Nolt (15.3.1) Logics, '[Basic technique of supervaluation],’ summary

 

by Corry Shores

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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own. As proofreading is incomplete, you will find typos and other districting errors. I apologize in advance.]

 

 

 

Summary of

 

John Nolt

 

Logics

 

Part 5: Nonclassical Logics

 

Chapter 15: Mildly Nonclassical Logics

 

15.3 Supervaluations

 

15.3.1 [Basic technique of supervaluation]

 

 

 

 

Brief summary:

A three-valued semantics was one way to deal with a number of situations where bivalence was unsatisfactory. A third value, I or indeterminate, was used. But instead of that third value, we can keep just T or F, although certain formulas can be assigned neither of these two values. These cases of no value are called truth value gaps. To evaluate the truth value of formulas with component truth value gaps, we can use a technique invented by Bas van Fraassen called supervaluation. First we begin by making a truth table and putting in the values we know, but we place gaps where there is no value. This is called a partial valuation. Next, we fill those gaps with T or F such that every possible combination of values is given. These are called classical completions. Finally, we make a supervaluation of these classical completions in the following way. If all the classical completions compute the formula as true, then the supervaluation is true. If all the classical completions compute the formula as false, then the supervaluation is false. And if not all the classical completions are either entirely T or F, then the supervaluation does not assign a value to the formula. Here is the formal definition of supervaluation:

DEFINITION A supervaluational model of a formula or set of formulas consists of

1. a partial valuation S, which assigns to each sentence letter of that formula or set of formulas the value T, or the value F, or no value. We use the notation ‘S(Φ)’ to denote the value (if any) assigned to Φ by S.

2. A supervaluation V_S of S that assigns truth values V_S (Φ) to formulas Φ according to these rules:

V_S (Φ) = T iff for all classical completions S′ of S, S′(Φ) = T;

V_S (Φ) = F iff for all classical completions S′ of S, S′(Φ) = F;

V_S  assigns no truth value to Φ otherwise.

(Nolt 415-416, boldface his)

 

 

 

Summary

 

Nolt recalls that we can have certain motivations to reject simply two logical values, true and false, and to add some third one [see section 15.2]. But we have further motivations that would make us want to hold on to just the truth and falsity without adding a third or more. One solution would be to say that propositions can be either true, false, or neither true nor false. Sentences with this third option “are said to exhibit truth-value gaps” (Nolt 414 boldface his). [The idea here might be the following. We have reason to think that certain propositions do not fall neatly under the valuations of true or false. But we also have the intuition that truth is something that admits of no quantitative variation. It is absolute in some sense. So the notion of a third value might be mathematically modelable, but philosophically speaking it is too difficult to conceive and believe strongly in the reality of. So this solution keeps the idea that there are no other logical values than true and false. But it deals with the problematic cases by saying that they simply have neither of these values, rather than saying they have a third value.] Nolt will now examine how we can calculate truth values when a component proposition lacks a value (414).

 

[Recall Bochvar’s three-valued semantics from section 15.2 where any instance of a constituent indeterminate value would make the whole formulation indeterminate.] Nolt notes that one option “would be to declare all such formulas truth-valueless. The result would be just like Bochvar's three-valued semantics, except that in the truth tables where Bochvar’s logic has an ‘I’, this semantics would have a blank” (415). [There were other ways to arrange the truth values for three-valued semantics,] or instead we “could create truth tables like those for other forms of three-valued logic” (415). But, Nolt says, “this approach is hardly novel, differing in no essential respect from three-valued logics themselves” (415). [The problem with that might be the following. Suppose we simply mimicked the three-value truth tables, only replacing the ‘I’ value with a blank. We might then ask what really the difference would be between them other than how we philosophically interpret the meaning of the third option. In other worlds, this “gap” option would be little more than a different way to interpret the indeterminately three-valued semantics.]

 

Nolt says that instead we should consider Bas van Fraassen’s technique of supervaluation, which “unlike most multivalued logics, preserves the validity of all classical inference patterns” (Nolt 415). The first thing we note about supervaluations is partial value assignment, where we assign to sentences letters either T, F, or no value.

In propositional logic, a supervaluational semantics assigns to sentence letters the value T, or the value F, or no value at all. We shall call such an assignment a partial valuation. (Note that at one extreme some partial valuations assign truth values to all the sentence letters of a formula or set of formulas and, at the other, some assign no truth values at all.)

(Nolt 415, boldface his)

[In other cases, we compute the truth value of complex formulas on the basis of the truth values of their component formulas.] In supervaluations, we will calculate the values of complex formulas by means of classical completions of partial valuations, rather than by calculating by means of “truth tables directly from the truth values of their components” (415).

 

[The basic idea with this technique seems to be the following. Although we might have truth gaps for certain constituent parts of a formula, that may not in the end have any effect on its overall value, were we to give it every possible classical valuation. And thus for some cases, we can determinately say that it maintains classical truth or falsity, despite the gaps in its internal values. So we begin with a partial valuation of a complex formula, meaning that some of the sentence letters will not originally be assigned a T or F value. The second step is a classical completion, where we exhaust every possible arrangement for these blank values to receive either T or F. The next thing we do is we see if regardless of every possible substitution, does the whole formula compute as either T or F, in all cases? If so, we say that the formula is T or F, depending on what all the evaluations compute it as.  If instead they are not all of one or the other value, then we say that it has no value. Giving it this final valuation based on the classical completions of its partial valuation is called its supervaluation.]

A more interesting method – and one that, unlike most multivalued logics, preserves the validity of all classical inference patterns-is the technique of supervaluations invented by Bas van Fraassen. In propositional logic, a supervaluational semantics assigns to sentence letters the value T, or the value F, or no value at all. We shall call such an assignment a partial valuation. (Note that at one extreme some partial valuations assign truth values to all the sentence letters of a formula or set of formulas and, at the other, some assign no truth values at all.) The truth values of complex formulas, however, are not calculated by truth tables directly from the truth values of their components. Rather, the calculation takes into account all of what are called the classical completions of a given partial valuation.

 

Let S be any partial valuation. Then a classical completion of S is a classical valuation (one that assigns each sentence letter in the relevant formula or set of formulas one and only one of the values T or F) that fills in all the truth-value gaps left by S. In other words, a classical completion of S does not change any truth-value assignment to a sentence letter that has already been made by S, but merely supplements assignments made by S, giving each sentence letter of the given formula or set of formulas a truth value. Since each truth-value gap can be filled in by the assignment either of T or of F, each partial valuation, unless it is classical to begin with, has more than one classical completion. Consider, for example, the formula ‘(P∨Q)&(R∨S)’, and let S be the partial valuation of this formula that assigns the value T to both ‘P’ and ‘R’ but no value to ‘Q’ or ‘S’.

(Nolt 415, emphases his)

[Let us first try to give the partial valuation for this situation in a table format.

As we can see, we have only entered the data that we are sure of. The rest is left out, and thus this is a partial valuation. In the next step, we will fill in every possible combination of values for Q and S, and then we evaluate them on the basis of normal classical valuation rules, in this case for disjunction and finally for conjunction. The result will be that all possible classical completions of this partial valuation will make it true, which means that the gaps would not interfere with classical evaluation of this formula, as it will not matter what values the gaps might have.]

Then S has with respect to this formula four classical completions, corresponding to the four ways of assigning truth value to the sentence letters ‘Q’ and ‘R’. Each classical completion of S is represented by a horizontal line in the truth table below:

The columns under ‘P’ and ‘R’ list only T’s, because these are the values assigned by the partial valuation S, and they are retained in each classical completion.

 

To determine the truth value of the compound formula ‘(P∨Q)&(R∨S)’, we expand S into a new nonclassical valuation V_S called the supervaluation of S. | This is done by calculating the truth value of ‘(P∨Q)&(R∨S)’ on each of the classical completions of S, using the valuation rules of classical logic, as in the table above. If ‘(P∨Q)&(R∨S)’ is true on each of these classical completions (as the table shows that it is), then ‘(P∨Q)&(R∨S)’ is true on the supervaluation V_S. If it had been true on none of them, then it would have been false on V_S. And if it had been true on some but not others, then it would have been assigned no truth value on V_S.

 

A supervaluation, then, is constructed in two stages. First, we define a partial valuation S, which assigns to each sentence letter the value T, or the value F, or no value. Next, at the second stage, we construct all the classical completions of S and use the classical valuation rules to calculate the truth values of complex formulas on each of these classical completions. A formula Φ (whether atomic or complex) is then assigned a truth value by the supervaluation V_S if and only if all the classical completions of S agree on that truth value; if not, V_S assigns no value to Φ. More formally:

DEFINITION A supervaluational model of a formula or set of formulas consists of

1. a partial valuation S, which assigns to each sentence letter of that formula or set of formulas the value T, or the value F, or no value. We use the notation ‘S(Φ)’ to denote the value (if any) assigned to Φ by S.

2. A supervaluation V_S of S that assigns truth values V_S (Φ) to formulas Φ according to these rules:

V_S (Φ) = T iff for all classical completions S′ of S, S′(Φ) = T;

V_S (Φ) = F iff for all classical completions S′ of S, S′(Φ) = F;

V_S  assigns no truth value to Φ otherwise.

(Nolt 415-416, boldface his)

 

As we can see, supervaluation may leave truth-value gaps in complex formulas. For example, if we have a disjunction, and one term is assigned false but the other a gap, then the classical completion will not all have the same value for the disjunction (416).

 

[For the next point, recall Kleene’s three-valued semantics from section 15.2.

(Nolt 412)

] We might be tempted to say that supervaluations follow the pattern of Kleene’s semantics, where we can simply take the value ‘I’ to be equivalent to a gap. So consider disjunction in the Kleene system, where both disjuncts are indeterminate. That makes the whole disjunction indeterminate. Similarly, the classical completion for a partial evaluation where both disjuncts are valueless would one giving false (where both disjuncts are false) and the rest giving true, and thus the whole would be valueless, just as in Kleene the whole is indeterminate. However, this does not hold in all cases, as for example with the formula ‘P∨~P’, where ‘P’ is either indeterminate or has no truth-value. [As we see in the Kleene system, when a formula is I, then its negation is I. Thus both P and ~P would be I in our example. And when we have a disjunction where both disjuncts are I, as in our case here, then the whole disjunction is I. However, for supervaluations, the negation of a formula with no value also has no value. For, supposing it is true, then the negation is false. And supposing it is false, then the negation is true. There is not a consistent outcome, so P has no value and ~P has no value. Then, when we make a classical completion of ‘P∨~P’, we see that it is always true, no matter what we assign it (given that both P and ~P are gaps).

Thus supervaluations are not equivalent to Kleene’s three-valued semantics.]

 

Nolt then defines validity and consistency for supervaluations. “a formula is valid iff it is true on all supervaluations; and a sequent is valid iff there is no supervaluation on which its premises are true and its conclusion is untrue. (And, of course, there are two ways of being untrue: being false and being truth-valueless.)” (Nolt 417).

 

Nolt then shows how “the logic that results from these stipulations is just classical propositional logic – even though, as we have seen, its semantics differs significantly from that of classical logic. Thus, for example, an inference is valid on supervaluational semantics if and only if it is valid on classical bivalent semantics, despite the fact that the former, but not the latter, permits truth-value gaps” (Nolt 417).

 

[Nolt next proves this with a complicated metatheorem and proof (417-418). After that he extends supervaluation semantics to predicate logic (418-419). These parts are left for another summary.]

 

 

 

 

 

 

From:

 

Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.

 

 

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Peirce (CP1.427-1.440) Collected Papers of Charles Sanders Peirce, Vol1/Bk3/Ch4/§3, 'Fact', summary

 

by Corry Shores

 

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[The following is summary. Boldface and bracketed commentary are mine. Proofreading is incomplete, so please forgive my typos.]

 

 

Summary of

 

Charles Sanders Peirce

 

Collected Papers of Charles Sanders Peirce

 

Volume 1: Principles of Philosophy

 

Book 3: Phenomenology

 

Chapter 4: The Logic of Mathematics; An Attempt to Develop My Categories from within

 

§3: Fact [1.427-1.440]

 

 

 

 

Brief summary:

The elemental phenomenal structure of fact has the following characteristics: it is logically contingent, accidentally actual, unconditionally necessary, and brute in the sense of being force without law or reason. This notion of fact can still fit within a deterministic view, so long as we only consider the elemental structure of fact by isolating simple actions from their context of other related actions that might be said to cause them. Facts involve accidental pairings of things in which the brute force exerted by the one upon the other places them into a dyadic relationship, meaning that from the perspective of the first, it is acting upon the second, and from the perspective of the second, it is being acted upon by the first. We learn of facts either directly, when they resist our exertions and thereby inform us of an external world outside our ego, or indirectly, as when others give us testimony of their own direct relation to facts or when we observe the physical effects of resistance between the actions of external objects. And, a fact’s existence consists in the existence of its effects in the world. This mode of ‘being over against something else’ is existence itself, which is fact’s mode of being. The fact is individual in two senses, namely, by being over against something else through brute force, it is independent of that other thing, and by being independent of the qualities or determinations it may or may not contingently have, depending on the circumstances of the interaction. So far, these observations cover the first seven characteristics of fact, to which Peirce adds five more. Facts {1} have distinct features, {2} are either accidentally actual or involve brute force, {3} have a here and now, {4} are dyadic, {5} are the sum of their consequences, {6} exist by fighting, {7} are determinate with respect to qualities they could possess, {8} have a subject, which is the grammatical subject of the sentence asserting the existence of the fact, {9} are always connected with reciprocal facts (which may or may not be separable from the first fact), {10} are naturally classifiable into dichotomies, especially into dichotomies of reciprocal facts, {11} are such that, whenever  there is a variation in a reciprocal fact, this means there has been a change in the qualities of the subjects and not the creation or destruction of either subject, {12} are accidental and simply happen to come into an encounter rather than depending on each other before that for their existence. While this list is not necessarily exhaustive, it is sufficient for our purposes of comparing fact and duality and of explaining why all phenomena are composed of quality, fact, and law.

 

 

 

 

 

 

 

Summary

 

1.427

[We may characterize the elemental phenomenal structure of fact as being logically contingent, accidentally actual, unconditionally necessary, and brute. It is a matter of brute force in the sense that it is force without law or reason. But it is not general. So it is not the negative generality of quality’s potentiality nor is the positive generality of law’s conditional necessity. It furthermore is thus not permanent or eternal, as these are matters of generality too.]

 

[Peirce will now define fact in a way that properly places it in the context of the other elemental phenomenal structures of law and quality. Law is a matter of generality, which includes things which are permanent or eternal and also things that are conditional. Thus fact is not these things. Peirce says that there is a negative sort of generality, as what characterizes the potentiality of quality, and also there is a positive sort of generality, namely, conditional necessity, which is characteristic of law. Fact, however, is a matter of the contingent (in the logical sense) and thus of the accidentally actual, and it is also a matter of brute force, which is force without law or reason, and which is anything involving an unconditional necessity.]

Next, what is fact?

As before, it is not the usage of language which we seek to learn, but what must be the description of fact in order that our division of the elements of phenomena into the categories of quality, fact, and law may not only be true, but also have the utmost possible value, being governed by those same characteristics which really dominate the phenomenal world. It is first requisite to point out something which must be excluded from the category of fact. This is the general, and with it the permanent or eternal (for permanence is a species of generality), and the conditional (which equally involves generality). Generality is either of that negative sort which belongs to the merely potential, as such, and this is peculiar to the category of quality; or it is of that positive kind which belongs to conditional necessity, and this is peculiar to the category of law. These exclusions leave for the category of fact, first, that which the logicians call the contingent, that is, the accidentally actual, and second, whatever involves an unconditional necessity, that is, force without law or reason, brute force.

(233)

 

 

 

1.428

[The notion that there is brute fact goes against a deterministic view that sees a broad context where all events operate according to fixed laws. However, we can analyze situations and phenomenal experiences so to isolate simple actions taken without context to others, and as such we find that accidental factuality and brute force (that is, not law-guided force) are elemental parts of phenomena and events.]

 

[Peirce then addresses the objection that there is no such thing as brute force, freedom of the will, and the accidental. Peirce disagrees. But for the sake of argument he will consider these claims as true. (I am not entirely sure I understand what ‘brute’ means in this context. It seems the thinking here is the following. Some people object, because they believe the world is purely deterministic. So a brute force or action for some reason cannot be one happening in a deterministic system. Perhaps this is because a deterministic system operates according to regularities and laws, but something brute has no reason behind it. Rather, it just is what it is and does what it does, without any cause or reason. Peirce next seems to be saying that the ones who object see the system of actions and reactions on a large scale, as if taking a birds-eye-view on the world. However, were we to only consider each action on its own, without placing it into relation with other actions, then it would be brute. He says that it would be brute, even if it does not show brute force. So perhaps the idea is that it would have no cause or reason, and also that it may not be affecting anything else.) Peirce continues to note that there is still a way to understand phenomena as exhibiting force without reference to law. (Perhaps the idea is that force would normally operate according to laws, but brute force would not.) Our own sense of our exercise of our will is one such case. (Peirce’s next point seems to be that although we might normally place the state or actions of something into a broader context where law, regularity, and generality seem at play, we can also decompose such a context down to the singular elements involved, thereby stripping away the other law-guided elements. As such, we can say that accidental factuality is an elemental part of larger phenomenal structures, and it can be seen as such by means of some sort of analysis.)]

It may be said that there is no such phenomenon in the universe as brute force, or freedom of will, and nothing accidental. I do not assent to either opinion; but granting that both are correct, it still remains true that considering a single action by itself, apart from all others and, therefore, | apart from the governing uniformity, it is in itself brute, whether it show brute force or not. I shall presently point out a sense in which it does display force. That it is possible for a phenomenon in some sense to present force to our notice without emphasizing any element of law, is familiar to everybody. We often regard our own exertions of will in that way. In like manner, if we consider any state of an individual thing, putting aside other things, we have a phenomenon which is actual, but in itself is not necessitated. It is not pretended that what is here termed fact is the whole phenomenon, but only an element of the phenomenon – so much as belongs to a particular place and time. That when more is taken into account, the observer finds himself in the realm of law in every case, I fully admit. (Nor does that conflict with tychism.)

(233-234)

 

 

 

1.429

[Facts are accidental coincidences of pairs of things in which the brute force exerted by one upon the other places them into a dyadic relationship, meaning that from the perspective of the first, it is acting upon the second, and from the perspective of the second, it is being acted upon by the first. Regularity and law cannot be established between just two things. It requires a third which shows there to be some regularity imposed upon the relations of all the parts. Without that third thing, two parts are merely coincidental. Yet simply being coincidental does not mean that the two parts are really dyadic, as they might merely be the combination of two independent things. For there to be a dyad, the dual relation between the two needs to involve the brute force of the one acting upon the other asymmetrically.]

 

[So in order to arrive upon fact, we need to analyze the situation to strip away contextual factors which are matters of generality and law. However, if we analyze the situation too much so that we arrive upon a monadic structure with no parts, then we are examining quality and not fact. Thus, we need to attend to something more than a monad or quality but less than a triad or law. Peirce then says that among instances of fact, we consider some of them to be especially accidental in nature. While the other cases are no less factual, we will examine first these special instances, because in them the accidental nature of fact is most obvious. Peirce notes that we call these highly accidental facts “coincidences” because in them we see “the coming together of two things”. We need at least two things in order to form a coincidence, and if more things are involved, that does not change the essential coincidental nature of the situation. Peirce’s next point is remarkable. Regularity cannot be found in only two things. It requires at least three. (Perhaps the idea here is the following. Two things may have similarities are connections which could be construed as governing the way that such parts relate to one another. But there is no such confirmation or obvious evidence of there being such a regularity between them. It could be that other parts would relate differently than these two have. However, were there a third part, then the similarities in how all three relate can tell us for example that the way one pair relate bears a similarity to the way another pair relate, and thus there may be something beyond the coincidental combinations governing the way that such parts relate.) He gives the example of dots. Were there only two dots, there is no regularity yet. However, were their three dots, they could be arranged in some regular way, as for example were they arranged in a straight line or if they formed the vertices of an equilateral triangle. (So perhaps the three dots could be arranged in an irregular sort of way. But as soon as they form a straight line, we find that the way the third relates to the other two is in accordance with a regularity that is in place for all the dots, namely, that none may vary at a different angle from the others. And if the points are the vertices of an equilateral triangle, there again is a regularity that is in force on all three, because they each are set off from one another at the same angle. But just two dots, no matter how they are positioned in relation to one another, have no apparent regularity, because the third term is needed to confirm that some consistency in the manner of relation is imposed on the parts.) Peirce notes some possible exceptions, which in fact still hold to this schema. Two dots on the surface of the earth may be placed antipodally, with that antipodality being the regularity. However, here the third term is the earth itself, which establishes the spherical regularity governing the antipodal positioning of the points. (The next example I do not understand, but I will explain why. He writes: “So two straight lines in a plane can be set at right angles, which is a sort of regularity. But this is another rule proving exception, since <AOB is made equal to <BOC. Now those angles are distinguished by being formed of two different parts of the line AC; so that really three things, OA, OB, and OC are considered.” As far as I understand, “<AOB” would be an angle where O is the vertex, and A and B are points on the different rays emanating from that vertex. So I can conceive of a right angle <AOB. But then what is <BOC? It would seem to have the same vertex (O) and the same point on one line (B) and thus that same ray, but there is now point C instead of A. I would have understood if he wrote “<AOB is made equal to <BOA”, but I do not know what the C is here. Possible Peirce is using a different notation convention for angles, but probably I just miss an obvious idea. Possibly also he means that one right angle is to be understood as sharing the angular properties of any other right angle, but then I do not know why he would use two of the same letters in the notation, and also then we have added more terms than just two or three. He says further that angles <AOB and <BOC are made by forming two different parts of the line AC. This is further confusing for me. Was there originally a long line AC, and it was bent or taken apart? If so, how was that done so to produce <AOB and <BOC? Here a diagram would have been helpful or some clarification on the notation.) (He then seems to conclude that accidental actuality is thus the coincidental and not law-governed coming together of two things.) Peirce now turns to the issue of brute force. He writes that “The type of brute force is the exertion of animal strength.” So we can formulate in our mind an action we want to take. But the actual execution of that act can only happen when we exert the force necessary to commit it. He further notes that strength cannot be exerted unless there is something to resist that exertion. (For, without that resistance, no effort would be needed and thus no force would be exerted). This means that we have a duality of units that are not symmetrically related. Suppose simply there is a coincidence of two phenomena with no active or interactive relation between them. Here we simply have a monoidal dyad. (Perhaps the idea here is that it is not a dyadic dyad, because there is no dyadic relation. It is the simple sum of two monads. But I am not sure.) However, were a first object to act on a second one, while the second resists or reacts upon the first, then we have more than two things coincidentally related. We also have them in an oppositional relation, (making the relation dyadic rather than the bare combination of monads. I may not be following this part properly. The overall notion seems to be that Peirce shows that simple accidental coincidence is not enough for a dyadic relation. There needs to be two perspectives, one upon the other, which are inversely related such that one part can be said to be receiving action and the other giving it. For more on this conception, see especially sections 1.326-1.328.)]

On the other hand, if the view be limited to any part of the phenomenal world, however great, and this be looked upon as a monad, entirely regardless of its parts, nothing is presented to the observer but a quality. How much, then, must we attend to, in order to perceive the pure element of fact? There are certain occurrences which, when they come to our notice, we set down as “accidental.” Now, although there is really no more of the factual element in these than in other facts, yet the circumstance that we call them par excellence contingent, or “accidental,” would lead us to expect that which distinguishes the realm of fact from the realms of quality and of law, to be particularly prominent in them. We call such facts “coincidences,” a name which implies that our attention is called in them to the coming together of two things. Two phenomena, and but two, are required to constitute a coincidence; and if there are more than two no new form of relationship appears further than a complication of pairs. Two phenomena, whose parts are not attended to, cannot display any law, or regularity. Three dots may be placed in a straight line, which is a kind of regularity; or they may be placed at the vertices of an equilateral triangle, which is another kind of regularity. But two dots cannot be placed in any particularly regular way, since there is but one way in which they can be placed, unless they were set together, when they would cease to be two. It is true that on the earth two dots may be placed | antipodally. But that is only one of the exceptions that prove the rule, because the earth is a third object there taken into account. So two straight lines in a plane can be set at right angles, which is a sort of regularity. But this is another rule proving exception, since <AOB is made equal to <BOC. Now those angles are distinguished by being formed of two different parts of the line AC; so that really three things, OA, OB, and OC are considered. So much for accidental actuality. The type of brute force is the exertion of animal strength. Suppose I have long ago determined how and when I will act. It still remains to perform the act. That element of the whole operation is purely brute execution. Now observe that I cannot exert strength all alone. I can only exert my strength if there be something to resist me. Again duality is prominent, and this time in a [more] obtrusively dual way than before, because the two units are in two different relations the one to the other. In the coincidence the two phenomena are related in one way to one another. It is a monoidal dyad. But in the exertion of strength, although I act on the object and the object acts on me, which are two relations of one kind and joined in one reaction, yet in each of these two relations there is an agent and a patient, a doer and a sufferer, which are in contrary attitudes to one another. So that the action consists of two monoid dyads oppositely situated.

(234-235)

 

 

1.430

[The formal dyadic structure of fact might also mean that there is a correspondence between the contents and combinations of particular phenomena with particular ideas. In pursuit of this possibility and to learn more about fact in general, we will examine it further.]

 

[Thus fact is a matter of two, and law a matter of three or more. And we saw already how quality is a matter of one. (The next point I do not follow well enough. The first idea seems to be that we should not be surprised to find that certain kinds of phenomenal parts have their own sort of manner of forming relations with other parts. The next idea might be elaborating on this, saying that some categories of parts are too complex to be included among the simpler types, but I am not sure. The following point might be that the categories of types come to be strongly associated in our minds with such formal conceptions regarding their manner of relation. He next point is that we in fact should be surprised to find that the material parts of phenomena are coextensive with formal ideas. I am not sure what he means here exactly. As a result of this fact, we should try to understand better what the dyad’s connection is. I did not follow the reasoning in these sentences, so please see the quotation below. Maybe his point is that the fact that types of phenomenal parts have formal structures to their manners of relation could make us wonder if the phenomenal parts and their relations constitute or coincide with certain formal ideas, as if one such combination produces or correlates with one idea, and another combination with another idea. So we should examine the dyadic structure to see if and how ideas might correspond to it. That is a guess.) Peirce will exclude for now cases of fact where there seems to be a triad formation, “such as a process with beginning, middle, and end”. We can return to those cases after we examine law, as there might be an element of generality involved in this cases. So Peirce will first examine fact more closely to determine more of its properties. Secondly he will consider the element of duality in fact.]

All this renders it quite certain that the nature of fact is in some way connected with the number two, and that of law with three or some higher number or numbers, just as we have already seen that quality is described by means of the number one. But although it is hardly more than might be expected to find that a particular form – that some are too complex to suit this matter, while others [are] too simple to call into action its distinctive powers – and that in that way that category comes to have an intimate affinity with a certain formal conception, yet it would certainly be astonishing if it should turn out that material constituents of phenomena were coextensive with formal ideas. We consequently wish to discover just what the connection of the dyad with fact is. We shall do well to postpone the consideration of those facts which seem to involve a triad, such as a process with beginning, middle and end, until we have examined the nature of law. For we naturally suspect, after what has been pointed out above, that where there is a threeness in a fact, there an element of generality may lurk. Putting aside then, for the present, triadic facts, we may add to the properties of fact already noticed such others as may seem worth mention, and may then turn to the consideration of duality, its properties and different formal types, so as to compare these with what is to be remarked in regard to fact.

(235-236)

 

 

 

1.431

[We learn of facts directly when they resist our exertions. They give us undeniable evidence of an external world and of something outside our own ego. When we are interrupted, this is a matter of resistance, and it is only different in degree to exertions of our will. We can also learn of facts indirectly, as when others give testimony (and thus with them having experienced the resistance directly) or when we see physical effects in the world (thus with that resistance being directly experienced by physical objects).]

 

[The way that we come to know facts is by their resisting us. We can question the external world as much as we want, but we cannot deny it or that there is something more than our ego when we anger someone else and they knock us down. This resistance that meets us from without shows us that there is something independent to us. Even when we have a sensation, there is resistance, because our mind’s train of thought has been interrupted. Now, since exerting our will and being interrupted both involve resistance, they are different really only in the degree of the resistance involved. Peirce then notes that facts can be learned without oneself undergoing resistance to them. This happens when we learn facts indirectly. One way this can take place is if someone tells us about some fact. (In that case someone still had to encounter resistance to learn the fact in the first place). The other way is by observing the physical effects of a fact (and in that case still something is meeting with resistance in a physically observable way).]

Whenever we come to know a fact, it is by its resisting us. A man may walk down Wall Street debating within himself the existence of an external world; but if in his brown study he jostles up against somebody who angrily draws off and knocks him down, the sceptic is unlikely to carry his scepticism so far as to doubt whether anything beside the ego was concerned in that phenomenon. The resistance shows him that something independent of him is there. When anything strikes upon the senses, the mind's train of thought is always interrupted; for if it were not, nothing would distinguish the new observation from a fancy. Now there is always a resistance to interruption; so that on the whole the difference between the operation of receiving a sensation and that of exerting the will is merely a difference of degree. We may, however, learn of a fact indirectly. Either the fact was experienced directly by some other person whose testimony comes to us, or else we know it by some physical effect of it. Thus we remark that the physical effects of a fact can take the place of experience of the fact by a witness. Hence, when we pass from the consideration of the appearance of a fact in experience to its existence in the world of fact, we pass from regarding the appearance as depending on opposition to our will to regarding the existence as depending on physical effects.

(236)

 

 

 

 

1.432

[A fact’s existence consists in the existence of its effects. Otherwise it is a mere potentiality and quality. To be a fact, something must encounter resistance to its exertions and thereby have an effect in the world. The mode of being of ‘being over against something else’ is existence, which is fact’s mode of being.]

 

[Peirce then says that the existence of a fact consists in the existence of its effects. (His thinking here might be that certain facts may not be directly evident or at least not yet verified, but they are evinced in their effects. So we consider what at first is just a “supposed” fact, and it is shown to be a real fact if all its consequences reveal it to be such.) His example is something that is supposed to have the fact of being hard, and indeed it does act in every respect like a hard thing. Its actual fact of being hard consists then in its behaving in such a way that its actions have the consequences that result from hard things. (What is odd here is that its factuality is not something that can be real unless it is having some effect. Perhaps the idea is that it could not possibly be having no effect, and thus the facts of something are always inherent to it. Or perhaps the idea is that it can have certain qualities, like potentialities (as firsts), and they are only actualized when they have consequences. And thus the object could have the quality of hardness on its own and without having any effects, but it can only have the factuality of hardness if that hardness has certain effects in the world.) Peirce says that “the fact fights its way into existence,” because “it exists by virtue of the oppositions which it involves”. (In other words, without opposition, there can be no fact.) He continues, “It does not exist, like a quality, by anything essential, by anything that a mere definition could express”. Rather,  facts “take place” here and now. We know facts personally when they resist our brute will. (Peirce means brute in the sense of unreasoning. See paragraphs above.) We thus can conceive of a fact as something “gaining reality by actions against other realities”. Peirce calls existence the mode of being that comes from something not simply being itself but rather “being over against a second thing”, and thus existence is the mode of being that belongs to fact.]

There can hardly be a doubt that the existence of a fact does consist in the existence of all its consequences. That is to say, if all the consequences of a supposed fact are real facts, that makes the supposed fact to be a real one. If, for example, something supposed to be a hard body acts in every respect like such a body, that constitutes the reality of that hard body; and if two seeming particles act in every respect as if they were attracting particles, that makes them really so. | This may be expressed by saying that the fact fights its way into existence; for it exists by virtue of the oppositions which it involves. It does not exist, like a quality, by anything essential, by anything that a mere definition could express. That does not help its mode of being. It might hinder it; because where there is not a unit there cannot be a pair; and where there is not a quality there cannot be a fact; or where there is not possibility there cannot be actuality. But that which gives actuality is opposition. The fact “takes place.” It has its here and now; and into that place it must crowd its way. For just as we can only know facts by their acting upon us, and resisting our brute will (I say brute will, because after I have determined how and when I will exert my strength, the mere action itself is in itself brute and unreasoning), so we can only conceive a fact as gaining reality by actions against other realities. And further to say that something has a mode of being which lies not in itself but in its being over against a second thing, is to say that that mode of being is the existence which belongs to fact.

(236-237)

 

 

 

1.433

[Since there are many types of existence in the universe, that means every existence is a matter of being something that is over against  other things (namely, the other types of existence). This is another way to arrive upon the conclusion that the being of fact is existence, which is ‘being over against a second thing’.]

 

[Peirce it seems will argue a different way for this notion that fact’s being is existence, which is being over against a second thing. (I will not be able to summarize or explain this part, so you will need to read the quotation to follow. Somehow the fact that there are different types of existence means that each kind has a place among the others, and thus it is a second to other objects, with the universe itself being a first. Please consult the quotation below.)]

The same conclusion can be reached by another line of thought. There are different kinds of existence. There is the existence of physical actions, there is the existence of psychical volitions, there is the existence of all time, there is the existence of the present, there is the existence of material things, there is the existence of the creations of one of Shakespeare's plays, and, for aught we know, there may be another creation with a space and time of its own in which things may exist. Each kind of existence consists in having a place among the total collection of such a universe. It consists in being a second to any object in such universe taken as first. It is not time and space which produce this character. It is rather this character which for its realization calls for something like time and space.

(237)

 

 

 

 

1.434

[The fact is individual in two senses: 1) it exists by means of brute force that places something over against other things, thus making it independent of them, and 2) it is independent of many qualities, which it determinately either has or does not have. In other words, it is individual because it is independent and separate from other things and because it is independent from determinations it may or may not have.]

 

[Unlike law, fact is not general. Rather, it is individual. Peirce says that ‘individual’ expresses two things peculiar to facts: namely, 1) its brute force of being over against something opposing it, and 2) its being independent of qualities and determinations. (I may not have followed his wording right in how I summarize the distinction, so please consult the quotation below.) The first character, Peirce says, is the one that we just discussed above (on fact’s existence as opposition), and it includes the idea of the fact consisting in brute force or self-assertion. Facts have no reason, and this is what distinguishes the individual fact from the general fact or law as well as distinguishing the individual fact from quality or possibility, “which only hopes it won’t be intruding”. The second character of fact that ‘individual’ expresses is that the fact is determinate in the sense that it either possesses or does not posses some or another possibility or quality. (Above Peirce seemed to have said that this is a matter of fact being independent of qualities and determinations. Perhaps the idea is that a fact is independent of certain qualities and determinations, but not of others, which it possesses. Or maybe the idea is that since fact either can have certain determinations or not, that makes facts in general something more than their determinations, which are contingent. I am not sure.) This is unlike law, which always remains partly indeterminate (perhaps because it applies to instances that may not exist yet). ((Another possibility is that the idea here is that only individuals can take qualities, but facts are dyads.)).]

When we speak of a fact as individual, or not general, we mean to attribute to it two characters each of which is altogether peculiar to facts. One of these is the character just described, the other having a mode of being independent of any qualities or determinations, or, as we may say, having brute fighting force, or self-assertion. The individual fact insists on being here irrespective of any reason, whether it be true or not | that when we take a broader view we are able to see that, without reason, it never could have been endowed with that insistency. This character makes a gulf between the individual fact and the general fact, or law, as well as between the individual fact and any quality, or mere possibility, which only mildly hopes it won’t be intruding. But besides that character, individuality implies another, which is that the individual is determinate in regard to every possibility, or quality, either as possessing it or as not possessing it. This is the principle of excluded middle, which does not hold for anything general, because the general is partially indeterminate; and any philosophy which does not do full justice to the element of fact in the world (of which there are many, so remote is the philosopher's high walled garden from the market place of life, where fact holds sway), will be sure sooner or later to become entangled in a quarrel with this principle of excluded middle.

(172)

 

 

1.435

[We have seen seven characteristics of facts: they {1} have distinct features, {2} are either accidentally actual or involve brute force, {3} have a here and now, {4} are dyadic, {5} are the sum of their consequences, {6} exist by fighting, and {7} are determinate with respect to qualities they could possess.]

 

[We have so far seen that fact has six characteristic features (Peirce will give seven in fact): {1} it has distinct features (unlike quality but like law), {2} facts are either accidentally actual or involve brute force, {3} every fact has a here an now, {4} facts are dyadic, {5} every fact is the sum of its consequences, {6} fact’s existence is a matter of fighting, and {7} every fact is determinate in reference to every ‘character’. Peirce will now mention other characteristics of fact.]

Thus far, in this section, attention has been called successively (but in no philosophical sequence) to six characteristic features of fact. In recollecting them, we may place at their head the circumstance that fact has distinct features, for this distinguishes it from quality although not from law. The others already examined have been as follows: second, facts are either accidentally actual or involve brute force; third, every fact has a here and now; fourth, fact is intimately associated with the dyad; fifth, every fact is the sum of its consequences; sixth, the existence of facts consists in fight; seventh, every fact is determinate in reference to every character. But in making our distribution of the elements of phenomena into quality, fact, and law, we were led to notice additional features of fact. I continue to take them up promiscuously.

(238)

 

 

 

 

1.436

[The eighth feature of fact is that every fact has a subject, which is the grammatical subject of the sentence asserting the existence of the fact. There are technically two subjects, namely, the thing in question and its existence. Although this subject is a physical, material thing, this will not contradict idealism or panpsychism.]

 

[Peirce turns now to the eighth feature of fact. We are to think of the sentence that would assert the existence of the fact. That sentence has a grammatical subject. So the eighth feature of fact is that every fact has a subject, which is this grammatical subject. (I am not sure how such a sentence would be formulated. Is it something like, “There is a rock” or “The rock exists”? Or “The rock (that exists) is hard”?). Peirce says in fact that there are two subjects. The first subject is the existence itself of the subject. (Please consult the text below, as I am not sure I follow where the distinction lies between the two subjects, and I do not know how best to characterize them). The second subject is the thing in question that exists here and now and on the basis of its reacting against other things. This subject either has or does not have any quality, and it is of a material or physical substance. Even though this subject of fact is understood as physical, this will not contradict idealism or the view that material things have a psychical substratum.]

The eighth feature of fact is that every fact has a subject, which is the grammatical subject of the sentence that asserts the existence of the fact. Indeed, in a logical sense, there are two subjects; for the fact concerns two things. One of these two subjects, at least, is a thing itself of the nature of fact, or we may express this in other words by saying that the existence of this subject is a fact. This subject is a thing. It has its here and now. It is the sum of all its characters, or consequences. Its existence does not depend upon any definition, | but consists in its reacting against the other things of the universe. Of it every quality whatever is either true or false. That this subject, whose actions all have single objects, is material, or physical substance, or body, not a psychical subject, we shall see when we come to consider psychical subjects in discussing the nature of law. This does not in the least contradict idealism, or the doctrine that material bodies, when the whole phenomenon is considered, are seen to have a psychical substratum.

(238-239)

 

 

1.437

[The ninth feature of fact is that every fact is connected with a reciprocal fact. This is because any fact requires that it has some effect on something else. However, the second fact may or may not be separable from the first one. It is inseparable if its own factuality depends on that of the first, as in cases where both facts evince themselves by interacting with similar facts, as for example the hardness of one body requires interaction with the hardness of a second, and therefore the second requires the hardness of the first. It is inseparable however if the first fact can manifest equally whether or not any of its potential reciprocal facts are present, as for example were a hard object to melt in a vessel and fill the open spaces, either displacing the gasses already at the bottom or not if it is a vacuum (however there will still be some other reciprocal fact, perhaps somehow changing the balance or center of gravity of the vessel).]

 

[Peirce turns to the ninth feature of fact, which is that every fact is connected with a reciprocal fact. And, the reciprocal fact may or may not be inextricably bound up with the first fact. He gives the example of two colliding objects. The first object striking the second is one fact, and the second object striking upon the first is the other fact. (The next idea seems to be the following. Suppose that one of the striking bodies has some hardness to it, but the other body does not. Perhaps the first body would pass through the second, and there would be no striking to begin with, and the second body would perhaps even be unable to offer some resistance to the first. Thus,] “if one body is hard, there must be a second body of some degree of hardness for the former to resist”. (You will have to consult the quotation below, as I may misinterpret this paragraph. It seems his next idea is that certain facts are separable from their reciprocal facts. So the second hard body could disappear, but that will not do away with the first body’s fact of hardness, because some third hard body might be affected by it. He next gives the example of a hard body melting and filling the empty places in its vessel. He says at the beginning of this change there will be another change reciprocal to the first one. But I am not sure what that could be. Perhaps for example it might displace air in those vacant spaces, pushing it upward while the melted material falls downward. He also says there may not be such a consequence. So perhaps for example the melting happens in a vacuum. So he concludes that us distinguishing different types of facts that are inseparable from reciprocal facts does not thereby divide the facts into ones whose reciprocal facts are separable and those that are not. The first example he gave of hard bodies striking was supposed to exemplify inseparable reciprocal facts. But he also said that the annihilation of the second body does not destroy the hardness of the first. I am not following so well, but perhaps he is saying that the hardness of the second body requires the hardness of the first body for its own hardness, just as the first requires the second, and is therefore an inseparable fact. Let me quote.)]

The ninth feature of fact is that every fact is connected with a reciprocal fact, which may, or may not, be inextricably bound up with it. If one body strikes upon another, that second body reciprocally strikes upon it; and the two facts are inseparable. But if one body is hard, there must be a second body of some degree of hardness for the former to resist. Yet the annihilation of the second body would [not] destroy the hardness of the first. It would not affect it; for any other body that might grow hard at any time and the first body, remaining unaffected, would realize its hardness whenever the impact with the other should happen to occur. Here, therefore, the reciprocal fact is not so inseparable from the other. If a solid body suddenly melts, it will at once flow into the vacant parts of its vessel; and the beginning to any such consequent fact will be a change reciprocal to the first change. But there is no particular consequence which will be inseparable from the melting, perhaps. There may or may not be. So we see that the division between facts inseparable from reciprocal facts is not coincident with a division of facts into those whose reciprocal facts are separable and those whose reciprocal facts are inseparable.

(239)

 

 

 

1.438

[The tenth feature of fact is that facts are naturally classifiable into dichotomies, especially into dichotomies of reciprocal facts.]

 

[The tenth feature of fact is that “its natural classification takes place by dichotomies”. Peirce says he just illustrated this feature. So he perhaps means that since every fact is connected with a reciprocal fact, that every fact involves some dichotomy of fact pairings.]

The tenth feature of fact, which has just been illustrated is that its natural classification takes place by dichotomies.

(239)

 

 

 

1.439

[The eleventh feature of fact is that whenever there is a variation in a reciprocal fact, this means there has been a change in the qualities of the subjects of the fact, and it does not mean that there was the creation or destruction of either subject.]

 

[The eleventh feature of fact Peirce qualifies as the eleventh feature of dual fact. (But I am not sure if that makes it different from fact in general or if dual fact is special case of fact. It seems that although all facts have their reciprocal facts, we can still think of them independently, but here we will think of them in their pairings, hence the qualification ‘dual fact’. However, it could also be that a dual fact is one whose reciprocal fact is inseparable, because in the following Peirce will make that distinction. In all, I do not follow Peirce’s point in this paragraph very well, so please consult the quotation to follow. I will guess. Perhaps he is making the following points. Let us begin with his example, and I will offer first an interpretation of what it might be illustrating. A star suddenly bursts into view, but there is no external subject that caused it to do so. Since nothing caused it to come into existence, the star must already have been in existence. However, it varied over time, as it was previously not visibly bright but now it is. So the fact of its pre-existence is shown in its becoming bright, but so too is its brightness being shown. And the third fact it shows is that previously there was dark. Probably this illustration is making some other point than that. Peirce says that this is the only way we can deduce metaphysical truths. Perhaps he is saying that we deduce the metaphysical truth of the star’s pre-existence on the basis that nothing else could have caused it to come into being, and also that nothing can arise out of nothing. Peirce earlier said also that when there is an action that destroys or produces something, then “either a third subject will be concerned, so that the fact is one of these the study of which we have expressly postponed, or that which is produced or destroyed will be one of those facts whose reciprocal facts are separable”. For the first case, perhaps he is saying that the third fact would be the cause of the destruction or production. Regarding if they are separable or not, perhaps the idea is that the existence of the second fact does not depend on the existence of the first, so it can be destroyed or produced by other causes. But I do not follow, so please check the quotation. Finally Peirce concludes that the subjects of facts are permanent and eternal. Let me further guess. He might be saying that in this example there are two inseparable facts, the fact of the darkness of one thing and the fact of the brightness of the other. The increasing brightness of the star does not destroy the darkness of the dark thing nor does it attest to the creation of the bright thing. It rather shows the existence of these reciprocal facts and only shows there was a variation in their reciprocal qualities. The point might be then that the variation which attests to the existence of one fact attests to the existence of the other, but it does not attest to the creation or destruction of either, since that would for some reason require a third thing. But I do not know why one fact cannot destroy another, unless the idea is that it would destroy itself in the process. Yet even then I would not know why that would disqualify it as an instance of destruction of reciprocal fact.) ]

The eleventh feature of dual fact is that if it involves any variation in time, this variation consists of a change in the qualities of its subjects, but never the annihilation or production of those subjects. We may, indeed, conceive of an action by which something is produced or destroyed. But either a third subject will be concerned, so that the fact is one of those the study of which we have expressly postponed, or that which | is produced or destroyed will be one of those facts whose reciprocal facts are separable. If a star suddenly bursts into view, when no external subject caused it to do so, then, just as the appearance will be irrefragable demonstration that something dark was there before, so the fact itself will constitute the previous existence of its subject. For this is the only method by which we can deduce metaphysical truths. Consequently, bodies, and the subjects of facts generally, are permanent and eternal.

(239-240)

 

 

 

 

1.440

[The twelfth feature of fact is that it is accidental in the sense that it simply happens. The subjects involved in the fact do not depend on one another for their existence but rather just happen to come into an encounter.]

 

[The twelfth feature of fact is that it is accidental and is something that simply happens. (Peirce elaboration may not be entirely clear but is perhaps the following. We might think that because facts involve brute force, that the continuance of this force requires some law. For, were it to persist, it would need some generality in the situation rather than a simple instantaneous reaction. However, this law involved in the continued exertion of the brute force does not mean that the existence of the action depends upon the existence of what resists the action. Rather, they coincidentally or accidentally come into interaction. Please read the quotation, as I am not certain this is the best interpretation.)]

The twelfth feature of fact is that it is accidental. That is to say, even if it involves brute force, and though that force be governed by a law which requires the acting body continually to exert this force, yet nevertheless the individual action is not involved in the existence of the fact, but on the contrary is something that can only happen by having a subject with an independent mode of being not dependent upon this nor upon any determination whatsoever. It is something which happens.

(240)

 

 

 

1.440

[This list is not complete but is sufficient for our purposes of comparing fact and duality and of explaining why all phenomena are composed of quality, fact, and law.]

 

[Peirce says that this list is not necessarily exhaustive, but it should be sufficient for his task of comparing fact with duality and with explaining why all phenomena are composed of quality, fact, and law.]

I have taken no pains to make this promiscuous list of properties of fact complete, having only cared that it should be sufficient to enable us to compare the characters of fact with those of duality and thus ultimately to attain an understanding of why all phenomena should be composed of quality, fact, and law.

(240)

 

 

 

 

 

 

 

 

 

 

 

From:

 

Peirce, C.S. Collected Papers of Charles Sanders Peirce, Vol 1: Principles of Philosophy.  In Collected Papers of Charles Sanders Peirce [Two Volumes in One], Vols. 1 and 2. Edited by Charles Hartshorne and Paul Weiss. Cambridge, Massachusetts: 1965 [1931].

 

 

 

 

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Nolt (15.2) Logics, ‘Multivalued Logics,’ summary

by Corry Shores

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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own. As proofreading is incomplete, you will find typos and other districting errors. I apologize in advance.]

 

Summary of

John Nolt

 

Logics

Part 5: Nonclassical Logics

Chapter 15: Mildly Nonclassical Logics

15.2 Multivalued Logics

Brief summary:

There are a number of reasons to be dissatisfied with the bivalence of classical logic, that is, with the limitation to just two values, true (T) and false (F). Among these reasons are: 1) some sentences are unintelligible and would thus seem to be neither T nor F, 2) some sentences have terms that fail to refer to an object, which might make us want to say they are neither T nor F, 3) semantic paradoxes, such as the liar’s paradox, would seem to have neither a T nor a F value, 4) there are metaphysical issues that could compel us to consider certain situations as being neither T nor F, as for example when making statements now about an unknowable future, 5) certain practical concerns, as for example in computer database creation, where designating certain formulas as neither T nor F is more useful, and 6) the vagueness of situations can make clear-cut propositions about their real status difficult on account of ambiguities, and hence we would not want such statements to be entirely T or F. Thus we have cause for devising logics with three or more values. Three such multivalued semantics are Bochvar’s, Kleene’s, and Łukasiewicz’s. In all three, a third value, namely indeterminate or ‘I’, is added, but how propositions containing indeterminate values are evaluated varies in each system.

Truth Tables for Bochvar’s Three-Valued Semantics

(Nolt 408)

Truth Tables for Kleene’s Three-Valued Semantics

(Nolt 412)

Truth Tables for Łukasiewicz’s Three-Valued Semantics

(Nolt 413)

Each will make certain classical tautologies valid or invalid, depending on how they evaluate in each semantic. And each one in some way may make certain counter-intuitive tautologies in classical logic become invalid, and certain intuitive classical tautologies become invalid. In each semantics, an operator for ‘it is true that’, symbolized T, may be used for establishing tautologies, since it converts values to just T or F. (Without this operator, classical tautologies in these three-valued semantics can be valued as indeterminate. Thus not all valuations would make it true, and hence they would no longer be tautologous). Its truth table is the following:

(Nolt 411)

Summary

So far in this book (and as announced in section 3.1), we have only considered the two truth values, T and F, and we also assumed “that each statement had exactly one of these truth values in every possible situation” (Nolt 406). Nolt now says that “there are reasons not to be satisfied with it” (406). He gives as one sort of case grammatically well formed sentences that “seem to have no intelligible meaning” (406). He offers as an example, “Development digests incredulity” (406). It is not clear what it means. Suppose we simply conclude that it is false. But that means “Development does not digest incredulity” would have to be true. But since it is not more intelligible that the non-negated form, we seem to have the same reason to say that it also should be false. [Nolt’s next ideas seem to be the following. Since the sentences unintelligibility makes it impossible to determine its truth value, we can say that it is a sentence but also that it is one whose value is neither true nor false. Or, we can go further and say that it is not even a sentence and thus truth and falsity (or its lack of them both) is not even a relevant issue. Let me quote:]

We might, then, decide to rule them both out of court; they both, we might conclude, are neither true nor false. Against this conclusion, some have objected that since neither sentence is really intelligible, neither makes a statement, and where no statement is made there is nothing that can be either true or false. The question of truth or falsity simply does not arise. On this view, unintelligible sentences do not challenge the principle of bivalence because that principle applies only to statements.

(406)

[So one situation where bivalence is problematic is with sentences that are unintelligible.] Another situation where bivalence is problematic are cases of reference failure [where the sentence is intelligible but also where certain terms in it do not refer to something, and thus the conditions for determining its truth value are lacking] (Nolt 406). Nolt offers as an example the following sentence: “The Easter Bunny is not a vegetarian”. We might be inclined to deem the sentence as not true, because the Easter Bunny does not exist (406). If we deem it not true, then we would conclude it is false, in a bivalent system. [So now, “The Easter Bunny is a vegetarian” is false, and thus in a bivalent system, “The Easter Bunny is a not a vegetarian” would be true.] But then, Nolt asks, would it be true that the Easter Bunny is a carnivore? As we can see, when a subject that is being predicated does not refer to some object, it would seem that the sentence can be neither true nor false.

The problem, of course, is that since the term ‘The Easter Bunny’ does not pick out an object of which we may predicate either vegetarianism or nonvegetarianism, it seems misleading to think of these sentences as either true or false. We might reasonably conclude, then, that because of the reference failure they are neither. Notice that here it is less plausible to argue that no statement has been made. We understand perfectly well what it means to say that the Easter Bunny is a vegetarian. But what is asserted seems not to be either true or false.

(406)

Another case that challenges bivalence are the semantic paradoxes, and in fact they present “even stronger arguments against the principle of bivalence” (407). He has us consider a sentence called “S” which says:

Sentence S is false.

(407)

Nolt says that with this sentence,

we can actually offer a metalinguistic proof that it is neither true nor false. For suppose for reductio that it is true. Then what it says (that it is false) is correct, and so it is false. It is, then, on this supposition both true and false, which contradicts the principle of bivalence. Suppose, on the other hand, that it is false. Then, since it says of itself that it is false, it is true. Hence once again it is both true and false, in contradiction to the principle of bivalence. Hence, from the principle of bivalence itself, we derive by reductio both the conclusion that this sentence is not true and the conclusion that it is not false. It is, then, certainly not bivalent.

(407)

[Nolt will now have us consider another sort of logical situation that challenges bivalence. For this, we should first recall the model of time that we discussed in section 13.2.1. Here, time was nondeterministic, but statements about the future could still be true in the present, if in fact they are going to be true in the future. Another view argues that were this so, it is not really an nondeterministic system. To be such, contingent statements about the future should be neither true nor false until their value is determined at the relevant future moment. This view was put forth by Jan Łukasiewicz. Were one to take up this position, one could thus object to bivalence for metaphysical reasons.]

One might also reject bivalence on metaphysical grounds. Jan Łukasiewicz, who constructed the first multivalued semantics early in the twentieth century, held that contingent statements about the future do not become true until made true by events. Suppose, for example, that a year from now you decide to write a novel. Still it is not true now that a year from now you will decide to write a novel; the most that is true now is that it is possible that you will and possible that you won't. Only when you actually do decide a year hence does it become true that a year earlier you were going to decide to write a novel a year hence. Obviously, Łukasiewicz’s conception of time is different from that presented in Section 13.2, where we modeled a nondeterministic time in which contingent statements about the future may be true at present. Łukasiewicz assumed that the present truth of contingent statements about the future implies determinism. In any case, the idea that the truth of a contingent statement does not “happen” until a specific moment in time, whether right or not, is of logical interest. It implies that many statements about the future are neither true nor false now so that there is some third semantic status, which Łukasiewicz called ‘possible’ or ‘indeterminate’, in addition to truth and falsity.

(407, boldface mine)

[The reasons for rejecting bivalence so far are unintelligible sentences, reference failure, semantic paradoxes, and metaphysical issues like the indeterminacy of presently made statements about the unknowable future.] The next reason Nolt gives for rejecting bivalence is that particular statements would for certain practical reasons be best considered as having an unknown truth value in computer databases. [The idea seems to be that the database will function better, or that the programming can be more efficient or effective, were certain statements considered neither true nor false, even if in actual reality they are one of the two.]

There may also be more mundane, practical grounds for rejecting bivalence. The designers of a computer database of propositions, for example, might want to list some propositions as true, some as false, and others as unknown. There is, of course, no metaphysical basis for the third value in this case. The propositions listed as unknown may all in fact be true or false. But in practice the inferential routines used with the database may work best if they embody a non-bivalent logic.

(407)

The final motivation for rejecting bivalence that Nolt offers are cases of vagueness, where truth is perhaps better understood as having different degrees. Nolt gives the example of the sentence “This is a car” in reference to a midsize sedan. This is entirely true. But if we said, “This is a car” in reference to an eighteen-wheeler, then it would be entirely false. Yet, what if we said this in reference to a van?

Many people feel that such assertions are “sort of true, but not exactly true.” Since, like ‘car’, virtually all words are somewhat vague, for virtually all statements there are borderline situations in which we are hesitant to say that the statement is either true or false. But the notion that truth comes in degrees leads beyond consideration of a mere third alternative to truth and falsity into the realm of infinite valued and fuzzy logics ( see Section 16.1).

[So we have the following reasons to consider an alternative to bivalent logic:

1) unintelligible sentences,

2) reference failure,

3) semantic paradoxes,

4) metaphysical issues (like the indeterminacy of presently made statements about the unknowable future)

5) practical concerns (as for example in computer database creation)

6) vagueness

] On account of these issues, we might want to consider that perhaps there exists, or at least that it would be useful to incorporate, a third truth value in addition to truth and falsity (Nolt 408). Nolt for now will posit just one such value, the indeterminate, notated as “I” (Nolt 408). But as we might imagine, we will need to revise our standard truth tables to accommodate the properties of this value. When we assign truth values to sentences, we can now assign T, F, or I. Yet, there is not just one way to assign values. We can at least say that when formulas that are “governed by the operators” and also that have components that are all either true or false, then they will take their usual truth values on the whole (408). But we still need to determine what to do when complex sentences have at least one component with the value I. Nolt says there are two general policies for these cases (408), [namely,  that even one internal instance of I should make the whole sentence I, or instead such a sentence (with one instance of I) could have the value T or F, depending on how the semantics is constructed.]

Two general policies suggest themselves:

1. Indeterminacy of the part should infect the whole so that, if a complex formula has an indeterminate component, then the formula as a whole should be indeterminate.

2. If the truth value of the whole is determined on a classical truth table by the truth or falsity of some components, even if other components are indeterminate, then the whole formula should have the value so determined.

(Nolt 408)

Nolt illustrates the difference in how these policies would work by having us consider a disjunction ‘P∨Q’, where ‘P’ has the value T and ‘Q’ has the value I. We now wonder, what is the value of the entire disjunction? The first policy would say that it is I, because part of it is I. However, the second policy would note that even though one term is indeterminate, it still fulfills the requirements for a true disjunction in the classical truth table, because at least one disjunct is true (Nolt 408).

Nolt says it is “not obvious which of these policies is preferable,” and in fact “one may be preferable for some applications, the other for others” (Nolt 408). For this reason, Nolt will discuss both policies, and at the end of the section, he will consider yet a third policy (408).

In line with the first policy, Russian logician D.A. Bochvar, in the late 1930’s “proposed a three-valued semantics for propositional logic” where “the indeterminacy of a part infects the whole” (Nolt 408). Such a semantics is shown in these truth tables [taken from Nolt 408]:

(Nolt 408)

Nolt has us recall how in classical logic where there are only 2 values, there are 2ⁿ valuations, which would be 2ⁿ horizontal lines in the truth tables, for sentences containing n sentence letters (409). [So the first table above for negation, in a bivalent logic, would only have 2 horizontal rows, and the set of tables below it would have 4]. But since here we have three values, there are 3ⁿ valuations, and thus in the second set of tables above, there are 9 rows (409).

Nolt then shows how this additional complexity factors into the valuation rules in Bochvar’s logic. Here Nolt just gives the rules for negation and conjunction.

v(~Φ) = T iff v(Φ) = F

v(~Φ) = F iff v(Φ) = T

v(~Φ) = I iff v(Φ) = I

v(Φ&Ψ) = T iff v(Φ) = T and v(Φ) = T

v(Φ&Ψ) = F iff either v(Φ) = F and v(Ψ) = T, or v(Φ) = T and v(Ψ) = F, or v(Φ) = F and v(Ψ) = F.

v(Φ&Ψ) = I iff either v(Φ) = I or v(Ψ) = I, or both.

(Nolt 409)

Nolt then notes some of the “striking features” of this sort of logic. 1) all classically tautologous formulas are not tautologous in Bochvar’s semantics. We consider for example the tautology ‘P→P’. [Here, whether P is true or false, the formula will be true regardless.] But in Bochvar’s semantics, if P is I, then the formula is I. That means it is no longer true on all evaluations and is thus not tautologous. Nolt adds however that we can at least say of the classical tautologies that in Bochvar’s logic they are never false (Nolt 409).

If we wanted the classical tautologies to remain distinguished as such in Bochvar’s semantics, then we could define a tautology as “any formula which is not false on any line of its truth table – that is, which is either true or indeterminate on all lines” (409). [Nolt introduces a term here, designated. A truth value is designated if it counts towards a formula being tautologous.]

it comes down to a question of which truth values we shall accept as designated – that is, which values count toward tautologousness. If a statement must be true on all lines of its truth table to count as a tautology, then T is the only designated value. If a statement need merely be either true or indeterminate on all valuations, then both T and I are designated values. For Bochvar, only T was designated.

(409, boldface his)

Nolt turns now to the notion of validity, which also involves different options for how to define it. When there are just two values, T and F, then it does not matter which of the following definitions we use, as they will both suffice to identify the valid formulas.

a sequent is valid iff:

1) there is no valuation on which its premises are all true and its conclusion is untrue.

2) there is no valuation on which its premises are all true and its conclusion is false.

(409)

As Nolt explains,

Given bivalence, untruth and falsity are the same thing. In a multivalued logic, however, the difference between the two definitions is substantial, for there may | be valuations on which the premises are true and the conclusion is indeterminate.

(409-410, boldface mine)

[We should here consult Nolt’s definition of counterexample from earlier in the book: “A possible situation in which an argument’s premises are true and its conclusion is not true is called a counterexample to the argument. We may define validity more briefly simply by saying that a valid argument is one without a counterexample” (6, boldface his). Notice the wording of “its conclusion is not true). Thus using this definition, we would have a counterexample if the premises are true and the conclusion is indeterminate.] Nolt asks if we should consider instances where the premises are true and the conclusion indeterminate as counterexamples, and he replies:

If we think so, we will adopt the first definition of validity. If we think not, we will adopt the second. Bochvar adopted the first, and it is the one we shall use here. Indeed, we stipulate now that we shall for the purposes of this section (and particularly the exercise at the end) retain the wording of all the definitions of semantic concepts presented in Chapter 3.

(410)

Nolt then notes how Bochvar’s semantics can invalidate certain sequents “that at least some logicians have regarded as suspect” (410). He offers for example the “paradoxes of material implication” [but he does not explain why one might object to these arguments]:

Q ⊢ P → Q

~P ⊢ P → Q

[Nolt does not make the truth tables, so the ones I provide in the following may be incorrect.

As we can see, there are no horizontal lines where the premise is true but the conclusion not true. Now let us examine these sequents using Bochvar’s semantics.

Here we can see that there is a line in each where the premise is true but the conclusion is not true. Thus in this semantics they are not valid arguments.]

Q ⊢ P → Q

~P ⊢ P → Q

though valid in classical logic, are on Bochvar's semantics invalid. In the first case, the counterexample is the valuation on which ‘Q’ is true and ‘P’ indeterminate, which makes the premise true and the conclusion indeterminate (hence untrue). In the second, the counterexample is the valuation on which ‘P’ is false and ‘Q’ indeterminate.

(Nolt 410)

[Nolt then turns to a metatheorem, which covers cases like these. The metatheorem says that a sequent is invalid if it introduces a sentence letter not found in the argument. The proof seems to do the following. We consider a sequence of premises that is consistent, and a conclusion that contains a sentence letter not found in the premises. We next assume that there is some valuation that makes all premises true. Next, we consider another valuation. Like the first, it makes all the premises true. But in addition, it makes all the sentence letters of the conclusion not found in the premises have the value I. This of course will do nothing to change the fact that all the premises are true. It will only make the conclusion I, since even one instance of an I value will “contaminate” the formula or formulas of the conclusion. Thus we would have a valuation that makes the premises true and the conclusion not true. Hence furthermore, when there is a sentence letter in the conclusion not found in the premises, the argument will be invalid in Bochvar’s semantics.]

METATHEOREM: Let Φ₁, ..., Φ_n ⊢ Ψ be a sequent of ordinary propositional logic (as defined by the formation rules of Chapter 2). Then on Bochvar's semantics, if {Φ₁, ..., Φ_n} is consistent and Ψ contains a sentence letter not found in Φ₁, ..., Φ_n then that sequent is invalid.

PROOF: Suppose that {Φ₁, ..., Φ_n} is consistent and Ψ contains a sentence letter not found in Φ₁, ..., Φ_n Since {Φ₁, ..., Φ_n} is consistent, there is some Bochvar valuation v of Φ₁, ..., Φ_n that makes each of these formulas true. But now consider the valuation v′, which is just like v except that in addition it assigns the value I to each of the sentence letters that appear in Ψ but not in Φ₁, ..., Φ_n. v′ makes Φ₁, ..., Φ_n true but Ψ indeterminate, and so v′ is a counterexample, which proves the sequent invalid.

Therefore, if {Φ₁, ..., Φ_n} is consistent and Ψ contains a sentence letter not found in Φ₁, ..., Φ_n, then that sequent is invalid. QED

(Nolt 410)

Nolt then notes a complication. In the metatheorem he specified that the formulas must be those of “ordinary propositional logic” (410). He explains, “The reason for this qualification is that Bochvar added a novel operator to his logic, and the metatheorem does not apply to formulas containing this novel operator” (Nolt 410).

The operator means something like, “it is true that”, and Nolt will symbolize it as ‘T’, following Susan Haack (citing: Philosophy of Logics. Cambridge: Cambridge University Press, 1978, p. 207.) T is a monadic operator like negation. We need first to fashion a formation rule to allow for its inclusion in our language:

If Φ is a formula, so is TΦ.

(Nolt 411)

And Nolt shows the truth table for this as:

(Nolt 411)

We notice, then, that only if a formula Φ is true then is it true that Φ. And thus “if Φ is false or indeterminate, then it is not true that Φ” (411).

[So as we can see from the table, whenever a formula has the T operator, its value can only be either T or F, and thus it is bivalent. So by using the T operator, we can now formulate tautologies in Bochvar’s semantics. (It seems, however, that the formulation expressing the law of excluded middle is not tautologous).]

This “truth operator” gives Bochvar’s logic a new twist, for any formula of which it is the main operator is bivalent. And though, as we saw above, Bochvar’s logic contains no tautologies among the formulas of ordinary propositional logic, it does have tautologies. These, however, are all formulas containing the truth operator. Here are some examples:

TP → TP

TP ∨ ~TP

T(P ∨ ~P) ∨ (~TP & ~T~P)

Notice, by contrast, that ‘TP ∨ T~P’ is not tautologous, for it is false when ‘P’ is indeterminate.

(Nolt 411)

Nolt explains that it was Bochvar’s hope that his T operator would resolve the semantic paradoxes [like the liar paradox] (Nolt 411). Above we called the following sentence S:

Sentence S is false.

(Nolt 411)

We had reason to think that it was neither true nor false. With Bochvar’s semantics, we can designate it as I, which means that TS would be F. [I am not certain, but it seems that this avoids the paradox for the following reason. As we know, we cannot simply designate S as true or false, because this leads to a contradiction. I do not know however what it means if we simply say that S is I, without also adding the T operator. I am just guessing, but perhaps the problem is the following. We are now saying that S is I, but S says of itself that it is false. Since what it says of itself is not what it is (as it says it is false but it was designated as I), that makes S false. But we said at first it was I, hence the contradiction. At any rate, the idea is that TS eliminates the problem, when S is I, because it makes TS simply be false in a non-problematic way. There is no paradox for TS, because TS means, ‘it is true that S is false’. The truth table tells us that when S is I, then TS is false. This means that ‘it is not true that S is false’ is true. Here there is no contradiction, because in fact it is not true that S is false, as it is I and not false. But I am just guessing at the reasoning here, so please consult the quoted text that follows after the next summarization.]

But, there is still a problem with this semantic paradox. Suppose we fashion the formula U:

Sentence U is untrue.

(411)

It cannot be true, because then it says of itself that it is untrue, which is a contradiction. [Let us first consider the difference between untrue and false in this three-valued system. To be false means to be neither true nor I. To be untrue means to be either false or I.] [The next step here is to suppose that it is either false or indeterminate. I am not sure if we also add the T operator. First let us consider it without that operator. Suppose we say that this sentence is false. That means it is also not-true and thus untrue. However, it says of itself that it is untrue, and so it would be true as well (because what it says is what is the case). So if we say it is false, it leads to a contradiction. Suppose instead its value is I. That means again it is untrue and thus (given what is says) is also true. Hence if we designate U as I, then we obtain a contradiction. Let us further consider the T operator. Suppose U is true. The formula TU, which means, ‘It is true that Sentence U is untrue’, would be true according to the truth table for T. But it says of itself that it is untrue, hence a contradiction. Suppose now that U is false. That makes TU, which means, ‘It is true that Sentence U is untrue’ be false according to the truth tables. But that would mean ‘It is not true that sentence U is untrue’ is true. And if U is not untrue, it would have to be either true or I. But we first designated U as false, so we have a contradiction. So suppose instead we designate U as I. That means, TU, which again would read ‘It is true that sentence U is untrue’ is false, according to the truth tables. But that means ‘It is not true that sentence U is untrue’ is true. And if U is not untrue, it cannot be I or false and thus must be true. However, we originally designated U as I and thus as not being true. (I had trouble following the reasoning in this section, so please consult the quotation to follow):]

Bochvar hoped that his new semantics would solve the problem of semantic paradox. Consider, for example, the semantically paradoxical sentence that we have called S:

Sentence S is false.

We argued above that S is neither true nor false. But if we adopt Bochvar's semantics, there is a third option: It might have the value I. Suppose, then, that it does. In that case, using the sentence letter ‘S’ to represent sentence S, though the formula ‘S’ has the value I, the formula ‘TS’ has the value F, for it is in fact not true that sentence S is false.

This three-valued approach seems neatly to dissolve the paradox. Unfortunately, however, a new paradox emerges to take its place. Let U be the sentence:

Sentence U is untrue.

As before, suppose for reductio that this sentence is true. Then what it says is correct, and so it is untrue. Hence it is both true and untrue – a contradiction. Therefore it is not true. It follows, on Bochvar’s semantics, that it has one of the | values F or I. But in either case it is untrue, and so what it says is correct; hence it is true. Once again we have a contradiction – despite the third value.

Bochvar’s semantics does not, therefore, provide a general solution to semantic paradoxes – nor does any other three-valued or multivalued semantics. If semantic paradox is the problem, multivalued semantics is not the solution.

(Nolt 411-412)

Nolt next turns to the second policy, where indeterminacy in a formula’s parts does not necessarily affect the whole, so long as the other parts still suffice to produce a T or F valuation under the normal rules. [So it seems that under this policy, an indeterminate value that is part of a composite formula will cause the whole formula to be indeterminate, unless the conditions for truth or falsity are sufficiently met by the other terms. So consider conjunction, which was defined on p.50 as T if both conjuncts are T, and it is F if either or both conjuncts is not true. But in these new policy tables, it seems that the rule would be that it is false iff either or both conjuncts is false (and not just untrue). In the new truth tables, we see that the conjunction’s value is true only when both conjuncts are true, and it is only false when either conjunct is false. But it is indeterminate when one is true and the other indeterminate. For, here it fulfills neither qualification. It cannot be true, because not both conjuncts are true. And it cannot be false, because neither of the conjuncts are false. It is also indeterminate when both conjuncts are indeterminate, for the same reason. Disjunctions under this new policy are true so long as at least one disjunct is true, and false only when both are false. It is indeterminate when one disjunct is false and the other indeterminate, or when both are indeterminate. Thus the criteria seems to be that a disjunct is true only when both disjuncts are true; it is false only when both are false, and it is indeterminate either when one is false and the other indeterminate or when both are indeterminate. Next is the conditional. It is false only when the antecedent is true but the consequent false. What determines whether it is true or indeterminate seems to be something like the following. If the consequent is indeterminate and the antecedent is not false, then it is indeterminate, and true otherwise. (The thinking might be that the “indeterminate” could potentially mean false, and so any situation where it is ambiguous where potentially the antecedent is true and the consequent is false means that the whole conditional is indeterminate.)  Finally, consider the biconditional. This is true only if both terms are true, and it is only false if one is true and the other false. It is indeterminate if either one or both are indeterminate.]

(Nolt 412)

Nolt writes that the “three-valued propositional semantics expressed by these tables was first proposed by S. C . Kleene”, and that we can use the T operator, “which has the same truth table as before” (Nolt 412). Like with Bochvar’s semantics, Kleene’s three-valued semantics also makes all classically tautologous formulations non-tautologous, because for any of them there we can assign all the formula’s component letters as I, thereby making all formulas not true, even if they are necessarily true in classical bivalent logic.

On Kleene’s semantics, as on Bochvar’s, classically tautologous formulas are nontautologous. For, as on Bochvar’s semantics, any statement of ordinary propositional logic all of whose atomic components have the truth value I has itself the truth value I; hence, for any formula, there is always a valuation (namely, the valuation that assigns I to all of its sentence letters) on which that formula is not true.

(412)

One important difference between Kleene’s and Bochvar’s semantics is that Kleene’s “makes most of the classical inference patterns, including paradoxes of material implication, valid” (412). However, some of the classical valid inferences are invalid in Kleene’s semantics. “In particular, inferences to what are classically tautologies from irrelevant premises still fail. The sequent ‘P ⊢ Q→Q’, for example, though classically valid, is invalid on Kleene’s semantics, for the valuation on which ‘P’ is true and ‘Q’ indeterminate is a counterexample” (Nolt 412).

Łukasiewicz’s semantics makes the conditional and the biconditional true when both terms are indeterminate.

Kleene’s semantics assigns the classical values T and F to more complex formulas than Bochvar’s semantics does. Łukasiewicz, who was the first to explore three-valued logic, proposed a semantics that goes even further in this direction. Łukasiewicz’s semantics is like Kleene’s, except that where Kleene makes the conditional and biconditional indeterminate when both their components are indeterminate, Łukasiewicz makes them true. Łukasiewicz’s semantics is thus expressed by the following truth tables: |

(Nolt 412-413)

And as in the other semantics, we can add the T operator to Łukasiewicz’s system (413).

Nolt then discusses which classical tautologies are made valid by Łukasiewicz’s semantics and which are not. Some examples of classical tautologies that are made true are:

P  → P

P  → (P ∨ Q)

(P & Q) → P

P  ↔ ~~P

(Nolt 413)

[If we look at the truth tables for Łukasiewicz’s semantics, we see that all of these would be true regardless if the terms are T, F, or I.]  However, the law of excluded middle and the law of noncontradiction are not tautologous.

P ∨ ~P

~(P & ~P)

(Nolt 413)

[If we look at the truth tables for Łukasiewicz’s semantics, we see that] when we assign P as I, in the first one we have an indeterminate value, and the same for the second. And thus they are not always true and hence are not tautologous. Nolt says that certain classical tautologies could become false in Łukasiewicz’s semantics, as for example:

~(P ↔ ~P)

(Nolt 413)

Łukasiewicz’s semantics makes invalid certain sequents in which there is an inference of classical tautologies from unrelated premises. “For example, ‘P ⊢ Q ∨ ~Q’ is invalid since its premise is true while its conclusion is indeterminate in the case in which ‘P’ has the value T and ‘Q’ the value I. Anomalously, however, ‘P ⊢ Q → Q’ remains valid” (413).

There are other cases in Łukasiewicz’s semantics that are potentially problematic, because they invalidate certain logical equivalences [that we intuitively would want to keep]:

Moreover, Łukasiewicz’s semantics dispenses with the classical logical equivalences between ~(Φ & ~Ψ) or ~Φ ∨ Ψ and Φ → Ψ, precisely because of this case. For although Φ → Ψ is true when both Φ and Ψ are indeterminate, both ~(Φ & ~Ψ) and ~Φ ∨ Ψ are indeterminate in that case. Some logicians find these features inelegant.

(Nolt 413)

 

Nolt says that more than three values are possible, even infinitely many. He gives an example for four-valued logic: “Some of the early semantics for modal logic, for example, used four values: contingently true, contingently false, necessarily true, and necessarily false, with the first two as designated values” (Nolt 413). But he says there are too many such variants to mention here (413).

 

Nolt says that we can also construct multivalued predicate logics, which often involves making some choices regarding the way atomic formulas are to be evaluated.

One can also, of course, create multivalued predicate logics. This generally requires some adjustment of the definition of a valuation and of the valuation rules for atomic formulas. We might, for example, as in free logics, allow names to lack referents. Then either all or some of the formulas containing these names could be stipulated to have the truth value I. Or we could attempt to dichotomize atomic | formulas into those that are meaningful and those that are meaningless, assigning the latter the value I. Or, if we are doing tense logic, we might design our models so that atomic statements about the future always receive the value I. But again, we shall not bother with the details of these variations.

(413-414)

 

 

From:

Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.

.