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Natural evolution of a propositional calculus using modulo-2 arithmetic

and the bare bones of logic exposed, showing that logic is an empirical activity





The procedures described here rest on the premise that the mechanism of logic consists of sequences of marks or signs on a piece of paper, or some analogous contrivance in, say, a computer. These signs are open to various (not necessarily logical) interpretations, but it is usually some preconceived, specific type of interpretation that drives the way the system of signs unfolds, that is, the potential behaviour of the signs, which eventually becomes formalised as rules. The system so developed then becomes a guide for the interpretive aspect – sometimes a useful guide, at other times replete with paradox.


A further premise is that logic as such cannot exist in the absence of meaning; all logic is “relevance” logic, empirical and not analytic in nature. It is the behaviour of marks on paper, not their interpretation, for which rules can be formulated, and which by some stretch of imagination may be deemed “analytic”.  


The elementary calculus introduced here is manifestly a system of marks on paper, and its proper development is by “demonstrative construction” only. For the sake of brevity, however, I shall also make use of rules and some already familiar concepts. This procedure could be called a character build as opposed to a diagrammatic build aiming towards, for example, Euler or Venn diagrams. A justification and some advantages of the system are summarised here. However I attach very little importance to this particular scheme as such; doubtless many others like it could be devised. After all, it only provides another way of telling whether any elementary propositional formula is soluble or insoluble (tautologous, inconsistent or contingent).


Since logic, even as a system of signs, is essentialy paradoxical (see below) and formal procedures are circular, demonstration and practice appear to offer an alternative, more direct way of understanding how logic works. An illustration and comments on a very basic demonstrative system can be found here.


At least two other systems based on mod-2 arithmetic are possible. One mirrors the system described here in that the roles of 0 and 1 are reversed. The other has been used in computer science and includes the function 1-x to signify the negation of x. Both show calculational merit - the first because of its affinity with Boolean algebra, the second because it leads into probability computation, but neither is quite as "natural" as the present system.


There’s no connection between the “circles” in this system and Euler’s circles, used in the system described in the logic of natural language (see here).




The fundamental entity is empty 2-dimensional space



The basic elements are designated zones of space, represented by circular areas ("circles")


A circle may also be said to represent or signify 2-dimensional space in general




The basic operational unit is a circle containing 3 spaces, namely 2 contained circles and the surrounding contained space (called the “surround”, or simply “space”)




Any given circle can contain 2 circles, or be contained, along with an accompanying circle, within a circle


Any circle can be represented by 2 opposed part-circles or brackets  (  )


Any contained circle, symbolised “0”, may be called  “zero”, “even” or “true”


            “0” may be replaced by certain other symbols (see below)


Any surround may be called “plus”, “add” or “equals”, symbolised “+” or “=”. These signs are never obligatory but may at times be useful


Each of the 2 brackets representing a circle may be concomitantly replaced by a 1, and in that case the resultant pair of 1’s is normally re-surrounded by brackets


All the foregoing expressions have identical logical significance, called “valid”. All these expressions are reversible (eliminable), i.e. they can be reduced to empty 2-dimensional space, so as they stand they are redundant. (In a certain sense, all valid arguments and the concept of truth are redundant – see below)




An operational unit may contain just one part-circle or 1 (alongside an accompanying circle), but in that case the unit is “incomplete” or “unbalanced” and has a different logical significance called “void”


Any part-circle standing alone is normally replaced by a 1, and called “one” or  “odd”


“1” can be read in conjunction with its surround or “+” symbol and called “not” or “is false”, symbolised  “~“. For example (~0) means the same as (0 1) or (0 + 1)


The space (surround) within an operational unit may be substituted by a non-space or dot “.” (or asterisk “*” if preferred), but in that case the unit is “converted” and has a different logical significance. The dot (or asterisk) is normally obligatory unless the connected terms can be juxtaposed in a way that clearly distinguishes the non-space connection from a space (as in column 2 of the first truth table below)


“.” is called “convert” or “multiply” and has the effect of restoring the imbalance caused by a single 1, or destroying the balance resulting from the presence of two 1’s. Thus (0.1) is valid while (1.1) is void. (0.0) has the same significance as (0 0)


            The combination ( . )  may be considered either void or not well-formed


This completes the development of the basic arithmetic of “even” and “odd” (mod-2 arithmetic), with a measure of logical interpretation.


The subsequent evolution of the system as a logical calculus depends partly on philosophical outlook and partly on calculational expediency. My view is that propositional elements are not variables. A proposition, p, even a negative one, always purports to be true. If it is false, its falseness is in the form of a denial, external to the proposition itself. However, the development of this approach as a calculus is problematic.


A more common view is that propositional elements are variables, i.e. capable of adopting different values (0 or 1). As this approach leads to a certain neatness in calculations it is the one we shall adopt here.




Any contained 0 or 1 may be replaced irreversibly by any alphabetical symbol from the range p-w, called a propositional element, and in that case the formula changes its logical significance.

We will now skip straight to a form of truth table. A complete table for 2 propositional elements might look like this:





normal form (bracket & dot free)


summative normal form

Alternative form

Classical formula

p     0  0  1  1





q     0  1  0  1










       0  0  0  0




1 or T

       0  0  0  1




p V q

       0  0  1  0

pq  p

p.q + p

q ≥ p

q ⊃ p

       0  1  0  0

pq  q

p.q + q

p ≥ q

p ⊃ q

       1  0  0  0

pq  p  q  1

p.q + p+q+1

~(p & q)

~(p & q)

       0  0  1  1





       0  1  0  1





       1  0  0  1

p q

p + q

p = q

p ≡ q

       0  1  1  0

p  q  1

p + q + 1

~(p = q)

~(p ≡ q)

       1  0  1  0

q  1

q + 1



       1  1  0  0

p  1

p + 1



       0  1  1  1

pq  p  q

p.q + p + q

p & q

p & q

       1  0  1  1

pq  q  1

p.q + q + 1

~(p ≥ q)

~(p ⊃ q)

       1  1  0  1

pq  p 1

p.q + p + 1

~(q ≥ p)

~(q ⊃ p)

       1  1  1  0

pq  1

p.q + 1


~(p V q)

       1  1  1  1




0 or F


Analogues in classical logic:


0 corresponds to 1 or T

“.” (multiply) is the same as “inclusive either/or”

“+” and its analogues are the same as equivalence

“≥” means precisely “greater than or equal to” in the arithmetic system; its classical analogue is the implication sign


The usual purpose of logical calculus is to find out whether an argument or complex expression is valid, and if not valid, whether it is void (inconsistent) or reducible to a more compact consistent form. This is done by putting the entire argument in summative form, which complies with the rules of mod-2 algebra, and then applying a cancellation procedure. For example the simple argument  (p & q) ⊃ q  becomes:


Example 1:

(p & q).q + q  (by substituting in line 4 of the table), and expanding this to:

(p.q + p + q).q + q

= (p.q + p.q + q) + q

and finally cancelling all pairs of identical terms, which leaves nothing or 0,

so the argument is valid.


Example 2:

p & ~p

= p.(p+1) + p + (p+1)

= p + p + p + p + 1

= 1  (void)


Example 3:

(p & q) ⊃ ~(p.q)

= (p.q+p+q).(p.q +1) + (p.q +1)

= (p.q+p.q+p.q) +  (p.q+p+q) + (p.q +1)

= p.q + p + q   (consistent)


The most compact summative form to which an expression can be reduced is called the summative normal form.


Sumary of operations:


Op    Result

0+0     0

0.0      0

1+0     1

0.1       0

1+1      0

1.1       1

p+0      p

p+1     p+1

p.0       0

p.1       p

p+p      0

p.p       p


Operational rules:


+0 and .1 have no effect on whatever they are applied to

.0 applied to anything gives 0

The sum of any 2 identical terms yields 0

p.p reduces to p

p+1 remains unchanged


Other modifications


Bracket elimination


Let x, y, z be place markers, each for any of 0, 1, p, q, r …


Matched pairs of brackets in ((x y) z),  ((x+y)+z),  ((x.y).z) and indefinitely extended formulae of similar form can be eliminated ad hoc


In the absence of brackets, “.” takes precedence over space or “+” (i.e. “.” is calculated first)


Multi-term equivalence


The nominal behaviour of +, = and the equivalence sign of classical logic in expanded bracket-free formulae is the same:  they mean precisely that an even number of terms have the value 1 (odd) in this system, or 0 or F in the classical system. It is suggested, however, that  the usage of = be reserved to mean that all terms (within brackets if any) connected by = have the same value, or that all terms reduce to the same normal form.


Advantages and “psychology” of the system


1. '0' seems a more appropriate representation of truth than 1, since the idea of redundancy applies naturally to both 0 and "true". Indeed one could also say '0' is the sign for the "nothingness" out of which the system springs!


2. Given that the notation of logic (whether afforded a logical interpretation or not) is usually all that actually exists, in a rigorous sense of existence, the evolution of that notation out of empty space could hardly be achieved more naturally any other way.


3. Since logic is essentialy paradoxical (see below) and its exposition through formal procedures (definitions, proofs, inferences etc) seems circular, in the sense that the logic must be known before it can be systematized, a natural or empirical method of construction, and familiarisation through demonstration and practice, provide a more plausible method of understanding logical principles.


4. Because the system uses already familiar arithmetical principles of calculation, there’s no need to learn a Boolean style of calculation. There’s nothing especially distinctive about Boolean algebra compared to arithmetical systems, as (presumably) all 2-value systems are analogous.


5. The summative normal form of this system replaces both the disjunctive and conjunctive normal forms of Boolean logic.


6. The system can be expanded into a limited first order predicate calculus by using exponent variables.



The bare bones of logic and the ultimate paradox


(Most of the following section is lifted straight out of a notebook written around 1986)


Consider an expression (in the current system) such as  (p ≥ q).(q ≥ p) = 0.  Here, the “= 0”  might be interpreted as “is valid”; yet it is totally redundant.  One might say, what makes the formula valid is just that it can be seen to be valid, after converting it to a more transparent form or using a validation procedure, such as the cancellation method described above.


An inconsistent expression, on the other hand, seems to demand completion by appending a “1”, e.g.  (p&~p) = 1, or, using that redundant 0 again,  (p&~p)+1 = 0. It seems that, after all, we need to make a “wrong” expression look “right” again, in a formal sense. Why then can’t we just see that it’s wrong, and then mark it wrong, with a cross or whatever? Well, marking it “wrong” serves exactly the same (futile) purpose as completing the expression with “+1” or “=1”. Our final move, whether formalised or not, is always against a background of truth, validity or “being right”.


This is most evident when we look back at the initial stages of development of the system, using only the constants 0 and 1 and no (so-called) variables. Any expression containing a single 1, or indeed any odd number of 1’s, could be interpreted as being not simply void or inconsistent, but irregular or illegitimate. Which begs the question, what’s the difference between inconsistency, illegitimacy and “not being well-formed”?


Taking another tack, one might say: the non-redundancy of “+1” or “=1” is not due to either a formal requirement or a psychological compulsion to turn a void expression into a valid one. Rather, it consists only in the fact that “+1” has a logical function while “+0” has none.  “+1” acts like a switch that changes a state to its opposite or back again, as in:


EVEN            ODD            EVEN           ODD …….

              +1                +1                +1              +1 ……….


The only states that concern us at present are the 0 and 1 signs themselves:

0               1               0              1 …….

        +1            +1             +1           +1 ………. 


But the states 0 and 1 are not necessarily connected with the operation “+1”, since the latter can be replaced by another sign, such as “-” :

0               1               0              1 …….

         -                -               -              - ………. 


And then it might be claimed that one of the states can be depicted in terms of the “-“ operation on the other, e.g. :

1               -1              1              -1 …….

         -                -               -               - ………. 


Furthermore, the same sign could be used for the “-“ and the “1” :

-               --               -              -- …….

         -                -               -              - ………. 


and we could interpret the resultant system (containing just one recognisable type of element) as merely stipulating that every “correct” expression must consist of an even number of elements.


Indeed it’s easy to construct a recognisably “logical” (though incomplete) system in which all legal formulae comprise only an even number of straight lines, e.g.:

l + l = L

l + L = l

L - L = l

-l + l = l

-l ≠ -L


Now suppose we re-arrange our simple line scheme thus:





so that a “correct” formula comprises paired lines with no remainder and an “incorrect” formula contains an unmatched line:





As noted above, we can put various interpretations on the “incorrect” construction. We could say it’s a permissible formula but represents an inconsistent or false proposition (but what would that mean at this simple level?). Or we could say it’s not a well-formed formula, in which case it shouldn’t be written at all, but if it is written, it might be taken to signify any malformed construction, such as $%#@. (But how does it do that? It doesn’t – we have to do it ourselves!). Or we could say it’s “incomplete”, i.e. it shouldn’t be written by itself but must be accompanied by another odd-numbered set of dashes to complete it, e.g.:





The difference between










might be supposed to represent the difference between seeing that the expression (printed in black) is correct and demonstrating or declaring it to be correct. But the problem with the first expression is that it isn’t a correct formula within the system, while the trouble with the second is just that it is correct and doesn’t possess the wherewithal to proclaim the incorrectness of any part of itself. The addendum in red is a redundant fill-in, it’s necessary to write it in red so we can distinguish it from the original expression in black, and this situation is no different from the first formula – we can just see that the black part is incorrect.

And that, essentially, is the root paradox of propositional calculus. It's impossible to tell the ultimate lie in the language designed for telling it. Well, if you can't see that, I can help you no further. There's no correct way of articulating it.

One last thing I feel compelled to add, though, stemming directly from this demonstration: Any logical expression which expands to a longer formula than the entity which purports to be derived from it is phoney. A case in point is Russell's set-theoretical derivation/definition of natural numbers.


..........Dave Robinson............15/03/2010...........................HOME