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Revision as of 20:56, 29 July 2009
, 20:56, 29 July 2009→Themes and variations: center displays
For example, consider the following expression:
For example, consider the following expression:
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<pre>
<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
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</pre>
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We may regard this algebraic expression as a general expression for an infinite set of arithmetic expressions, starting like so:
We may regard this algebraic expression as a general expression for an infinite set of arithmetic expressions, starting like so:
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<pre>
<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
</pre>
</pre>
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Now consider what this says about the following algebraic law:
Now consider what this says about the following algebraic law:
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<pre>
<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
</pre>
</pre>
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It permits us to understand the algebraic law as saying, in effect, that every one of the arithmetic expressions of the contemplated pattern evaulates to the very same canonical expression as the upshot of that evaluation. This is, as far as I know, just about as close as we can come to a conceptually and ontologically minimal way of understanding the relation between an algebra and its corresponding arithmetic.
It permits us to understand the algebraic law as saying, in effect, that every one of the arithmetic expressions of the contemplated pattern evaulates to the very same canonical expression as the upshot of that evaluation. This is, as far as I know, just about as close as we can come to a conceptually and ontologically minimal way of understanding the relation between an algebra and its corresponding arithmetic.
To begin with a concrete case that's as easy as possible, let's examine this extremely simple algebraic expression:
To begin with a concrete case that's as easy as possible, let's examine this extremely simple algebraic expression:
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<pre>
<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
</pre>
</pre>
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In this context the variable name "''a''" appears as an ''operand name''. In functional terms, "''a''" is called an ''argument name'', but we are probably well advised to avoid the confusing connotations of the word "argument" here, as it also refers in logical discussions to a more or less specific pattern of reasoning. In syntactic terms, this same "''a''" would be classified as a ''terminal sign''.
In this context the variable name "''a''" appears as an ''operand name''. In functional terms, "''a''" is called an ''argument name'', but we are probably well advised to avoid the confusing connotations of the word "argument" here, as it also refers in logical discussions to a more or less specific pattern of reasoning. In syntactic terms, this same "''a''" would be classified as a ''terminal sign''.
As we've already discussed, the algebraic variable name indicates the contemplated absence or presence of any arithmetic expression taking its place in the surrounding template, which expression is proxied well enough by its value, of which values we know but two. Thus, the given algebraic expression varies between these choices:
As we've already discussed, the algebraic variable name indicates the contemplated absence or presence of any arithmetic expression taking its place in the surrounding template, which expression is proxied well enough by its value, of which values we know but two. Thus, the given algebraic expression varies between these choices:
{| align="center" style="text-align:center; width:90%"
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<pre>
<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
</pre>
</pre>
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The above selection of arithmetic expressions is what it means to contemplate the absence or presence of the operand "a" in the algebraic expression "(a)". But what would it mean to contemplate the absence or presence of the operator "(_)" in the algebraic expression "(a)"?
The above selection of arithmetic expressions is what it means to contemplate the absence or presence of the operand "a" in the algebraic expression "(a)". But what would it mean to contemplate the absence or presence of the operator "(_)" in the algebraic expression "(a)"?
We had been contemplating the penultimately simple algebraic expression "(a)" as a name for a set of arithmetic expressions, namely, (a) = {(), (())}, taking the equality sign in the appropriate sense.
We had been contemplating the penultimately simple algebraic expression "(a)" as a name for a set of arithmetic expressions, namely, (a) = {(), (())}, taking the equality sign in the appropriate sense.
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<pre>
<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
</pre>
</pre>
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Then we asked the corresponding question about the operator "(_)": The above selection of arithmetic expressions is what it means to contemplate the absence or presence of the operand "a" in the algebraic expression "(a)". But what would it mean to contemplate the absence or presence of the operator "(_)" in the algebraic expression "(a)"?
Then we asked the corresponding question about the operator "(_)": The above selection of arithmetic expressions is what it means to contemplate the absence or presence of the operand "a" in the algebraic expression "(a)". But what would it mean to contemplate the absence or presence of the operator "(_)" in the algebraic expression "(a)"?
Clearly, a variation between the absence and the presence of the operator "(_)" in the algebraic expression "(a)" refers to a variation between the algebraic expressions "a" and "(a)", respectively, somewhat as pictured here:
Clearly, a variation between the absence and the presence of the operator "(_)" in the algebraic expression "(a)" refers to a variation between the algebraic expressions "a" and "(a)", respectively, somewhat as pictured here:
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` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
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</pre>
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But how shall we signify such variations in a coherent calculus?
But how shall we signify such variations in a coherent calculus?
Here is how we might suggest an algebraic expression of the form "(q)" where the absence or presence of the operator "(_)" depends on the value of the algebraic expression "p", the operator "(_)" being absent whenever p is unmarked and present when whenever p is marked.
Here is how we might suggest an algebraic expression of the form "(q)" where the absence or presence of the operator "(_)" depends on the value of the algebraic expression "p", the operator "(_)" being absent whenever p is unmarked and present when whenever p is marked.
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` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
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</pre>
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It was obvious to me from the very outset that this sort of tactic would need some work to become a usable calculus, especially when it became time to feed those punchcards back into the computer.
It was obvious to me from the very outset that this sort of tactic would need some work to become a usable calculus, especially when it became time to feed those punchcards back into the computer.
One of the other tactics of syntax that I tried at this time — somewhere in the 70's ... when did we quit using punchcards? — by way of porting operator variables into logical graphs and the laws of form, was to hollow out a leg of Spencer-Brown's crosses, gnomons, markers, whatever you call them, like this:
One of the other tactics of syntax that I tried at this time — somewhere in the 70's ... when did we quit using punchcards? — by way of porting operator variables into logical graphs and the laws of form, was to hollow out a leg of Spencer-Brown's crosses, gnomons, markers, whatever you call them, like this:
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<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
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The initial idea that I had in mind here was the same as before, that the operator over q is counted as absent when p evaluates to a space and counted as present when p evaluates to a cross.
The initial idea that I had in mind here was the same as before, that the operator over q is counted as absent when p evaluates to a space and counted as present when p evaluates to a cross.
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 of 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:
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 of 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:
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<pre>
<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
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</pre>
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The funny thing that just happened — it's an acquired sense of humor — is that our contemplation of an operator variable just as quickly got converted into the contemplation of a newly introjected but otherwise quite ordinary operand variable, albeit within a newly-fanged formula. In its interpretation for logic, the new form of operation that forms here may be viewed as an extension of ordinary negation, specifically, a negation of the first variable that is "controlled" by the value of the second variable. Thus, we may regard this development as marking a form of "controlled reflection", or a form of "reflective control". By way of an inline syntax for this, I will employ the form "(a, b)".
The funny thing that just happened — it's an acquired sense of humor — is that our contemplation of an operator variable just as quickly got converted into the contemplation of a newly introjected but otherwise quite ordinary operand variable, albeit within a newly-fanged formula. In its interpretation for logic, the new form of operation that forms here may be viewed as an extension of ordinary negation, specifically, a negation of the first variable that is "controlled" by the value of the second variable. Thus, we may regard this development as marking a form of "controlled reflection", or a form of "reflective control". By way of an inline syntax for this, I will employ the form "(a, b)".
Writing out a formal operation table yields the following summary:
Writing out a formal operation table yields the following summary:
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o-------------------o-------------------o-------------------o
o-------------------o-------------------o-------------------o
o-------------------o-------------------o-------------------o
o-------------------o-------------------o-------------------o
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</pre>
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* En, for which blank = false and cross = true, calls this "equivalence".
* En, for which blank = false and cross = true, calls this "equivalence".
The step of controlled reflection that we just took can be iterated just as far as we wish to take it, as suggested by the following set:
The step of controlled reflection that we just took can be iterated just as far as we wish to take it, as suggested by the following set:
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` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
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By way of inline syntax, I will transliterate these expressions as "(a)", "(a, b)", "(a, b, c)", "(a, b, c, d)", and so on, capturing the general style of expression in the form "(''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x''<sub>''k''</sub>)". With this move we have passed beyond the graph-theoretical form of rooted trees to what graph theorists generally call ''rooted cacti''.
By way of inline syntax, I will transliterate these expressions as "(a)", "(a, b)", "(a, b, c)", "(a, b, c, d)", and so on, capturing the general style of expression in the form "(''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x''<sub>''k''</sub>)". With this move we have passed beyond the graph-theoretical form of rooted trees to what graph theorists generally call ''rooted cacti''.
The following Table will suffice to suggest the syntactic correspondences among the "streamer-cross" forms that Peirce used in his essay on "Qualitative Logic" and Spencer Brown used in his book ''Laws of Form'', as they become extended by successive steps of controlled reflection, the plaintext string syntax, and the rooted cactus graphs:
The following Table will suffice to suggest the syntactic correspondences among the "streamer-cross" forms that Peirce used in his essay on "Qualitative Logic" and Spencer Brown used in his book ''Laws of Form'', as they become extended by successive steps of controlled reflection, the plaintext string syntax, and the rooted cactus graphs:
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o-----------------------------o-----------------o-----------o
o-----------------------------o-----------------o-----------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` ` ` |
o-----------------------------o-----------------o-----------o
o-----------------------------o-----------------o-----------o
</pre>
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<pre>
| Reference:
| Reference:
|
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Let us examine the formal operation table for the next in our series of reflective operations to see if we can elicit the general pattern.
Let us examine the formal operation table for the next in our series of reflective operations to see if we can elicit the general pattern.
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o-------o-------o-------o-----------o
o-------o-------o-------o-----------o
o-------o-------o-------o-----------o
o-------o-------o-------o-----------o
</pre>
</pre>
|}
Or, thinking in terms of the graphic equivalents, writing "o" for a blank node and "|" for an edge:
Or, thinking in terms of the graphic equivalents, writing "o" for a blank node and "|" for an edge:
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<pre>
o-------o-------o-------o-----------o
o-------o-------o-------o-----------o
o-------o-------o-------o-----------o
o-------o-------o-------o-----------o
</pre>
</pre>
|}
Evidently, the rule is that "(''a'', ''b'', ''c'')" denotes the value denoted by "o" if and only if exactly one of the variables has the value denoted by "|", otherwise it denotes the value denoted by "|". Examination of the whole sequence of reflective negations will show that this is the general rule.
Evidently, the rule is that "(''a'', ''b'', ''c'')" denotes the value denoted by "o" if and only if exactly one of the variables has the value denoted by "|", otherwise it denotes the value denoted by "|". Examination of the whole sequence of reflective negations will show that this is the general rule.
The formal rule of evaluation for a "''k''-lobe" or "''k''-operator" is:
The formal rule of evaluation for a "''k''-lobe" or "''k''-operator" is:
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o-----------------------------------------------------------o
o-----------------------------------------------------------o
o-----------------------------------------------------------o
o-----------------------------------------------------------o
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</pre>
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The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:
The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:
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o-----------------------------------------------------------o
o-----------------------------------------------------------o
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
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==Case analysis-synthesis theorem==
==Case analysis-synthesis theorem==
