sentence “x2>0 forx >0 or forx <0” (E. V. Paduˇceva). In Latin, the func- tions of exclusive and inclusive “or” are expressed by two different words, aut andvel. “And” can sometimes express a time sequence: compare the sentences
“Jane got married and had a baby” with “Jane had a baby and got married”
(S. Kleene). The conjunction∧can be expressed in different languages by juxtaposition: Chinese: ma mo—horse and donkey
Swahili: shika kitabu usome—take a book and read a preposition: Russian: Petya s Ma˘seˇı—Peter and Masha
a conjunction: and, i, et
a postpositional particle: Latin: senatus populusque—the senate and the people
two conjunctions: Russian: kak. . . tak.
D¨ohmann has catalogued the ways of expressing 16 logical polynomials in two variables in several languages of the world.
3. Curious as all this material may be, it should be regarded critically; in such comparisons with logic, the subtleties of usage often elude us. As an example, let us analyze the natural semantics of “if. . .then.” We have already mentioned that in languages ofL1this connective corresponds not only to “⇒” but also to the rule of deduction modus ponens. Moreover, MP more adequately represents the meaning of “if. . . then.”
Actually, the rule that any conditional is true if its antecedent is known to be false has almost no parallel in natural logic. Examples of the type “if snow is black, then 2×2 = 5,” which keep cropping up in textbooks, are capable only of confusing the student, since no natural subsystem in our language has expressions with this semantics. A possible exception is certain poetic and ex- pressive formulas with extremely limited usage (“If she be false, O, then heaven mocks itself!”). Formal mathematics, in which a single contradiction destroys the entire system, clearly has the features of poetic hyperbole.
Finally, in the logic of predicates there is no place at all for the modal aspect of the use of “if. . . then” in instructions of the type “if this hap- pens, do that.” On the other hand, this aspect can easily be expressed by the semantics of the connective “if. . . then. . .else” in algorithmic languages such as Algol. Unless one uses techniques suggested by algorithmic languages, any attempt to find a model for modality in languages based on L1 is doomed to failure (compare: A. A. Ivin,The Logic of Norms, MGU Press, 1973).
4. We have mentioned several times that the choice of the primitive modes of expression in the logic of predicates does not reflect psychological reality.
Elementary logical operations, even one-step deductions, may require a highly trained intellect; yet, logically complicated operations can often be performed as a single elementary act of thought even by a damaged brain.
Sublieutenant Zasetsky, aged twenty-three, suffered a head injury 2 March 1943 that penetrated the left parieto-occipital area of the cranium. The
injury. . .was further complicated, by inflammation that resulted in adhesions of the brain to the meninges and marked changes in the adjacent tissues.
Professor A. R. Luria met Zasetsky at the end of May 1943, and observed his condition for the next 26 years. In this time Zasetsky wrote nearly 3000 pages, describing with agonizing effort his life and illness as he struggled to regain his reason. His notebooks, which provided the material for Luria’s book The Man with a Shattered World(Basic Books, Inc., New York, 1972, translated by L. Solotaroff), not only show his perseverance and determination, but are also revealing from a psychological point of view.
At first, the destruction of Zasetsky’s psyche was overwhelming. The pre- dominant disorder was asemia, the inability to connect symbols with their meaning. Luria describes his first meeting with Zasetsky:
“ ‘Try reading this page,” I suggested to him.
“What’s this?. . .No, I don’t know. . .don’t understand. . .whatisthis?. . ..”
I suggested he try to do something simple with numbers, like add six and seven.
“Seven. . .six. . .what’s it? No, I can’t . . .just don’t know.”
The ability to understand the simplest predicates was lost: “What season is there before winter? Before winter? After winter?. . . Summer?. . . Orsomething !No, I can’t get it. Before spring? It’s spring now . . .and . . . and before. . .I’ve already forgotten, just can’t remember.”
Zasetsky lost the ability to interpret the syntactic devices for organizing mean- ing: “In the school where Dunya studied a woman worker from the factory came to give a report.” What did this mean to him? Who gave the report—
Dunya or the factory worker? And where was Dunya studying? Who came from the factory? Where did she speak?
This is a fairly difficult example composed by Professor Luria, but here is what Zasetsky himself writes:
I also had trouble with expressions like: “Is an elephant bigger than a fly?”
and “Is a fly bigger than an elephant?” All I could figure out was that a fly is small and an elephant is big, but I didn’t understand the wordsbigger andsmaller. The main problem was I couldn’t understand which word they referred to.
What attracts our attention is the complexity of Zasetsky’s metalinguistic text describing his linguistic difficulties. The subtlety of the analysis seems incompatible with the crude errors being analyzed. This could be explained by the retrospective nature of the analysis, but the following even more complicated description was written concurrently with the experience of the mental defect being described:
Sometimes I’ll try to make sense out of those simple questions about the elephant and the fly, decide which is right or wrong. I know that when you rearrange the words, the meaning changes. At first I didn’t think it did, it didn’t seem to make any difference whether or not you rearranged the words.
But after I thought about it a while I noticed that the sense of the four words (elephant, fly, smaller, larger) did change when the words were in a different order. But my brain, my memory, can’t figure out right away what the word smaller(orlarger) refers to. So I always have to think about them for a while . . .So sometimes ridiculous expressions like “a fly is bigger than an elephant”
seem right to me, and I have to think about it a while longer.
We can also see how complicated mental abilities were preserved while
“simple” ones were lost from examples of Zasetsky’s creative imagination, which resemble literary-psychological studies:
Say I’m a doctor examining a patient who is seriously ill. I’m terribly worried about him, grieve for him with all my heart. (After all, he’s human too, and helpless. I might become ill and also need help. But right now it’s him I’m worried about—I’m the sort of person who can’t help caring.) But say I’m another kind of doctor—someone who is bored to death with patients and their complaints. I don’t know why I took up medicine in the first place, because I don’t really want to work and help anyone. I’ll do it if there’s something in it for me, but what do I care if a patient dies? It’s not the first time people have died, and it won’t be the last.
All of this shows that there is no basis whatsoever for Rosser’s opinion that
“once the proof is discovered, and stated in symbolic logic, it can be checked by a moron.” The human mind is not at all well suited for analyzing formal texts.