**RELATIONSHIP BETWEEN THE UNIVERSAL CATEGORIES OF PEIRCE**

by **Robert Marty**, Honorary Professor, University of Perpignan, France, incorporating suggestions by **James Crombie**, PhD, Professeur agrégé de philosophie, Université Sainte-Anne, Nova Scotia, Canada.

**Abstract **: This note calls attention to the symmetry between the Frege-Strawson conditions for “presupposition” and the relationships of categorized phenomena (termed primans, secundans and tertians) under the Peircean categories of First, Second and Third.

**1-Clarifications on the notation and the terminology**

Peirce’s universal categories are *Firstness*, *Secondness* and *Thirdness* as specified in the following definitions:

*« Firstness** is the mode of being of that which is such as it is, positively and without reference to anything else.** *

*Secondness** is the mode of being of that which is such as it is, with respect to a second but regardless of any third.** *

*Thirdness is the mode of being of that which is such as it is, in bringing a second and third into relation to each other.**« *(CP 8.328)

* *We will redraft these definitions in order to avoid a confusion which frequently occurs between the categories themselves and the elements of phenomena which are categorized under them. We call « priman » an element belonging to the Firstness. Peirce uses this term for example in CP 1.295, CP 1.318, CP 1.351, SS. pp. 192-193 (Letter to Lady Welby (draft) probably from July 1905), MS 793 (Four versions of a certain page 11, 5 occurrences). Peirce also uses other equivalent terms and in particular « Possible » in his letter to Lady Welby of December 23, 1908. We call « secundan » an element belonging to the Secondness (1.319, MS 793a,) and « tertian » an element belonging to the Thirdness (1,297, 1,319) called Necessitant in the afore mentioned letter.

For the record, primans are « qualities of feeling », secundans are existing things and the facts concerning them while tertians are concepts, laws and more generally regularities of an indefinite future that « govern relations between secundans and / or primans « . On this basis, we will allow ourselves to reformulate the above-quoted definitions from CP 8.328 as follows. Note that additions or modifications are underlined and the term « thing » means « anything »:

« Firstness is the mode of being of that which is such as it is, positively and without reference to anything else. » This is the mode of being of a priman element.

« Secondness is the mode of being of that which is such as it is in relation to another thing, but without regard to a third thing, whatever it may be. » This is the mode of being of a secundan element.

« Thirdness is the mode of being that which is such as it is, by putting two things in reciprocal relation. » This is the mode of being of a tertian element«

**2- The logic of categories.**

Peirce refers many times to the relations between elements of categorized phenomena since they intervene in all of his classifications of signs. It is clear that when we have a secundan we necessarily have two constitutive primans because if we have two things in relation we have two times a thing, each having its Firstness. Other way of saying this is to say that a dyad degenerates into two monads when one « forgets » the relation which connect them. Likewise, when we have a tertian we necessarily have three secundans and three primans. This makes it possible to affirm that there exists between the three categories relations of presupposition: Thirdness presupposes Secondness, which presupposes Firstness. We state this on the basis of an appeal to what “seems obvious”. But we can provide a more satisfactory defense of our thesis as follows.

As far as presuppositions are concerned, two frameworks are available to us, logical and semantico-pragmatic. The latter approach covers discursive utterances and speakers. The interlocutors are taken into account, along with their particular beliefs, and along with the background information on which the dialogues are built. It is obviously not this framework which is suitable for relations between universal categories, since semantico-pragmatics inevitably leads us back to particularity. It is therefore the logical framework that should be chosen if we want to validate the basis of a model. Further advantages of the logical approach will appear as we proceed.

Recall that since G. Frege a proposition q is considered as presupposed by a proposition p if q is preserved under the negation of p. Here is a very simple illustration:

Consider the proposition (p*) John knows that Peter is visiting Barcelona *and the proposition (q) *Peter is visiting Barcelona*.

We form the negation of p noted ¬ p.

(¬ p) *Jean does not know that Peter is visiting Barcelona.*

The statement (p) *John knows that Peter is visiting Barcelona *indicates that (q) *Peter is visiting Barcelona* is true; the statement (¬ p) *John does not know that Peter is visiting Barcelona* also indicates that (q) *Peter is visiting Barcelona* is true. The proposition q is therefore a presupposition of p. Note, however, that from a logical point of view, q must be true so that a truth value can be assigned to the statement p (Strawson 1950).

Our starting point will be what Peirce writes in his letter to Lady Welby of December 23, 1908: *« It is obvious that a Possible cannot determine anything other than a Possible, and likewise a Necessitant cannot be determined by anything other than a Necessitant « .* We rewrite it as follows:

(a) « It is obvious that a Priman cannot determine anything other than a Priman, and likewise a Tertian cannot be determined by anything but a Tertian. » |

Some additional references:

*« […] But we need not, and must not, banish the idea of the first from the second ; on the contrary, the second is precisely that which cannot be without the first ».** *(CP 1.358)

transformed as follows:

« […] But we need not, and must not, banish the idea of the priman from the secundan ; on the contrary, the secundan is precisely that which cannot be without the priman« . (CP 1.358)

And also:

*« […] The first is that whose being is simply in itself, not referring to anything nor lying behind anything. The second is that which is what it is by force of something to which it is second. The third is that which is what it is owing to things between which it mediates and which it brings into relation to each other.* (CP 1.356)

and :

* » […] **for what involves a second is itself a second to that second.**« * (CP 1.357)

transformed as follows:

« […] The priman is that whose being is simply in itself, not referring to anything nor lying behind anything. The secundan is that which is what it is by force of something to which it is secundan. The tertian is that which is what it is owing to things between which it mediates and which it brings into relation to each other*.* (CP 1.356)

and

» […] for what involves a secundan is itself a secundan to that secundan. » (CP 1.357)

*We will establish that there is an equivalence between the proposition (a) above and the proposition:*

(b) « Thirdness presupposes Secondness which presupposes Firstness » |

** 2.1 Formalization of (a).**

Let S be the set {1, 2, 3} where 1 represents a priman, 2 a secundan and 3 a tertian. Let R be the relation *« can determine ».*

We look at the propositions « x can determine y », x Ꞓ{1, 2, 3}, y Ꞓ{1, 2, 3}.

We now consider propositions p* _{ij}* of the form “x can determine y” where x Ꞓ{1, 2, 3} and y Ꞓ{1, 2, 3}.

There are 9 of them. The first part of (a): « It is obvious that a priman cannot determine anything other than a priman » is strictly equivalent to:

(p_{11}) 1 R 1 is true, (p_{12}) 1 R 2 is false, (p_{13}) 1 R 3 is false.

The second part of (a): « A tertian cannot be determined by anything but a tertian » is strictly equivalent to:

(p_{33}) 3 R 3 is true, (p_{23}) 2 R 3 is false, (p_{13}) 1 R 3 is false

We record these results in a table (incomplete):

Let’s look at the 4 remaining propositions, namely (p_{21}), (p_{23}), (p_{22}), (p_{32}). (p_{22}) is a tautology and with regard to (p_{21}), (p_{23}), (p_{32}), since no one is forbidden by the premises. Hence the completed table (T) :

**2.2 Frege-Strawson tests**

It is therefore assumed that (a) amounts to asserting the validity of the above table. We will now show that 2 presupposes 1.

For this we consider the following propositions:

(p) « x is a secundan » and (q) « x can determine a priman »

and we test assuming that

(¬ p) « x is not a secondan »

so x is either a priman or a tertian.

If x is a priman, then according to (a) it can only determine a priman. Thisverifies (q) which becomes a tautology; therefore (q) is true; Strawson’s same condition is verified. If x is a tertian, we know only that it cannot be determined by anything but a tertian; it therefore has the possibility of determining a priman. So in both cases the negation of (p) does not affect (q) which remains true. It follows that (q) is a presupposition of (p).

Since every secundan can determine a priman, it is established that Secondness presupposes Firstness.

Now

(p’) « x is a tertian » and (q’) « x can determine a secundan or a priman »

we test assuming that

(¬ p’) « x is not a tertian »,

from which it follows that x is either a priman or a secundan.

If x is a priman by (a) it can determine a priman and (q’) is true and if it is a secundan then (q’) becomes a tautology and therefore Strawson’s condition is verified. In both cases (q’) is true. It follows that (q ‘) is a presupposition of (p’).

Since every tertian can determine a secundan or a priman, it is established that Thirdness presupposes Secondness and/or Firstness.

Proposition ** (b) « Thirdness presupposes Secondness which presupposes Firstness » **is thus established in the sense of Frege-Strawson.

Conversely, if we assume that proposition (b) is true, then on the first line of the table (T) we read the first part of (a) and the second part reads on the third column, which establishes the equivalence of (a) and (b).

Obviously these logical presuppositions can then be considered as the morphisms of an algebraic category whose objects are these categories confirming the symmetry between the Frege-Strawson conditions for “presupposition” and the relationships of categorized phenomena under the Peircean categories of First, Second and Third.