From 9acf6ca60ec8355b99f620ddc8405f2fe5ff2c08 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Thu, 18 Jul 2024 20:13:52 -0600 Subject: [PATCH] Beta-equality and dialetheism. --- .../plugins/obsidian-to-anki-plugin/data.json | 19 +- notes/_journal/2024-07-18.md | 12 + notes/_journal/{ => 2024-07}/2024-07-17.md | 0 notes/algebra/set.md | 6 +- notes/lambda-calculus/alpha-conversion.md | 8 + notes/lambda-calculus/beta-reduction.md | 144 ++++++++- notes/logic/equality.md | 15 - notes/logic/equiv-trans.md | 170 ----------- notes/logic/pred-logic.md | 257 +++++++++++++++- notes/logic/prop-logic.md | 281 +++++++++++++++++- notes/logic/quantification.md | 268 ----------------- notes/ontology/dialetheism.md | 31 ++ notes/set/graphs.md | 2 +- 13 files changed, 734 insertions(+), 479 deletions(-) create mode 100644 notes/_journal/2024-07-18.md rename notes/_journal/{ => 2024-07}/2024-07-17.md (100%) delete mode 100644 notes/logic/equality.md delete mode 100644 notes/logic/quantification.md create mode 100644 notes/ontology/dialetheism.md diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index bbdbcb4..e242293 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -183,7 +183,7 @@ "_journal/2024-02-02.md": "a3b222daee8a50bce4cbac699efc7180", "_journal/2024-02-01.md": "3aa232387d2dc662384976fd116888eb", "_journal/2024-01-31.md": "7c7fbfccabc316f9e676826bf8dfe970", - "logic/equiv-trans.md": "64b6110fdfb3d2c254de6d2dc6d7f312", + "logic/equiv-trans.md": "f910dc13cb20db291b1d1241e8046bee", "_journal/2024-02-07.md": "8d81cd56a3b33883a7706d32e77b5889", "algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2", "algorithms/loop-invariant.md": "3b390e720f3b2a98e611b49a0bb1f5a9", @@ -325,7 +325,7 @@ "_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b", "set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53", "set/index.md": "ea5e92b9792a8e093bac259f85f1f829", - "set/graphs.md": "55298be7241906cb6b61673cf0a2e709", + "set/graphs.md": "1a0c09f643829dae6a101b96de31eb40", "_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb", "_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391", "awk/variables.md": "e40a20545358228319f789243d8b9f77", @@ -456,8 +456,8 @@ "_journal/2024-05/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5", "_journal/2024-05-16.md": "580c7ec61ec56be92fa8d6affcf0a5f6", "_journal/2024-05/2024-05-15.md": "4e6a7e6df32e93f0d8a56bc76613d908", - "logic/pred-logic.md": "c23c3da8756ac0ef17b9710a67440d84", - "logic/prop-logic.md": "5f20f5c27c7b59c59fc125ba78e37bd8", + "logic/pred-logic.md": "a709cb45e7554ffc578cba0eb1e86e57", + "logic/prop-logic.md": "daa927c5eb31813fcf21bcb18ba0f1ec", "_journal/2024-05-17.md": "fb880d68077b655ede36d994554f3aba", "_journal/2024-05/2024-05-16.md": "9fdfadc3f9ea6a4418fd0e7066d6b10c", "_journal/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c", @@ -485,7 +485,7 @@ "_journal/2024-05/2024-05-25.md": "3e8a0061fa58a6e5c48d12800d1ab869", "_journal/2024-05-27.md": "b36636d10eab34380f17f288868df3ae", "_journal/2024-05/2024-05-26.md": "abe84b5beae74baa25501c818e64fc95", - "algebra/set.md": "ecf6aef8bc64fc14a73178adcdd3594e", + "algebra/set.md": "843b092a980133ba7bd61b1750a6df08", "algebra/boolean.md": "ee41e624f4d3d3aca00020d9a9ae42c8", "git/merge-conflicts.md": "761ad6137ec51d3877f7d5b3615ca5cb", "_journal/2024-05-28.md": "0f6aeb5ec126560acdc2d8c5c6570337", @@ -515,7 +515,7 @@ "_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381", "_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8", "_journal/2024-06/2024-06-07.md": "c6bfc4c1e5913d23ea7828a23340e7d3", - "lambda-calculus/alpha-conversion.md": "c9b7d60602e13e8a60d8784a859d4655", + "lambda-calculus/alpha-conversion.md": "007828faf9b4ace5bd30b87a36a90dcf", "lambda-calculus/index.md": "64efe9e4f6036d3f5b4ec0dc8cd3e7b9", "x86-64/instructions/condition-codes.md": "efb0a3244139e91461c1b327d897206f", "x86-64/instructions/logical.md": "818428b9ef84753920dc61e5c2de9199", @@ -539,7 +539,7 @@ "set/functions.md": "b41c04a596a7e711801c32eff9333a3e", "_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4", "_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307", - "lambda-calculus/beta-reduction.md": "e40e64be380bbe7852cfe6a310e400bf", + "lambda-calculus/beta-reduction.md": "6c9a9f4983b0974e0184acaee7c27a22", "_journal/2024-06-16.md": "ded6ab660ecc7c3dce3afd2e88e5a725", "_journal/2024-06/2024-06-15.md": "c3a55549da9dfc2770bfcf403bf5b30b", "_journal/2024-06-17.md": "63df6757bb3384e45093bf2b9456ffac", @@ -610,7 +610,10 @@ "_journal/2024-07/2024-07-15.md": "462fb4294cbbe8855071c638351df147", "ontology/nominalism.md": "46245c644238157e15c7cb6def27d90a", "_journal/2024-07-17.md": "e0371a91e99f131e7258cc82c2a04cc8", - "_journal/2024-07/2024-07-16.md": "149222eab7a7f58993b8e4dc8a3fb884" + "_journal/2024-07/2024-07-16.md": "149222eab7a7f58993b8e4dc8a3fb884", + "_journal/2024-07-18.md": "a9d26ce938228973f07178a15128a681", + "_journal/2024-07/2024-07-17.md": "0c816cd6110bdd14d3eac4e5b82510cf", + "ontology/dialetheism.md": "4d14b681d6913bd775450c52c2417767" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-07-18.md b/notes/_journal/2024-07-18.md new file mode 100644 index 0000000..b9bd0c1 --- /dev/null +++ b/notes/_journal/2024-07-18.md @@ -0,0 +1,12 @@ +--- +title: "2024-07-18" +--- + +- [x] Anki Flashcards +- [x] KoL +- [x] OGS +- [ ] Sheet Music (10 min.) +- [ ] Korean (Read 1 Story) + +* Notes on [[beta-reduction#β-equality|β-equality]]. +* Brief notes on [[dialetheism]]. \ No newline at end of file diff --git a/notes/_journal/2024-07-17.md b/notes/_journal/2024-07/2024-07-17.md similarity index 100% rename from notes/_journal/2024-07-17.md rename to notes/_journal/2024-07/2024-07-17.md diff --git a/notes/algebra/set.md b/notes/algebra/set.md index 6d65fb5..ec6699b 100644 --- a/notes/algebra/set.md +++ b/notes/algebra/set.md @@ -142,7 +142,7 @@ END%% %%ANKI Basic -Let $I$ be an index set and $H$ a function $I \subseteq \mathop{\text{dom}}H$. How is $\bigtimes_{i \in I} H(i)$ defined? +Let $I$ be an index set and $H$ a function such that $I \subseteq \mathop{\text{dom}}H$. How is $\bigtimes_{i \in I} H(i)$ defined? Back: $\bigtimes_{i \in I} H(i) = \{ f \mid f \text{ is a function with domain } I \text { and } \forall i \in I, f(i) \in H(i) \}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). @@ -825,8 +825,8 @@ Let $A$, $B$, and $C$ be arbitrary sets. Then %%ANKI Basic -What kind of propositional logical statement are the antimonotonicity properties of $\subseteq$? -Back: An implication. +What kind of propositional logical statements are the antimonotonicity properties of $\subseteq$? +Back: Implications. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% diff --git a/notes/lambda-calculus/alpha-conversion.md b/notes/lambda-calculus/alpha-conversion.md index ae450e8..a506a43 100644 --- a/notes/lambda-calculus/alpha-conversion.md +++ b/notes/lambda-calculus/alpha-conversion.md @@ -59,6 +59,14 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi END%% +%%ANKI +Basic +Is $\alpha$-conversion a symmetric relation? +Back: Yes. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + %%ANKI Basic $\alpha$-conversion is most related to what kind of $\lambda$-term? diff --git a/notes/lambda-calculus/beta-reduction.md b/notes/lambda-calculus/beta-reduction.md index 3c9da5f..b72ae12 100644 --- a/notes/lambda-calculus/beta-reduction.md +++ b/notes/lambda-calculus/beta-reduction.md @@ -497,6 +497,126 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi END%% +## β-equality + +We say $P$ is **$\beta$-equal** or **$\beta$-convertible** to $Q$ ($P =_\beta Q$) iff $Q$ can be obtained from $P$ by a finite series of $\beta$-contractions, reversed $\beta$-contractions, and changes of bound variables. That is, $P =_\beta Q$ iff there exist $P_0, \ldots, P_n$ ($n \geq 0$) such that $P_0 \equiv P$, $P_n \equiv Q$, and $$\forall i \leq n - 1, (P_i \,\triangleright_{1\beta}\, P_{i+1}) \lor (P_{i+1} \,\triangleright_{1\beta}\, P_i) \lor (P_i \equiv_\alpha P_{i+1}).$$ + +%%ANKI +Basic +$\triangleright_\beta$ denotes what relation? +Back: $\beta$-reduction. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +$\triangleright_{1\beta}$ denotes what relation? +Back: $\beta$-contraction. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +$=_{\beta}$ denotes what relation? +Back: $\beta$-equality. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Cloze +{$\beta$-equality} is also known as {$\beta$-convertibility}. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +Is $\beta$-reduction a symmetric relation? +Back: No. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +Is $\beta$-equality a symmetric relation? +Back: Yes. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Cloze +{$\beta$-equality} is the symmetric generalization of {$\beta$-reduction}. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +What does it mean for $P$ to be $\beta$-equal to $Q$? +Back: $Q$ can be obtained from $P$ by a finite series of $\beta$-contractions, reversed $\beta$-contractions, and $\alpha$-conversions. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +How is "$P$ is $\beta$-equal to $Q$" denoted? +Back: $P =_\beta Q$ +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Cloze +$P =_\beta Q$ iff $\exists P_0, \ldots, P_n$ s.t. $P_0 \equiv P$, $P_n \equiv Q$, and $\forall i \leq n - 1$, {$P_i \,\triangleright_{1\beta}\, P_{i+1}$} or {$P_{i+1} \,\triangleright_{1\beta}\, P_i$} or {$P_i \equiv_\alpha P_{i+1}$}. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +$\beta$-reduction constitute what two operations? +Back: $\beta$-contractions and $\alpha$-conversions. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +$\beta$-equality constitute what three operations? +Back: $\beta$-contractions, reversed $\beta$-contractions, and $\alpha$-conversions. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Cloze +{$M =_\beta M' \land N =_\beta N'$} $\Rightarrow [N/x]M =_\beta [N'/x]M'$ +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +How would Hindley et al. describe the following implication? $$M =_\beta M' \land N =_\beta N' \Rightarrow [N/x]M =_\beta [N'/x]M'$$ +Back: As "substitution is well-defined with respect to $\beta$-equality." +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +If $P =_\beta Q$, how do $P$ and $Q$'s $\beta$-normal forms relate to one another? +Back: Either $P$ and $Q$ have the same $\beta$-normal form or neither $P$ nor $Q$ have a $\beta$-normal form. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + ## Church-Rosser Theorem If $P \,\triangleright_\beta\, M$ and $P \,\triangleright_\beta\, N$, then there exists a term $T$ such that $M \,\triangleright_\beta\, T$ and $N \,\triangleright_\beta\, T$. As an immediate corollary, if $P$ has a $\beta$-normal form then it it is unique modulo $\equiv_\alpha$. @@ -567,7 +687,7 @@ END%% %%ANKI Basic -What does the Church-Rosser theorem for $\triangleright_\beta$ state in terms of confluence? +What does the Church-Rosser theorem state in terms of confluence? Back: $\beta$-reduction is confluent. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). @@ -577,7 +697,7 @@ END%% Basic The following diagram is a representation of what theorem? ![[church-rosser.png]] -Back: The Church-Rosser theorem for $\triangleright_\beta$. +Back: The Church-Rosser theorem. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). END%% @@ -600,13 +720,31 @@ END%% %%ANKI Basic -In the following Church-Rosser diagram, what do the arrows represent? +In the following diagram of the Church-Rosser theorem, what do the arrows represent? ![[church-rosser.png]] Back: $\beta$-reductions. Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). END%% +Likewise, if $P =_\beta Q$, then there exists a term $T$ such that $P \,\triangleright_\beta\, T$ and $Q \,\triangleright_\beta\, T$. + +%%ANKI +Basic +What does the Church-Rosser theorem state in terms of $=_\beta$? +Back: If $P =_\beta Q$ then there exists a term $T$ such that $P \,\triangleright_\beta\, T$ and $Q \,\triangleright_\beta\, T$. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + +%%ANKI +Basic +What theorem encourages giving $\beta$-equality its name? +Back: The Church-Rosser theorem. +Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). + +END%% + ## Bibliography * Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). \ No newline at end of file diff --git a/notes/logic/equality.md b/notes/logic/equality.md deleted file mode 100644 index 05450e9..0000000 --- a/notes/logic/equality.md +++ /dev/null @@ -1,15 +0,0 @@ ---- -title: Equality -TARGET DECK: Obsidian::STEM -FILE TAGS: equality -tags: - - equality ---- - -## Overview - - - -## Bibliography - -* Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). \ No newline at end of file diff --git a/notes/logic/equiv-trans.md b/notes/logic/equiv-trans.md index 8f1fc26..c9530d1 100644 --- a/notes/logic/equiv-trans.md +++ b/notes/logic/equiv-trans.md @@ -90,176 +90,6 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in END%% -* Commutative Laws - * $(E1 \land E2) = (E2 \land E1)$ - * $(E1 \lor E2) = (E2 \lor E1)$ - * $(E1 = E2) = (E2 = E1)$ - -%%ANKI -Basic -Which of the basic logical operators do the commutative laws apply to? -Back: $\land$, $\lor$, and $=$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What do the commutative laws allow us to do? -Back: Reorder operands. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What is the commutative law of e.g. $\land$? -Back: $E1 \land E2 = E2 \land E1$ - -END%% - -* Associative Laws - * $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$ - * $E1 \lor (E2 \lor E3) = (E1 \lor E2) \lor E3$ - -%%ANKI -Basic -Which of the basic logical operators do the associative laws apply to? -Back: $\land$ and $\lor$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What do the associative laws allow us to do? -Back: Remove parentheses. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What is the associative law of e.g. $\land$? -Back: $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -* Distributive Laws - * $E1 \lor (E2 \land E3) = (E1 \lor E2) \land (E1 \lor E3)$ - * $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$ - -%%ANKI -Basic -Which of the basic logical operators do the distributive laws apply to? -Back: $\land$ and $\lor$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What do the distributive laws allow us to do? -Back: "Factor" propositions. -Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What is the distributive law of e.g. $\land$ over $\lor$? -Back: $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -* De Morgan's Laws - * $\neg (E1 \land E2) = \neg E1 \lor \neg E2$ - * $\neg (E1 \lor E2) = \neg E1 \land \neg E2$ - -%%ANKI -Basic -Which of the basic logical operators do De Morgan's Laws apply to? -Back: $\neg$, $\land$, and $\lor$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What is De Morgan's Law of e.g. $\land$? -Back: $\neg (E1 \land E2) = \neg E1 \lor \neg E2$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -* Law of Negation - * $\neg (\neg E1) = E1$ - -%%ANKI -Basic -What does the Law of Negation say? -Back: $\neg (\neg E1) = E1$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -* Law of the Excluded Middle - * $E1 \lor \neg E1 = T$ - -%%ANKI -Basic -Which of the basic logical operators does the Law of the Excluded Middle apply to? -Back: $\lor$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What does the Law of the Excluded Middle say? -Back: $E1 \lor \neg E1 = T$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -Which equivalence schema is "refuted" by sentence, "This sentence is false." -Back: Law of the Excluded Middle -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -* Law of Contradiction - * $E1 \land \neg E1 = F$ - -%%ANKI -Basic -Which of the basic logical operators does the Law of Contradiction apply to? -Back: $\land$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What does the Law of Contradiction say? -Back: $E1 \land \neg E1 = F$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Cloze -The Law of {1:the Excluded Middle} is to {2:$\lor$} whereas the Law of {2:Contradiction} is to {1:$\land$}. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -Gries lists other "Laws" but they don't seem as important to note here. - %%ANKI Basic How is $\Rightarrow$ written in terms of other logical operators? diff --git a/notes/logic/pred-logic.md b/notes/logic/pred-logic.md index 96c92e3..adf167f 100644 --- a/notes/logic/pred-logic.md +++ b/notes/logic/pred-logic.md @@ -50,7 +50,262 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n END%% -## Sets +## Quantification + +A **quantifier** refers to an operator that specifies how many members of a set satisfy some formula. The most common quantifiers are $\exists$ and $\forall$, though others (such as the counting quantifier) are also used. + +%%ANKI +Basic +What are the most common first-order logic quantifiers? +Back: $\exists$ and $\forall$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What term refers to operators like $\exists$ and $\forall$? +Back: Quantifiers. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +* **Existential quantification** ($\exists$) asserts the existence of at least one member in a set satisfying a property. + +%%ANKI +Basic +What symbol denotes existential quantification? +Back: $\exists$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How many members in the domain of discourse must satisfy a property in existential quantification? +Back: At least one. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +$\exists x : S, P(x)$ is shorthand for what? +Back: $\exists x, x \in S \land P(x)$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What term refers to $S$ in $\exists x : S, P(x)$? +Back: The domain of discourse. +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +What is the identity element of $\lor$? +Back: $F$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +* **Universal quantification** ($\forall$) asserts that every member of a set satisfies a property. + +%%ANKI +Basic +What symbol denotes universal quantification? +Back: $\forall$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How many members in the domain of discourse must satisfy a property in universal quantification? +Back: All of them. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +$\forall x : S, P(x)$ is shorthand for what? +Back: $\forall x, x \in S \Rightarrow P(x)$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What is the identity element of $\land$? +Back: $T$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Cloze +{1:$\exists$} is to {2:$\lor$} as {2:$\forall$} is to {1:$\land$}. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is $\forall x : S, P(x)$ equivalently written in terms of existential quantification? +Back: $\neg \exists x : S, \neg P(x)$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +How is $\exists x : S, P(x)$ equivalently written in terms of universal quantification? +Back: $\neg \forall x : S, \neg P(x)$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. +END%% + +* **Counting quantification** ($\exists^{=k}$ or $\exists^{\geq k}$) asserts that (at least) $k$ (say) members of a set satisfy a property. + +%%ANKI +Basic +What symbol denotes counting quantification (of *exactly* $k$ members)? +Back: $\exists^{=k}$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What symbol denotes counting quantification (of *at least* $k$ members)? +Back: $\exists^{\geq k}$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is $\exists x : S, P(x)$ written in terms of counting quantification? +Back: $\exists^{\geq 1}\, x : S, P(x)$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is $\forall x : S, P(x)$ written in terms of counting quantification? +Back: Assuming $S$ has $k$ members, $\exists^{= k}\, x : S, P(x)$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Cloze +Propositional logical operator: $\forall x, \forall y, P(x, y)$ {$\Leftrightarrow$} $\forall y, \forall x, P(x, y)$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +Propositional logical operator: $\forall x, \exists y, P(x, y)$ {$\Leftarrow$} {$\exists y, \forall x, P(x, y)$}. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +Propositional logical operator: $\exists x, \forall y, P(x, y)$ {$\Rightarrow$} $\forall y, \exists x, P(x, y)$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Cloze +Propositional logical operator: $\exists x, \exists y, P(x, y)$ {$\Leftrightarrow$} $\exists y, \exists x, P(x, y)$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +When does $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$ hold true? +Back: Always. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +When does $\forall x, \exists y, P(x, y) \Rightarrow \exists y, \forall x, P(x, y)$ hold true? +Back: When there exists a $y$ that $P(x, y)$ holds for over all quantified $x$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +### Identifiers + +Identifiers are said to be **bound** if they are parameters to a quantifier. Identifiers that are not bound are said to be **free**. A first-order logic formula is said to be in **prenex normal form** (PNF) if written in two parts: the first consisting of quantifiers and bound variables (the **prefix**), and the second consisting of no quantifiers (the **matrix**). + +%%ANKI +Basic +Prenex normal form consists of what two parts? +Back: The prefix and the matrix. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is the prefix of a formula in PNF formatted? +Back: As only quantifiers and bound variables. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is the matrix of a formula in PNF formatted? +Back: Without quantifiers. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +Which identifiers in the following are bound? $\exists x, P(x) \land P(y)$ +Back: Just $x$. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +Which identifiers in the following are free? $\exists x, P(x) \land P(y)$ +Back: Just $y$. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is the following rewritten in PNF? $(\exists x, P(x)) \land (\exists y, Q(y))$ +Back: $\exists x \;y, P(x) \land Q(y)$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is the following rewritten in PNF? $(\exists x, P(x)) \land (\forall y, Q(y))$ +Back: N/A. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +## As Sets A **state** is a function that maps identifiers to values. A predicate can be equivalently seen as a representation of the set of states in which it is true. diff --git a/notes/logic/prop-logic.md b/notes/logic/prop-logic.md index 91c2031..861e115 100644 --- a/notes/logic/prop-logic.md +++ b/notes/logic/prop-logic.md @@ -104,14 +104,7 @@ END%% ## Implication -Implication is denoted as $\Rightarrow$. It has truth table - -$p$ | $q$ | $p \Rightarrow q$ ---- | --- | ----------------- -$T$ | $T$ | $T$ -$T$ | $F$ | $F$ -$F$ | $T$ | $T$ -$F$ | $F$ | $T$ +Implication is denoted as $\Rightarrow$. It has truth table $$\begin{array}{c|c|c} p & q & p \Rightarrow q \\ \hline T & T & T \\ T & F & F \\ F & T & T \\ F & F & T \end{array}$$ Implication has a few "equivalent" English expressions that are commonly used. Given propositions $P$ and $Q$, we have the following equivalences: @@ -321,7 +314,273 @@ Reference: Reference: Gries, David. *The Science of Programming*. Texts and Mon END%% -## Sets +## Laws + +### Commutativity + +For propositions $E1$ and $E2$: + +* $(E1 \land E2) = (E2 \land E1)$ +* $(E1 \lor E2) = (E2 \lor E1)$ +* $(E1 = E2) = (E2 = E1)$ + +%%ANKI +Basic +Which of the basic logical operators do the commutative laws apply to? +Back: $\land$, $\lor$, and $=$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What do the commutative laws allow us to do? +Back: Reorder operands. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What is the commutative law of e.g. $\land$? +Back: $E1 \land E2 = E2 \land E1$ + +END%% + +### Associativity + +For propositions $E1$, $E2$, and $E3$: + +* $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$ +* $E1 \lor (E2 \lor E3) = (E1 \lor E2) \lor E3$ + +%%ANKI +Basic +Which of the basic logical operators do the associative laws apply to? +Back: $\land$ and $\lor$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What do the associative laws allow us to do? +Back: Remove parentheses. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What is the associative law of e.g. $\land$? +Back: $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +### Distributivity + +For propositions $E1$, $E2$, and $E3$: + +* $E1 \lor (E2 \land E3) = (E1 \lor E2) \land (E1 \lor E3)$ +* $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$ + +%%ANKI +Basic +Which of the basic logical operators do the distributive laws apply to? +Back: $\land$ and $\lor$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What do the distributive laws allow us to do? +Back: "Factor" propositions. +Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What is the distributive law of e.g. $\land$ over $\lor$? +Back: $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +### De Morgan's + +For propositions $E1$ and $E2$: + +* $\neg (E1 \land E2) = \neg E1 \lor \neg E2$ +* $\neg (E1 \lor E2) = \neg E1 \land \neg E2$ + +%%ANKI +Basic +Which of the basic logical operators do De Morgan's laws involve? +Back: $\neg$, $\land$, and $\lor$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is De Morgan's law (distributing $\land$) expressed as an equivalence? +Back: $\neg (E1 \land E2) = \neg E1 \lor \neg E2$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. +Tags: programming::equiv-trans + +END%% + +### Law of Negation + +For any proposition $E1$, it follows that $\neg (\neg E1) = E1$. + +%%ANKI +Basic +How is the law of negation expressed as an equivalence? +Back: $\neg (\neg E1) = E1$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. +Tags: programming::equiv-trans + +END%% + +### Law of Excluded Middle + +For any proposition $E1$, it follows that $E1 \lor \neg E1 = T$. + +%%ANKI +Basic +Which of the basic logical operators does the law of excluded middle involve? +Back: $\lor$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is the law of excluded middle expressed as an equivalence? +Back: $E1 \lor \neg E1 = T$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. +Tags: programming::equiv-trans + +END%% + +%%ANKI +Basic +Which equivalence schema is "refuted" by sentence, "This sentence is false." +Back: The law of excluded middle +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +### Law of Contradiction + +For any proposition $E1$, it follows that $E1 \land \neg E1 = F$. + +%%ANKI +Basic +Which of the basic logical operators does the law of contradiction involve? +Back: $\land$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How is the law of contradiction expressed as an equivalence? +Back: $E1 \land \neg E1 = F$ +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. +Tags: programming::equiv-trans + +END%% + +%%ANKI +Cloze +The law of {1:excluded middle} is to {2:$\lor$} whereas the law of {2:contradiction} is to {1:$\land$}. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +What does the principle of explosion state? +Back: That any statement can be proven from a contradiction. +Reference: “Principle of Explosion,” in _Wikipedia_, July 3, 2024, [https://en.wikipedia.org/w/index.php?title=Principle_of_explosion](https://en.wikipedia.org/w/index.php?title=Principle_of_explosion&oldid=1232334233). + +END%% + +%%ANKI +Basic +How is the principle of explosion stated in first-order logic? +Back: $\forall P, F \Rightarrow P$ +Reference: “Principle of Explosion,” in _Wikipedia_, July 3, 2024, [https://en.wikipedia.org/w/index.php?title=Principle_of_explosion](https://en.wikipedia.org/w/index.php?title=Principle_of_explosion&oldid=1232334233). + +END%% + +%%ANKI +Basic +What does the law of contradiction say? +Back: For any proposition $P$, it holds that $\neg (P \land \neg P)$. +Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. + +END%% + +%%ANKI +Basic +How does the principle of explosion relate to the law of contradiction? +Back: If a contradiction could be proven, then anything can be proven. +Reference: “Principle of Explosion,” in _Wikipedia_, July 3, 2024, [https://en.wikipedia.org/w/index.php?title=Principle_of_explosion](https://en.wikipedia.org/w/index.php?title=Principle_of_explosion&oldid=1232334233). + +END%% + +%%ANKI +Basic +Suppose $P$ and $\neg P$. Show schematically how to use the principle of explosion to prove $Q$. +Back: $$\begin{align*} P \\ \neg P \\ P \lor Q \\ \hline Q \end{align*}$$Reference: “Principle of Explosion,” in _Wikipedia_, July 3, 2024, [https://en.wikipedia.org/w/index.php?title=Principle_of_explosion](https://en.wikipedia.org/w/index.php?title=Principle_of_explosion&oldid=1232334233). + +END%% + +%%ANKI +Cloze +The law of {contradiction} and law of {excluded middle} create a dichotomy in "logical space". +Reference: “Law of Noncontradiction,” in _Wikipedia_, June 14, 2024, [https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction](https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction&oldid=1229006759). + +END%% + +%%ANKI +Basic +Which property of partitions is analagous to the law of contradiction on "logical space"? +Back: Disjointedness. +Reference: “Law of Noncontradiction,” in _Wikipedia_, June 14, 2024, [https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction](https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction&oldid=1229006759). + +END%% + +%%ANKI +Basic +Which property of partitions is analagous to the law of excluded middle on "logical space"? +Back: Exhaustiveness. +Reference: “Law of Noncontradiction,” in _Wikipedia_, June 14, 2024, [https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction](https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction&oldid=1229006759). + +END%% + +%%ANKI +Cloze +The law of {1:contradiction} is to "{2:mutually exclusive}" whereas the law of {2:excluded middle} is "{1:jointly exhaustive}". +Reference: “Law of Noncontradiction,” in _Wikipedia_, June 14, 2024, [https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction](https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction&oldid=1229006759). + +END%% + +%%ANKI +Basic +Which logical law proves equivalence of the law of contradiction and excluded middle? +Back: De Morgan's law. +Reference: “Law of Noncontradiction,” in _Wikipedia_, June 14, 2024, [https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction](https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction&oldid=1229006759). + +END%% + +## As Sets A **state** is a function that maps identifiers to $T$ or $F$. A proposition can be equivalently seen as a representation of the set of states in which it is true. @@ -494,4 +753,6 @@ END%% ## Bibliography * Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. -* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). \ No newline at end of file +* “Law of Noncontradiction,” in _Wikipedia_, June 14, 2024, [https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction](https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction&oldid=1229006759). +* * Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). +* “Principle of Explosion,” in _Wikipedia_, July 3, 2024, [https://en.wikipedia.org/w/index.php?title=Principle_of_explosion](https://en.wikipedia.org/w/index.php?title=Principle_of_explosion&oldid=1232334233). \ No newline at end of file diff --git a/notes/logic/quantification.md b/notes/logic/quantification.md deleted file mode 100644 index 4818434..0000000 --- a/notes/logic/quantification.md +++ /dev/null @@ -1,268 +0,0 @@ ---- -title: Quantification -TARGET DECK: Obsidian::STEM -FILE TAGS: logic::quantification -tags: - - logic - - quantification ---- - -## Overview - -A **quantifier** refers to an operator that specifies how many members of a set satisfy some formula. The most common quantifiers are $\exists$ and $\forall$, though others (such as the counting quantifier) are also used. - -%%ANKI -Basic -What are the most common first-order logic quantifiers? -Back: $\exists$ and $\forall$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What term refers to operators like $\exists$ and $\forall$? -Back: Quantifiers. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -* **Existential quantification** ($\exists$) asserts the existence of at least one member in a set satisfying a property. - -%%ANKI -Basic -What symbol denotes existential quantification? -Back: $\exists$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -How many members in the domain of discourse must satisfy a property in existential quantification? -Back: At least one. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -$\exists x : S, P(x)$ is shorthand for what? -Back: $\exists x, x \in S \land P(x)$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What term refers to $S$ in $\exists x : S, P(x)$? -Back: The domain of discourse. -Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). - -END%% - -%%ANKI -Basic -What is the identity element of $\lor$? -Back: $F$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -* **Universal quantification** ($\forall$) asserts that every member of a set satisfies a property. - -%%ANKI -Basic -What symbol denotes universal quantification? -Back: $\forall$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -How many members in the domain of discourse must satisfy a property in universal quantification? -Back: All of them. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -$\forall x : S, P(x)$ is shorthand for what? -Back: $\forall x, x \in S \Rightarrow P(x)$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What is the identity element of $\land$? -Back: $T$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Cloze -{1:$\exists$} is to {2:$\lor$} as {2:$\forall$} is to {1:$\land$}. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -How is $\forall x : S, P(x)$ equivalently written in terms of existential quantification? -Back: $\neg \exists x : S, \neg P(x)$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -How is $\exists x : S, P(x)$ equivalently written in terms of universal quantification? -Back: $\neg \forall x : S, \neg P(x)$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. -END%% - -* **Counting quantification** ($\exists^{=k}$ or $\exists^{\geq k}$) asserts that (at least) $k$ (say) members of a set satisfy a property. - -%%ANKI -Basic -What symbol denotes counting quantification (of *exactly* $k$ members)? -Back: $\exists^{=k}$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -What symbol denotes counting quantification (of *at least* $k$ members)? -Back: $\exists^{\geq k}$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -How is $\exists x : S, P(x)$ written in terms of counting quantification? -Back: $\exists^{\geq 1}\, x : S, P(x)$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -How is $\forall x : S, P(x)$ written in terms of counting quantification? -Back: Assuming $S$ has $k$ members, $\exists^{= k}\, x : S, P(x)$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Cloze -Propositional logical operator: $\forall x, \forall y, P(x, y)$ {$\Leftrightarrow$} $\forall y, \forall x, P(x, y)$. -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - -%%ANKI -Cloze -Propositional logical operator: $\forall x, \exists y, P(x, y)$ {$\Leftarrow$} {$\exists y, \forall x, P(x, y)$}. -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - -%%ANKI -Cloze -Propositional logical operator: $\exists x, \forall y, P(x, y)$ {$\Rightarrow$} $\forall y, \exists x, P(x, y)$. -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - -%%ANKI -Cloze -Propositional logical operator: $\exists x, \exists y, P(x, y)$ {$\Leftrightarrow$} $\exists y, \exists x, P(x, y)$. -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - -%%ANKI -Basic -When does $\exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)$ hold true? -Back: Always. -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - -%%ANKI -Basic -When does $\forall x, \exists y, P(x, y) \Rightarrow \exists y, \forall x, P(x, y)$ hold true? -Back: When there exists a $y$ that $P(x, y)$ holds for over all quantified $x$. -Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). - -END%% - -## Identifiers - -Identifiers are said to be **bound** if they are parameters to a quantifier. Identifiers that are not bound are said to be **free**. A first-order logic formula is said to be in **prenex normal form** (PNF) if written in two parts: the first consisting of quantifiers and bound variables (the **prefix**), and the second consisting of no quantifiers (the **matrix**). - -%%ANKI -Basic -Prenex normal form consists of what two parts? -Back: The prefix and the matrix. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -How is the prefix of a formula in PNF formatted? -Back: As only quantifiers and bound variables. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -How is the matrix of a formula in PNF formatted? -Back: Without quantifiers. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -Which identifiers in the following are bound? $\exists x, P(x) \land P(y)$ -Back: Just $x$. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -Which identifiers in the following are free? $\exists x, P(x) \land P(y)$ -Back: Just $y$. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -How is the following rewritten in PNF? $(\exists x, P(x)) \land (\exists y, Q(y))$ -Back: $\exists x \;y, P(x) \land Q(y)$ -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -%%ANKI -Basic -How is the following rewritten in PNF? $(\exists x, P(x)) \land (\forall y, Q(y))$ -Back: N/A. -Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. - -END%% - -## Bibliography - -* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. -* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). \ No newline at end of file diff --git a/notes/ontology/dialetheism.md b/notes/ontology/dialetheism.md new file mode 100644 index 0000000..9023037 --- /dev/null +++ b/notes/ontology/dialetheism.md @@ -0,0 +1,31 @@ +--- +title: Dialetheism +TARGET DECK: Obsidian::H&SS +FILE TAGS: ontology::dialetheism +tags: + - dialetheism + - ontology +--- + +## Overview + +A **dialetheia** is a sentence $A$ such that both it and its negation ($\neg A$) are true. **Dialetheism** is the view that there are dialetheias. In other words, dialetheism admits the existence of true contradictions. + +%%ANKI +Cloze +A {dialetheia} is a {sentence such that both it and its negation are true}. +Reference: Graham Priest, Francesco Berto, and Zach Weber, “Dialetheism,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta and Uri Nodelman, Summer 2024 (Metaphysics Research Lab, Stanford University, 2024), [https://plato.stanford.edu/archives/sum2024/entries/dialetheism/](https://plato.stanford.edu/archives/sum2024/entries/dialetheism/). + +END%% + +%%ANKI +Cloze +{Dialetheism} is the view that {dialetheia} exist. +Reference: Graham Priest, Francesco Berto, and Zach Weber, “Dialetheism,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta and Uri Nodelman, Summer 2024 (Metaphysics Research Lab, Stanford University, 2024), [https://plato.stanford.edu/archives/sum2024/entries/dialetheism/](https://plato.stanford.edu/archives/sum2024/entries/dialetheism/). + +END%% + +## Bibliography + +* Graham Priest, Francesco Berto, and Zach Weber, “Dialetheism,” in _The Stanford Encyclopedia of Philosophy_, ed. Edward N. Zalta and Uri Nodelman, Summer 2024 (Metaphysics Research Lab, Stanford University, 2024), [https://plato.stanford.edu/archives/sum2024/entries/dialetheism/](https://plato.stanford.edu/archives/sum2024/entries/dialetheism/). +* Nikk Effingham, _An Introduction to Ontology_ (Cambridge: Polity Press, 2013). \ No newline at end of file diff --git a/notes/set/graphs.md b/notes/set/graphs.md index f317aa6..ae8da4b 100644 --- a/notes/set/graphs.md +++ b/notes/set/graphs.md @@ -563,7 +563,7 @@ END%% %%ANKI Cloze -Given a directed graph, incident {1:to} is to {1:in}-degrees whereas incident {1:from} is to {1:out}-degrees. +Given a directed graph, incident {1:to} is to {2:in}-degrees whereas incident {2:from} is to {1:out}-degrees. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). END%%