User:Jon Awbrey/TABLE
Cactus Language
Ascii Tables
o-------------------o | | | @ | | | o-------------------o | | | o | | | | | @ | | | o-------------------o | | | a | | @ | | | o-------------------o | | | a | | o | | | | | @ | | | o-------------------o | | | a b c | | @ | | | o-------------------o | | | a b c | | o o o | | \|/ | | o | | | | | @ | | | o-------------------o | | | a b | | o---o | | | | | @ | | | o-------------------o | | | a b | | o---o | | \ / | | @ | | | o-------------------o | | | a b | | o---o | | \ / | | o | | | | | @ | | | o-------------------o | | | a b c | | o--o--o | | \ / | | \ / | | @ | | | o-------------------o | | | a b c | | o o o | | | | | | | o--o--o | | \ / | | \ / | | @ | | | o-------------------o | | | b c | | o o | | a | | | | o--o--o | | \ / | | \ / | | @ | | | o-------------------o |
Table 13. The Existential Interpretation o----o-------------------o-------------------o-------------------o | Ex | Cactus Graph | Cactus Expression | Existential | | | | | Interpretation | o----o-------------------o-------------------o-------------------o | | | | | | 1 | @ | " " | true. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | o | | | | | | | | | | 2 | @ | ( ) | untrue. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a | | | | 3 | @ | a | a. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a | | | | | o | | | | | | | | | | 4 | @ | (a) | not a. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b c | | | | 5 | @ | a b c | a and b and c. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b c | | | | | o o o | | | | | \|/ | | | | | o | | | | | | | | | | 6 | @ | ((a)(b)(c)) | a or b or c. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | | | a implies b. | | | a b | | | | | o---o | | if a then b. | | | | | | | | 7 | @ | ( a (b)) | no a sans b. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b | | | | | o---o | | a exclusive-or b. | | | \ / | | | | 8 | @ | ( a , b ) | a not equal to b. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b | | | | | o---o | | | | | \ / | | | | | o | | a if & only if b. | | | | | | | | 9 | @ | (( a , b )) | a equates with b. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b c | | | | | o--o--o | | | | | \ / | | | | | \ / | | just one false | | 10 | @ | ( a , b , c ) | out of a, b, c. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b c | | | | | o o o | | | | | | | | | | | | | o--o--o | | | | | \ / | | | | | \ / | | just one true | | 11 | @ | ((a),(b),(c)) | among a, b, c. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | | | genus a over | | | b c | | species b, c. | | | o o | | | | | a | | | | partition a | | | o--o--o | | among b & c. | | | \ / | | | | | \ / | | whole pie a: | | 12 | @ | ( a ,(b),(c)) | slices b, c. | | | | | | o----o-------------------o-------------------o-------------------o |
Table 14. The Entitative Interpretation o----o-------------------o-------------------o-------------------o | En | Cactus Graph | Cactus Expression | Entitative | | | | | Interpretation | o----o-------------------o-------------------o-------------------o | | | | | | 1 | @ | " " | untrue. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | o | | | | | | | | | | 2 | @ | ( ) | true. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a | | | | 3 | @ | a | a. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a | | | | | o | | | | | | | | | | 4 | @ | (a) | not a. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b c | | | | 5 | @ | a b c | a or b or c. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b c | | | | | o o o | | | | | \|/ | | | | | o | | | | | | | | | | 6 | @ | ((a)(b)(c)) | a and b and c. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | | | a implies b. | | | | | | | | o a | | if a then b. | | | | | | | | 7 | @ b | (a) b | not a, or b. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b | | | | | o---o | | a if & only if b. | | | \ / | | | | 8 | @ | ( a , b ) | a equates with b. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b | | | | | o---o | | | | | \ / | | | | | o | | a exclusive-or b. | | | | | | | | 9 | @ | (( a , b )) | a not equal to b. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b c | | | | | o--o--o | | | | | \ / | | | | | \ / | | not just one true | | 10 | @ | ( a , b , c ) | out of a, b, c. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a b c | | | | | o--o--o | | | | | \ / | | | | | \ / | | | | | o | | | | | | | | just one true | | 11 | @ | (( a , b , c )) | among a, b, c. | | | | | | o----o-------------------o-------------------o-------------------o | | | | | | | a | | | | | o | | genus a over | | | | b c | | species b, c. | | | o--o--o | | | | | \ / | | partition a | | | \ / | | among b & c. | | | o | | | | | | | | whole pie a: | | 12 | @ | (((a), b , c )) | slices b, c. | | | | | | o----o-------------------o-------------------o-------------------o |
Table 15. Existential & Entitative Interpretations of Cactus Structures o-----------------o-----------------o-----------------o-----------------o | Cactus Graph | Cactus String | Existential | Entitative | | | | Interpretation | Interpretation | o-----------------o-----------------o-----------------o-----------------o | | | | | | @ | " " | true | false | | | | | | o-----------------o-----------------o-----------------o-----------------o | | | | | | o | | | | | | | | | | | @ | ( ) | false | true | | | | | | o-----------------o-----------------o-----------------o-----------------o | | | | | | C_1 ... C_k | | | | | @ | C_1 ... C_k | C_1 & ... & C_k | C_1 v ... v C_k | | | | | | o-----------------o-----------------o-----------------o-----------------o | | | | | | C_1 C_2 C_k | | Just one | Not just one | | o---o-...-o | | | | | \ / | | of the C_j, | of the C_j, | | \ / | | | | | \ / | | j = 1 to k, | j = 1 to k, | | \ / | | | | | @ | (C_1, ..., C_k) | is not true. | is true. | | | | | | o-----------------o-----------------o-----------------o-----------------o |
o-----------------o-----------------o-----------------o-----------------o | Graph | String | Entitative | Existential | o-----------------o-----------------o-----------------o-----------------o | | | | | | @ | " " | untrue. | true. | o-----------------o-----------------o-----------------o-----------------o | | | | | | o | | | | | | | | | | | @ | ( ) | true. | untrue. | o-----------------o-----------------o-----------------o-----------------o | | | | | | r | | | | | @ | r | r. | r. | o-----------------o-----------------o-----------------o-----------------o | | | | | | r | | | | | o | | | | | | | | | | | @ | (r) | not r. | not r. | o-----------------o-----------------o-----------------o-----------------o | | | | | | r s t | | | | | @ | r s t | r or s or t. | r and s and t. | o-----------------o-----------------o-----------------o-----------------o | | | | | | r s t | | | | | o o o | | | | | \|/ | | | | | o | | | | | | | | | | | @ | ((r)(s)(t)) | r and s and t. | r or s or t. | o-----------------o-----------------o-----------------o-----------------o | | | | r implies s. | | | | | | | o r | | | if r then s. | | | | | | | | @ s | (r) s | not r, or s | no r sans s. | o-----------------o-----------------o-----------------o-----------------o | | | | r implies s. | | r s | | | | | o---o | | | if r then s. | | | | | | | | @ | (r (s)) | | no r sans s. | o-----------------o-----------------o-----------------o-----------------o | | | | | | r s | | | | | o---o | | |r exclusive-or s.| | \ / | | | | | @ | (r , s) | |r not equal to s.| o-----------------o-----------------o-----------------o-----------------o | | | | | | r s | | | | | o---o | | | | | \ / | | | | | o | | |r if & only if s.| | | | | | | | @ | ((r , s)) | |r equates with s.| o-----------------o-----------------o-----------------o-----------------o | | | | | | r s t | | | | | o--o--o | | | | | \ / | | | | | \ / | | | just one false | | @ | (r , s , t) | | out of r, s, t. | o-----------------o-----------------o-----------------o-----------------o | | | | | | r s t | | | | | o o o | | | | | | | | | | | | | o--o--o | | | | | \ / | | | | | \ / | | | just one true | | @ | ((r),(s),(t)) | | among r, s, t. | o-----------------o-----------------o-----------------o-----------------o | | | | genus t over | | r s | | | species r, s. | | o o | | | | | t | | | | | partition t | | o--o--o | | | among r & s. | | \ / | | | | | \ / | | | whole pie t: | | @ | ( t ,(r),(s)) | | slices r, s. | o-----------------o-----------------o-----------------o-----------------o |
Wiki TeX Tables
\(\text{Cactus Graph}\!\) | \(\text{Cactus Expression}\!\) | \(\text{Interpretation}\!\) |
o-------------------o | | | @ | | | o-------------------o |
\({}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}\) | \(\operatorname{true}.\) |
o-------------------o | | | o | | | | | @ | | | o-------------------o |
\(\texttt{(~)}\) | \(\operatorname{false}.\) |
o-------------------o | | | a | | @ | | | o-------------------o |
\(a\!\) | \(a.\!\) |
o-------------------o | | | a | | o | | | | | @ | | | o-------------------o |
\(\texttt{(} a \texttt{)}\) |
\(\begin{matrix} \tilde{a} \'"`UNIQ-MathJax1-QINU`"' '''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd. '"`UNIQ--pre-00000040-QINU`"' '"`UNIQ--pre-00000041-QINU`"' '"`UNIQ--pre-00000042-QINU`"' '"`UNIQ-MathJax2-QINU`"' ===='"`UNIQ--h-38--QINU`"'[[Logical implication]]==== The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false. The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical Implication''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ⇒ q |- | F || F || T |- | F || T || T |- | T || F || F |- | T || T || T |} <br> ===='"`UNIQ--h-39--QINU`"'[[Logical NAND]]==== The '''NAND operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are true. In other words, it produces a value of ''true'' if and only if at least one of its operands is false. The [[truth table]] of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical NAND''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ↑ q |- | F || F || T |- | F || T || T |- | T || F || T |- | T || T || F |} <br> ===='"`UNIQ--h-40--QINU`"'[[Logical NNOR]]==== The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' if and only if both of its operands are false. In other words, it produces a value of ''false'' if and only if at least one of its operands is true. The [[truth table]] of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:mintcream; font-weight:bold; text-align:center; width:45%" |+ '''Logical NOR''' |- style="background:aliceblue" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ↓ q |- | F || F || T |- | F || T || F |- | T || F || F |- | T || T || F |} <br> =='"`UNIQ--h-41--QINU`"'Relational Tables== ==='"`UNIQ--h-42--QINU`"'Factorization=== {| align="center" style="text-align:center; width:60%" | {| align="center" style="text-align:center; width:100%" | \(\text{Table 7. Plural Denotation}\!\) |
|- |
\(\text{Object}\!\) | \(\text{Sign}\!\) | \(\text{Interpretant}\!\) |
\(\begin{matrix} o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots \end{matrix}\) |
\(\begin{matrix} s \\ s \\ s \\ \ldots \\ s \\ \ldots \end{matrix}\) |
\(\begin{matrix} \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \end{matrix}\) |
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Sign Relations
O | = | Object Domain | |
S | = | Sign Domain | |
I | = | Interpretant Domain |
O | = | {Ann, Bob} | = | {A, B} | |
S | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} | |
I | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} |
Object | Sign | Interpretant |
---|---|---|
A | "A" | "A" |
A | "A" | "i" |
A | "i" | "A" |
A | "i" | "i" |
B | "B" | "B" |
B | "B" | "u" |
B | "u" | "B" |
B | "u" | "u" |
Object | Sign | Interpretant |
---|---|---|
A | "A" | "A" |
A | "A" | "u" |
A | "u" | "A" |
A | "u" | "u" |
B | "B" | "B" |
B | "B" | "i" |
B | "i" | "B" |
B | "i" | "i" |
Triadic Relations
Algebraic Examples
X | Y | Z |
---|---|---|
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
X | Y | Z |
---|---|---|
0 | 0 | 1 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
Semiotic Examples
Object | Sign | Interpretant |
---|---|---|
A | "A" | "A" |
A | "A" | "i" |
A | "i" | "A" |
A | "i" | "i" |
B | "B" | "B" |
B | "B" | "u" |
B | "u" | "B" |
B | "u" | "u" |
Object | Sign | Interpretant |
---|---|---|
A | "A" | "A" |
A | "A" | "u" |
A | "u" | "A" |
A | "u" | "u" |
B | "B" | "B" |
B | "B" | "i" |
B | "i" | "B" |
B | "i" | "i" |
Dyadic Projections
LOS | = | projOS(L) | = | { (o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I } | |
LSO | = | projSO(L) | = | { (s, o) ∈ S × O : (o, s, i) ∈ L for some i ∈ I } | |
LIS | = | projIS(L) | = | { (i, s) ∈ I × S : (o, s, i) ∈ L for some o ∈ O } | |
LSI | = | projSI(L) | = | { (s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O } | |
LOI | = | projOI(L) | = | { (o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S } | |
LIO | = | projIO(L) | = | { (i, o) ∈ I × O : (o, s, i) ∈ L for some s ∈ S } |
Method 1 : Subtitles as Captions
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Method 2 : Subtitles as Top Rows
projOS(LA)
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projOS(LB)
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projSI(LA)
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projSI(LB)
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projOI(LA)
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projOI(LB)
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Relation Reduction
Method 1 : Subtitles as Captions
X | Y | Z |
---|---|---|
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
X | Y | Z |
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0 | 0 | 1 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
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projXY(L0) = projXY(L1) | projXZ(L0) = projXZ(L1) | projYZ(L0) = projYZ(L1) |
Object | Sign | Interpretant |
---|---|---|
A | "A" | "A" |
A | "A" | "i" |
A | "i" | "A" |
A | "i" | "i" |
B | "B" | "B" |
B | "B" | "u" |
B | "u" | "B" |
B | "u" | "u" |
Object | Sign | Interpretant |
---|---|---|
A | "A" | "A" |
A | "A" | "u" |
A | "u" | "A" |
A | "u" | "u" |
B | "B" | "B" |
B | "B" | "i" |
B | "i" | "B" |
B | "i" | "i" |
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projXY(LA) ≠ projXY(LB) | projXZ(LA) ≠ projXZ(LB) | projYZ(LA) ≠ projYZ(LB) |
Method 2 : Subtitles as Top Rows
X | Y | Z |
---|---|---|
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
X | Y | Z |
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0 | 0 | 1 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
projXY(L0)
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projXZ(L0)
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projYZ(L0)
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projXY(L1)
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projXZ(L1)
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projYZ(L1)
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projXY(L0) = projXY(L1) | projXZ(L0) = projXZ(L1) | projYZ(L0) = projYZ(L1) |
Object | Sign | Interpretant |
---|---|---|
A | "A" | "A" |
A | "A" | "i" |
A | "i" | "A" |
A | "i" | "i" |
B | "B" | "B" |
B | "B" | "u" |
B | "u" | "B" |
B | "u" | "u" |
Object | Sign | Interpretant |
---|---|---|
A | "A" | "A" |
A | "A" | "u" |
A | "u" | "A" |
A | "u" | "u" |
B | "B" | "B" |
B | "B" | "i" |
B | "i" | "B" |
B | "i" | "i" |
projXY(LA)
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projXZ(LA)
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projYZ(LA)
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projXY(LB)
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projXZ(LB)
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projYZ(LB)
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projXY(LA) ≠ projXY(LB) | projXZ(LA) ≠ projXZ(LB) | projYZ(LA) ≠ projYZ(LB) |
Formatted Text Display
- So in a triadic fact, say, the example
A gives B to C |
- we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object. We say that the proposition has three logical subjects. We regard it as a mere affair of English grammar that there are six ways of expressing this:
A gives B to C | A benefits C with B |
B enriches C at expense of A | C receives B from A |
C thanks A for B | B leaves A for C |
- These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).
Work Area
x0 | x1 | 2f0 | 2f1 | 2f2 | 2f3 | 2f4 | 2f5 | 2f6 | 2f7 | 2f8 | 2f9 | 2f10 | 2f11 | 2f12 | 2f13 | 2f14 | 2f15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Draft 1
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Draft 2
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Inquiry and Analogy
Test Patterns
1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Table 10
\(x\): | 1 0 | \(f\) | \(m_0\) | \(m_1\) | \(m_2\) | \(m_3\) | \(m_4\) | \(m_5\) | \(m_6\) | \(m_7\) | \(m_8\) | \(m_9\) | \(m_{10}\) | \(m_{11}\) | \(m_{12}\) | \(m_{13}\) | \(m_{14}\) | \(m_{15}\) |
\(f_0\) | 0 0 | \(0\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
\(f_1\) | 0 1 | \((x)\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
\(f_2\) | 1 0 | \(x\!\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
\(f_3\) | 1 1 | \(1\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
\(x:\) | 1 0 | \(f\!\) | \(m_0\) | \(m_1\) | \(m_2\) | \(m_3\) | \(m_4\) | \(m_5\) | \(m_6\) | \(m_7\) | \(m_8\) | \(m_9\) | \(m_{10}\) | \(m_{11}\) | \(m_{12}\) | \(m_{13}\) | \(m_{14}\) | \(m_{15}\) |
\(f_0\) | 0 0 | \(0\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
\(f_1\) | 0 1 | \((x)\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
\(f_2\) | 1 0 | \(x\!\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
\(f_3\) | 1 1 | \(1\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Table 11
Measure | Happening | Exactness | Existence | Linearity | Uniformity | Information |
\(m_0\!\) | Nothing happens | |||||
\(m_1\!\) | Just false | Nothing exists | ||||
\(m_2\!\) | Just not \(x\!\) | |||||
\(m_3\!\) | Nothing is \(x\!\) | |||||
\(m_4\!\) | Just \(x\!\) | |||||
\(m_5\!\) | Everything is \(x\!\) | \(f\!\) is linear | ||||
\(m_6\!\) | \(f\!\) is not uniform | \(f\!\) is informed | ||||
\(m_7\!\) | Not just true | |||||
\(m_8\!\) | Just true | |||||
\(m_9\!\) | \(f\!\) is uniform | \(f\!\) is not informed | ||||
\(m_{10}\!\) | Something is not \(x\!\) | \(f\!\) is not linear | ||||
\(m_{11}\!\) | Not just \(x\!\) | |||||
\(m_{12}\!\) | Something is \(x\!\) | |||||
\(m_{13}\!\) | Not just not \(x\!\) | |||||
\(m_{14}\!\) | Not just false | Something exists | ||||
\(m_{15}\!\) | Anything happens |
Table 12
\(x:\) \(y:\) |
1100 1010 |
\(f\!\) | \(m_0\) | \(m_1\) | \(m_2\) | \(m_3\) | \(m_4\) | \(m_5\) | \(m_6\) | \(m_7\) | \(m_8\) | \(m_9\) | \(m_{10}\) | \(m_{11}\) | \(m_{12}\) | \(m_{13}\) | \(m_{14}\) | \(m_{15}\) | \(m_{16}\) | \(m_{17}\) | \(m_{18}\) | \(m_{19}\) | \(m_{20}\) | \(m_{21}\) | \(m_{22}\) | \(m_{23}\) |
\(f_0\) | 0000 | \((~)\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
\(f_1\) | 0001 | \((x)(y)\!\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ||
\(f_2\) | 0010 | \((x) y\!\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | ||||
\(f_3\) | 0011 | \((x)\!\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||
\(f_4\) | 0100 | \(x (y)\!\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||||||||
\(f_5\) | 0101 | \((y)\!\) | ||||||||||||||||||||||||
\(f_6\) | 0110 | \((x, y)\!\) | ||||||||||||||||||||||||
\(f_7\) | 0111 | \((x y)\!\) | ||||||||||||||||||||||||
\(f_8\) | 1000 | \(x y\!\) | ||||||||||||||||||||||||
\(f_9\) | 1001 | \(((x, y))\!\) | ||||||||||||||||||||||||
\(f_{10}\) | 1010 | \(y\!\) | ||||||||||||||||||||||||
\(f_{11}\) | 1011 | \((x (y))\!\) | ||||||||||||||||||||||||
\(f_{12}\) | 1100 | \(x\!\) | ||||||||||||||||||||||||
\(f_{13}\) | 1101 | \(((x) y)\!\) | ||||||||||||||||||||||||
\(f_{14}\) | 1110 | \(((x)(y))\!\) | ||||||||||||||||||||||||
\(f_{15}\) | 1111 | \(((~))\!\) |
\(u:\) \(v:\) |
1100 1010 |
\(f\!\) | \(m_0\) | \(m_1\) | \(m_2\) | \(m_3\) | \(m_4\) | \(m_5\) | \(m_6\) | \(m_7\) | \(m_8\) | \(m_9\) | \(m_{10}\) | \(m_{11}\) | \(m_{12}\) | \(m_{13}\) | \(m_{14}\) | \(m_{15}\) | \(m_{16}\) | \(m_{17}\) | \(m_{18}\) | \(m_{19}\) | \(m_{20}\) | \(m_{21}\) | \(m_{22}\) | \(m_{23}\) |
\(f_0\) | 0000 | \((~)\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
\(f_1\) | 0001 | \((u)(v)\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
\(f_2\) | 0010 | \((u) v\!\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
\(f_3\) | 0011 | \((u)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_4\) | 0100 | \(u (v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
\(f_5\) | 0101 | \((v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_6\) | 0110 | \((u, v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_7\) | 0111 | \((u v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_8\) | 1000 | \(u v\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_9\) | 1001 | \(((u, v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_{10}\) | 1010 | \(v\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_{11}\) | 1011 | \((u (v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_{12}\) | 1100 | \(u\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_{13}\) | 1101 | \(((u) v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_{14}\) | 1110 | \(((u)(v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_{15}\) | 1111 | \(((~))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Table 13
\(u:\) \(v:\) |
1100 1010 |
\(f\!\) | \(\alpha_0\) | \(\alpha_1\) | \(\alpha_2\) | \(\alpha_3\) | \(\alpha_4\) | \(\alpha_5\) | \(\alpha_6\) | \(\alpha_7\) | \(\alpha_8\) | \(\alpha_9\) | \(\alpha_{10}\) | \(\alpha_{11}\) | \(\alpha_{12}\) | \(\alpha_{13}\) | \(\alpha_{14}\) | \(\alpha_{15}\) |
\(f_0\) | 0000 | \((~)\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_1\) | 0001 | \((u)(v)\!\) | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_2\) | 0010 | \((u) v\!\) | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_3\) | 0011 | \((u)\!\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_4\) | 0100 | \(u (v)\!\) | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_5\) | 0101 | \((v)\!\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_6\) | 0110 | \((u, v)\!\) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_7\) | 0111 | \((u v)\!\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_8\) | 1000 | \(u v\!\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_9\) | 1001 | \(((u, v))\!\) | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
\(f_{10}\) | 1010 | \(v\!\) | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
\(f_{11}\) | 1011 | \((u (v))\!\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
\(f_{12}\) | 1100 | \(u\!\) | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
\(f_{13}\) | 1101 | \(((u) v)\!\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
\(f_{14}\) | 1110 | \(((u)(v))\!\) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
\(f_{15}\) | 1111 | \(((~))\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Table 14
\(u:\) \(v:\) |
1100 1010 |
\(f\!\) | \(\beta_0\) | \(\beta_1\) | \(\beta_2\) | \(\beta_3\) | \(\beta_4\) | \(\beta_5\) | \(\beta_6\) | \(\beta_7\) | \(\beta_8\) | \(\beta_9\) | \(\beta_{10}\) | \(\beta_{11}\) | \(\beta_{12}\) | \(\beta_{13}\) | \(\beta_{14}\) | \(\beta_{15}\) |
\(f_0\) | 0000 | \((~)\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
\(f_1\) | 0001 | \((u)(v)\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
\(f_2\) | 0010 | \((u) v\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
\(f_3\) | 0011 | \((u)\!\) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
\(f_4\) | 0100 | \(u (v)\!\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
\(f_5\) | 0101 | \((v)\!\) | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
\(f_6\) | 0110 | \((u, v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
\(f_7\) | 0111 | \((u v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
\(f_8\) | 1000 | \(u v\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
\(f_9\) | 1001 | \(((u, v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
\(f_{10}\) | 1010 | \(v\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
\(f_{11}\) | 1011 | \((u (v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
\(f_{12}\) | 1100 | \(u\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
\(f_{13}\) | 1101 | \(((u) v)\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
\(f_{14}\) | 1110 | \(((u)(v))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
\(f_{15}\) | 1111 | \(((~))\!\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Figure 15
Table 16
\(\begin{array}{clcl} \mathrm{A} & \mathrm{Universal~Affirmative} & \mathrm{All}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u (v) = 0 \\ \mathrm{E} & \mathrm{Universal~Negative} & \mathrm{All}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u \cdot v = 0 \\ \mathrm{I} & \mathrm{Particular~Affirmative} & \mathrm{Some}\ u\ \mathrm{is}\ v & \mathrm{Indicator~of}\ u \cdot v = 1 \\ \mathrm{O} & \mathrm{Particular~Negative} & \mathrm{Some}\ u\ \mathrm{is}\ (v) & \mathrm{Indicator~of}\ u (v) = 1 \\ \end{array}\) |
Table 17
\(u:\) \(v:\) |
1100 1010 |
\(f\!\) | \((\ell_{11})\) \(\text{No } u \) \(\text{is } v \) |
\((\ell_{10})\) \(\text{No } u \) \(\text{is }(v)\) |
\((\ell_{01})\) \(\text{No }(u)\) \(\text{is } v \) |
\((\ell_{00})\) \(\text{No }(u)\) \(\text{is }(v)\) |
\( \ell_{00} \) \(\text{Some }(u)\) \(\text{is }(v)\) |
\( \ell_{01} \) \(\text{Some }(u)\) \(\text{is } v \) |
\( \ell_{10} \) \(\text{Some } u \) \(\text{is }(v)\) |
\( \ell_{11} \) \(\text{Some } u \) \(\text{is } v \) |
\(f_0\) | 0000 | \((~)\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
\(f_1\) | 0001 | \((u)(v)\!\) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
\(f_2\) | 0010 | \((u) v\!\) | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
\(f_3\) | 0011 | \((u)\!\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
\(f_4\) | 0100 | \(u (v)\!\) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
\(f_5\) | 0101 | \((v)\!\) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
\(f_6\) | 0110 | \((u, v)\!\) | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
\(f_7\) | 0111 | \((u v)\!\) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
\(f_8\) | 1000 | \(u v\!\) | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
\(f_9\) | 1001 | \(((u, v))\!\) | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
\(f_{10}\) | 1010 | \(v\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
\(f_{11}\) | 1011 | \((u (v))\!\) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
\(f_{12}\) | 1100 | \(u\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
\(f_{13}\) | 1101 | \(((u) v)\!\) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
\(f_{14}\) | 1110 | \(((u)(v))\!\) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
\(f_{15}\) | 1111 | \(((~))\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
Table 18
\(u:\) \(v:\) |
1100 1010 |
\(f\!\) | \((\ell_{11})\) \(\text{No } u \) \(\text{is } v \) |
\((\ell_{10})\) \(\text{No } u \) \(\text{is }(v)\) |
\((\ell_{01})\) \(\text{No }(u)\) \(\text{is } v \) |
\((\ell_{00})\) \(\text{No }(u)\) \(\text{is }(v)\) |
\( \ell_{00} \) \(\text{Some }(u)\) \(\text{is }(v)\) |
\( \ell_{01} \) \(\text{Some }(u)\) \(\text{is } v \) |
\( \ell_{10} \) \(\text{Some } u \) \(\text{is }(v)\) |
\( \ell_{11} \) \(\text{Some } u \) \(\text{is } v \) |
\(f_0\) | 0000 | \((~)\) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
\(f_1\) | 0001 | \((u)(v)\!\) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
\(f_2\) | 0010 | \((u) v\!\) | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
\(f_4\) | 0100 | \(u (v)\!\) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
\(f_8\) | 1000 | \(u v\!\) | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
\(f_3\) | 0011 | \((u)\!\) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
\(f_{12}\) | 1100 | \(u\!\) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
\(f_6\) | 0110 | \((u, v)\!\) | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
\(f_9\) | 1001 | \(((u, v))\!\) | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
\(f_5\) | 0101 | \((v)\!\) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
\(f_{10}\) | 1010 | \(v\!\) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
\(f_7\) | 0111 | \((u v)\!\) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
\(f_{11}\) | 1011 | \((u (v))\!\) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
\(f_{13}\) | 1101 | \(((u) v)\!\) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
\(f_{14}\) | 1110 | \(((u)(v))\!\) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
\(f_{15}\) | 1111 | \(((~))\) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
Table 19
\(\text{Mnemonic}\) | \(\text{Category}\) | \(\text{Classical Form}\) | \(\text{Alternate Form}\) | \(\text{Symmetric Form}\) | \(\text{Operator}\) |
\(\text{E}\!\) \(\text{Exclusive}\) |
\(\text{Universal}\) \(\text{Negative}\) |
\(\text{All}\ u\ \text{is}\ (v)\) | \(\text{No}\ u\ \text{is}\ v \) | \((\ell_{11})\) | |
\(\text{A}\!\) \(\text{Absolute}\) |
\(\text{Universal}\) \(\text{Affirmative}\) |
\(\text{All}\ u\ \text{is}\ v \) | \(\text{No}\ u\ \text{is}\ (v)\) | \((\ell_{10})\) | |
\(\text{All}\ v\ \text{is}\ u \) | \(\text{No}\ v\ \text{is}\ (u)\) | \(\text{No}\ (u)\ \text{is}\ v \) | \((\ell_{01})\) | ||
\(\text{All}\ (v)\ \text{is}\ u \) | \(\text{No}\ (v)\ \text{is}\ (u)\) | \(\text{No}\ (u)\ \text{is}\ (v)\) | \((\ell_{00})\) | ||
\(\text{Some}\ (u)\ \text{is}\ (v)\) | \(\text{Some}\ (u)\ \text{is}\ (v)\) | \(\ell_{00}\!\) | |||
\(\text{Some}\ (u)\ \text{is}\ v\) | \(\text{Some}\ (u)\ \text{is}\ v\) | \(\ell_{01}\!\) | |||
\(\text{O}\!\) \(\text{Obtrusive}\) |
\(\text{Particular}\) \(\text{Negative}\) |
\(\text{Some}\ u\ \text{is}\ (v)\) | \(\text{Some}\ u\ \text{is}\ (v)\) | \(\ell_{10}\!\) | |
\(\text{I}\!\) \(\text{Indefinite}\) |
\(\text{Particular}\) \(\text{Affirmative}\) |
\(\text{Some}\ u\ \text{is}\ v\) | \(\text{Some}\ u\ \text{is}\ v\) | \(\ell_{11}\!\) |