User:Jon Awbrey/Figures and Tables 54
Syntax and Semantics of a Calculus for Propositional Logic
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Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.
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Wiki + LaTeX + JPG
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable \(k\)-ary scope.
| \(\text{Graph}\) | \(\text{Expression}\) | \(\text{Interpretation}\) | \(\text{Other Notations}\) |
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\(~\) | \(\mathrm{true}\) | \(1\) |
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\(\texttt{(}~\texttt{)}\) | \(\mathrm{false}\) | \(0\) |
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\(a\) | \(a\) | \(a\) |
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\(\texttt{(} a \texttt{)}\) | \(\mathrm{not}~ a\) | \(\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime\) |
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\(a ~ b ~ c\) | \(a ~\mathrm{and}~ b ~\mathrm{and}~ c\) | \(a \land b \land c\) |
(B)(C))_Big.jpg)
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\(\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\) | \(a ~\mathrm{or}~ b ~\mathrm{or}~ c\) | \(a \lor b \lor c\) |
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\(\texttt{(} a \texttt{(} b \texttt{))}\) |
\(\begin{matrix} a ~\mathrm{implies}~ b \\[6pt] \mathrm{if}~ a ~\mathrm{then}~ b \end{matrix}\) |
\(a \Rightarrow b\) |
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\(\texttt{(} a \texttt{,} b \texttt{)}\) |
\(\begin{matrix} a ~\mathrm{not~equal~to}~ b \\[6pt] a ~\mathrm{exclusive~or}~ b \end{matrix}\) |
\(\begin{matrix} a \neq b \\[6pt] a + b \end{matrix}\) |
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\(\texttt{((} a \texttt{,} b \texttt{))}\) |
\(\begin{matrix} a ~\mathrm{is~equal~to}~ b \\[6pt] a ~\mathrm{if~and~only~if}~ b \end{matrix}\) |
\(\begin{matrix} a = b \\[6pt] a \Leftrightarrow b \end{matrix}\) |
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\(\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}\) |
\(\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~false}. \end{matrix}\) |
\(\begin{matrix} & \bar{a} ~ b ~ c \\ \lor & a ~ \bar{b} ~ c \\ \lor & a ~ b ~ \bar{c} \end{matrix}\) |
,(B),(C))_Big.jpg)
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\(\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}\) |
\(\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~true}. \\[6pt] \mathrm{partition~all} \\ \mathrm{into}~ a, b, c. \end{matrix}\) |
\(\begin{matrix} & a ~ \bar{b} ~ \bar{c} \\ \lor & \bar{a} ~ b ~ \bar{c} \\ \lor & \bar{a} ~ \bar{b} ~ c \end{matrix}\) |
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\(\texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}\) |
\(\begin{matrix} \mathrm{oddly~many~of} \\ a, b, c \\ \mathrm{are~true}. \end{matrix}\) |
\(a + b + c\) \(\begin{matrix} & a ~ b ~ c \\ \lor & a ~ \bar{b} ~ \bar{c} \\ \lor & \bar{a} ~ b ~ \bar{c} \\ \lor & \bar{a} ~ \bar{b} ~ c \end{matrix}\) |
,(B),(C))_Big.jpg)
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\(\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}\) |
\(\begin{matrix} \mathrm{partition}~ x \\ \mathrm{into}~ a, b, c. \\[6pt] \mathrm{genus}~ x ~\mathrm{comprises} \\ \mathrm{species}~ a, b, c. \end{matrix}\) |
\(\begin{matrix} & \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c} \\ \lor & x ~ a ~ \bar{b} ~ \bar{c} \\ \lor & x ~ \bar{a} ~ b ~ \bar{c} \\ \lor & x ~ \bar{a} ~ \bar{b} ~ c \end{matrix}\) |
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Table 1 is the first of several “Rosetta Stones” we'll use in this discussion to translate between different languages for the same subject matters. The present Table displays equivalent expressions for frequently encountered propositional forms in four notations for propositional calculus. The Table has four columns, labeled “Graph”, “Expression”, “Interpretation”, and “Other Notations”, respectively.
• Column 1 “Graph” exhibits a logical graph for a commonly occurring propositional form.
• Column 2 “Expression” exhibits the text string transcription of the graph in Column 1.
• Column 3 “Interpretation” gives one or more verbal formulas for the graph in Column 1.
• Column 4 “Other Notations” shows several ways of notating the graph's logical meaning.
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~ | true | 1 |
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( ) | false | 0 |
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a | a | a |
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( a ) | not a | ¬a ā ã a′ |
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abc | a and b and c | a ∧ b ∧ c |
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((a)(b)(c)) | a or b or c | a ∨ b ∨ c |
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( a ( b )) |
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a ⇒ b |
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( a , b ) |
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(( a , b )) |
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( a , b , c ) | just one of a, b, c is false. |
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(( a ),( b ),( c )) |
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( a ,( b , c )) | oddly many of a, b, c are true. |
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( x ,( a ),( b ),( c )) |
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Mathstodon Versions
Differential Logic and Dynamic Systems • Overview
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
❝Stand and unfold yourself.❞ — Hamlet • Francisco • 1.1.2
In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade-off between dynamic paradigms and symbolic paradigms. Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system’s state through time. Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system’s description or an agent’s state of information. Logic-based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus. The work laid out in this report is intended to address that lack.
This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms.
The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.
Differential Logic and Dynamic Systems • Review and Transition 1
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Review_and_Transition
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable \(k\)-ary scope.
• A bracketed list of propositional expressions in the form \(\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}\) indicates that exactly one of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) is false.
• A concatenation of propositional expressions in the form \(e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k\) indicates that all of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) are true, in other words, that their logical conjunction is true.
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.
Review and Transition (OEIS Version)
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable \(k\)-ary scope.
- A bracketed list of propositional expressions in the form \(\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}\) indicates that exactly one of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) is false.
- A concatenation of propositional expressions in the form \(e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k\) indicates that all of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) are true, in other words, that their logical conjunction is true.
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.
| \(\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}\) |
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Table 1 is the first of several “Rosetta Stones” we'll use in this discussion to translate between different languages for the same subject matters. In this case the Table displays equivalent expressions for simple examples of propositional forms in four notations for propositional calculus.
- Column 1 shows the logical graphs used to represent a number of simple propositional forms.
- Column 2 shows the traverse strings corresponding to the logical graphs in Column 1.
- Column 3 interprets the graph and string by means of conventional verbal formulas.
- Column 4 translates the interpretation into a number of symbolic notations.
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes “teletype” parentheses \(\texttt{(} \ldots \texttt{)}\) or barred parentheses \((\!| \ldots |\!)\) may be used for logical operators.
The briefest expression for logical truth is the empty word, usually denoted by \({}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}\) or \({}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}\) in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression \({}^{\backprime\backprime} \texttt{((} ~ \texttt{))} {}^{\prime\prime},\) or, especially if operating in an algebraic context, by a simple \({}^{\backprime\backprime} 1 {}^{\prime\prime}.\) Also when working in an algebraic mode, the plus sign \({}^{\backprime\backprime} + {}^{\prime\prime}\) may be used for exclusive disjunction. For example, we have the following paraphrases of algebraic expressions by bracket expressions:
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\(\begin{matrix} x + y ~=~ \texttt{(} x, y \texttt{)} \\[6pt] x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))} \end{matrix}\) |
It is important to note that the last expressions are not equivalent to the triple bracket \(\texttt{(} x, y, z \texttt{)}.\)
Note. The usage that one often sees, of a plus sign "\(+\)" to represent inclusive disjunction, and the reference to this operation as boolean addition, is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:
The expression \(x + y\) seems indeed uninterpretable, unless it be assumed that the things represented by \(x\) and the things represented by \(y\) are entirely separate; that they embrace no individuals in common. (Boole, 66).
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177–263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.