(A page from the Loglan web site.)

(From Lognet 92/2. Used with the permission of The Loglan Institute, Inc.)

# Sau La Lodtua (From the Logic-Worker = Logician)

by M. Randall Holmes

My last column inspired all kinds of interesting reactions. As I feared, my sentences were judged to be truly tortured! There was at least one actual (minor) grammatical error. In a subsequent column, I will do a complete discussion of the various responses; unfortunately, I don't have enough time to prepare that column before my current publication deadline.

This time, I'm going to start talking about basic concepts of first-order predicate logic, the mathematical theory on which Loglan is partially based. I'm going to avoid symbols by using a subset of English adorned with parentheses; I will also exhibit Loglan equivalents as I go. The advantages of Loglan will probably become evident.

We start with propositional logic. Simple sentences (also called "propositions", thence "propositional logic") will be represented by capital letters like P, Q, R. A sentence like P may mean anything at all; sometimes I will give a temporary definition like

• P = "Snow is white",
• Q = "La Djan, ga mrenu" (John is a man)
usually preparatory to giving the English or Loglan translation of some complex sentence. Please note that quotation marks are being used to delimit sentences; there is no distinction of use and mention implied.

We will present propositional logic as having two basic ways of constructing new sentences from old. In presenting these constructions, I will make an additional distinction between the meanings of letters: I have said that P, Q, R stand for simple propositions (without logical structure); I will let letters X, Y, Z... "stand in" for any sentence at all (including the complex ones we are about to show how to construct).

The first of these constructions: if X is any sentence, not(X) is a sentence. not(X) is called the "negation" of X. not(X) is false if X is true, and true if X is false. Notice the use of X rather than P or Q in this construction; this indicates that I can build sentences like not(not(P)), negations of complex sentences, as well as sentences like not(P), the negation of a simple sentence. If we temporarily define P:

• P = "Snow is white",
we discover that
• not(P) = "Snow is not white".
On the other hand, if we want to form not(not(P)), we get something more complex, say
• not(not(P)) = "It is not the case that snow is not white."
The English speaker need not feel too handicapped, since the occasions on which one would wish to say not(not(P)) (instead of P) are very limited in number; they are neatly dealt with by the logical principle of "double negation", which can be written (a little loosely) as
• not(not(P)) = P
A more alarming situation arises with
• P = "Everything is perfect"
where we might think that not(P) would be
• *not(P) = "Everything is not perfect"
but further reflection should convince us that
• not(P) = "Something is not perfect"
is the correct form. The last example involves logical issues which we are not ready to discuss, but it should make it apparent that the negation operation on English sentences is quite complicated. In Loglan, it is fairly easy to form negations of any sentence; one uses the little word no. Grammatically, no attaches itself to the predicate or to an argument of the sentence, not to the sentence itself as in propositional logic, but the effect on the meaning of the sentence is the same. Let us suppose that
• P = "La Djan, ga blanu" (John is blue);
then not(P) is either of the following:
• not(P) = "La Djan, ga no blanu";
• not(P) = "No la Djan, ga blanu";
not(not(P)) could be one of
• not(not(P)) = "La Djan, ga no no blanu"
• not(not(P)) = "No la Djan, ga no blanu".
Loglan has the advantage that it is grammatically simple to negate any sentence. There is actually a sentence-negating form analogous to our logical construction:
• P = "La Djan, ga sadji" (John is wise)
• not(P) = "No gu la Djan, ga sadji"
However, if we wish to negate P when
• P = "Ra ba gudbi" (Everything is good)
we discover that the correct form is
• not(P) = "Ba no gudbi" (Something is not good).
The issue here is more profound, and Loglan shares the problem, whatever it is, with English.

The second basic construction: if X and Y are any sentences, (X and Y) is a sentence, called the "conjunction" of X and Y. (X and Y) is true exactly if X is true and Y is true; it is false if either X or Y or both is false. In English, there is very little to say about the equivalent of (P and Q); we give an example:

• P = "Snow is white";
• Q = "Grass is green";
• (P and Q) = "Snow is white and grass is green".
A common way of making claims (P and Q) which have similar structure is illustrated:
• P = "John is wise";
• Q = "Mary is wise";
• R = "John is tall";
• (P and Q) = "John and Mary are wise";
• (P and R) = "John is wise and tall";
In logic, logical combinations of arguments with propositional operations like conjunction are never formed; such operations on predicates are sometimes used, although one has usually slipped into set theory at this point. (The set of things which are "wise and tall" is the "intersection" of the set of wise things and the set of tall things.)

Loglan looks like English in this respect, with one slight difference. The primary little word with the sense of "and" is e, which is used to connect predicates or arguments as in the previous example:

• P = "La Djan, ga sadji" (John is wise);
• Q = "La Meris, ga sadji" (Mary is wise);
• R = "La Meris, ga gudbi";
• (P and Q) = "La Djan, e la Meris, ga sadji";
• (Q and R) = "La Meris, ga sadji, e gudbi".
The difference is that Loglan uses a variant ice to connect complete sentences:
• P = "La Djan, ga sadji"
• Q = "La Meris, ga gudbi"
• (P and Q) = "La Djan, ga sadji, ice la Meris, ga gudbi"
Another form is the "forethought" form
• (P and Q) = "Ke la Djan, ga sadji, ki la Meris, ga gudbi"
Although English has little trouble with simple conjunctions, we can represent sentences in our logic notation with little difficulty which cause English to have fits. The first logical operation which we will define is an excellent example:

Definition:

• (P or Q) is defined as not(not(P) and not(Q)).
First, what does it mean? not(not(P) and not(Q)) asserts that (not(P) and not(Q)) is false. (not(P) and not(Q)) is true exactly if not(P) and not(Q) are both true, i.e., if P and Q are both false. So (P or Q) asserts that it is not the case that P and Q are both false, i.e., that one or both of P and Q are true. A precise way of saying this in English is "P and/or Q"; the English word "or" has a competing "exclusive" sense, as in "The child will have cake or pie for dessert", where it is understood that the child will have one dessert. Now try to make a claim not(not(P) or not(Q)) in English:
• P = "The sun is shining"
• Q = "The wind is blowing"
• not(not(P) and not(Q)) = "It is not the case both that the sun is not shining and that the wind is not blowing".
Notice the difference in grammatical expression between the inner occurrences of negation and the outermost negation. Notice also the appearance of the word "both", the English device for "forethought" conjunction; if it were left out, we might read the sentence as being of the form (not(not(P)) and not(Q)). Loglan does a little better:
• P = "La Djan, ga sadji"
• Q = "La Meris, ga gudbi"
• not(not(P) and not(Q)) = No ke la Djan, ga no gudbi, ki la Meris, ga no sadji"
With the word "or" available (though ambiguous) in English, and the little word a and its variant ica-- with the precise meaning of the operation in logic--available in Loglan, it is interesting to see how we actually negate sentences in "and" and "or" (sentences in "or" are called "disjunctions"):
• (P and Q) = "Snow is white and grass is green".
• not(P and Q) = "Snow is not white or grass is not green" = (not(P) or not(Q))
• (P or Q) = "I will surrender or flee"
• not(P or Q) = "I will not surrender and not flee" = (not(P) and not(Q))
Although psychology is not my line, I doubt that our mental operation of negation ever builds a sentence like not(P and Q); I suspect that the mental rule for negating (P and Q) is "Change the 'and' to 'or' and negate the parts". Similar considerations apply to negating disjunctions. The fact that this simple transformation is possible indicates why we do not need to utter sentences like not(not(P) and not(Q)), and thus find them unfamiliar and difficult when we encounter them.

The two logical principles illustrated above are called "DeMorgan's Laws", and can be written:

• not(P and Q) = (not(P) or not(Q))
• not(P or Q) = (not(P) and not(Q)).
I'll stop at this point; there is more to come along this line (unless I get screams of pain from the audience). Another way to stop me from continuing in this vein is to ask questions...

--Hue Randall Holmes 