Inquiry and Analogy • Higher Order Propositional Expressions
Interpretive Categories for Higher Order Propositions (n = 1)
Table 12 presents a series of interpretive categories for the higher order propositions in Table 11. I’ll leave those for now to the reader’s contemplation and discuss them when we get two variables into the mix. The lower dimensional cases tend to exhibit condensed or degenerate structures and their full significance will become clearer once we get beyond the 1‑dimensional case.
Resources
- Logic Syllabus
- Boolean Function
- Boolean-Valued Function
- Logical Conjunction
- Minimal Negation Operator
- Introduction to Inquiry Driven Systems
- Functional Logic • Part 1 • Part 2 • Part 3
- Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History
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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
then the next higher order of propositions consists of maps of type 
There are exactly four propositions one can make about the elements of 
Each proposition has the concrete type 
and the abstract type 
From that beginning there are exactly sixteen higher order propositions one can make about the initial set of four propositions. Each higher order proposition has the abstract type 
for 
= 1 to 4, while the entries in Column 2 show the value each proposition takes on the argument value listed in the corresponding column head.
for 
= 0 to 15. The entries in the body of the Table show the value each measure assigns to each proposition 
and 
involves a subscription to such notions, as shown by the membership relations invoked in their indices.
In the Current situation the Air is cool.
Just Before it rains, the Air is cool.
The Current situation is just Before it rains.
Just Before it rains, a Dark cloud will appear.
In the Current situation, a Dark cloud will appear.
are a random sample of some undefined class 
of whose characters 
are samples,![\begin{matrix} T ~\text{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}; \\[4pt] S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime}, ~\text{are}~ Q\text{'s}; \\[4pt] \text{Hence,}~ T ~\text{is a}~ Q. \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++T+%7E%5Ctext%7Bis%7D%7E+P%5E%7B%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%5Cprime%7D%3B++%5C%5C%5B4pt%5D++S%5E%7B%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%5Cprime%7D%2C+%7E%5Ctext%7Bare%7D%7E+Q%5Ctext%7B%27s%7D%3B++%5C%5C%5B4pt%5D++%5Ctext%7BHence%2C%7D%7E+T+%7E%5Ctext%7Bis+a%7D%7E+Q.++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![\begin{array}{l|l} \text{Every}~ X ~\text{is, for example,}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}, ~\text{etc.}, & S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime}, ~\text{etc., are samples of the}~ X\text{'s}, \\[4pt] T ~\text{is found to be}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}, ~\text{etc.}; & S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime}, ~\text{etc., are found to be}~ Q\text{'s}; \\[4pt] \text{Hence, hypothetically,}~ T ~\text{is an}~ X. & \text{Hence, inductively, every}~ X ~\text{is a}~ Q. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D++%5Ctext%7BEvery%7D%7E+X+%7E%5Ctext%7Bis%2C+for+example%2C%7D%7E+P%5E%7B%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%5Cprime%7D%2C+%7E%5Ctext%7Betc.%7D%2C++%26++S%5E%7B%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%5Cprime%7D%2C+%7E%5Ctext%7Betc.%2C+are+samples+of+the%7D%7E+X%5Ctext%7B%27s%7D%2C++%5C%5C%5B4pt%5D++T+%7E%5Ctext%7Bis+found+to+be%7D%7E+P%5E%7B%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%5Cprime%7D%2C+%7E%5Ctext%7Betc.%7D%3B++%26++S%5E%7B%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%5Cprime%7D%2C+%7E%5Ctext%7Betc.%2C+are+found+to+be%7D%7E+Q%5Ctext%7B%27s%7D%3B++%5C%5C%5B4pt%5D++%5Ctext%7BHence%2C+hypothetically%2C%7D%7E+T+%7E%5Ctext%7Bis+an%7D%7E+X.++%26++%5Ctext%7BHence%2C+inductively%2C+every%7D%7E+X+%7E%5Ctext%7Bis+a%7D%7E+Q.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=-1&c=20201002)
are taken at random from such a class that their characters at random are such as 
![\begin{matrix} T ~\text{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}, \\[4pt] S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\text{are}~ Q; \\[4pt] \therefore T ~\text{is}~ Q. \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++T+%7E%5Ctext%7Bis%7D%7E+P%5E%7B%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%5Cprime%7D%2C++%5C%5C%5B4pt%5D++S%5E%7B%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%5Cprime%7D+%7E%5Ctext%7Bare%7D%7E+Q%3B++%5C%5C%5B4pt%5D++%5Ctherefore+T+%7E%5Ctext%7Bis%7D%7E+Q.++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![\begin{matrix} 1. \\[4pt] S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\text{are taken as being}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}, \\[4pt] S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\text{are}~ Q; \\[4pt] \therefore ~(\text{By induction})~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime} ~\text{is}~ Q, \\[4pt] T ~\text{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}; \\[4pt] \therefore ~(\text{Deductively})~ T ~\text{is}~ Q. \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++1.++%5C%5C%5B4pt%5D++S%5E%7B%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%5Cprime%7D+%7E%5Ctext%7Bare+taken+as+being%7D%7E+P%5E%7B%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%5Cprime%7D%2C++%5C%5C%5B4pt%5D++S%5E%7B%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%5Cprime%7D+%7E%5Ctext%7Bare%7D%7E+Q%3B++%5C%5C%5B4pt%5D++%5Ctherefore+%7E%28%5Ctext%7BBy+induction%7D%29%7E+P%5E%7B%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%5Cprime%7D+%7E%5Ctext%7Bis%7D%7E+Q%2C++%5C%5C%5B4pt%5D++T+%7E%5Ctext%7Bis%7D%7E+P%5E%7B%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%5Cprime%7D%3B++%5C%5C%5B4pt%5D++%5Ctherefore+%7E%28%5Ctext%7BDeductively%7D%29%7E+T+%7E%5Ctext%7Bis%7D%7E+Q.++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![\begin{matrix} 2. \\[4pt] S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\text{are, for instance,}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}, \\[4pt] T ~\text{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}; \\[4pt] \therefore ~(\text{By hypothesis})~ T ~\text{has the common characters of}~ S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime}, \\[4pt] S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\text{are}~ Q; \\[4pt] \therefore ~(\text{Deductively})~ T ~\text{is}~ Q. \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++2.++%5C%5C%5B4pt%5D++S%5E%7B%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%5Cprime%7D+%7E%5Ctext%7Bare%2C+for+instance%2C%7D%7E+P%5E%7B%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%5Cprime%7D%2C++%5C%5C%5B4pt%5D++T+%7E%5Ctext%7Bis%7D%7E+P%5E%7B%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%7D%2C+P%5E%7B%5Cprime%5Cprime%5Cprime%7D%3B++%5C%5C%5B4pt%5D++%5Ctherefore+%7E%28%5Ctext%7BBy+hypothesis%7D%29%7E+T+%7E%5Ctext%7Bhas+the+common+characters+of%7D%7E+S%5E%7B%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%5Cprime%7D%2C++%5C%5C%5B4pt%5D++S%5E%7B%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%7D%2C+S%5E%7B%5Cprime%5Cprime%5Cprime%7D+%7E%5Ctext%7Bare%7D%7E+Q%3B++%5C%5C%5B4pt%5D++%5Ctherefore+%7E%28%5Ctext%7BDeductively%7D%29%7E+T+%7E%5Ctext%7Bis%7D%7E+Q.++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
is read as 
that is, 
or 
Inquiry Into Inquiry 