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Revision as of 12:46, 14 December 2008

32,917 bytes added ,  12:46, 14 December 2008
→‎Tables: back to ordinary parentheses for function notation
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==Inquiry Into Intuitionism==
 
==Inquiry Into Intuitionism==
   −
Notes on a discussion with "Gribskoff" ([http://planetmath.org/?op=getuser&id=21395 Manuel S. Lourenço]) about his artile on [http://planetmath.org/encyclopedia/IntuitionisticLogic.html Intuitionistic Logic] at [http://planetmath.org/ PlanetMath].
+
Notes on a discussion with "Gribskoff" ([http://planetmath.org/?op=getuser&id=21395 Manuel S. Lourenço]) about his article on [http://planetmath.org/encyclopedia/IntuitionisticLogic.html Intuitionistic Logic] at [http://planetmath.org/ PlanetMath].
    
<br>
 
<br>
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Notes on a couple of discussions that I found in the Foundations Of Mathematics Archives (FOMA) about building bridges between classical-apagogical and constructive-intuitionsitic mathematics.
 
Notes on a couple of discussions that I found in the Foundations Of Mathematics Archives (FOMA) about building bridges between classical-apagogical and constructive-intuitionsitic mathematics.
   −
<pre>
+
===Background===
−
Background --
     −
AM = A Mani
+
: AM = A Mani
−
HF = Harvey Friedman
+
: HF = Harvey Friedman
−
NT = Neil Tennant
+
: NT = Neil Tennant
−
SS = Stephen G Simpson
+
: SS = Stephen G Simpson
−
TF = Torkel Franzen
+
: TF = Torkel Franzen
−
VP = Vaughan Pratt
+
: VP = Vaughan Pratt
   −
Feb 1998, Intuitionistic Mathematics and Building Bridges
+
====Feb 1998 : Intuitionistic Mathematics and Building Bridges====
−
http://www.cs.nyu.edu/pipermail/fom/1998-February/thread.html#1160
  −
NT: http://www.cs.nyu.edu/pipermail/fom/1998-February/001160.html
  −
TF: http://www.cs.nyu.edu/pipermail/fom/1998-February/001162.html
  −
SS: http://www.cs.nyu.edu/pipermail/fom/1998-February/001246.html
  −
VP: http://www.cs.nyu.edu/pipermail/fom/1998-February/001248.html
     −
Oct 2008, Classical/Constructive Mathematics
+
* http://www.cs.nyu.edu/pipermail/fom/1998-February/thread.html#1160
−
http://www.cs.nyu.edu/pipermail/fom/2008-October/thread.html#13127
+
# NT: http://www.cs.nyu.edu/pipermail/fom/1998-February/001160.html
−
HF: http://www.cs.nyu.edu/pipermail/fom/2008-October/013127.html
+
# TF: http://www.cs.nyu.edu/pipermail/fom/1998-February/001162.html
−
AM: http://www.cs.nyu.edu/pipermail/fom/2008-October/013142.html
+
# SS: http://www.cs.nyu.edu/pipermail/fom/1998-February/001246.html
 +
# VP: http://www.cs.nyu.edu/pipermail/fom/1998-February/001248.html
   −
Foreground --
+
====Oct 2008 : Classical &and;&or; Constructive Mathematics====
   −
Re: Classical/Constructive Mathematics
+
* http://www.cs.nyu.edu/pipermail/fom/2008-October/thread.html#13127
−
    Harvey Friedman (15 Oct 2008, 00:36:36 EDT)
+
# HF: http://www.cs.nyu.edu/pipermail/fom/2008-October/013127.html
 +
# AM: http://www.cs.nyu.edu/pipermail/fom/2008-October/013142.html
   −
HF: There seems to be a resurgence of interest in comparisons between
+
===Foreground===
−
    classical and constructive (foundations of) mathematics.  This is
  −
    a topic that has been discussed quite a lot previously on the FOM.
  −
    I have been an active participant in prior discussions.
     −
HF: There was a lot of basic information presented earlier, and I think
+
Harvey Friedman (15 Oct 2008), "Classical/Constructive Mathematics", FOMA.
−
    that it would be best to restate some of this, so that the discussion
  −
    can go forward with its benefit.
     −
HF: In this message, I would like to focus on some important ways in which
+
<blockquote>
−
    classical and constructive foundations are alike or closely related.
+
<p>There seems to be a resurgence of interest in comparisons betweenclassical and constructive (foundations of) mathematics.  This is a topic that has been discussed quite a lot previously on the FOM.  I have been an active participant in prior discussions.</p>
   −
HF: For many formal systems for fragments of classical mathematics, T,
+
<p>There was a lot of basic information presented earlier, and I think that it would be best to restate some of this, so that the discussion can go forward with its benefit.</p>
−
    there is a corresponding system T' obtained by merely restricting
  −
    the classical logical axioms to constructive logical axioms - where
  −
    the resulting system is readily acceptable as a formal system for
  −
    a "corresponding" fragment of constructive mathematics. Of course,
  −
    there may be good ways of restating the axioms in the classical system,
  −
    which do NOT lead to any reasonable fragment of constructive mathematics
  −
    in this way.
     −
HF: The most well known example of this is PA = Peano Arithmetic.  Suppose
+
<p>In this message, I would like to focus on some important ways in which classical and constructive foundations are alike or closely related.</p>
−
    we formalize PA in the most common way, with the axioms for successor,
  −
    the defining axioms for addition and multiplication, and the axiom
  −
    scheme of induction, with the usual axioms and rules of classical
  −
    logic. Then HA = Heyting Arithmetic, is simply PA with the axioms
  −
    and rules of classical logic weakened to the axioms and rules of
  −
    constructive logic.
     −
HF: Why do we consider HA as being a reasonable constructive system?
+
<p>For many formal systems for fragments of classical mathematics, T, there is a corresponding system T' obtained by merely restricting the classical logical axioms to constructive logical axioms &mdash; where the resulting system is readily acceptable as a formal system for a "corresponding" fragment of constructive mathematics. Of course, there may be good ways of restating the axioms in the classical system, which do NOT lead to any reasonable fragment of constructive mathematics in this way.</p>
−
    A common answer is simply that a constructivist reads the axioms
  −
    as "true" or "valid".
     −
HF: An apparently closely related fact about HA is purely formal.
+
<p>The most well known example of this is PA = Peano Arithmetic. Suppose we formalize PA in the most common way, with the axioms for successor, the defining axioms for addition and multiplication, and the axiom scheme of induction, with the usual axioms and rules of classical logic. Then HA = Heyting Arithmetic, is simply PA with the axioms and rules of classical logic weakened to the axioms and rules of constructive logic.</p>
−
    HA possesses a great number of properties that are commonly
  −
    associated with "constructivism".  The early pioneering work
  −
    along these lines is, if I remember correctly, due to S.C. Kleene.
  −
    Members of this list should be able to supply really good references
  −
    for this work, better than I can. PA possesses NONE of these properties.
     −
HF: RESEARCH PROBLEM: Is there such a thing as a complete list of such
+
<p>Why do we consider HA as being a reasonable constructive system? A common answer is simply that a constructivist reads the axioms as "true" or "valid".</p>
−
    formal properties? Is there a completeness theorem along these lines?
  −
    I.e., can we state and prove that HA obeys all such (good from the
  −
    constructive viewpoint) properties?
     −
HF: On the other hand, we can formalize PA, equivalently, using the
+
<p>An apparently closely related fact about HA is purely formal. HA possesses a great number of properties that are commonly associated with "constructivism". The early pioneering work along these lines is, if I remember correctly, due to S.C. Kleene.  Members of this list should be able to supply really good references for this work, better than I can.  PA possesses NONE of these properties.</p>
−
    *least number principle scheme* instead of the induction scheme.
  −
    If a property holds of n, then that property holds of a least n.
  −
    Then, when we convert to constructive logic, we get a system PA#
  −
    that is equivalent to PA - thus possessing none of these properties!
     −
HF: For many of these T,T' pairs, some very interesting relationships
+
<p>RESEARCH PROBLEM: Is there such a thing as a complete list of such formal properties?  Is there a completeness theorem along these lines?  I.e., can we state and prove that HA obeys all such (good from the constructive viewpoint) properties?</p>
−
    obtain between the T and T'. Here are three important ones.
     −
HF: 1.  It can be proved that T is consistent if and only if T'
+
<p>On the other hand, we can formalize PA, equivalently, using the *least number principle scheme* instead of the induction scheme.  If a property holds of n, then that property holds of a least n.  Then, when we convert to constructive logic, we get a system PA# that is equivalent to PA &mdash; thus possessing none of these properties!</p>
−
        is consistent.
     −
HF: 2.  Every A...A sentence, whose matrix has only bounded quantifiers,
+
<p>For many of these T,T' pairs, some very interesting relationships obtain between the T and T'. Here are three important ones.</p>
−
        that is provable in T, is already provable in T'.
     −
HF: 3.  More strongly, every A...AE...E sentence, whose matrix has only
+
<p>1.  It can be proved that T is consistent if and only if T' is consistent.</p>
−
        bounded quantifiers, that is provable in T, is already provable
  −
        in T'.
     −
HF: The issue arises as to just where these proofs are carried out - e.g.,
+
<p>2. Every A&hellip;A sentence, whose matrix has only bounded quantifiers, that is provable in T, is already provable in T'.</p>
−
    constructively or classically. This is particularly critical in the
  −
    case of 1. The situation is about as "convincing" as possible:
     −
HF: Specifically, for each of these results, one can use weak quantifier
+
<p>3.  More strongly, every A&hellip;AE&hellip;E sentence, whose matrix has only bounded quantifiers, that is provable in T, is already provable in T'.</p>
−
    free systems K of arithmetic, where constructive and classical amount
  −
    to the same. E.g., for 1, there is a primitive operation in K which,
  −
    provably in K, converts any inconsistency in T to a corresponding
  −
    inconsistency in T'.
     −
HF: Results like 1 point in the direction of there being no difference
+
<p>The issue arises as to just where these proofs are carried out &mdash; e.g., constructively or classically.  This is particularly critical in the case of 1.  The situation is about as "convincing" as possible:</p>
−
    between the "safety" of classical and constructive mathematics.
     −
HF: Results like 2,3 point in the direction of there being no difference
+
<p>Specifically, for each of these results, one can use weak quantifier free systems K of arithmetic, where constructive and classical amount to the same.  E.g., for 1, there is a primitive operation in K which, provably in K, converts any inconsistency in T to a corresponding inconsistency in T'.</p>
−
    between the "applicability" of classical and constructive mathematics,
  −
    in many contexts.
     −
HF: CAUTION:  For AEA sentences, PA and HA differ. There are some
+
<p>Results like 1 point in the direction of there being no difference between the "safety" of classical and constructive mathematics.</p>
−
    celebrated A...AE...EA...A theorems of PA which are not known
  −
    to be provable in HA. Some examples were discussed previously
  −
    on the FOM.
     −
HF: RESEARCH PROBLEM: Determine, in some readily intelligible terms
+
<p>Results like 2,3 point in the direction of there being no difference between the "applicability" of classical and constructive mathematics, in many contexts.</p>
−
    (perhaps classical), necessary and sufficient conditions for
  −
    a sentence of a given form is provable in HA and PA. Matters
  −
    get delicate when there are several quantifiers and arrows (-->)
  −
    present.
     −
HF: I will continue with this if sufficient responses are generated.
+
<p>CAUTION: For AEA sentences, PA and HA differ.  There are some celebrated A&hellip;AE&hellip;EA&hellip;A theorems of PA which are not known to be provable in HA.  Some examples were discussed previously on the FOM.</p>
   −
I, too, find myself returning to questions about classical v. constructive logic
+
<p>RESEARCH PROBLEM:  Determine, in some readily intelligible terms (perhaps classical), necessary and sufficient conditions for a sentence of a given form is provable in HA and PA.  Matters get delicate when there are several quantifiers and arrows (-->) present.</p>
−
lately, partly in connection with Peirce's Law, the Propositions As Types (PAT)
  −
analogy, the question of a PAT analogy for classical propositional calculus,
  −
and the eternal project of integrating functional, relational, and logical
  −
styles of programming as much as possible.
     −
I am still in the phase of chasing down links between the various questions
+
<p>I will continue with this if sufficient responses are generated.</p>
−
and I don't have any news or conclusions to offer, but my web searches keep
+
</blockquote>
−
bringing me back to this old discussion on the FOM list:
     −
http://www.cs.nyu.edu/pipermail/fom/1998-February/thread.html#1160
+
I, too, find myself returning to questions about classical v. constructive logic lately, partly in connection with Peirce's Law, the Propositions As Types (PAT) analogy, the question of a PAT analogy for classical propositional calculus, and the eternal project of integrating functional, relational, and logical styles of programming as much as possible.
   −
I find one comment by Vaughan Pratt to be especially e-&/or-pro-vocative:
+
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:
   −
VP: It has been my impression from having dealt with a lot of lawyers over the
+
[http://www.cs.nyu.edu/pipermail/fom/1998-February/thread.html#1160 Intuitionistic Mathematics and Building Bridges].
−
    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
+
I find one comment by Vaughan Pratt to be especially e-&and;/&or;-pro-vocative:
−
    that go along with whichever scientific method is currently in
  −
    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,
+
Vaughn Pratt (27 Feb 1998), "Intuitionistic Mathematics and Building Bridges", FOMA.
−
    ((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,
+
<blockquote>
−
here is one of my current write-ups on
+
<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>
−
Peirce's Law that led me back into this
+
 
−
old briar patch:
+
<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>The very *thought* of a lawyer or scientist appealing to Peirce's law, ((P&rarr;Q)&rarr;P)&rarr;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>
 +
</blockquote>
 +
 
 +
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==
 +
 
 +
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:
   −
====Tables====
+
{| align="center" cellpadding="8"
 +
| <math>\Upsilon (u) = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{X \to \mathbb{B}}.</math>
 +
|}
   −
The auxiliary notations:
+
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>\alpha_i f = \Upsilon (f_i, f),\!</math>
+
{| align="center" cellpadding="8"
 +
| <math>\Upsilon (u, v) = 1 \quad \Leftrightarrow \quad u \Rightarrow v.</math>
 +
|}
   −
: <math>\beta_i f = \Upsilon (f, f_i),\!</math>
+
===Tables===
   −
define two series of measures:
+
Define two families of measures:
   −
: <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B},</math>
+
{| align="center" cellpadding="8"
 +
| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 0 \ldots 15,</math>
 +
|}
   −
incidentally providing compact names for the column headings of the next two Tables.
+
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&nbsp;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:&nbsp; <math>\alpha_i f = \Upsilon (f_i \Rightarrow f)</math>'''
+
|+ '''Table 1.  Qualifiers of Implication Ordering:&nbsp; <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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0000
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0001
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0010
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0011
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0100
−
|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0101
−
|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0110
−
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 1000
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
| 1001
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
| 1010
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1    || 1    || 1
+
| 1011
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 1100
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;|| 1   || 1    ||&nbsp;||&nbsp;|| 1   || 1
+
| 1101
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 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>
−
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| 1110
−
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 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&nbsp;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:&nbsp; <math>\beta_i f = \Upsilon (f \Rightarrow f_i)</math>'''
+
|+ '''Table 2.  Qualifiers of Implication Ordering:&nbsp; <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 626: 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>
−
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| 0001
−
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 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>
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| 0010
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 0011
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| 0100
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 1
+
| 0101
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1
+
| 0110
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 0111
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1001
−
|&nbsp;|| 1   ||&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1010
−
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1011
−
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1100
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1101
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1110
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1111
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 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 767: 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>
 
| &nbsp;
 
| &nbsp;
−
| <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>
 
| &nbsp;
 
| &nbsp;
−
| <math>\text{No}\  p\ \text{is}\ (q)</math>
+
| <math>\text{No}\  u\ \text{is}\ (v)</math>
 
| <math>(\ell_{10})</math>
 
| <math>(\ell_{10})</math>
 
|-
 
|-
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
−
| <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>
 
|-
 
|-
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
−
| <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>
 
|-
 
|-
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
−
| <math>\text{Some}\ (p)\ \text{is}\ (q)</math>
+
| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
 
| &nbsp;
 
| &nbsp;
−
| <math>\text{Some}\ (p)\ \text{is}\ (q)</math>
+
| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
 
| <math>\ell_{00}\!</math>
 
| <math>\ell_{00}\!</math>
 
|-
 
|-
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
−
| <math>\text{Some}\ (p)\ \text{is}\ q</math>
+
| <math>\text{Some}\ (u)\ \text{is}\ v</math>
 
| &nbsp;
 
| &nbsp;
−
| <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>
 
| &nbsp;
 
| &nbsp;
−
| <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>
 
| &nbsp;
 
| &nbsp;
−
| <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=====
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====Exercise 2====
    
<blockquote>
 
<blockquote>
Line 848: Line 1,668:  
</blockquote>
 
</blockquote>
   −
=====Exercise 3=====
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====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>
Jon Awbrey
12,080

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