Hopes for the stereoscopic gamma went a-glimmering. Peirce was not able to pull the system together as he wanted to. But he did try to make a start; we restate here a passage we quoted in the Introduction:

In endeavoring to begin the construction of the gamma part of the system of existential graphs, what I had to do was to select, from the enormous mass of ideas thus suggested, a small number convenient to work with. It did not seem to be convenient to use more than one actual sheet at one time; but it seemed that various different kinds of cuts would be wanted (4.514).

The "different cut" to which Peirce seems to have paid most attention is the "broken cut," which we will recall from the Introduction. This is., in fact, the only of the gamma cuts to which any real attention is paid in the writings of Peirce appearing in the Collected Papers. The bulk of this chapter will be devoted to a study of systems including the broken cut among their signs.

In 4.516, Peirce states two rules for the use of the broken cut; one of these rules is our 0.13 of the Introduction. The other rule has two clauses and is stated as our 0.14 and 0.15 in the Introduction. We may at this point repeat these rules as we stated them here:

At this point we shall simply remark that these rules are open to amplification and interpretation, as we shall eventually see; and depending on how we amplify and interpret, they will yield us a variety of logical systems. In this they are like the rule of "strict implication introduction," for example, which is used by Anderson and Johnstone(n1) to set up a system of natural deduction which is basically equivalent to the Lewis-modal S4. Although the rule as they present it enables us to derive the theorems of S4 within their system, there would be no trouble in so modifying the restrictions on the application of the rule as to enable us to derive the theorems of, say, the system S5.

We shall, then, construct several "broken cut" systems by slightly modifying certain rules for transformations involving the broken cut. As we will recall from the Introduction, the graph

may be interpreted as asserting, "It is possibly not the case that X." Systems containing the broken cut among their signs, then, may be considered systems of modal logic.

In chapters I and II we compared the alpha and beta systems to ordinary logical calculi. The project for gamma will be similar. We shall compare the broken cut systems which we will define with ordinary systems of modal logic. We shall define four different broken cut systems, and shall compare them to four different standard modal systems. Three of the systems we shall thus use belong to the family of "Lewis-modal" systems, the "classical" modal systems of contemporary logic. The fourth system is a rather unusual modal system invented by Łukasiewicz .

The usual axiomatizations of modal systems ordinarily take either "necessity" (with 'Lp' read as "necessarily p" ) or "possibility" (with 'Mp' read as "possibly p") as primitive modal operators. Since the primitive modal operator in the gamma systems is the broken cut, which states "possibly not,'' it may be worthwhile to provide axiomatizations of our ordinary modal systems taking "possibly not" as primitive rather than the usual "possibly" or "necessarily." We thus shall state some rather neat axiomatizations of several Lewis-modal systems taking "possibly not" as primitive -- these are, so far as I am aware, new bases for these systems.

We shall then present the axiomatic basis for that unusual system of Lukasiewicz, and then make some remarks about that calculus.

The next step will be to describe each of the four broken cut systems we have mentioned, and to compare each in turn to an appropriate standard modal system. We shall, in fact, show that our broken cut systems are equivalent to these standard systems in the same sense that alpha, say, is equivalent to the CPC.

3.1 Remarks on Some Standard Modal Systems

If the alpha cut is to be considered a negation sign, then the broken cut will be a sign that states "possibly not.'' The broken cut then represents a "weak negation." We shall presently be comparing the systems involving the broken cut to certain standard modal calculi. As we mentioned, these systems are ordinarily formulated using either 'M' (possibility) or 'L' (necessity) as a primitive modal operator. It is easily possible, however, to get nice axiomatizations of the standard modal systems with which we will be concerned using "possibly not" as the only primitive modal concept. We may read 'Rp' as "possibly-not p." Then we have

The axioms and rule will be considered to be subjoined to a complete CPC base, including the rules of detachment and substitution of variables. The specific formulations of the systems are as follows

The axioms are, in general, quite expressive of the nature of 'R' as a sign of "weak negation." Axiom 2 indicates that this weak negation of a statement follows from the ordinary negation; this axiom is equivalent in deductive power to the theses 'CLpp' and 'CpMp' . Axioms 1, 1¢, and 1¢¢' are "laws of transposition" for the weak negation, analogous to ordinary PC laws of transposition like ' CCNpNqCqp' . By applications of appropriate CPC laws and the definition of 'L' , axiom 1 becomes 'CLCpqCLpLq', and axiom 1¢ becomes 'CLCpqLCLpLq'. It is well-known that CPC, RL, and the two theses:

together are an axiomatic basis for the system T, while a replacement of above by

yields the system S4. Our axioms 1 and 1¢, again-, are easily shown to be theses of  T and S4 respectively. We may state without hesitation, then, that the axioms in 'R' as we have given them form bases for T and S4. The system S4.2 is ordinarily formulated by subjoining the formula 'CMLpLMp' to a basis sufficient for S4; our axiom 3 is definitionally equivalent to this last formula, so we have also provided a sufficient base for S4.2.

The iterated modality 'RR' is equivalent to 'ML' ("possibly-not possibly-not" is the same as "possibly necessarily"). With this in mind, and transposing the consequent of 1¢¢, 1¢¢ becomes:

This latter formula is, of course, also easily transformable back into 1¢¢. If *1¢¢ is a thesis of a system, and *2 is also a thesis, then *1 is easily provable. Now substitute 'Cpp' for 'p' in *1¢¢; the result is:

By PC and RL, the following strict equivalence holds:

By PC and the substiitutivity of strict equivalence, which holds in the system containing PC, RL, and *2, all of which we have, and using the strict equivalence at 5, 4 becomes

detaching now with the thesis 'LCpp' , which holds in all Lewis-modal systems, we get:

This is a characteristic reduction formula for S5, the strongest of the Lewis-modal systems; CPC, RL, *2, and 7, in fact, constitute a standard axiomatic base for S5. The formula *1¢¢

is itself an S5 thesis, and easily shown to be such. It follows, then, that CPC, RL, *1¢¢, and *2 form a sufficient base for S5; and then so too will CPC, RL, 1¢¢, and 2 -- our axioms in 'R' -- as we stated earlier.

	8.	CdpCdNpdq
	9.	CLpp

with standard definition "M =_df NLN" ; this is the system with 'L' primitive. It might also be stated with 'M' primitive, or even with 'R' primitive: Lukasiewicz also lists two "axiomatic rejections" for the system:

These will not be of too much interest for our purposes, for they do not actually enter into the generation of the set of theorems of the system.

which are also incompatible with many characteristic Lewis-modal theses. This last is a reduction formula, but it differs from, say, the reduction theses of S5. Where the S5 reduction formulas "reduce" a string of modal operators to the rightmost member of the string, as indicated by 7, the Ł-modal reduction theses reduce such a string to the leftmost member, as indicated by 11. As an example, 'MLLMMMLMLp' is equivalent in S5 to the simple 'Lp', while in Ł-modal it is equivalent to 'Mp'.

3.2 The Broken Cut Systems

We now turn our attention back to the broken cut. Earlier in this chapter, we repeated from the Introduction the rules 0.13, 0.14, and 0.15 for the use of the broken cut. There is not too much doubt about how the latter two of these rules are to be interpreted and applied; 0.13,

however, begs for study. The question raised by 0.13 is this: Just what is to be the over-all function of the alpha rules of transformation in a system containing the gamma cut? For example, the transformation

would involve the iteration of a graph "across a broken cut." Do we wish to permit such transformations at all? Do we wish to permit them with certain restrictions? Or do we wish to permit them in unlimited fashion? The rule 0.13 is in itself a very limited rule, as it is stated. But it does involve a certain "cross-breeding" of alpha and gamma concepts. Although it is really just a weak "gamma-version" of our alpha R_ins, it somewhat coercively turns our attention to the question of alpha and gamma "cross-breeding" in the rules in general.

Well, we now propose to open an experimental farm for the investigation of some of the possible hybrids, and for the comparison of them with some of the earlier mentioned standard breeds of modal logic. I think we shall find the alpha-gamma progeny, unlike most hybrids, relatively fertile.

3.21 Gamma-MR: Broken Cuts with Minimal Restrictions

The first broken cut system we shall develop and study is one which allows the most liberal possible interpretation within gamma of the alpha rules of inference; we shall call this system "Gamma-MR." The basis for Gamma-MR will be as follows:

The single axiom for Gamma-MR will be b, the blank alpha SA.

The rules R_ins and R_ers will apply just as they do in alpha, and for purposes of their application, broken cuts will be counted as if they were alpha cuts to determine "oddness" or "eveness" of enclosure.

The rules R_itr and R_dit will apply just as they do in alpha; any graph may be iterated (or deiterated), across any kind of cut or combination or cuts, just as if they were alpha cuts.

The rules R_bcl and R_nbc may be applied just as they are in alpha, in any area at all. But it is understood that the only cuts that may be inserted or removed by these rules are alpha cuts.

In addition, there shall be two other rules which shall apply in this system; these rules correspond to our 0.14 and 0.15.

Before we go any further, let us state -- without explicit proof, however -- two theses which will be useful in what follows. These theses will be analogs for our modal systems of lemmas 1.03 and 1.04:

*3.01	When QA(B) and  D É B are theorems of L-modal, so too is  QA(D). When QA(B) and  D B are theorems of S4, S4.2, or S5, so too is  QA(D).
*3.02	When QA(B) and  B É D are theorems of L-modal, so too is  QA(D). When QA(B) and  B D are theorems of S4, S4.2, or S5, so too is  QA(D).

The formula 'p q', of course, is read, ''p strictly implies q''; it is equivalent; to the formula -- in Polish notation -- 'LCpq'. The notation used above is used simply for the sake of consistency with that of lemmas 1.03 and 1.04. We may assume that the systems for which these theses are stated are "L-primitive') systems; and 'L' affects the A-pos or C-pos of subformulas in its scope in no way. The theses above are provable by an induction on the number of L's (belonging to the formula Q^A(B) or Q^C(B)) within whose score B is located. But without going through the tedium of an explicit proof, I submit that the theses are intuitively quite acceptable. Recall the relationship of lemmas 1.03 and 1.04 to the rule of substitutivity of material equivalence in CPC. It may then help our intuition if we note that substitutivity of material equivalence holds in the Ł-modal system, while substitutivity of strict equivalence holds in all the Lewis-modal systems. That the above theses would hold in "M-primitive" or "R-primitive" systems is evident. Just note that since ERpNLp, an R will affect the A-pos or C-pos of formulas in its scope just as if it were an 'N'.

We also showed in chapter I that when R(Y, X) is one of the rules of inference of alpha, and is true for alpha graphs X and Y, then if f(X) is a CPC theorem, so too is f(Y).

The rules of inference of alpha hold unrestricted in Gamma-MR But *3.01 and *3.02 hold for Ł-modal, and the schema

CdpCdNpdq

holds unrestricted in Ł-modal, as we have mentioned (it is, in fact, an axiom-schema of that system). There would then be no trouble in extending the proof s of chapter I to show that when R(Y, X) is one of the alpha rules as employed in Gamma-MR, and is true for Gamma-MR graphs X and Y, then if f*(X) is a theorem of Ł-modal, then so too is f*(Y).

Gamma-MR also includes two rules for the broken cut -- R_gam, which permits an evenly enclosed alpha cut to be transformed to a broken cut, and R_ngm, which permits an oddly enclosed broken cut to be transformed to an alpha cut. Ł-modal, of course, contains the law 'CNpRp'; in the presence of *3.01 and *3.02, this means -- since the broken cut is correlated by our functions to 'R' just as the alpha cut is to 'N' -- that if either R_gam(Y, X ) or R_ngm(Y, X) is true for Gamma-MR graphs X and Y, then if f*(X) is an Ł-modal theorem, so too is f*(Y).

In summary, all of the above means that

If X is a theorem of Gamma-MR, then f*(X) is a theorem of Ł-modal.

Now turning to the "other direction" of the proof, recall again that the rules of insertion and erasure, iteration and deiteration, and positive and negative biclosure hold unrestricted in Gamma-MR. This means that the graph

The graph

is trivially provable in Gamma-MR by R_gam and the alpha rules. This graph corresponds by g* (or by h*) to the Ł-modal axiom 'CNpRp' (which is, of course, equivalent to 'CLpp').

It is clear that the analogs of the rules of substitution and detachment hold in Gamma-MR as "derived rules of. inference" just as they do in alpha.

Given the above, we submit that the following hold, as analogs of *1.17 and *1.21 of chapter I:

If f*(X) is a theorem of Ł-modal, then X is a theorem of Gamma-MR

and

A is a theorem of Ł-modal iff g*(X) is a theorem of Gamma-MR.

To drive the point home a little harder, we shall now engage in a few graphical derivations within Gamma-MR. We shall prove within Gamma-MR some typical graphs of the system; they shall be seen to correspond to characteristic theses of the Ł-modal system.

By alpha rules

		(1)
(1), Ritr, Rbcl		(2)
(2), Ritr (note the dependence on unrestricted Ritr)		(3)
(3), Rbcl		(4)

alpha rules		(5)
alpha rules		(6)
(6), Rbcl		(7)
(7), Rgam		(8)
(8), Ritr		(9)
(9), Rdit		(10)
(10), (5). Rdit		(11)
(11), Rnbc		(12)

The graphs at (11) and (12) express, equivalently, the thesis 'RR1' [equivalent to RRCpp] or 'ML1' ; they assert that the "true" is "possibly necessary." The thing one might find surprising is that the simple "broken cut biclosure" with a broken cut enclosed oddly is a thesis of Gamma-MR, derivable only by the rules. It is not immediately apparent from an examination of R_gam, for example, that this is possible. Yet this theorem should not be too surprising considering that Gamma-MR and Ł-modal are equivalent to each other, and 'ML1' is a law of Ł-modal.

We set about the proof, now, of one more characteristic graph of Gamma-MR, again, one that is equivalent to a characteristic thesis of Ł-modal.

(12), Rbcl, Rins		(13)
(13), Ritr		(14)
alpha rules		(15)
(15), Ritr, Rins		(16)
(16), Rbcl. Ritr		(17)
(17), Rngm, Rnbc		(18)
(18), Rnbc		(19)

CLCpqCLCqpCdpdq (n1)

which is a statement of the substitutivity of strict equivalence, a principle which, as a rule of inference, is characteristic of all the Lewis modal systems, and which is considerably weaker than the Ł-modal theses of extensionality.

In Gamma-MR, the "extensionalizing" is accomplished by the unrestricted permission to use the rules R_itr and R_dit. The theses *3.01 and *3.02 show that the rules of insertion and erasure hold unrestricted even for gamma systems which might be called "Lewis-type," as we shall see; the rules of biclosure and for the broken cut also hold without restriction. The strange nature of Gamma-MR may then be credited to its unrestricted rules of iteration and deiteration.

3.22 Gamma-4: A "Classical" Broken Cut System

which is what we might call 'the necessity biclosure." Needless to say, the presence of this graph does not give us the right to use this pair of cuts as if it were a pair of alpha cuts in the application of the rules R_bcl and R_nbc. These rules will still apply just to the insertion or deletion of alpha biclosures. Note too that with R_ngm and R_nbc there is no trouble in getting from this graph as axiom to the null graph b as theorem.

We shall now state the axiom and the rules as they apply to Gamma-4, just as we did for Gamma-MR.

The sole axiom of Gamma-4 is the graph

Rules R_ins and R_ers apply just as they did in Gamma-MR, again counting alpha and broken cuts alike to determine oddness or evenness of enclosure.

The rules R_itr and R_dit will apply as before, except for the following restriction: The only graphs that may be iterated or deiterated across broken cuts are those of form

The rules R_bcl and R_nbc apply as in Gamma-MR.

The rules R_gam and R_ngm (for the broken cut) apply as in Gamma-MR.

The restriction on the rules of iteration and deiteration prevents an iteration in Gamma-4 like that leading from step (2) to step (3) in the deductions of Section 3.21.

We turn our attention to the status of these rules in the system S4. First of all, the strict implication

LCNpRp

This indicates that whenever, for Gamma-4 graphs X and Y, either R_gam(Y, X) or R_ngm(Y, X) is true, then if f*(X) is ari S4 theorem, then so too is f*(Y) .

The strict implication

LCKpqp

is also a law of S4; in the presence of *3.01 and *3.02 this means that the rules of insertion in odd and erasure in even have analogous derived rules of inference in S4, just as do the above mentioned rules for the broken cut.

A strict equivalence that holds in S4 is

LEpNNp.

Since the system S4 contains the rule of substitutivity of strict equivalence, this means that the rules of biclosure have analogous derived rules of inference in S4.

We move now to the rules of iteration and deiteration as stated for Gamma-4. It will be recalled that the proof in chapter I that the CPC contains rules analogous to R_itr and R_dit pivots about the existence in CPC of a certain theorem schema which for our purposes here we may express as:

If this schema is in a system, and the system contains the rule of substitutivity of material equivalence, then that system contains derived rules of inference analogous to R_itr and R_dit. The schema at (1) quite definitely does not hold in the general case in S4; however, recalling the nature of the restriction on the rules of iteration and deiteration in Gamma-4, it is possible t o write a schema which does hold in S4 and whose presence makes possible proofs of the versions of iteration and deiteration appropriate to S4.

A little thought will tell us what this schema must be; we will recall that the restriction on iteration and deiteration for Gamma-4 states that only graphs of form may be iterated across broken cuts, that is, iterated "into modal contexts." This suggests that the schema we want in the general case in S4 is:

Actually, the simple

EKpd1Kpdp

does hold in S4 just as it does in CPC.

We shall now show that (3) is indeed a theorem schema in the general case in S4. The following schema holds in S4:(n1)

The following is an easily provable S4 thesis

	CLpLCqp	(7)
(6). (7). PC	CLpCd1dp	(8)
(8), p / Lp	CLLpCd1dp	(9)

But 'LELLpLp' is a theorem of S4; by (9) and substitutivity of strict equivalence, then:

	CLpCd1dLp	(10)
(10), PC	CKLpd1KLpdLp	(11)

But (11) is one half of (3), which is the formula we wished to prove. The converse of (11) is provable along much the same lines that (11) was. We may then say that (3) holds in the general case in S4. But this means that when, for Gamma-4 graphs X and Y, R_itr(Y, X) or R_dit(Y, X) is true with the Gamma-4 restriction, then if f*(X) is an S4 theorem, so too is f*(Y).

Now we note that where X is the sole axiom of Gamma-4, then f*(X) is 'NRNKp₀Np₀', which is the same as 'LCp₀p₀' , and is clearly a theorem of S4. We may say, then, that:

If X is a theorem of Gamma-4, f*(X) is a theorem of S4.

We shall now go on to show both that the converse of the above statement holds, and that the set of theorems of S4 maps into the set of theorems of Gamma-4 by the function g*: (f* and g*, we should recall, are the functions f and g of chapter I extended to account for the broken cut). Gamma-4 contains all that alpha contains and more. We shall have accomplished our purpose, then, if we can show that the axioms and rule of S4 beyond those of the CPC map appropriately into the set of theorems of Gamma-4. First of all, we may easily prove the following:

alpha rules		(12)
(12),  Rgam		(13)

The graph at (13) is clearly a correlate by g* or h* of our axiom 2 for S4, 'CNpRp'. Now:

Gamma-4 axiom, Rbcl		(14)
(14), Rins		(15)
(15), Ritr		(16)
(16), Rbcl		(17)
(17), Ritr		(18)
(18), Rngm		(19)
(19), Rbcl, Rnbc		(20)

Now in (20), take as and as  ; the graph resulting from these substitutions contains a "replica" of (20) itself. Deiterate this "replica" with (20) and apply R_nbc and R_ngm to obtain

		(21)
by alpha rules		(22)
(22), Rins		(23)
(23), Ritr		(24)

We may now move, by purely alpha methods, employing (21), (24), and the alpha-provable graph [this is like transitivity of implication in PC]

(25)

to

(26)

The graph at (26) will be found, upon examination, to correspond to our axiom 1¢ for S4, 'CRCRpRqCpq'. The graph of (24) corresponds to 'CRNRpRp', which becomes, on transposition of antecedent and consequent, 'CLpLLp', the characteristic reduction formula of S4.

Note that with as an axiom of Gamma-4, the equivalent of RL holds in Gamma-4. This is so because any graph derivable as a theorem upon the blank SA alone will be derivable as a theorem within the "necessity biclosure" which is the Gamma-4  axiom; this point is illustrated by the deduction leading to the graph of (20) above. It is now fairly clear that both:

EKLpd1KLpdLp

EKMLpd1KMLpdMLp

A little reflection will show us that this is indeed the case. Let us turn to step (8) in the deductions of the preceding section, and instead of substituting Lp for p, substitute MLp (we begin the numbering anew in this section):

CLMLpCd1dMLp

(1)

Now, in S4.2 the following thesis holds -- it is not a law of S4:

	CMLpLMLp	(2)
(1), (2), PC	CMLpCd1dMLp	(3)
(3), PC	CKMLpd1KMLpdMLp	(4)

We noted that the converse of the analogous formula in Section 3.22 is proven much as is that formula itself; so too will the converse of (4) follow, and we may state that

EKMLpd1KMLpdMLp

(5).

The formula at (5) is the one we wanted, and we are safe in saying that appropriate analogs of the rules of iteration and deiteration with their Gamma-4.2 restrictions hold in S4.2. Since all else in Gamma-4.2 is as in Gamma-4, we may conclude that

A graph X is a theorem of Gamma-4.2 only if f*(X) is a theorem of S4.2.

To show that the converse of the above holds, and also that the set of S4.2 theorems maps into the set of Garmna-4.2 theorems by g*, we must show only that there is a theorem of Gamma-4.2 which corresponds by g* and h* to the special axiom of S4.2, 'CMLpLMp' -- or, in 'R', 'CRRpNRRNp'.

Gamma-4 axiom		(6)
(6), Rins		(7)
(6), Ritr, (in Gamma-4.2 version)		(8)
(8), Rngm		(9)
(8), Rbcl		(10)

The graph of (10) states 'CRRpNRRNp', which is the same as 'CMLpLMp', Note the characteristic Gamma-4.2 operation at step (8). It is now safe for us to state, on the basis of the above, our work in Section 3.22, and the relationship between S4 and S4.2, that both:

If f*(X) is an S4.2 theorem, then X is a theorem of Gamma-4.2,

and

A is a theorem of S4.2 iff g*(A) is a theorem of Gamma-4-.2.

The systems S4.2 and Gamma-4.2 are then translatable into each other, and also may be considered equivalent in the same sense that alpha is equivalent to the CPC,

3.24 Gamma-5: The Limiting "Lewis-Modal" Broken Cut System

In many respects, the system S5 may be considered a "limiting" system among the Lewis-modal calculi -- to go into an e extensive study of this, however, would really be outside the scope of this paper.(n1)

The system Gamma-5 is to be a "broken-cut" version of  S5; this is why we have called it a "limiting" broken cut system. As might be suspected, Gamma-5 differs from Gamma-4 and Gamma-4.2 only in having a different, more lenient restriction upon its application of the rules of iteration and deiteration. The statement of this restriction is quite simple:

In Gamma-5, the rules R_itr and R_dit apply as they do in alpha, except that a graph X may be iterated or deiterated across a broken cut only if each of X's minimal graphs is in the "scope" of a broken cut belonging to X, that is, provided each of the minimal graphs of X is part of a subgraph of X whose "outermost sign" is a broken cut.

As was the case in Sections 3.22 and 3.23, we shall state a formula whose theoremhood in S5 will guarantee the existence in S5 of derived rules of inference analogous to the rules of iteration and deiteration with the above restriction. The formula is

All the theorems of S4 and S4.2, of course, hold in S5; the schema at (1) is one which does not hold in general in S4 and S4.2, but which will be shown to hold in S5. As was the case in Section 3.23, we shall first turn to step (8) in the deductions of Section 3.22, and now, instead of substituting ' Lp' for 'p', we shall substitute a, where a is any wff having each of its propositional variables in the scope of a modal operator belonging to a . The result is

CLaCd1da

(2)

Now, there is a pair of rules of inference, due to Prior, which hold in S5 and which -- when subjoined to the CPC -- actually form a basis sufficient for S5.(n1) The rules are:

R1:	If Cab is a theorem, then. CLab is a theorem, and
R2:	If Cab is a theorem, then. CaLb is a theorem, provided each of the variables of a is part of a subformula of a beginning with a modal operator.

We have, by PC alone:

Caa

(3)

But let a be of such a form that each of its propositional variables is part of a subformula of a beginning with a modal operator; this means both that the proviso of R2 is fulfilled and that a is the same as the a of the schema at (2). Then:

Again, the converse of (6) will be easily provable, and we may state:

where a, of course, is as required in (1).

To show that the converse of the above holds and that the set of S5 theorems maps into the set of Gamma-5 theorems by it would be sufficient to show that a graphical correlate of our axiom 1¢¢ is a theorem of Gamma-5; 1¢¢ is, of  course, 'CRRCpqCRqRp'. But actually it will not be necessary to develop a proof for this particular graph. We know that if the formula 'CMLpLp' is added to a base sufficient for S4, the result is the system S5, Another formula which will do the job, and which, in fact, is just a transposed version of 'CMLpLp' is

CNLpLNLp.

Another way of writing this formula is

CRpNRRp.

Since Gamma-5, with a more liberal restriction on its rules of iteration and deiteration than either of the other such systems we have discussed, will contain all the theorems of both of these systems, it is clear that if a graph corresponding to the last mentioned axiom can be proven in Gamma-5, then graphs corresponding to all S5 theses will be derivable. The proof is as follows:

Gamma-4 axiom		(8)
(8), Rins		(9)
(9), Ritr, (Gamma-5 version)		(10)
(10), Rbcl		(11)

The graph of (11), a theorem of Gamma-5, corresponds by g* or h* to 'CRpNRRp', We are then safe in asserting that both:

If f*(X) is an S5 theorem, then X is a theorem of Gamma-5,

and

A is a theorem of S5 iff g*(A) is a theorem of Gamma-5.

The systems S5 and Gamma-5 are then translatable into each other, in our technical sense of translatable; and we may consider them equivalent in the same sense that alpha and the CPC are equivalent.

3.25 A Summary of the Broken Cut Systems Here Presented

In the preceding sections we have shown how systems equivalent to some contemporary modal systems may be formulated within the "broken cut" notation of Peirce's gamma graphs. The method of setting up these systems is quite simple. We add to the notation and rules of alpha the broken cut and the rules which permit us to change an evenly enclosed or unenclosed alpha cut into a broken cut, and an oddly enclosed broken cut into an alpha cut. We allow the rules of insertion in odd and erasure in even and of biclosure to hold just as they do in alpha, counting broken cuts as if they were alpha cuts to determine whether an enclosure is odd or even. If we add no new graphical axiom and allow the rules of iteration and deiteration to apply with no restrictions, the system obtained, Gamma-MR, is equivalent to Ł-modal.

If we add as an axiom the graph and place certain restrictions on the rules of iteration and deiteration, we shall obtain several different systems, depending on what the restrictions are.

If we state that only graphs of form may be iterated or reiterated through broken cuts, we get the system Gamma-4, which is equivalent to the Lewis-modal S4.

If we permit also graphs of form to be iterated or deiterated through broken cuts, the system yielded is Gamma-4.2, which is equivalent to the Lewis-modal S4.2.

If we permit any graph all of whose minimal graphs are within the scope of broken cuts belonging to that graph to be so iterated or deiterated, we get Gamma-5, which is equivalent to the system S5.

It is not by accident that the Lewis-modal broken cut systems are so formulable. In an article on these systems(n1) I have shown that S4 and S4.2 may be formulated by the subjunction of Prior' s rules R1 and R2 (which we saw in Section 3.24) to the CPC, changing the proviso of R2 to read, "Provided a is completely modalized." I distinguish between S4, S4.2, and S5 by stating for each of these systems what it is to be completely modalized. In S4 we find that a formula is completely modalized if it begins with an 'L' or is a conjunction of formulas each of which begins with an 'L'. In S4.2 we find that in addition formulas beginning with 'ML' will be completely modalized. For S5, complete modalization is as in Prior's version of the proviso.