelevant to whether we are stating genuine necessities when the predicates are fully grounded physically or fully grounded in other observable or verifiable ways. "Fully grounded physically" is a predicate that promises reducibility without remainder; often we can "see" it, without providing it: e.g., the "shine" of paint is visible, but most of us have no idea what physical basis it has (which vary, of course, with the kind of paint).
What assures "coming out right," "verification under all relevant conditions," differs in the two extreme cases and also in the intermediate ones. But at the one extreme "coming out right" has nothing to do with what happens, but with how we do the thinking; in intermediate cases, it has to do with both thinking and happening in different proportions: for instance, whether a base-runner is "safe" in a given play, and at the other extreme, what we think, to be true, is what happens.
4. Necessities of nature exhaust the relevant content.
Necessities of nature, like "humans can think" earn truth by saying what is so, and have necessity by being so "no matter what" (even though they rely on what ought to be as much as on what is). They are not made so by some conceptual or linguistic nesting or by being the structure of "the mind" [see Chapter 3, "analyticity" and "a prioricity," below]. Rather, analytic and conceptual inclusions are results of purported cognition, not the explanation of it, and have no sure grip on truth. Nor are necessities of nature (G = 6.7x10-8cm3/gm-sec2) so "no matter what," having objects that exist "no matter what"; but instead, they are so no matter what because their objects (and situations) are all the relevant possibility with content. The natures of actual things and what is potential to them, are all the relevant possibility with content [see Chapter 2]. That is, what iron can do and can have done to it is all the relevant possibility with content for iron. Anything beyond that fails to involve iron, even if we say it does.
It makes only verbal sense to say "suppose iron were not magnetizable"; there is nothing that determines "all the way down" what the overflow necessities would be, so nothing definite has been supposed. That explains why we can't just make-up real possibilities by making-up consistent descriptions; for if the conceptions are defective or deficient, no definite overflow necessities will be incorporated by reference, and so no determinate real possibility will have been supposed. Now one might ask why real possibilities have to have every necessity determinate. The reason is: real things, like iron, do, and, so, a situation supposed to involve iron but indeterminate as to natural necessities will fail to involve iron. If every necessity were not determinate, the supposition can be satisfied by situations that not only differ in their necessities but are opposed. Which situation then will have been supposed? By exhausting all the potentiality with content, necessities of nature have verbal contraries that are referentially vacuous [see Chapter 2]. That will explain why the opposites of the necessities of nature are impossible.
With formal truths, complete certification (that is, "that everything authorized to be said holds of its object under every relevant condition") is enough, because the designata have no traits that are not constructed by what is true, the authorized statements. That is because formal objects are constituted as objects by the authorized assertions, both actual and potential. What is authorized to be said and is fully verified by the designata is what turns out to be so, and what turns out is all there is, no matter what. Counter-situations are inconsistent for the same objects (where "the same" is determined by the original conceptions). Not so with necessities of nature; there are verbally consistent counter-situations but they have no content. That distinction is the basis for two kinds of impossibility, the inconsistent, and the empty [see Chapter 2]. Someone says, "Suppose that tree had been made of plastic" or "suppose iron were not magnetizable"; it is easy to see that those suppositions are impossible, but not because of inconsistency. They turn out to be vacuous.
5. Transcendent determinacy.
Real things overflow whatever we can say. It takes a lakefull of reality to make a drop of truth.
Real objects are transcendently determinate in four ways, and in two respects: (i) they have two kinds of "de re overflow"; and (ii) the compliant reality is infinitely more determinate than what is said ("my car started"), yet (iii) incompossible physical realities are equally compliant realities for the same statement. So, on a cold day, when I might mention my car's starting, whether it is right in front of my house, or across the street, with the door open or closed, are all equally compliant, though mutually incompossible. By contrast, formal objects have no necessities except for what is contained in the thought system; and have no accidents at all. Certainly, there are no incompossible situations that are equally compliant with a formal truth. Formal objects cannot, then, be empirical, material things. They cannot be individuals, either. As a result, quantification in logic over formal objects is equivocal with quantification over real individuals. It is a harmful fiction to say scalene triangles or numbers can have proper names and, thus, be objects of quantification for first-order logic; "right triangle with 1-inch, 2-inch, and 3-inch sides" is a kind. Even hands of cards have no accidents as hands of cards (arrangements under a set of rules), whereas "a double play" in baseball has overflow necessities (e.g., it can't be done after the second out, or by the team at bat) and accidents, e.g., what stadium, what players, what game, what inning.
Beyond the two classes (and, thus, four kinds) of transcendent determinacy so far mentioned, there is another, of three kinds. All empirical truths, whether necessary or contingent, have truth-conditions that transcend verification. Trivially, each implies an unending list of statements most of which we cannot practically check, including whatever is necessary that we may never even know about. More significantly, for any predicate applicable to a thing, there is another, incompatible with the first, though perhaps far less "natural" to use, equally well justified or verified on all the data, so that no matter how far along in time, or space, there are, in principle, incompatible competitors between which the data cannot, at that point, choose (that is cosmic grueness).
There is a more interesting facet of that point that Duns Scotus noticed (c.1300): that an ordinary color or shape predicate, as well as a dispositional one, applied to a physical object attributes not a momentary but a continuing and indefinite condition which exceeds the period of observation, for which fully verifying data cannot be present at the time-of-predication. Thus, physical object predicates raise the same difficulties as the general problem of induction and the under-determination of hypotheses: they cannot be conclusively verified.
Formal objects have no features not entailed, or otherwise contained, by their thought systems; and the entailment is not just by paradoxes of implication, because arithmetic would then contain geometry, but by restriction to some "relevance" relation. The objects conclusively verify what is authorized to be said of them. There is no overflow of necessities or of intrinsic accidents. What is true of formal objects, after the initial postulates, is true by way of a logical connection to the initial conceptions and assertions. That, of course, opens questions about hidden connections in incomplete systems, since belonging to the system requires that the unprovable theorems be necessary conditions for all other truths, and yet not be generable by logical transformations from an initial axiomatization.
Real objects (other than God) are not like that. Real objects have features, even necessary ones, not included in the meanings (or logical consequences thereof), of their common names. Furthermore, for any empirical truth, there has to be an infinity of things to hold of its objects that are not contained, either conceptually or by logical implication in what is said. Therefore, for anything you say to be true (empirically) an infinity of things not contained in what you say have to be so. Moreover, as I said above, mutually incompatible infinitely determinate realities would equally well comply with any empirical truth.
Those features are the transcendent determinacy of real being by which reality overflows whatever we can say.
6. Conceptual nesting.
Certification for both kinds of necessities has to be complete, in that each in its way exhausts all alternatives. But how they do that, and what counts to certify or de-certify differs. Formal truths stand and fall in entire systems because truth-making for individual cases is from the truth-makers for all and for all together (from the form of thinking). So when one goes, they all go.
Natural necessities are clustered, too, ("iron rusts," "aluminum pits," "glass breaks," "copper corrodes") because the explanatory microstructures are frequently common or similar, and for other reasons [see next], but not because there is a common, single form of thinking that causes their truth or a single cause of their exhausting all the alternatives (necessity).
Natural necessities once firmly believed, get bound together conceptually. Unbound terms [q.v. Ross, 1981, Chapter 7] like "iron rusts" get craft-bound meanings, ("rust is hydrated oxide of iron formed by exposure to moisture and air"), so "iron" comes to mean "a lustrous, ductile, malleable, silver-gray metal, good conductor of heat and electricity, attracted by magnets and easily magnetizable, atomic number 26, atomic weight 55.847, with 2/8/14/2 electronic configuration, that flows at 1538 degrees Centigrade." And "rusting" is a kind of oxidation ("any reaction involving the loss of electrons to oxygen, in this case from an active metal in contact with air and water"). "Rusting is oxidation" is first a hypothesis, then an empirical truth, then "obvious" and eventually regarded as analytic; finally, to the educated "iron oxidizes" seems to be observable, as does the oxidation of battery contacts and tarnished silver. Development from new truth to meaning-inclusion happens frequently and very quickly, especially in new technologies, like computer-talk [see Chapter 3, Part I, Sec. 6]. For instance, is it empirical or analytic that "computers require a machine language"? Like Quine, I think there is no definite line between the analytic and the empirical, though I think there are clear cases on both sides of the mushy band, and that all a clear case of analyticity amounts to is the encoding of a conviction (that may not even be true, or even be common anymore) into a meaning-inclusion [see Chapter 3].
Purported natural necessities stand or fall in families, or at least in clumps, because they have common explanations, common observational bases, and because natural necessities (as thought contents) come to be linked in meaning to one another. Thus the theory of caloric and the theory of the aether fell as wholes. Contingent truths cluster the same ways. They stand or fall, are revised or replaced, as a group and as individuals in response to tests, measurements, photographs, x-rays, experiments, and other observations of things. Nevertheless, it is not easy to say what all those observations have in common by which they depend on the real natures of things; still, as we renew what we mean by real natures, thinking more of software or a program for material behavior, we will see such tests as ways of detecting promised behavior, or being sure it has failed to appear. Those tests, all of them, are irrelevant to settling whether something is a formal truth, and are intimately connected with whether something is a natural necessity.
Although formal truths are made true, and their objects are made to be, by human thinking, it is not just anyone's thought that counts. Miscalculations don't affect arithmetic, nor do mistaken proofs of supposed theorems, nor does the discovery of a statement we don't know how to prove (Goldbach's conjecture) or even the discovery of an unprovable theorem (though how that is to be done in a particular case is unclear to me). It seems that we have to regard all the objects of a given kind of formal thinking as in being potentially (as entia rationis), whenever enough of that kind of thinking is actual to settle the features of the rest of the objects even if they are unknown: so the next prime number is settled even though we don't know which number it is yet (the next one after the latest gigantic calculation) and it was settled and was potential as soon as the system was determinate enough for the notions of prime and real numbers to be articulate.
7. Web of belief.
Truth-bearers are entangled. They belong to clusters that are legitimated or banished as groups, with "core" truth-bearers, better insulated from being "picked off" by a clashing experience than "fringe" beliefs. Some people arrange their most misguided, and aggressive and depressive beliefs so that nothing can pick them off: "Poverty is the fault of the poor"; "Crime is caused by moral corruption, or mental illness, or poverty, or lack of religion, or too lax punishment." The best insulated and most intolerant beliefs are not typically reduced, as Quine seemed to think [1991 and 1951], to meaning-inclusions, but are regarded as genuine insights without which another person is considered stupid or gullible. So, in my view, the central strands of various webs of conviction are a mixture of genuine insights and outright prejudices with ample emotional and non-conceptual content. In any case, as Chapter 3 makes clear, I agree with Quine [1991:272] that it is an empirical matter that determines which are the elements of word meanings and which meaning inclusions there are; and, I add, such relations can come about without tracking what is true.
Still, the "web of belief," a fabric whose center is protected from individual falsification and whose edges are more easily clipped or torn by experience and test, is a revealing metaphor, especially if we think of our convictions as multiply inter-tangled webs. The kinds of insulation of the centers are quite varied, emotional, and peculiar to the subjects of discourse, as are what counts for decertification or clipping at the edges. So, the resistance to antisepsis in medicine during the last century, and the resistance to procedures strict enough to prevent 50% of the patients entering a hospital nowadays from contracting a new infection there, or in the refusal to let patients control their own pain medications, are based on attitudes that amount to beliefs that cannot be traced to any particular convictions that evidence can unseat directly. Broad changes in societal awareness and evaluations (as diet consciousness and reduction of adult cigarette smoking illustrate) change behavior, not directly in response to evidence, but in response to peer pressure, fashion, and social and political evaluations, sometimes supported by reported evidence. As far as there are analogues, that I know of, in the formal sciences, there are fashions of proof, elementariness [Rota, 1991, pp. 490-493] and simplicity, not differences of conclusion.
8. Immunity to observation.
When a false formal proposition is derived the whole system goes dead. That is because inconsistency is "shorting out" and a syntactical defect, if it does not cause an inconsistency, will turn the system into gibberish. With the made-up systems in between, like some card games with indeterminate situations, monetary systems, and accounting or legal systems with internal conflicts, we sometimes keep the system, as we keep a legal system in which killing a viable fetus by attacking the mother is murder, though the fetus is not legally a person until born, or as in some sports, we empower referees (or use T.V. replays) to re-decide the problematic situations.
Formal truths hover above the web of empirical belief, providing structures and links (often semantic networks) among empirical beliefs, and are themselves impervious to tests, measurements, experiments, statistics, standard deviations, x-rays, thermography, ultrasound --and all the other ways we settle whether a claim is true (except for tests of consistency). Formal truths have "immunity to observation."
Nevertheless, formal truths can come under indirect observation. An abstraction, Euclidean geometry, may not fit experience, say space-ship navigation to the moon, as well as some other device. Traditional symbolic logic, for example, needs to be replaced with more supple devices that don't incorporate obvious falsehoods, for instance, that you can have proper names for abstract objects that can't be individuated. We may notice something that needs a fitting quantificational abstraction, say, the pattern of smoke, and thus invent a formality for it, e.g., the mathematics of chaos. But that's all indirect and connected with utility, not truth or necessity. We invent formal truths for utility as well as beauty. Whatever our objectives, formal truths are not observationally falsified or discredited.
9. Packed virtuality.
Natural necessities (and all empirical truths) are "pregnant with consequences," not in the focus of thinking, but relevant, even necessary for truth. The consequences are not confined to what they contain syntactically and semantically (or entail logically). Pragmatic outcomes known to experience, and many unnoticed, are crucial: ice is slippery; lots of coffees and teas taste good. There can't be a cat on your couch if it is not made of cells, has not living insides, and actual brain states. Yet, ordinarily you would not think of these things because they have no role in our discourse. Nor are these things "contained" semantically or, otherwise, formally in the situation thought of.
Formal truths are quite otherwise. The objects have no features other than those implied or otherwise contained. They do not have pragmatic consequences or background behavioral contexts. Empirical thoughts do not "contain" everything that is logically implied by them, and, as I said, require much more than is "contained." Formal truths of suitable complexity, however, like arithmetic, do contain everything that is implied, and more; but they do not have pragmatic outcomes presupposed in discourse.
Here is an example. When I say, "That table is strong enough for you to stand on it," among the thought-consequences not thought-of are that the table will not change much within the time it takes for you to step up on it, nor will your weight. All physical-object judgments are projective that way. They mean practically (not linguistically) whatever we rely upon for their being so. "It is blue," referring to a book cover, means it will, under normal conditions stay that way, though how long is a matter of variation among materials. Frequently we discover we were relying upon some feature only when our assertion fails because that background feature, which we had not even adverted to, is missing; some accidents happen that way, as do some failings at cooking.
Empirical thinking is more than the crisp thoughts [see Quine, 1991 and 1951] that bear truth or falsity. Every crisp thought needs an envelope of expectation and reliance. Every "foreground thought" has an unfocused background of what you expect and rely upon when you think as you do [see M. Polanyi, 1966]. A whole family of crisp thoughts can drain away if the underlying convictions are punctured, as happens when people lose faith, fall out of love, or feel betrayed.
10. Forms of thought that make their objects, make their truths.
No one doubts that imagination can make objects and make judgements true of such objects. Thus, Walt Disney made Mickey Mouse and the rest of the stories.
It used to be thought that axioms, like Euclid's, were "about" some ideal objects, namely, plane figures, that "made them true," where the axioms were also self-evident and where "a triangle is an enclosed three-sided figure" was thought to be no more obvious than "the sum of the interior angles of a plane triangle is 180 degrees." That was a mistake. The theorem that gives the sum of the angles is not only less obvious than the definition of a triangle, it can be replaced. Secondly the axioms and definitions are not truths about objects with pre-given features, but truths that make their objects. Thirdly the objects are not "ideal," in the sense of being really existent, outside space and time, but only ideal in the sense of being made by human thinking and clearly defined; and the "objects" are not individuals, even when images, sketches or drawings for them are.
Self-evidence is just transparent obviousness, caused by our own thinking, rather than by some feature of the objects. The more completely one understands a formal system, or a quasi-formal one like chess, the more statements will be transparently obvious in that one sees all at once why they are so, without any reasoning being required.
Formal necessities and natural necessities differ in what is relevant to certification, in what is contained in thought, in the sorts of objects referred to, and in what counts as overflow conditions not contained in the meanings of common names. Natural necessities, like all empirical statements, are transcendently determinate in all four ways I mentioned, whereas genuinely formal truths are not. Between the extremes of formal necessity and natural necessity, there are varied intermediates sharing some features of the one extreme and some of the other. All the features mentioned are meaning relevant to "true" and "necessary" [Ross, 1981, Chapter 8]. Thus pairs of statements that differ in such features, though they are true and necessary in a very broad sense, will have to have different particular accounts of truth-making and necessity (and impossibility) to accommodate natural necessities and real impossibilities and the variety of intermediates like the necessities and impossibilities concerning poker, baseball, law, imaginary objects, the future, and what might have been.