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-\chapter{The Logic of Admissible Contradictions (Work in Progress)}
-
-\section{Chapter III. A Provisional Axiomatic Treatment}
-
-
-In the first and second chapters, we developed our intuitions 
-concerning perceptions of the logically impossible in as much detail as we 
-could. We decided, on intuitive grounds, which contradictions were 
-admissible and which were not. As we proceeded, it began to appear that the 
-results suggested by intuition were cases of a few general principles. In this 
-chapter, we will adopt these principles as postulates. The restatement of our 
-theory does not render the preceding chapters unnecessary. Only by 
-beginning with an exhaustive, intuitive discussion of perceptual illusions 
-could we convey the substance underlying the notations which we call 
-admissble contradictions, and motivate the unusual collection of postulates 
-which we will adopt. 
-
-All properties will be thought of as "parameters," such as time, 
-location, color, density, acidity, etc. Different parameters will be represented 
-by the letters x, y, z, .... Different values of one parameter, say x, will be 
-represented by $x_1$, $x_2$, .... Each parameter has a domain, the set of all values 
-it can assume. An ensembie ($x_0$, $y_0$, $z_0$, ...) will stand for the single possible 
-phenomenon which has x-value $x_0$, y-value $y_0$, etc. Several remarks are in 
-order. My ensembles are a highly refined version of Rudolph Carnap's 
-intensions or intension sets (sets of all possible entities having a given 
-property). The number of parameters, or properties, must be supposed to be 
-indefinitely large. By giving a possible phenomenon fixed values for every 
-parameter, I assure that there will be only one such possible phenomenon. In 
-other words, my intension sets are all singletons. Another point is that if we 
-specify some of the parameters and specify their ranges, we limit the 
-phenomena which can be represented by our "ensembles." If our first 
-parameter is time and its range is $R$, and our second parameter is spatial 
-location and its range is $R^2$, then we are limited to phenomena which are 
-point phenomena in space and time. If we have a parameter for speed of 
-motion, the motion will have to be infinitesimal. We cannot have a 
-parameter for weight at all; we can only have one for density. The physicist 
-encounters similar conceptual problems, and does noi find them 
-insurmountable. 
-
-Let ($x_1$, $y$, $z$, ...), ($x_2$, $y$, $z$, ...), etc. stand for possible phenomena 
-which all differ from each other in respect to parameter x but are identical in 
-respect to every other parameter $y$, $z$, ... . (If the ensembles were intension 
-sets, they would be disjoint precisely because $x$ takes a different value in 
-each.) A "simple contradiction family" of ensembles is the family [($x_1$,$y$,$z$, 
-...), ($x_2$, $y$, $z$, ...), ...]. The family may have any number of ensembles. It 
-actually represents many families, because $y$, $z$, ... are allowed to vary; but 
-each of these parameters must assume the same value in all ensembles in any 
-one family. $x$, on the other hand, takes different values in each ensemble in 
-any one family, values which may be fixed. A parameter which has the same 
-value throughout any one family will be referred to as a consistency 
-parameter. A parameter which has a different value in each ensemble in a 
-given family will be referred to as a contradiction parameter. 
-"Contradiction" will be shortened to "con." A simple con family is then a 
-family with one con parameter. The consistency parameters may be dropped 
-from the notation, but the reader must remember that they are implicitly 
-present, and must remember how they function. 
-
-A con parameter, instead of being fixed in every ensemble, may be 
-restricted to a different subset of its domain in every ensemble. The subsets 
-must be mutually disjoint for the con family to be well-defined. The con 
-family then represents many families in another dimension, because it 
-represents every family which can be formed by choosing a con parameter 
-value from the first subset, one from the second subset, etc. 
-
-Con families can be defined which have more than one con parameter, 
-i.e. more than one parameter satisfying all the conditions we put on x. Such 
-con families are not "simple." Let the cardinality of a con family be 
-indicated by a number prefixed to "family," and let the number of con 
-parameters be indicated by a number prefixed to "con." Remembering that 
-consistency parameters are understood, a 2-con $\infty$-family would appear as 
-[($x_1$, $y_1$). ($x_2$, $y_2$), ...].
-
-A "contradiction" or "$\varphi$-object" is not explicitly defined, but it is 
-notated by putting "$\varphi$" in front of a con family. The characteristics of $\varphi$-objects, 
-or cons, are established by introducing additional postulates in the 
-theory. 
-
-In this theory, every con is either "admissible" or "not admissible." 
-"Admissible" will be shortened to "am." The initial amcons of the theory 
-are introduced by postulate. Essentially, what is postulated is that cons with 
-a certain con parameter are am. (The cons directly postulated to be am are 
-on 1-con families.) However, the postulate will specify other requirements for 
-admissibility besides having the given con parameter. The requisite 
-cardinality of the con family will be specified. Also, the subsets will be 
-specified to which the con parameter must be restricted in each ensemble in 
-the con. A con must satisfy all postulated requirements before it is admitted 
-by the postulate. 
-
-The task of the theory is to determine whether the admissibility of the 
-cons postulated to be am implies the admissibility of any other cons. The 
-method we have developed for solving such problems will be expressed as a 
-collection of posiulates for our theory. 
-
-\postulate{1} Given $\varphi[(x\in A),(x\in B),\ldots]$ am, where $x\in A$, $x\in B$, ... are the 
-restrictions on the con parameter, and given $A_1\subset A$, $B_1\subset B$, ..., where $A_1,B_1,...\neq\emptyset$, then 
-$\varphi[(x\in A_1),(x\in B_1),...]$ is am. This postulate is obviously 
-equivalent to the postulate that $\varphi[(x\in A\cap C),(x\in B\cap C),...]$ is am, where $C$ is 
-a subset of $x$'s domain end the intersections are non-empty. (Proof: Choose 
-$C=A_1\cup B_1\cup\ldots$ .) 
-
-\postulate{2} If $x$ and $y$ are simple amcon parameters, then a con with con 
-parameters $x$ and $y$ is am if it satisfies the postulated requirements 
-concerning amcons on $x$ and the postulated requirements concerning amcons 
-on $y$. 
-
-The effect of all our assumptions up to now is to make parameters 
-totally independent. They do not interact with each other at all. 
-
-We will now introduce some specific amcons by postulate. If $s$ is speed, 
-consideration of the waterfall illusion suggests that we postulate 
-$\varphi[(s>O),(s=O)]$ to be am. (But with this postulate, we have come a long way from 
-the literary description of the waterfall illusion!) Note the implicit 
-requirements that the con family must be a 2-family, and that $s$ must be 
-selected from $[O]$ in one ensemble and from ${s:s>O}$ in the other ensemble. 
-
-If $t$ is time, $t\in R$, consideration of the phrase "b years ago," which is an 
-amcon in the natural language, suggests that we postulate $\varphi[(t):a-b\leq t\leq v-b \&a\leq v]$ to be am,
-where $a$ is a fixed time expressed in years A.D., $b$ is a fixed 
-number of years, and $v$ is a variable---the time of the present instant in years 
-A.D. The implicit requirements are that the con family must have the 
-cardinality of the continuum, and that every value of $t$ from $a-b$ to $v-b$ must 
-appear in an ensemble, where $v$ is a variable. Ensembles are thus continually 
-added to the con family. Note that there is the non-trivial possibility of using 
-this postulate more than once. We could admit a con for $a=1964$, $b=\sfrac{1}{2}$
-then admit another for $a=1963$, $b=2$, and admit still another for $a=1963$,
-$b=1$; etc. 
-
-Let $p$ be spatial location, $p\in R^2$. Let $P_i$ be a non-empty, bounded, 
-connected subset of $R^2$. Restriction subsets will be selected from the $P_i$.
-Specifically, let $P_1\cap P_2=\emptyset$. Consideration of a certain dreamed illusion 
-suggests that we admit $\varphi[(p\in P_1),(p\in P_2)]$. The implicit requirements are 
-obvious. But in this case, there are more requirements in the postulate of 
-admissibility. May we apply the postulate twice? May we admit first 
-$\varphi[(p\in P_1),(p\in P_2)]$ and then $\varphi[(p\in P_3),(p\in P_4)]$, where $P_3$ and $P_4$ are arbitrary 
-$P_i$'s different from $P_1$ and $P_2$? The answer is no. We may admit 
-$\varphi[(p\in P_1),(p\in P_2)]$ for arbitrary $P_1$ and $P_2$, $P_1\cap P_2=\emptyset$, but having made this "initial 
-choice," the postulate cannot be reused for arbitrary $P_3$ and $P_4$. A second 
-con $\varphi[(p\in P_3),(p\in P_4)]$, $P_3\cap P_4=\emptyset$, may be postulated to be am only if 
-$P_1\cup P_3$,$P_2\cup P_3$,$P_1\cup P_4$, and $P_2\cup P_4$ are not connected. In other words, you 
-may postulate many cons of the form $\varphi[(p\in P_i),(p\in P_j)]$ to be am, but 
-your first choice strongly circumscribes your second choice, etc. 
-
-We will now consider certain results in the logic of amcons which were 
-established by extensive elucidation of our intuitions. The issue is whether 
-our present axiomization produces the same results. We will express the 
-results in our latest notation as far as possible. Two more definitions are 
-necessary. The parameter $\theta$ is the angle of motion of an infinitesimally 
-moving phenomenon, measured in degrees with respect to some chosen axis. 
-Then, recalling the set $P_1$, choose $P_5$ and $P_6$ so that $P_1=P_5\cup P_6$ and 
-$P_5\cap P_6=\emptyset$. 
-
-The results by which we will judge our axiomization are as follows. 
-
-\begin{enumerate} % TODO with colons?
-
-	\item $\varphi[S, C_1\cup C_2]$ can be inferred to be am. 
-
-Our present notation cannot express this result, because it does not 
-distinguish between different types of uniform motion throughout a finite 
-region, \ie the types $M$, $C_1$, $C_2$, $D_1$, and $D_2$. Instead, we have infinitesimal 
-motion, which is involved in all the latter types of motion. Questions such as 
-"whether the admissibility of $\varphi[M,S]$ implies the admissibility of $\varphi[C_1,S]$" 
-drop out. The reason for the omission in the present theory is our choice of 
-parameters and domains, which we discussed earlier. Our present version is 
-thus not exhaustive. However, the deficiency is not intrinsic to our method; 
-and it does not represent any outright falsification of our intuitions. Thus, 
-we pass over the deficiency. 
-
-\item $\varphi[(p\in P_1,s_0),(p\in P_2,S_0)]$ and other such cons can be inferred to be am. 
-With our new, powerful approach, this result is trivial. It is guaranteed by 
-what we said about consistency parameters. 
-
-\item There is no way to infer that $\varphi[C_1,C_2]$ is am; and no way to infer that 
-$\varphi[(45^\circ,s_0\greater O),(60^\circ,s=s_0)]$ is am. 
-
-The first part of the result drops out. The second part is trivial with our new 
-method as long as we do not postulate that cons on $\theta$ are am. 
-
-\item $\varphi[(p\in P_2),(p\in P_5)]$ can be inferred to be am. 
-
-Yes, by Postulate 1. 
-
-\item $\varphi[(s>O, p\in P_1),(s=O, p\in P_2)]$ and $\varphi[(s>O, p\in P_2),(s=O, p\in P_1)]$ can 
-be inferred to be am. 
-
-Yes, by Postulate 2. These two amcons are distinct. The question of whether 
-they should be considered equivalent is closely related to the degree to 
-which con parameters are independent of each other. 
-
-\item There is no way to infer that $\varphi[(p\in P_5),(p\in P_6)]$ or $\varphi[(p\in P_1),(p\in P_3)]$
-is am. Our special requirement in the postulate of admissibility for 
-$\varphi[(p\in P_1),(p\in P_2)]$ guarantees this result. 
-\end{enumerate}
-
-The reason for desiring this last result requires some discussion. In 
-heuristic terms, we wish to avoid admitting both location in New York in 
-Greensboro and location in Manhattan and Brooklyn. We also wish to avoid 
-admitting location in New York in Greensboro and location in New York in 
-Boston. If we admitted either of these combinations, then the intuitive 
-rationale of the notions would indicate that we had admitted triple location. 
-While we have a dreamed illusion which justifies the concept of double 
-location, we have no intuitive justification whatever for the concept of triple 
-location. It must be clear that admission of either of the combinations 
-mentioned would not imply the admissibility of a con on a 3-family with 
-con parameter p by the postulates of our theory. Our theory is formally safe 
-from this implication. However, the intuitive meaning of either combination 
-would make them proxies for the con on the 3-family. 
-
-A closely related consideration is that in the preceding chapter, it 
-appeared that the admission of $\varphi[(p\in P_1),(p\in P_2)]$ and $\varphi[(p\in P_5),(p\in P_6)]$
-would tend to require the admission of the object $\varphi[(p\in P_2),\varphi[(p\in P_5),(p\in P_6)]]$
-(a Type 1 chain). Further, it this implication held, then by the same 
-rationale the admission of $\varphi[(p\in P_1),(p\in P_2)]$ and $\varphi[(s>O,p_0\in P_1),(s=O,p=p_0)]$,
-		both of which are am, would require the admission of the object 
-$\varphi[(p\in P_2), \varphi[(s>O,p_0\in P_1),(s=O, p=p_0)]]$. 
-We may now say, however, 
-that the postulates of our theory emphatically do not require us to accept 
-these implications. If there is an intuitively valid notion underlying the chain 
-on s and p, it reduces to the amcons introduced in result 5. As for the chain 
-on p alone, we repeat that simultaneous admission of the two cons 
-mentioned would tend to justify some triple location concept. However, we 
-do not have to recognize that concept as being the chain. It seems that our 
-present approach allows us to forget about chains for now. 
-
-Our conclusion is that the formal approach of this chapter is in good 
-agreement with our intuitively established results. 
-
-\section*{Note on the overall significance of the logic of amcons:}
-
-When traditional logicians said that something was logically impossible, 
-they meant to imply that it was impossible to imagine or visualize. But this 
-implication was empirically false. The realm of the logically possible is not 
-the entire realm of connotative thought; it is just the realm of normal 
-perceptual routines. When the mind is temporarily freed from normal 
-perceptual routines---especially in perceptual illusions, but also in dreams and 
-even in the use of certain "illogical" natural language phrases---it can imagine 
-and visualize the "logically impossible." Every text on perceptual 
-psychology mentions this fact, but logicians have never noticed its immense 
-significance. The logically impossible is not a blank; it is a whole layer of 
-meaning and concepts which can be superimposed on conventional logic, but 
-not reduced or assimilated to it. The logician of the future may use a drug or 
-some other method to free himself from normal perceptual routines for a 
-sustained period of time, so he can freely think the logically impossible. He 
-will then perform rigorous deductions and computations in the logic of 
-amcons. 
-
-- 
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