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| # VP: http://www.cs.nyu.edu/pipermail/fom/1998-February/001248.html | | # VP: http://www.cs.nyu.edu/pipermail/fom/1998-February/001248.html |
| | | |
− | ====Oct 2008 : Classical/Constructive Mathematics==== | + | ====Oct 2008 : Classical ∧∨ Constructive Mathematics==== |
| | | |
| * http://www.cs.nyu.edu/pipermail/fom/2008-October/thread.html#13127 | | * http://www.cs.nyu.edu/pipermail/fom/2008-October/thread.html#13127 |
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| ===Foreground=== | | ===Foreground=== |
| | | |
− | Harvey Friedman (15 Oct 2008), "Classical/Constructive Mathematics", FOMA | + | Harvey Friedman (15 Oct 2008), "Classical/Constructive Mathematics", FOMA. |
| | | |
| <blockquote> | | <blockquote> |
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| I am still in the phase of chasing down links between the various questions and I don't have any news or conclusions to offer, but my web searches keep bringing me back to this old discussion on the FOM list: | | I am still in the phase of chasing down links between the various questions and I don't have any news or conclusions to offer, but my web searches keep bringing me back to this old discussion on the FOM list: |
| | | |
− | <pre>
| + | [http://www.cs.nyu.edu/pipermail/fom/1998-February/thread.html#1160 Intuitionistic Mathematics and Building Bridges]. |
− | http://www.cs.nyu.edu/pipermail/fom/1998-February/thread.html#1160 | |
| | | |
− | I find one comment by Vaughan Pratt to be especially e-&/or-pro-vocative: | + | I find one comment by Vaughan Pratt to be especially e-∧/∨-pro-vocative: |
| | | |
− | VP: It has been my impression from having dealt with a lot of lawyers over the
| + | Vaughn Pratt (27 Feb 1998), "Intuitionistic Mathematics and Building Bridges", FOMA. |
− | last twenty years that the logic of the legal profession is rarely Boolean,
| |
− | with a few isolated exceptions such as jury verdicts which permit only
| |
− | guilty or not guilty, no middle verdict allowed. Often legal logic
| |
− | is not even intuitionistic, with conjunction failing commutativity
| |
− | and sometimes even idempotence. But that aside, excluded middle
| |
− | and double negation are the exception rather than the rule.
| |
| | | |
− | VP: Lawyers aren't alone in this. The permitted rules of reasoning
| + | <blockquote> |
− | that go along with whichever scientific method is currently in
| + | <p>It has been my impression from having dealt with a lot of lawyers over the last twenty years that the logic of the legal profession is rarely Boolean, with a few isolated exceptions such as jury verdicts which permit only guilty or not guilty, no middle verdict allowed. Often legal logic is not even intuitionistic, with conjunction failing commutativity and sometimes even idempotence. But that aside, excluded middle and double negation are the exception rather than the rule.</p> |
− | vogue seem to have the same non-Boolean character in general.
| |
| | | |
− | VP: The very *thought* of a lawyer or scientist appealing to Peirce's law,
| + | <p>Lawyers aren't alone in this. The permitted rules of reasoning that go along with whichever scientific method is currently in vogue seem to have the same non-Boolean character in general.</p> |
− | ((P->Q)->P)->P, to prove a point boggles the mind. And imagine them
| |
− | trying to defend their use of that law by actually proving it: the
| |
− | audience would simply ssume this was one of those bits of logical
| |
− | sleight-of-hand where the wool is pulled over one's eyes by some
| |
− | sophistry that goes against common sense.
| |
| | | |
− | Anyway, to make a long story elliptic, | + | <p>The very *thought* of a lawyer or scientist appealing to Peirce's law, ((P→Q)→P)→P, to prove a point boggles the mind. And imagine them trying to defend their use of that law by actually proving it: the audience would simply ssume this was one of those bits of logical sleight-of-hand where the wool is pulled over one's eyes by some sophistry that goes against common sense.</p> |
− | here is one of my current write-ups on | + | </blockquote> |
− | Peirce's Law that led me back into this | + | |
− | old briar patch: | + | Anyway, to make a long story elliptic, here is one of my current write-ups on Peirce's Law that led me back into this old briar patch: |
| | | |
| http://www.mywikibiz.com/Peirce's_law | | http://www.mywikibiz.com/Peirce's_law |
| | | |
− | More to say on this later, but I just wanted to get | + | More to say on this later, but I just wanted to get a good chunk of the background set out in one place. |
− | a good chunk of the background set out in one place. | |
− | </pre>
| |
| | | |
| ==Logical Graph Sandbox : Very Rough Sand Reckoning== | | ==Logical Graph Sandbox : Very Rough Sand Reckoning== |
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| But the latter is not a theorem in anyone's philosophy, so there is really no disagreement here. | | But the latter is not a theorem in anyone's philosophy, so there is really no disagreement here. |
| | | |
− | ===Functional quantifiers===
| + | ==Functional Quantifiers== |
| | | |
− | ====Tables====
| + | The '''umpire measure''' of type <math>\Upsilon : (X \to \mathbb{B}) \to \mathbb{B}</math> links the constant proposition <math>1 : X \to \mathbb{B}</math> to a value of 1 and every other proposition to a value of 0. Expressed in symbolic form: |
| | | |
− | The auxiliary notations:
| + | {| align="center" cellpadding="8" |
| + | | <math>\Upsilon (u) = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{X \to \mathbb{B}}.</math> |
| + | |} |
| | | |
− | : <math>\alpha_i f = \Upsilon (f_i, f),\!</math>
| + | The '''umpire operator''' of type <math>\Upsilon : (X \to \mathbb{B})^2 \to \mathbb{B}</math> links pairs of propositions in which the first implies the second to a value of 1 and every other pair to a value of 0. Expressed in symbolic form: |
| | | |
− | : <math>\beta_i f = \Upsilon (f, f_i),\!</math>
| + | {| align="center" cellpadding="8" |
| + | | <math>\Upsilon (u, v) = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math> |
| + | |} |
| | | |
− | define two series of measures:
| + | ===Tables=== |
| | | |
− | : <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B},</math> | + | Define two families of measures: |
| | | |
− | incidentally providing compact names for the column headings of the next two Tables.
| + | {| align="center" cellpadding="8" |
| + | | <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 0 \ldots 15,</math> |
| + | |} |
| + | |
| + | by means of the following formulas: |
| + | |
| + | {| align="center" cellpadding="8" |
| + | | <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f),</math> |
| + | |- |
| + | | <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i).</math> |
| + | |} |
| + | |
| + | ====Table 1==== |
| + | |
| + | The values of the sixteen <math>\alpha_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table 1. Expressed in terms of the implication ordering on the sixteen functions, <math>\alpha_i f = 1\!</math> says that <math>f\!</math> is ''above or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\ge f_i\!</math> in the implication ordering. |
| | | |
| {| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | | {| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
− | |+ '''Table 1. Qualifiers of Implication Ordering: <math>\alpha_i f = \Upsilon (f_i \Rightarrow f)</math>''' | + | |+ '''Table 1. Qualifiers of Implication Ordering: <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)</math>''' |
| |- style="background:ghostwhite" | | |- style="background:ghostwhite" |
− | | align="right" | <math>p:</math><br><math>q:</math> | + | | align="right" | <math>u:</math><br><math>v:</math> |
| | 1100<br>1010 | | | 1100<br>1010 |
| | <math>f\!</math> | | | <math>f\!</math> |
| + | | <math>\alpha_0</math> |
| + | | <math>\alpha_1</math> |
| + | | <math>\alpha_2</math> |
| + | | <math>\alpha_3</math> |
| + | | <math>\alpha_4</math> |
| + | | <math>\alpha_5</math> |
| + | | <math>\alpha_6</math> |
| + | | <math>\alpha_7</math> |
| + | | <math>\alpha_8</math> |
| + | | <math>\alpha_9</math> |
| + | | <math>\alpha_{10}</math> |
| + | | <math>\alpha_{11}</math> |
| + | | <math>\alpha_{12}</math> |
| + | | <math>\alpha_{13}</math> |
| + | | <math>\alpha_{14}</math> |
| | <math>\alpha_{15}</math> | | | <math>\alpha_{15}</math> |
− | | <math>\alpha_{14}</math>
| |
− | | <math>\alpha_{13}</math>
| |
− | | <math>\alpha_{12}</math>
| |
− | | <math>\alpha_{11}</math>
| |
− | | <math>\alpha_{10}</math>
| |
− | | <math>\alpha_9</math>
| |
− | | <math>\alpha_8</math>
| |
− | | <math>\alpha_7</math>
| |
− | | <math>\alpha_6</math>
| |
− | | <math>\alpha_5</math>
| |
− | | <math>\alpha_4</math>
| |
− | | <math>\alpha_3</math>
| |
− | | <math>\alpha_2</math>
| |
− | | <math>\alpha_1</math>
| |
− | | <math>\alpha_0</math>
| |
| |- | | |- |
− | | <math>f_0</math> || 0000 || <math>(~)</math> | + | | <math>f_0</math> |
− | | || || || || || || || | + | | 0000 |
− | | || || || || || || || 1 | + | | <math>(~)</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_1</math> || 0001 || <math>(p)(q)\!</math> | + | | <math>f_1</math> |
− | | || || || || || || || | + | | 0001 |
− | | || || || || || || 1 || 1 | + | | <math>(u)(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_2</math> || 0010 || <math>(p) q\!</math> | + | | <math>f_2</math> |
− | | || || || || || || || | + | | 0010 |
− | | || || || || || 1 || || 1 | + | | <math>(u) v\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_3</math> || 0011 || <math>(p)\!</math> | + | | <math>f_3</math> |
− | | || || || || || || || | + | | 0011 |
− | | || || || || 1 || 1 || 1 || 1 | + | | <math>(u)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_4</math> || 0100 || <math>p (q)\!</math> | + | | <math>f_4</math> |
− | | || || || || || || || | + | | 0100 |
− | | || || || 1 || || || || 1 | + | | <math>u (v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_5</math> || 0101 || <math>(q)\!</math> | + | | <math>f_5</math> |
− | | || || || || || || || | + | | 0101 |
− | | || || 1 || 1 || || || 1 || 1 | + | | <math>(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_6</math> || 0110 || <math>(p, q)\!</math> | + | | <math>f_6</math> |
− | | || || || || || || || | + | | 0110 |
− | | || 1 || || 1 || || 1 || || 1 | + | | <math>(u, v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_7</math> || 0111 || <math>(p q)\!</math> | + | | <math>f_7</math> |
− | | || || || || || || || | + | | 0111 |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 | + | | <math>(u v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_8</math> || 1000 || <math>p q\!</math> | + | | <math>f_8</math> |
− | | || || || || || || || 1 | + | | 1000 |
− | | || || || || || || || 1 | + | | <math>u v\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_9</math> || 1001 || <math>((p, q))\!</math> | + | | <math>f_9</math> |
− | | || || || || || || 1 || 1 | + | | 1001 |
− | | || || || || || || 1 || 1 | + | | <math>((u, v))\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_{10}</math> || 1010 || <math>q\!</math> | + | | <math>f_{10}</math> |
− | | || || || || || 1 || || 1 | + | | 1010 |
− | | || || || || || 1 || || 1 | + | | <math>v\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_{11}</math> || 1011 || <math>(p (q))\!</math> | + | | <math>f_{11}</math> |
− | | || || || || 1 || 1 || 1 || 1 | + | | 1011 |
− | | || || || || 1 || 1 || 1 || 1 | + | | <math>(u (v))\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_{12}</math> || 1100 || <math>p\!</math> | + | | <math>f_{12}</math> |
− | | || || || 1 || || || || 1 | + | | 1100 |
− | | || || || 1 || || || || 1 | + | | <math>u\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_{13}</math> || 1101 || <math>((p) q)\!</math> | + | | <math>f_{13}</math> |
− | | || || 1 || 1 || || || 1 || 1 | + | | 1101 |
− | | || || 1 || 1 || || || 1 || 1 | + | | <math>((u) v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_{14}</math> || 1110 || <math>((p)(q))\!</math> | + | | <math>f_{14}</math> |
− | | || 1 || || 1 || || 1 || || 1 | + | | 1110 |
− | | || 1 || || 1 || || 1 || || 1 | + | | <math>((u)(v))\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_{15}</math> || 1111 || <math>((~))</math> | + | | <math>f_{15}</math> |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 | + | | 1111 |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 | + | | <math>((~))</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |}<br> | | |}<br> |
| + | |
| + | ====Table 2==== |
| + | |
| + | The values of the sixteen <math>\beta_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table 2. Expressed in terms of the implication ordering on the sixteen functions, <math>\beta_i f = 1\!</math> says that <math>f\!</math> is ''below or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\le f_i\!</math> in the implication ordering. |
| | | |
| {| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | | {| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
− | |+ '''Table 2. Qualifiers of Implication Ordering: <math>\beta_i f = \Upsilon (f \Rightarrow f_i)</math>''' | + | |+ '''Table 2. Qualifiers of Implication Ordering: <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math>''' |
| |- style="background:ghostwhite" | | |- style="background:ghostwhite" |
− | | align="right" | <math>p:</math><br><math>q:</math> | + | | align="right" | <math>u:</math><br><math>v:</math> |
| | 1100<br>1010 | | | 1100<br>1010 |
| | <math>f\!</math> | | | <math>f\!</math> |
Line 569: |
Line 832: |
| | <math>\beta_{15}</math> | | | <math>\beta_{15}</math> |
| |- | | |- |
− | | <math>f_0</math> || 0000 || <math>(~)</math> | + | | <math>f_0</math> |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 | + | | 0000 |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 | + | | <math>(~)</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_1</math> || 0001 || <math>(p)(q)\!</math> | + | | <math>f_1</math> |
− | | || 1 || || 1 || || 1 || || 1 | + | | 0001 |
− | | || 1 || || 1 || || 1 || || 1 | + | | <math>(u)(v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_2</math> || 0010 || <math>(p) q\!</math> | + | | <math>f_2</math> |
− | | || || 1 || 1 || || || 1 || 1 | + | | 0010 |
− | | || || 1 || 1 || || || 1 || 1 | + | | <math>(u) v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_3</math> || 0011 || <math>(p)\!</math> | + | | <math>f_3</math> |
− | | || || || 1 || || || || 1 | + | | 0011 |
− | | || || || 1 || || || || 1 | + | | <math>(u)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_4</math> || 0100 || <math>p (q)\!</math> | + | | <math>f_4</math> |
− | | || || || || 1 || 1 || 1 || 1 | + | | 0100 |
− | | || || || || 1 || 1 || 1 || 1 | + | | <math>u (v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_5</math> || 0101 || <math>(q)\!</math> | + | | <math>f_5</math> |
− | | || || || || || 1 || || 1 | + | | 0101 |
− | | || || || || || 1 || || 1 | + | | <math>(v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_6</math> || 0110 || <math>(p, q)\!</math> | + | | <math>f_6</math> |
− | | || || || || || || 1 || 1 | + | | 0110 |
− | | || || || || || || 1 || 1 | + | | <math>(u, v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_7</math> || 0111 || <math>(p q)\!</math> | + | | <math>f_7</math> |
− | | || || || || || || || 1 | + | | 0111 |
− | | || || || || || || || 1 | + | | <math>(u v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_8</math> || 1000 || <math>p q\!</math> | + | | <math>f_8</math> |
− | | || || || || || || || | + | | 1000 |
− | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 | + | | <math>u v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_9</math> || 1001 || <math>((p, q))\!</math> | + | | <math>f_9</math> |
− | | || || || || || || || | + | | 1001 |
− | | || 1 || || 1 || || 1 || || 1 | + | | <math>((u, v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{10}</math> || 1010 || <math>q\!</math> | + | | <math>f_{10}</math> |
− | | || || || || || || || | + | | 1010 |
− | | || || 1 || 1 || || || 1 || 1 | + | | <math>v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{11}</math> || 1011 || <math>(p (q))\!</math> | + | | <math>f_{11}</math> |
− | | || || || || || || || | + | | 1011 |
− | | || || || 1 || || || || 1 | + | | <math>(u (v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{12}</math> || 1100 || <math>p\!</math> | + | | <math>f_{12}</math> |
− | | || || || || || || || | + | | 1100 |
− | | || || || || 1 || 1 || 1 || 1 | + | | <math>u\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{13}</math> || 1101 || <math>((p) q)\!</math> | + | | <math>f_{13}</math> |
− | | || || || || || || || | + | | 1101 |
− | | || || || || || 1 || || 1 | + | | <math>((u) v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{14}</math> || 1110 || <math>((p)(y))\!</math> | + | | <math>f_{14}</math> |
− | | || || || || || || || | + | | 1110 |
− | | || || || || || || 1 || 1 | + | | <math>((u)(v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{15}</math> || 1111 || <math>((~))\!</math> | + | | <math>f_{15}</math> |
− | | || || || || || || || | + | | 1111 |
− | | || || || || || || || 1 | + | | <math>((~))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |}<br> | | |}<br> |
| + | |
| + | ====Table 3==== |
| | | |
| {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | | {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| |+ '''Table 3. Simple Qualifiers of Propositions (''n'' = 2)''' | | |+ '''Table 3. Simple Qualifiers of Propositions (''n'' = 2)''' |
| |- style="background:ghostwhite" | | |- style="background:ghostwhite" |
− | | align="right" | <math>p:</math><br><math>q:</math> | + | | align="right" | <math>u:</math><br><math>v:</math> |
| | 1100<br>1010 | | | 1100<br>1010 |
| | <math>f\!</math> | | | <math>f\!</math> |
− | | <math>(\ell_{11})</math><br><math>\text{No } p </math><br><math>\text{is } q </math> | + | | <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math> |
− | | <math>(\ell_{10})</math><br><math>\text{No } p </math><br><math>\text{is }(q)</math> | + | | <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math> |
− | | <math>(\ell_{01})</math><br><math>\text{No }(p)</math><br><math>\text{is } q </math> | + | | <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math> |
− | | <math>(\ell_{00})</math><br><math>\text{No }(p)</math><br><math>\text{is }(q)</math> | + | | <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math> |
− | | <math> \ell_{00} </math><br><math>\text{Some }(p)</math><br><math>\text{is }(q)</math> | + | | <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math> |
− | | <math> \ell_{01} </math><br><math>\text{Some }(p)</math><br><math>\text{is } q </math> | + | | <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math> |
− | | <math> \ell_{10} </math><br><math>\text{Some } p </math><br><math>\text{is }(q)</math> | + | | <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math> |
− | | <math> \ell_{11} </math><br><math>\text{Some } p </math><br><math>\text{is } q </math> | + | | <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math> |
| |- | | |- |
− | | <math>f_0</math> || 0000 || <math>(~)</math> | + | | <math>f_0</math> |
− | | 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0 | + | | 0000 |
| + | | <math>(~)</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_1</math> || 0001 || <math>(p)(q)\!</math> | + | | <math>f_1</math> |
− | | 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0 | + | | 0001 |
| + | | <math>(u)(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_2</math> || 0010 || <math>(p) q\!</math> | + | | <math>f_2</math> |
− | | 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0 | + | | 0010 |
| + | | <math>(u) v\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_3</math> || 0011 || <math>(p)\!</math> | + | | <math>f_3</math> |
− | | 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0 | + | | 0011 |
| + | | <math>(u)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_4</math> || 0100 || <math>p (q)\!</math> | + | | <math>f_4</math> |
− | | 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0 | + | | 0100 |
| + | | <math>u (v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_5</math> || 0101 || <math>(q)\!</math> | + | | <math>f_5</math> |
− | | 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0 | + | | 0101 |
| + | | <math>(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_6</math> || 0110 || <math>(p, q)\!</math> | + | | <math>f_6</math> |
− | | 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0 | + | | 0110 |
| + | | <math>(u, v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_7</math> || 0111 || <math>(p q)\!</math> | + | | <math>f_7</math> |
− | | 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0 | + | | 0111 |
| + | | <math>(u v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| |- | | |- |
− | | <math>f_8</math> || 1000 || <math>p q\!</math> | + | | <math>f_8</math> |
− | | 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1 | + | | 1000 |
| + | | <math>u v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_9</math> || 1001 || <math>((p, q))\!</math> | + | | <math>f_9</math> |
− | | 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1 | + | | 1001 |
| + | | <math>((u, v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{10}</math> || 1010 || <math>q\!</math> | + | | <math>f_{10}</math> |
− | | 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 | + | | 1010 |
| + | | <math>v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{11}</math> || 1011 || <math>(p (q))\!</math> | + | | <math>f_{11}</math> |
− | | 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1 | + | | 1011 |
| + | | <math>(u (v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{12}</math> || 1100 || <math>p\!</math> | + | | <math>f_{12}</math> |
− | | 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 | + | | 1100 |
| + | | <math>u\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{13}</math> || 1101 || <math>((p) q)\!</math> | + | | <math>f_{13}</math> |
− | | 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1 | + | | 1101 |
| + | | <math>((u) v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{14}</math> || 1110 || <math>((p)(q))\!</math> | + | | <math>f_{14}</math> |
− | | 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1 | + | | 1110 |
| + | | <math>((u)(v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |- | | |- |
− | | <math>f_{15}</math> || 1111 || <math>((~))</math> | + | | <math>f_{15}</math> |
− | | 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 | + | | 1111 |
| + | | <math>((~))</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| |}<br> | | |}<br> |
| + | |
| + | ====Table 4==== |
| + | |
| + | {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 4. Simple Qualifiers of Propositions (''n'' = 2)''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>u:</math><br><math>v:</math> |
| + | | 1100<br>1010 |
| + | | <math>f\!</math> |
| + | | <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math> |
| + | | <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math> |
| + | | <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math> |
| + | | <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math> |
| + | | <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math> |
| + | | <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math> |
| + | | <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math> |
| + | | <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math> |
| + | |- |
| + | | <math>f_0</math> |
| + | | 0000 |
| + | | <math>(~)</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_1</math> |
| + | | 0001 |
| + | | <math>(u)(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_2</math> |
| + | | 0010 |
| + | | <math>(u) v\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_4</math> |
| + | | 0100 |
| + | | <math>u (v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_8</math> |
| + | | 1000 |
| + | | <math>u v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_3</math> |
| + | | 0011 |
| + | | <math>(u)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{12}</math> |
| + | | 1100 |
| + | | <math>u\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_6</math> |
| + | | 0110 |
| + | | <math>(u, v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_9</math> |
| + | | 1001 |
| + | | <math>((u, v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_5</math> |
| + | | 0101 |
| + | | <math>(v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{10}</math> |
| + | | 1010 |
| + | | <math>v\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_7</math> |
| + | | 0111 |
| + | | <math>(u v)\!</math> |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | |- |
| + | | <math>f_{11}</math> |
| + | | 1011 |
| + | | <math>(u (v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{13}</math> |
| + | | 1101 |
| + | | <math>((u) v)\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{14}</math> |
| + | | 1110 |
| + | | <math>((u)(v))\!</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |- |
| + | | <math>f_{15}</math> |
| + | | 1111 |
| + | | <math>((~))</math> |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:white; color:black" | 0 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | | style="background:black; color:white" | 1 |
| + | |}<br> |
| + | |
| + | ====Table 5==== |
| | | |
| {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | | {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
− | |+ '''Table 4. Relation of Quantifiers to Higher Order Propositions''' | + | |+ '''Table 5. Relation of Quantifiers to Higher Order Propositions''' |
| |- style="background:ghostwhite" | | |- style="background:ghostwhite" |
| | <math>\text{Mnemonic}</math> | | | <math>\text{Mnemonic}</math> |
Line 710: |
Line 1,587: |
| | <math>\text{E}\!</math><br><math>\text{Exclusive}</math> | | | <math>\text{E}\!</math><br><math>\text{Exclusive}</math> |
| | <math>\text{Universal}</math><br><math>\text{Negative}</math> | | | <math>\text{Universal}</math><br><math>\text{Negative}</math> |
− | | <math>\text{All}\ p\ \text{is}\ (q)</math> | + | | <math>\text{All}\ u\ \text{is}\ (v)</math> |
| | | | | |
− | | <math>\text{No}\ p\ \text{is}\ q </math> | + | | <math>\text{No}\ u\ \text{is}\ v </math> |
| | <math>(\ell_{11})</math> | | | <math>(\ell_{11})</math> |
| |- | | |- |
| | <math>\text{A}\!</math><br><math>\text{Absolute}</math> | | | <math>\text{A}\!</math><br><math>\text{Absolute}</math> |
| | <math>\text{Universal}</math><br><math>\text{Affirmative}</math> | | | <math>\text{Universal}</math><br><math>\text{Affirmative}</math> |
− | | <math>\text{All}\ p\ \text{is}\ q </math> | + | | <math>\text{All}\ u\ \text{is}\ v </math> |
| | | | | |
− | | <math>\text{No}\ p\ \text{is}\ (q)</math> | + | | <math>\text{No}\ u\ \text{is}\ (v)</math> |
| | <math>(\ell_{10})</math> | | | <math>(\ell_{10})</math> |
| |- | | |- |
| | | | | |
| | | | | |
− | | <math>\text{All}\ q\ \text{is}\ p </math> | + | | <math>\text{All}\ v\ \text{is}\ u </math> |
− | | <math>\text{No}\ q\ \text{is}\ (p)</math> | + | | <math>\text{No}\ v\ \text{is}\ (u)</math> |
− | | <math>\text{No}\ (p)\ \text{is}\ q </math> | + | | <math>\text{No}\ (u)\ \text{is}\ v </math> |
| | <math>(\ell_{01})</math> | | | <math>(\ell_{01})</math> |
| |- | | |- |
| | | | | |
| | | | | |
− | | <math>\text{All}\ (q)\ \text{is}\ p </math> | + | | <math>\text{All}\ (v)\ \text{is}\ u </math> |
− | | <math>\text{No}\ (q)\ \text{is}\ (p)</math> | + | | <math>\text{No}\ (v)\ \text{is}\ (u)</math> |
− | | <math>\text{No}\ (p)\ \text{is}\ (q)</math> | + | | <math>\text{No}\ (u)\ \text{is}\ (v)</math> |
| | <math>(\ell_{00})</math> | | | <math>(\ell_{00})</math> |
| |- | | |- |
| | | | | |
| | | | | |
− | | <math>\text{Some}\ (p)\ \text{is}\ (q)</math> | + | | <math>\text{Some}\ (u)\ \text{is}\ (v)</math> |
| | | | | |
− | | <math>\text{Some}\ (p)\ \text{is}\ (q)</math> | + | | <math>\text{Some}\ (u)\ \text{is}\ (v)</math> |
| | <math>\ell_{00}\!</math> | | | <math>\ell_{00}\!</math> |
| |- | | |- |
| | | | | |
| | | | | |
− | | <math>\text{Some}\ (p)\ \text{is}\ q</math> | + | | <math>\text{Some}\ (u)\ \text{is}\ v</math> |
| | | | | |
− | | <math>\text{Some}\ (p)\ \text{is}\ q</math> | + | | <math>\text{Some}\ (u)\ \text{is}\ v</math> |
| | <math>\ell_{01}\!</math> | | | <math>\ell_{01}\!</math> |
| |- | | |- |
| | <math>\text{O}\!</math><br><math>\text{Obtrusive}</math> | | | <math>\text{O}\!</math><br><math>\text{Obtrusive}</math> |
| | <math>\text{Particular}</math><br><math>\text{Negative}</math> | | | <math>\text{Particular}</math><br><math>\text{Negative}</math> |
− | | <math>\text{Some}\ p\ \text{is}\ (q)</math> | + | | <math>\text{Some}\ u\ \text{is}\ (v)</math> |
| | | | | |
− | | <math>\text{Some}\ p\ \text{is}\ (q)</math> | + | | <math>\text{Some}\ u\ \text{is}\ (v)</math> |
| | <math>\ell_{10}\!</math> | | | <math>\ell_{10}\!</math> |
| |- | | |- |
| | <math>\text{I}\!</math><br><math>\text{Indefinite}</math> | | | <math>\text{I}\!</math><br><math>\text{Indefinite}</math> |
| | <math>\text{Particular}</math><br><math>\text{Affirmative}</math> | | | <math>\text{Particular}</math><br><math>\text{Affirmative}</math> |
− | | <math>\text{Some}\ p\ \text{is}\ q</math> | + | | <math>\text{Some}\ u\ \text{is}\ v</math> |
| | | | | |
− | | <math>\text{Some}\ p\ \text{is}\ y</math> | + | | <math>\text{Some}\ u\ \text{is}\ v</math> |
| | <math>\ell_{11}\!</math> | | | <math>\ell_{11}\!</math> |
| |}<br> | | |}<br> |
| | | |
− | ====Exercises====
| + | ===Exercises=== |
| | | |
| Express the following formulas in functional terms. | | Express the following formulas in functional terms. |
| | | |
− | =====Exercise 1=====
| + | ====Exercise 1==== |
| | | |
| <blockquote> | | <blockquote> |
− | <math>(\forall x \in X)(p(x) \Rightarrow q(x))</math> | + | <math>(\forall x \in X)(u(x) \Rightarrow v(x))</math> |
| </blockquote> | | </blockquote> |
| | | |
| <blockquote> | | <blockquote> |
− | <math>\prod_{x \in X} (p_x (q_x)) = 1</math> | + | <math>\prod_{x \in X} (u_x (v_x)) = 1</math> |
| </blockquote> | | </blockquote> |
| | | |
− | This is just the form <math>\operatorname{All}\ p\ \operatorname{are}\ q,</math> already covered here: | + | This is just the form <math>\operatorname{All}\ u\ \operatorname{are}\ v,</math> already covered here: |
| | | |
| : [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory#Application_of_Higher_Order_Propositions_to_Quantification_Theory|Application of Higher Order Propositions to Quantification Theory]] | | : [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory#Application_of_Higher_Order_Propositions_to_Quantification_Theory|Application of Higher Order Propositions to Quantification Theory]] |
| | | |
− | Need to think a little more about the proposition <math>p \Rightarrow q</math> as a boolean function of type <math>\mathbb{B}^2 \to \mathbb{B}</math> and the corresponding higher order proposition of type <math>(\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math> | + | Need to think a little more about the proposition <math>u \Rightarrow v</math> as a boolean function of type <math>\mathbb{B}^2 \to \mathbb{B}</math> and the corresponding higher order proposition of type <math>(\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math> |
| | | |
− | =====Exercise 2=====
| + | ====Exercise 2==== |
| | | |
| <blockquote> | | <blockquote> |
Line 791: |
Line 1,668: |
| </blockquote> | | </blockquote> |
| | | |
− | =====Exercise 3=====
| + | ====Exercise 3==== |
| | | |
| <blockquote> | | <blockquote> |
| <math>(\forall x \in X)(Px \Rightarrow Qx) \lor (\forall x \in X)(Qx \Rightarrow Px)</math> | | <math>(\forall x \in X)(Px \Rightarrow Qx) \lor (\forall x \in X)(Qx \Rightarrow Px)</math> |
| </blockquote> | | </blockquote> |