Cf: Animated Logical Graphs • 21 https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-21/
All, A funny thing just happened. Let’s see if we can tell where. We started with the algebraic expression “(a)”, in which the operand “a” suggests the contemplated absence or presence of any arithmetic expression or its value, then we contemplated the absence or presence of the operator “( )” in “(a)” to be indicated by a cross or a space, respectively, for the value of a newly introduced variable, “b”, placed in a new slot of a newly extended operator form, as suggested by this picture: Figure 1. Control Form (a)_b https://inquiryintoinquiry.files.wordpress.com/2019/07/box-a-quo-b.jpg What happened here is this. Our contemplation of a constant operator as being potentially variable gave rise to the contemplation of a newly introduced but otherwise quite ordinary operand variable, albeit in a newly-fashioned formula. In its interpretation for logic the newly formed operation may be viewed as an extension of ordinary negation, one in which the negation of the first variable is “controlled” by the value of the second variable. We may regard this development as marking a form of “controlled reflection”, or a form of “reflective control”. From here on out we'll use the inline syntax “(a, b)” for the corresponding operation on two variables, whose operation table is given below. Table 1. Operation Table for (a, b) https://inquiryintoinquiry.files.wordpress.com/2019/07/table-ab-space-cross.png • The Entitative Interpretation (En), where Space = False and Cross = True, calls this operation “equivalence”. • The Existential Interpretation (Ex), where Space = True and Cross = False, calls this operation “distinction”. Regards, Jon
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