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Logic Syllabus • 3
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Concepts
• https://oeis.org/wiki/Logic_Syllabus#Logical_concepts

#Logic #LogicSyllabus #Ampheck #BooleanDomain #BooleanFunction #BooleanValuedFunction
#DifferentialLogic #LogicalGraph #MinimalNegationOperator #MultigradeOperator
#ParametricOperator #PeircesLaw #PropositionalCalculus #SoleSufficientOperator
#TruthTable #UniverseOfDiscourse #ZerothOrderLogic

#LogicSyllabus
• https://inquiryintoinquiry.com/logic-syllabus/

This page serves as a focal node for a collection of related resources.

#Logic #LogicalOperators

#LogicalConjunction #LogicalDisjunction
#ExclusiveDisjunction #LogicalEquality
#LogicalImplication #LogicalNegation
#LogicalNAND #LogicalNNOR

#LogicalConcepts

#Ampheck #SoleSufficientOperator
#BooleanDomain #UniverseOfDiscourse
#LogicalGraph #PropositionalCalculus
#MinimalNegationOperator #TruthTable
#BooleanFunction #BooleanValuedFunction

#DifferentialPropositionalCalculus • 7.2
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

In a #UniverseOfDiscourse based on 3 #BooleanVariables \(p,q,r\) there are \(2^3=8\) #SingularPropositions. Their #VennDiagrams are shown in Figure 10.

\(\text{Figure 10. Singular Propositions} : \mathbb{B}^3 \to\mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-singular-propositions.jpg

The Figure shows the 8 venn diagrams corresponding to the 8 singular propositions on 3 variables p, q, r. There is 1 diagram at the top of the Figure, 3 in the 2nd row, 3 in the 3rd row, and 1 diagram at the bottom. These represent the singular propositions of rank 3, 2, 1, and 0, respectively.

Each of the 8 diagrams consists of a rectangular area representing the universe of discourse and 3 overlapping circular areas P, Q, R representing 3 subsets of that universe. The circular areas are labeled p, q, r respectively to indicate the subsets P, Q, R where the corresponding boolean variables equal 1.

The singular proposition of rank 3 corresponds to the boolean product or logical conjunction pqr. The function equals 1 in the intersection of the 3 circular areas and its venn diagram indicates that by coloring the corresponding cell a shade of blue, leaving the remaining area white.

The 3 singular propositionss of rank 2 are expressed, respectively, by the following three logical expressions:

1. p and r and not q.
2. q and r and not p.
3. p and q and not r.

The 3 singular propositions of rank 1 are expressed, respectively, by the following three logical expressions:

1. q and not p and not r.
2. p and not q and not r.
3. r and not p and not q.

The singular proposition of rank 0 is expressed by the logical expression “not p and not q and not r”. The corresponding venn diagram shades the region outside all 3 labeled circular areas, leaving the union of those areas white.

#DifferentialPropositionalCalculus • 7
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

In our #Model of #Propositions as #Mappings of a #UniverseOfDiscourse to a set of 2 values, in other words, #IndicatorFunctions of the form \(f:X\to\mathbb{B},\) #SingularPropositions are those singling out the #MinimalDistinctRegions of the universe, represented by single cells of the corresponding #VennDiagram.

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#ModelTheory #ProofTheory #Semiotics

#DifferentialPropositionalCalculus • 6.1
• https://inquiryintoinquiry.com/2020/03/02/differential-propositional-calculus-6/

In a #UniverseOfDiscourse based on 3 #BooleanVariables \(p, q, r\) there are \(2^3 = 8\) #PositivePropositions. Their #VennDiagrams are shown in Figure 9.

\(\text{Figure 9. Positive Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-positive-propositions.jpg

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#CactusSyntax #MinimalNegationOperators
#DiscreteDynamics #QualitativeDynamics

The Figure shows the 8 venn diagrams corresponding to the 8 positive propositions or positive boolean functions on 3 variables p, q, r. There is 1 venn diagram at the top of the Figure, 3 in the 2nd row, 3 in the 3rd row, and 1 venn diagram at the bottom. These represent the positive propositions of rank 3, 2, 1, and 0, respectively.

Each of the 8 venn diagrams consists of a rectangular area representing the universe of discourse and 3 overlapping circular areas P, Q, R representing 3 subsets of that universe. The circular areas are labeled p, q, r respectively to indicate the subsets P, Q, R where the corresponding boolean variables are equal to 1, in other words, true.

The positive proposition of rank 3 corresponds to the boolean product or logical conjunction pqr. The function equals 1 or the proposition is true in the intersection of the 3 circular areas & the venn diagram indicates that by coloring that cell a shade of blue, leaving the remaining area white.

The 3 positive propositions of rank 2 correspond to the 3 boolean products, pr, qr, pq, respectively. Their diagrams shade the intersections P∩R, Q∩R, P∩Q, respectively.

The 3 positive propositions of rank 1 are just the 3 basic propositions, p, q, r and so their diagrams shade the circular areas P, Q, R, respectively.

The positive proposition of rank 0 is just the everywhere true proposition or the constant 1 function, so its diagram shades the whole rectangular area representing the universe of discourse.

#DifferentialPropositionalCalculus • 5.2
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/

In a #UniverseOfDiscourse based on three #BooleanVariables \(p, q, r\) the #LinearPropositions take the shapes of the #VennDiagrams shown in Figure 8. Equivalent verbal & variant logical expressions are given in the next few posts.

\(\text{Figure 8. Linear Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/02/venn-diagrams-e280a2-p-q-r-e280a2-linear-propositions.jpg

#DifferentialPropositionalCalculus • 1.5
https://inquiryintoinquiry.com/2020/02/18/differential-propositional-calculus-1/

Figure 3. Back, To The Future
https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-figure-3.jpg

Figure 3 preserves the initial #UniverseOfDiscourse \(X\) and extends the basis of discussion to a set of two qualities \(\{q, \mathrm{d}q\}.\) In corresponding fashion, the initial #PropositionalCalculus is extended by means of the #EnlargedAlphabet \(\{``𝑞", ``d𝑞"\}.\)

#DifferentialPropositionalCalculus • 1.3
https://inquiryintoinquiry.com/2020/02/18/differential-propositional-calculus-1/

Figure 1 represents a #UniverseOfDiscourse \(X\) together with a basis of discussion \(\{q\}\) for expressing propositions about the contents of that universe. Once the quality \(q\) is given a name, say, the symbol \(``q",\) we have the basis for a #FormalLanguage specifically cut out for discussing \(X\) in terms of \(q.\) This language is more formally known as the #PropositionalCalculus with #Alphabet \(\{``q"\}.\)

#DifferentialPropositionalCalculus • Overview
• https://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-overview/

By way of introduction to #DifferentialLogic we begin a chapter on #DifferentialPropositionalCalculi, which augment #PropositionalCalculi with terms for describing aspects of change and difference, for example, processes taking place in a #UniverseOfDiscourse or #LogicalTransformations mapping a #SourceUniverse to a #TargetUniverse.

#Logic #LogicalGraphs #DifferentialLogic
#DiscreteDynamics #QualitativeDynamics

#UniverseOfDiscourse

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