Differential Propositional Calculus • 8
• https://inquiryintoinquiry.com/2024/12/07/differential-propositional-calculus-8-b/

Formal Development (cont.)

Before moving on, let's unpack some of the assumptions, conventions, and implications involved in the array of concepts and notations introduced above.

A universe of discourse A° = [a₁, …, aₙ] qualified by the logical features a₁, …, aₙ is a set A plus the set of all functions from the space A to the boolean domain B = {0, 1}. There are 2ⁿ elements in A, often pictured as the cells of a venn diagram or the nodes of a hypercube. There are 2^(2ⁿ) possible functions from A to B, accordingly pictured as all the ways of painting the cells of a venn diagram or the nodes of a hypercube with a palette of two colors.

A logical proposition about the elements of A is either true or false of each element in A, while a function f : A → B evaluates to 1 or 0 on each element of A. The analogy between logical propositions and boolean-valued functions is close enough to adopt the latter as models of the former and simply refer to the functions f : A → B as propositions about the elements of A.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 7.2
• https://inquiryintoinquiry.com/2024/12/06/differential-propositional-calculus-7-b/

Formal Development (cont.)

Table 7 summarizes the basic notations needed to describe ordinary propositional calculi in a systematic fashion.

Table 7. Propositional Calculus • Basic Notation
• https://inquiryintoinquiry.files.wordpress.com/2020/02/propositional-calculus-basic-notation.png

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 6.2
• https://inquiryintoinquiry.com/2024/12/04/differential-propositional-calculus-6-b/

Cactus Calculus (cont.)

The briefest expression for logical truth is the empty word, denoted ε or λ in formal languages, where it forms the identity element for concatenation. It may be given visible expression in textual settings by means of the logically equivalent form (()), or, especially if operating in an algebraic context, by a simple 1. Also when working in an algebraic mode, the plus sign “+” may be used for exclusive disjunction. For example, we have the following paraphrases of algebraic expressions.

• x + y = (x, y)

• x + y + z = ((x, y), z) = (x, (y, z))

It is important to note the last expressions are not equivalent to the triple bracket (x, y, z).

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 6.1
• https://inquiryintoinquiry.com/2024/12/04/differential-propositional-calculus-6-b/

Cactus Calculus —

Table 6 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable k‑ary scope.

• A bracketed sequence of propositional expressions (e₁, e₂, …, eₖ) is taken to mean exactly one of the propositions e₁, e₂, …, eₖ is false, in other words, their “minimal negation” is true.

• A concatenated sequence of propositional expressions e₁ e₂ … eₖ is taken to mean every one of the propositions e₁, e₂, …, eₖ is true, in other words, their “logical conjunction” is true.

Table 6. Syntax and Semantics of a Calculus for Propositional Logic
• https://inquiryintoinquiry.files.wordpress.com/2022/10/syntax-and-semantics-of-a-calculus-for-propositional-logic-4.0.png

All other propositional connectives may be obtained through combinations of the above two forms. As it happens, the concatenation form is dispensable in light of the bracket form but it is convenient to maintain it as an abbreviation for more complicated bracket expressions. While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for bracket forms. In contexts where parentheses are needed for other purposes “teletype” parentheses (…) or barred parentheses (|…|) may be used for logical operators.

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 5
• https://inquiryintoinquiry.com/2024/12/03/differential-propositional-calculus-5-b/

Casual Introduction (concl.)

Table 5 exhibits the rules of inference responsible for giving the differential proposition dq its meaning in practice.

Table 5. Differential Inference Rules
• https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-e280a2-differential-inference-rules.png

If the feature q is interpreted as applying to an object in the universe of discourse X then the differential feature dq may be taken as an attribute of the same object which tells it is changing “significantly” with respect to the property q — as if the object bore an “escape velocity” with respect to the condition q.

For example, relative to a frame of observation to be made more explicit later on, if q and dq are true at a given moment, it would be reasonable to assume ¬q will be true in the next moment of observation. Taken all together we have the fourfold scheme of inference shown above.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics

Differential Propositional Calculus • 4
• https://inquiryintoinquiry.com/2024/12/02/differential-propositional-calculus-4-b/

Casual Introduction (cont.)

In Figure 3 we saw how the basis of description for the universe of discourse X could be extended to a set of two qualities {q, dq} while the corresponding terms of description could be extended to an alphabet of two symbols {“q”, “dq”}.

Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Salient among those propositions in the present setting are the four which single out the individual sample points at the initial moment of observation. Table 4 lists the initial state descriptions, using overlines to express logical negations.

Table 4. Initial State Descriptions
• https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-e280a2-initial-state-descriptions.png

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
#BooleanFunctions #BooleanDifferenceCalculus #CalculusOfLogicalDifferences
#PropositionalCalculus #DifferentialPropositionalCalculus #LogicalDynamics