[11.] Synechism, Bivalence, and Triadic Logic.

 

In this last set of notes, we will see how Peirce’s synechism (the view that there is real continuity in the universe) is affected by his 1896 “shift” towards a form of modal realism.

 

As we will see, the change in his thinking about continuity caused a change in his thinking about truth. In particular, it resulted in Peirce’s rejection of the

 

principle of bivalence (df.): every declarative sentence is either true or else false.

 

And this in turn resulted in his pioneering work in three-valued (“triadic”) logic, a formal system of logic that includes three truth values: the traditional values True and False, but also a new, third value, which Peirce called the Limit (that is, the limit between true and false).

 

In this final section of notes, we will examine how Peirce’s work on three-valued logic and his rejection of the principle of bivalence interfaces with his previous work on truth and inquiry, and we will consider whether Peirce himself may have been guilty of, as he put it, blocking “the way of inquiry.”

 

 

[11.1.] Triadic Logic.

 

 

[11.1.1.] Sentential Logic.

 

There are many different systems of formal or symbolic logic, but all of them are motivated by the desire to represent arguments in a perspicuous way.

 

One of the simplest systems of formal logic is sentential logic, which represents arguments the logic of which depends on connections between entire sentences. For example,

 

            Andrew is married.

If Andrew is married, then William is single.

Therefore, Andrew is married.

 

is a deductively valid argument, and its validity depends on the words that “connect” the two atomic sentences imbedded in the argument: “Andrew is married” and “William is single.” The same is true of this argument:

 

Andrew is married or William is single.

It is not the case that Andrew is married.

Therefore, William is single.

 

The emboldened words “If ... then ...” and “or” are sometimes called sentence connectives, and any system of sentential logic will have the capacity to represent these connectives and the sentences which they connect. The above arguments might be represented as

 

A A É W \ W

A v W ~A \ W

 

Each of the sentences is symbolized by an upper-case letter, “A” or “W.”

 

Here the “if-then” relation, called material implication, is represented by the horseshoe (“É“).

 

The “or” relation, called disjunction, is here represented by the vel (“v”).

 

The phrase “It is not the case that” represents negation, and is here symbolized by the tilde (“~”).

 

Other relations represented in traditional sentential logic are conjunction (“...... and .......”), sometimes represented by the dot (“·“) and material equivalence, typically represented by the triple-bar (“º“).

 

 

[11.1.2.] Truth Values and Truth Tables.

 

In traditional sentential logic, any sentence is either true or false. These “values” (truth and falsity) are said to be the two truth values recognized by traditional logic. They are the only values that a sentence in the system can take. So if “Andy is married” is represented in a traditional system of logic as “A”, then “A” must take one of the two truth values; no third option is possible.

 

One way of defining a symbol that represents a sentence connective is to provide a truth table: a simple matrix that shows what the truth value of a sentence formed with that connective would be, given all the possible assignments of truth values to its constituent sentences.[1] For example, a truth-table definition of disjunction would display what truth value a disjunctive sentence has for each possible assignment of truth values to its constituent sentences:

 

 

 

Notice that each cell in the chart is occupied by either a “T” (for true) or an “F” (for false). In specifying what becomes of a sentence having the form “p v q” for every possible combination of truth values of “p” and “q”, this table provides a definition of “v”. It tells us everything we need to know about how that symbol works in the system of formal logic in question.

 

The other four symbols can be defined in the same way:

 

 

negation

 

 

conjunction

 

 

material implication

 

 

material equivalence

 

 

 

[11.1.3.] Adding a Third Value. [2]

 

Peirce was the first logician to define logical operators for a many-valued system of formal logic, i.e. a system that incorporates more than two values. [3]

 

In February 1909, on three pages of a notebook in which he recorded his thoughts on logic, he defined several three-valued connectives using the truth-table, or matrix, method.[4]

 

Peirce’s three values were:

• “V” (verum, or true)
• “F” (falusm, or false)
• “L” (the Limit).

 

Peirce defined six different two-place connectives, three of which seem to be alternative forms of disjunction, and three of which seem to be alternative forms of conjunction.

 

disjunction[5]

 

 

conjunction[6]

 

 

 

disjunction[7]

 

 

conjunction[8]

 

 

 

disjunction[9]

 

 

conjunction

 

 

 

Peirce also defined four different one-place connectives: [10]

 

 

It is plausible to construe these as three-valued variants of traditional negation, since negation is the only one-place connective in traditional sentential logic.

 

 

It is unclear from what Peirce wrote how satisfied he was with this work. On one page he wrote that “Triadic Logic is universally true”; but on another, he wrote that “All this is mighty close to nonsense.”[11]

 

 

 

[11.2.] Philosophical Motivations.

 

What were the philosophical motivations behind Peirce’s pioneering work on many-valued logic? In particular, what sort of proposition did Peirce intend to be taken by his third truth value, “L”?

 

We have already seen Peirce hinting that there may be good reason for abandoning the Principle of Bivalence...

 

Affirmation and denial are in themselves unaffected by these concepts [viz. determination, generality and vagueness], but it is to be remarked that there are cases in which we can have an apparently definite idea of a border line between affirmation and negation. Thus, a point of a surface may be in a region of that surface, or out of it, or on its boundary. This gives us an indirect and vague conception of an intermediary between affirmation and denial in general, and consequently of an intermediate, or nascent state, between determination and indetermination. There must be a similar intermediacy between generality and vagueness. Indeed, in an article in the seventh volume of The Monist [“The Logic of Relatives,” the 1897 article in which Peirce rejects the IR account of substantial possibility] there lies just beneath the surface of what is explicitly said, the idea of an endless series of such intermediacies. (“Issues of Pragmaticism,” EP 2:353, CP 5.450, emphasis added)

 

There may be questions concerning which the pendulum of opinion never would cease to oscillate, however favorable circumstances may be. But if so, those questions are ipso facto not real questions, that is to say, are questions to which there is no true answer to be given. (“Issues of Pragmaticism,” EP 2:358, CP 5.450, emphasis added)

 

 

[11.2.1.] Continuity Breaches.

 

Recall that one of the consequences of Peirce’s modal shift of 1896 was that he began to define continuity in terms of possibility:

 

a continuum is a collection of so vast a multitude that in the whole universe of possibility there is not room for them to retain their distinct identities; but they become welded into one another.  Thus, the continuum is all that is possible, in whatever dimension it be continuous. (NEM 4:343; RLT 160, 1898)

 

For example, a continuous line does not consist in any number of actual individual points, however numerous. Instead, it

 

contains no points until the continuity is broken by marking the points. In accordance with this it seems necessary to say that a continuum, where it is continuous and unbroken, contains no definite parts; that its parts are created in the act of defining them and the precise definition of them breaks the continuity. (6.168, 1903)[12]

 

To understand how Peirce went from this new, modal conception of continuity, to his development of three-valued logic, we need to consider what Peirce thought about continuity breaches:

 

continuity-breach (df.) a gap or break in something otherwise continuous.

 

E.g., consider the surface of the dry-erase board. This is a continuous, unbroken plane. Now suppose I make a mark on the dry-erase board with a marker. I have broken the continuity of the white surface. This is a breach in the continuity of the otherwise continuous board.

 

Now, the board itself is white (P), and the mark itself is black, and so non-white (not-P). But what of the boundary between the white board and the non-white mark? Is it white or non-white (P or not-P)?

 

The boundary between the white and the non-white areas of the board is a continuity-breach. It is a line in an otherwise uninterrupted, continuous surface.

 

In 1893, before the modal shift, Peirce answered the question by saying:

·         The boundary does “not exist in such a sense as to have entirely determinate characters attributed to” it (4.127, 1893).

·         It is not the boundary itself, but the area in the “immediate neighborhood” of the boundary, that is partly white and partly not-white (6.126, 1892; 4.127, 1893).

·         Generally, a continuity-breach is itself neither P nor not-P, where P is the boundary-property relative to the breach in question.

·         Only insofar as we consider the area within an infinitesimal distance from the continuity-breach does it makes sense to say that the breach itself is partly P and partly not-P.[13]

 

In 1898, relatively soon after the modal shift, he wrote that a continuity-breach is characterized by “the pairedness” of its boundary-properties:

 

I draw a chalk line on the board. ... What I have really drawn there is an oval line. For this white chalk-mark is not a line, it is a plane figure in Euclid[‘s] sense, a surface, and the only line [that] is there is the line which forms the limit between the black surface and the white surface. ... the boundary between the black and white is neither black, nor white, nor neither, nor both. It is the pairedness of the two. It is for the white the active Secondness of the black; for the black the active Secondness of the white. (6.203, 1898)

 

So here Peirce’s position was:

·         The boundary B between the black area and the non-black area of a surface is neither black nor non-black.

·         Rather, it is “the pairedness” of black and non-black.[14]

 

This response to the problem is not very clear. Unfortunately, what I quoted above is all Peirce had to say in the way of an elaboration of the idea of pairedness; and by itself it is far too vague to be satisfactory.

 

But by 11 years later, he had formulated a much clearer answer to the question about boundaries:

 

... a blot is made on the sheet.  Then every point of the sheet is unblackened or is blackened.  But there are points on the boundary line, and those points are insusceptible of being unblackened or of being blackened, since these predicates refer to the area about S and a line has no area about any point of it. (MS 339, p.344 recto, Feb. 23, 1909)

 

Peirce’s position here was as follows:

·         The boundary B between the black area and the non-black area of a surface is neither black nor non-black.

·         So, neither “B is black” nor “B is non-black” is true.

·         But because B is neither black nor non-black, neither “B is non-black” nor “B is black” is false.

·         So “B is black” is neither true nor false, and “B is non-black” is neither true nor false.

 

Although still somewhat opaque, this is at least clear with regard to the purported truth value of “B is black” and “B is non-black”: both propositions are neither true nor false, but “at the limit” between truth and falsity.

 

 

[11.2.2.] Bivalence and Boundary Propositions.

 

So by 1909, Peirce had come to believe that a specific sort of proposition was neither true nor false:

 

boundary proposition (df.): a proposition that predicates of a (mathematical, spatial or temporal) continuity-breach one of the properties that is a boundary-property relative to that breach.

 

As an illustration, consider an ink blot on a white piece of paper:

·         the boundary, B, between the white area and the black area is a continuity breach

·         white and black (non-white) are the boundary properties relative to that breach

·         so the following are boundary propositions:

 

“B is white”

“B is non-white”

“B is not white”

“B is not non-white”

 

Peirce thought that each of these propositions took a third truth value beyond the two traditional values of true and false.

 

The third truth value that he articulated was the limit between true and false. 

 

S has a lower mode of being such that it can neither be determinately P, nor determinately not-P, but is at the limit between P and not P. (MS 339, February 23, 1909)

 

He symbolized the limit with the letter “L.”

 

And in admitting that meaningful propositions can be neither true nor false, Peirce gave up the principle of bivalence.

 

 

[11.2.3.] PEM vs. LEM.

 

So in developing his three-valued logic, Peirce gave up the principle of bivalence.

 

But he also saw triadic logic as a threat to his Principle of Excluded Middle (PEM):

 

                Triadic logic is that logic which, though not rejecting entirely the Principle of Excluded Middle, nevertheless recognizes that every proposition, S is P, is either true, or false, or else S has a lower mode of being such that it can neither be determinately P, nor determinately not-P, but is at the limit between P and not P. (MS 339, February 23, 1909)

 

This was echoed in a letter to William James, written only three days later:

 

                I have long felt that it is a serious defect in existing logic that it takes no heed of the limit between two realms.  I do not say that the Principle of Excluded Middle is downright false; but I do say that in every field of thought whatsoever there is an intermediate ground between positive assertion and positive negation which is just as Real as they. (NEM 3:851, Feb. 26, 1909)

 

 

As we’ve already seen, there is a close connection between the Principle of Bivalence and the LAW of Excluded Middle:

 

Law of Excluded Middle (LEM)

·         Every instance of “p or not-p” is true.

·         Either p or not-p.

·         p Ú ~p

If you give up Bivalence, then it seems you will also have to give up LEM.

 

But as we’ve also seen, Peirce’s “excluded middle” was something different:

 

Principle of Excluded Middle (PEM)

Material mode: for any property and for any individual, either that individual possesses that property or that individual does not possess that property.

Formal mode: for any pair of contradictory predicates “P” and “not-P” and for any individual (non-general) subject-term “S”, either “S is P” or “S is not-P” is true.

 

Giving up the Principle of Bivalence does not seem to threaten Peirce’s PEM, so it is very surprising to see him Peirce questioning PEM in the context of his triadic logic.

 

But as it turns out, there is an explanation of this surprising claim. It lies in a distinction Peirce made, between:

·         a logical principle not applying to a proposition; and

·         a logical principle being false with regard to a proposition.

 

Peirce did not claim that PEM does not apply to propositions that take the value “L”.

 

Nor did he say that the PEM is “downright” false:

·         “ Triadic logic is that logic which, though not rejecting entirely the Principle of Excluded Middle…”

·         “I do not say that the Principle of Excluded Middle is downright false…”

 

Peirce seems to have believed that the assignment of a value other than “true” or “false” to a proposition required that PEM be weakened or qualified in some way.

 

But triadic logic would only require a weakening or qualification of PEM if it were intended to accommodate propositions to which PEM applies in the first place.

 

Peirce intended “L” to be taken by propositions to which PEM applies but with regard to which PEM is false.

 

If PEM is false with regard to “S is P,” then

(i)                  “S” refers to an individual rather to a general (if it didn’t, PEM wouldn’t apply to it);

(ii)                 “S is P” is neither true nor false--so the individual to which “S” refers neither has nor lacks the property represented by “P.”

 

And this is in harmony with what Peirce said about boundary propositions, which he thought took the value “L”:

 

S has a lower mode of being such that it can neither be determinately P, nor determinately not-P, but is at the limit between P and not P. (MS 339, February 23, 1909)

 

In summary:

·         Peirce was motivated to develop triadic logic to accommodate propositions, not to which PEM fails to apply, but to which PEM applies and with regard to which it is false.

·         The weakening or qualification of PEM that triadic logic requires is simply the acknowledgment that, with regard to some propositions to which PEM applies (viz. object-individual propositions which don’t express necessity), that principle is false, i.e., some such propositions take a truth value other than “true” or “false.”

·         This is why, even though Peirce held triadic logic to require a weakening of PEM, his claim that PEM does not apply to a proposition (e.g., object-general propositions, such as “Man is mortal,” or propositions expressing necessity, such as “Socrates must be mortal”) does not imply that the proposition is neither true nor false.

 

Why would Peirce care so much about this? Why, after all, would Peirce take boundary-propositions to be interesting or important enough to motivate him to introduce three-valued logical connectives?

 

The answer lies in something that we have already seen. The notion of continuity was itself of supreme philosophical importance for Peirce. That the question of continuity-breaches and their boundary-properties was for him not simply an afterthought or a relatively unimportant aspect of the broader issue of the nature of continuity, is indicated by the fact that each time he revised his definition of continuity in a significant way, his position regarding continuity-breaches and their boundary-properties changed as well.

 

 

[11.3.] Blocking the Way of Inquiry?[15]

 

We have already seen that, by late in his life, Peirce had come to believe that some propositions, viz. what I have called boundary propositions, are neither true nor false. He thus came to reject the principle of bivalence with regard to those propositions.

 

But there is an apparent tension among this denial of bivalence for boundary propositions and two other aspects of Peirce’s system: his pragmatic clarification of truth and his insistence that we not “block the way of inquiry.”

 

Today we will examine this tension and consider whether it is objectionable.

 

 

[11.3.1.] A Regulative Assumption Revisited.

 

Recall that Peirce’s pragmatic clarification of the concept of truth is that a true proposition is one that would be believed at the hypothetical conclusion of inquiry, i.e., at the end of an indefinitely extended investigation into how things are.

·         After the 1896 modal shift, Peirce became consistent in putting this pragmatic clarification in the subjunctive mood. A true proposition is not necessarily one that will be agreed to at any actual point in history. Rather, it is one that would be agreed to were inquiry into it carried out as far as it could possibly go.[16]

 

We have already seen that the later Peirce did not intend his pragmatic clarification of truth to serve as a biconditional definition, something like

 

“S is P” is true if and only if __________________.

 

Instead, what Cheryl Misak calls the Truth to Inquiry Conditional is a regulative assumption of inquiry, a hope that must be adopted by an inquirer with regard to the question she is investigating:

 

The Truth to Inquiry Conditional (T-I): If “S is P” is true, then, if inquiry relevant to whether S is P were pursued as far as it could fruitfully go, it would be agreed that S is P. [17]

 

Before Peirce’s work on triadic logic, his view of bivalence seems to have been the same. That is, he seemed to want, not to assert the principle of bivalence,[18] but to maintain that it is an assumption of inquiry that the question the inquirer is attempting to answer actually has an answer.

 

In other words, he viewed the principle of bivalence in the same way he viewed T-I: as a regulative principle that must be adopted by inquirers. He held that an inquirer into whether S is P must assume that “S is P” is either true or false. If we are set to investigate whether S is P, we must presume that it either is or is not the case that S is P. It would make no sense to conduct an investigation into whether S is P if we did not believe that it either is or is not the case that S is P. So on Peirce’s account of truth, the principle of bivalence is a “hope” adopted with regard to “S is P” by anyone investigating whether or not S is P.

 

Consequently, to assign a proposition a value other than “true” or “false” is to abandon this hope with regard to that proposition. To assert that “S is P” is neither true nor false is to assert that the question whether it is or is not the case that S is P would never be permanently settled, no matter how long inquiry were to be pursued.

 

 

[11.3.2.] Limits to Inquiry.

 

It seems that in order to be consistent, Peirce would have had to accept something like the following with regard to his L-propositions (call this the “Limit-to-Inquiry” conditional):

 

The Limit to Inquiry Conditional (L-I): If “S is P” is at the limit between truth and falsity, then, if inquiry relevant to whether S is P were pursued as far as it could go, it would be agreed neither that S is P nor that it is not the case that S is P.

 

L-I is fundamentally different from T-I. The truth of L-I is not something for which an inquirer can hope. To accept that L-I is true with regard to a proposition is to give up the hope embodied in T-I. It is to resign oneself to the belief that inquiry would never permanently settle opinion regarding whether S is P. Given his pragmatic clarification of truth, Peirce would have to admit that agreement about L-propositions would never be reached, no matter how far inquiry into them might be pushed.

 

This seems to be in tension with his own injunction against, as he put it, blocking “the way of inquiry.” As we have seen, Peirce wrote that “to set up a philosophy which barricades the road of further advance toward the truth is the one unpardonable offence in reasoning.”[19]

 

One way of blocking the road of inquiry is to presume that a given question is unanswerable, to maintain that “this, that, and the other never can be known”. (1.138, c.1898; cf. 1.170, c.1897).[20]

 

If “S is P” takes the value “L”, then there is no “yes” or “no” answer to the question whether S is P.

 

And this raises the following question: by assigning the value “L” to some propositions, and thus accepting L-I with regard to those propositions, wasn’t Peirce in effect blocking the way of inquiry with regard to those propositions?

 

In short, the answer is “yes.” By his own standard, Peirce was blocking the way of inquiry into propositions taking “L” as their value.

 

Had he claimed that a broad class of proposition (modal propositions, say, or, propositions containing “scientifically sound predicates”) fails to be either true or false, Peirce himself would have been guilty of blocking a relatively wide avenue of inquiry.

 

But he rejected bivalence only for a very narrow range of propositions: boundary-propositions. Thus, Peirce was guilty of blocking, not a wide avenue of inquiry, but only a narrow alley-way.

 

 

[11.3.3.] Reality as Indeterminate.[21]

 

Peirce’s pragmatism about truth and reality, when paired with his acknowledgment of propositions which are neither true nor false, implies that there are ways in which the world—not just our descriptions of the world, but the world itself—is indeterminate.

 

In some cases, the world is such that S is neither P nor not-P; with regard to S being P, the world is indeterminate, and the proposition that S is P is neither true nor false.

 

This willingness to acknowledge worldly indeterminacy appears in Peirce’s defense of the synechistic, anti-Parmenidean claim that “being is a matter of more or less, so as to merge insensibly into nothing”:

 

... to say that a thing is is to say that in the upshot of intellectual progress it will attain a permanent status in the realm of ideas. Now, as no experiential question can be answered with absolute certainty, so we never can have reason to think that any given idea will either become unshakably established or be forever exploded. But to say that neither of these two events will come to pass definitively is to say that the object has an imperfect and qualified existence. (EP 2:2, 7.569, 1893, emphasis added)

 

If no possible inquiry, no further experiences or reflections, would ever result in a consensus regarding whether or not S is P, then there is no fact of the matter about whether S is P, and in that respect the world itself is indeterminate.

 

 

 

Stopping point for Monday November 12. Next time, I will present a paper on which I am currently working, to serve as a model for your own paper presentations, which begin Friday with David’s presentation.