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Figure 12: Reflexive frame representations of the lattice expansions in Figure 9.

Figure 12 Reflexive frame representations of the lattice expansions in Figure 9.

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Next is the property that if gy Fr a, then ~~ Fr ay. Assuming we have a proof from y to w, we construct a proof from =~ to -=y as follows:
Next is the property that if gy Fr a, then ~~ Fr ay. Assuming we have a proof from y to w, we construct a proof from =~ to -=y as follows:
Figure 4: Given a proof from y A wv to L, which easily yields a proof from y A wv to =v, Reiteration would permit the construction of a proof from w to 7y.
Figure 4: Given a proof from y A wv to L, which easily yields a proof from y A wv to =v, Reiteration would permit the construction of a proof from w to 7y.
Figure 6: Properties of six types of negation.
Figure 6: Properties of six types of negation.
Figure 8: Ns equipped with a pseudocomplementation (left), a weak pseudocomplementation (middle), and a protocomplementation (right), indicated by dashed arrows. Arrows for =0 = 1 and —1 = 0 are omitted. Figure 8 shows the Ns; lattice equipped with a pseudocomplementation that is not an orthocomplemen- tation (left), a weak pseudocomplementation that is neither a pseudocomplementation nor an orthocomple- mentation (middle), and a protocomplementation that is not a weak pseudocomplementation (right). Figure 9 shows the Benzene ring Og equipped with an orthocomplementation that is not a pseudocomplementation (left) and a pseudocomplementation that is not an orthocomplementation (right). Ca a ee | 1. Se: IE: TE 7 : ae . 7 res i 1 a = a 7. a ae . a 7 ry og 7
Figure 8: Ns equipped with a pseudocomplementation (left), a weak pseudocomplementation (middle), and a protocomplementation (right), indicated by dashed arrows. Arrows for =0 = 1 and —1 = 0 are omitted. Figure 8 shows the Ns; lattice equipped with a pseudocomplementation that is not an orthocomplemen- tation (left), a weak pseudocomplementation that is neither a pseudocomplementation nor an orthocomple- mentation (middle), and a protocomplementation that is not a weak pseudocomplementation (right). Figure 9 shows the Benzene ring Og equipped with an orthocomplementation that is not a pseudocomplementation (left) and a pseudocomplementation that is not an orthocomplementation (right). Ca a ee | 1. Se: IE: TE 7 : ae . 7 res i 1 a = a 7. a ae . a 7 ry og 7
Figure 9: The Benzene ring Og equipped with an orthocomplementation (left) and pseudocomplementation (right), indicated by dashed arrows. Arrows for =0 = 1 and 71 = 0 are omitted.
Figure 9: The Benzene ring Og equipped with an orthocomplementation (left) and pseudocomplementation (right), indicated by dashed arrows. Arrows for =0 = 1 and 71 = 0 are omitted.
Thus, Fr is the logic of bounded lattices with weak pseudocomplementations. Figure 10 shows the numbers of algebras up to isomorphism of size up to 10, calculated using Mace4 (McCune 2010), for Fr, intuitionistic logic (i.e., finite distributive lattices, each of which can be equipped with a unique pseudocom- plementation), and orthologic. For comparison we also include the number of lattices and the number of pseudocomplemented lattices (i.e., lattices in which each element has a pseudocomplement). Figure 10: f(n) is the number of algebras of size n up to isomorphism in the given class.
Thus, Fr is the logic of bounded lattices with weak pseudocomplementations. Figure 10 shows the numbers of algebras up to isomorphism of size up to 10, calculated using Mace4 (McCune 2010), for Fr, intuitionistic logic (i.e., finite distributive lattices, each of which can be equipped with a unique pseudocom- plementation), and orthologic. For comparison we also include the number of lattices and the number of pseudocomplemented lattices (i.e., lattices in which each element has a pseudocomplement). Figure 10: f(n) is the number of algebras of size n up to isomorphism in the given class.
see a state that cannot be seen from w (note that the arrow between v and w is symmetric). Figure 11: Reflexive frame representations of the lattice expansions in Figure 8
see a state that cannot be seen from w (note that the arrow between v and w is symmetric). Figure 11: Reflexive frame representations of the lattice expansions in Figure 8
Then the definition of < by (a,b) < (c,d) if c € b yields the relational frame on the right of Figure 14 Figure 14: A lattice with weak pseudocomplementation (left) represented by a pseudosymmetric reflexive frame (right, with reflexive loops assumed but not shown) as in Theorem 4.24.4.
Then the definition of < by (a,b) < (c,d) if c € b yields the relational frame on the right of Figure 14 Figure 14: A lattice with weak pseudocomplementation (left) represented by a pseudosymmetric reflexive frame (right, with reflexive loops assumed but not shown) as in Theorem 4.24.4.
Figure 15: Fitch-style rules for the logic with quantifiers. Vhere [yp]? = {x € X | M,x lk, y}, an easy induction shows that [yp]? is a cq-fixpoint, and
Figure 15: Fitch-style rules for the logic with quantifiers. Vhere [yp]? = {x € X | M,x lk, y}, an easy induction shows that [yp]? is a cq-fixpoint, and
Figure 16: Introduction and elimination rules for >
Figure 16: Introduction and elimination rules for >
Figure 17: Modus Ponens (left) and another way to introduce negation (right). In the context of the proof system with the rules of Figure 16, one might argue for another way of introducing negation: if a hypothetical update with y leads to a contradiction, then conclude =y. Supposing we now have 1 as a primitive symbol in our language (interpreted as 0 in our lattices), this rule is shown on the right of Figure 17. Algebraically, this is just a > 0 < 7a again. With respect to our pseudosymmetric reflexive frames equipped with accessibility relations R.4 to define A — (-), the condition that A — 0 C 7A corresponds to the condition that if > y € A, then Ra(x) 4 ©.
Figure 17: Modus Ponens (left) and another way to introduce negation (right). In the context of the proof system with the rules of Figure 16, one might argue for another way of introducing negation: if a hypothetical update with y leads to a contradiction, then conclude =y. Supposing we now have 1 as a primitive symbol in our language (interpreted as 0 in our lattices), this rule is shown on the right of Figure 17. Algebraically, this is just a > 0 < 7a again. With respect to our pseudosymmetric reflexive frames equipped with accessibility relations R.4 to define A — (-), the condition that A — 0 C 7A corresponds to the condition that if > y € A, then Ra(x) 4 ©.
Related topics:
MathematicsModal LogicIntuitionistic LogicPhilosophyLogicLattice TheoryProof TheoryPhilosophy Of MathematicsPhilosophical LogicPhilosophy of Logic
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