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