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diff --git a/essays/admissable_contraditions.tex b/essays/admissable_contraditions.tex new file mode 100644 index 0000000..f66bb98 --- /dev/null +++ b/essays/admissable_contraditions.tex @@ -0,0 +1,246 @@ +\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. + |