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18The empty class is the only class simultaneously “included in” and “excluded from” every conceivable class that is, bCe and bc if and only if b = @. Proof:

Figure 2 18The empty class is the only class simultaneously “included in” and “excluded from” every conceivable class that is, bCe and bc if and only if b = @. Proof:

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The “smallest” independent regions in a Venn diagram (1880) are called minterms. Each minterm is either inhabited or empty, that is, it either has or does not have at east one element. Figure 1 shows the Venn diagram possibilities involving either an inhabitation or an emptiness mark for each minterm inside a universe of discourse!? I with one specially designated subclass/term b. Here the minterms happen to be b itself and its complement, b’. When we do not know whether a minterm is inhabited or empty, we leave it blank, in order to indicate lack of information on our part.
The “smallest” independent regions in a Venn diagram (1880) are called minterms. Each minterm is either inhabited or empty, that is, it either has or does not have at east one element. Figure 1 shows the Venn diagram possibilities involving either an inhabitation or an emptiness mark for each minterm inside a universe of discourse!? I with one specially designated subclass/term b. Here the minterms happen to be b itself and its complement, b’. When we do not know whether a minterm is inhabited or empty, we leave it blank, in order to indicate lack of information on our part.
Figure 4.: A logical hexagon of opposition modelling the interaction between comple mentary classes and their shared border. The interaction between complementary classes and their shared border can be mod- elled by a logical hexagon of opposition — Figure 4, loosely inspired by Béziau (2012, pp. 38-39, picture 53).
Figure 4.: A logical hexagon of opposition modelling the interaction between comple mentary classes and their shared border. The interaction between complementary classes and their shared border can be mod- elled by a logical hexagon of opposition — Figure 4, loosely inspired by Béziau (2012, pp. 38-39, picture 53).
Figure 5.: Emptiness and inhabitation markers for a minterm in a universe class that has two generating terms.
Figure 5.: Emptiness and inhabitation markers for a minterm in a universe class that has two generating terms.
Figure 6.: Proofs from Piesk (2017) of syllogisms by means of Euler diagrams (part 1 of 3). 4. Algebraic and relational representations
Figure 6.: Proofs from Piesk (2017) of syllogisms by means of Euler diagrams (part 1 of 3). 4. Algebraic and relational representations
The corresponding categorical relations are mutually connected by the De Morgan’s
The corresponding categorical relations are mutually connected by the De Morgan’s
Table 1.: Algebraic and relational representations of categorical relations. These converse representations can be employed to transform categorical syllogisms in De Morgan’s syllogistic into “perfect”, composition-friendly syllogisms in the “first” figure.
Table 1.: Algebraic and relational representations of categorical relations. These converse representations can be employed to transform categorical syllogisms in De Morgan’s syllogistic into “perfect”, composition-friendly syllogisms in the “first” figure.
Figure 7.: A sublattice of the Boolean lattice generated by the atoms {b’c’, b'c, bc’, bc} where complementation is not employed.
Figure 7.: A sublattice of the Boolean lattice generated by the atoms {b’c’, b'c, bc’, bc} where complementation is not employed.
“The subject inhabitation (LC6) lemma, convenient for the proof of this mood-figure pair, requires the axioms of predicative inhabitation (LC3) and commutativity (LC1). Table 4 summarizes the axioms that have been adopted in the proof of each valid categorical syllogism in the preceding subsection.
“The subject inhabitation (LC6) lemma, convenient for the proof of this mood-figure pair, requires the axioms of predicative inhabitation (LC3) and commutativity (LC1). Table 4 summarizes the axioms that have been adopted in the proof of each valid categorical syllogism in the preceding subsection.
Thus, we have shown that the ML axioms are sufficiently powerful for proving all 24 classic categorical syllogisms.
Thus, we have shown that the ML axioms are sufficiently powerful for proving all 24 classic categorical syllogisms.
“The Disamis-3 (ML6) lemma, convenient for the proof of this mood-figure pair, requires the axioms Darii- (ML5) and I-conversion (ML1). Table 5.: Syllogism-axiom matrix for the McColl-Ladd system.
“The Disamis-3 (ML6) lemma, convenient for the proof of this mood-figure pair, requires the axioms Darii- (ML5) and I-conversion (ML1). Table 5.: Syllogism-axiom matrix for the McColl-Ladd system.
Does LC |= ML? In other words, can we derive all ML axioms from LC axioms? Let’s derive each individual ML axiom from LC.
Does LC |= ML? In other words, can we derive all ML axioms from LC axioms? Let’s derive each individual ML axiom from LC.
Figure 9.: Euler diagram representing the relations between the Boolean lattice, Leibniz-Cayley, and McColl-Ladd axiomatic systems in terms of the set of theorems that can be proved in them.
Figure 9.: Euler diagram representing the relations between the Boolean lattice, Leibniz-Cayley, and McColl-Ladd axiomatic systems in terms of the set of theorems that can be proved in them.
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History of LogicAristotelian LogicDIGITAL LOGIC DESIGNDiagrammatic LogicFirst-Order LogicDigital LogicFirst Order LogicLogic GatesHistory and Philosophy of LogicLogic Gate
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