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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)

Revision as of 20:18, 24 March 2010

77 bytes added ,  20:18, 24 March 2010
→‎Analysis of contingent propositions: renumb figs
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| [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (30)
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| [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (α)
 
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Since the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation.  There are in fact two different ways to execute the picture.
 
Since the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation.  There are in fact two different ways to execute the picture.
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Figure&nbsp;31 indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}</math> has the value 1, here interpreted as the logical value <math>\operatorname{true}.</math>  In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance:  Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded.  In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},</math> but in the pattern of regions that they indicate.
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Figure&nbsp;&beta; indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}</math> has the value 1, here interpreted as the logical value <math>\operatorname{true}.</math>  In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance:  Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded.  In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},</math> but in the pattern of regions that they indicate.
    
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| [[Image:Venn Diagram (P (Q)) (P (R)).jpg|500px]] || (31)
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| [[Image:Venn Diagram (P (Q)) (P (R)).jpg|500px]] || (&beta;)
 
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| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}</math>
 
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}</math>
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The easiest way to see the sense of the venn diagram is to notice that the expression <math>\texttt{(} p \texttt{(} q \texttt{))},</math> read as <math>p \Rightarrow q,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ q {}^{\prime\prime}.</math>  Its assertion effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>Q.\!</math>  In a similar manner, the expression <math>\texttt{(} p \texttt{(} r \texttt{))},</math> read as <math>p \Rightarrow r,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ r {}^{\prime\prime}.</math>  Asserting it effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>R.\!</math>
 
The easiest way to see the sense of the venn diagram is to notice that the expression <math>\texttt{(} p \texttt{(} q \texttt{))},</math> read as <math>p \Rightarrow q,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ q {}^{\prime\prime}.</math>  Its assertion effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>Q.\!</math>  In a similar manner, the expression <math>\texttt{(} p \texttt{(} r \texttt{))},</math> read as <math>p \Rightarrow r,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ r {}^{\prime\prime}.</math>  Asserting it effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>R.\!</math>
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Figure&nbsp;32 shows the other standard way of drawing a venn diagram for such a proposition.  In this ''punctured soap film'' style of picture &mdash; others may elect to give it the more dignified title of a ''logical quotient topology'' &mdash; one begins with Figure&nbsp;31 and then proceeds to collapse the fiber of 0 under <math>X\!</math> down to the point of vanishing utterly from the realm of active contemplation, arriving at the following picture:
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Figure&nbsp;&gamma; shows the other standard way of drawing a venn diagram for such a proposition.  In this ''punctured soap film'' style of picture &mdash; others may elect to give it the more dignified title of a ''logical quotient topology'' &mdash; one begins with Figure&nbsp;31 and then proceeds to collapse the fiber of 0 under <math>X\!</math> down to the point of vanishing utterly from the realm of active contemplation, arriving at the following picture:
    
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| [[Image:Venn Diagram (P (Q R)).jpg|500px]] || (32)
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| [[Image:Venn Diagram (P (Q R)).jpg|500px]] || (&gamma;)
 
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| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math>
 
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math>
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| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)).jpg|500px]] || (33)
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| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)).jpg|500px]] || (&delta;)
 
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| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 1.jpg|500px]]
 
| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 1.jpg|500px]]
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| (34)
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| (&epsilon;)
 
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| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
 
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
 
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| (35)
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| (&zeta;)
 
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| [[Image:Logical Graph (P (Q)) (P (R)) DNF.jpg|500px]]
 
| [[Image:Logical Graph (P (Q)) (P (R)) DNF.jpg|500px]]
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| (36)
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| (&eta;)
 
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</pre>
 
</pre>
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| (37)
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| (&theta;)
 
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| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
 
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
 
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| (38)
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| (&iota;)
 
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| [[Image:Logical Graph (P Q R , (P)).jpg|500px]] || (39)
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| [[Image:Logical Graph (P Q R , (P)).jpg|500px]] || (&kappa;)
 
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| [[Image:Logical Graph ((P , P Q R)).jpg|500px]] || (40)
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| [[Image:Logical Graph ((P , P Q R)).jpg|500px]] || (&lambda;)
 
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</pre>
 
</pre>
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| (41)
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| (&mu;)
 
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</pre>
 
</pre>
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| (42)
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| (&nu;)
 
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</pre>
 
</pre>
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| (43)
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| (&xi;)
 
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| [[Image:Logical Graph (( (P (Q)) (P (R)) , (P (Q R)) )).jpg|500px]]
 
| [[Image:Logical Graph (( (P (Q)) (P (R)) , (P (Q R)) )).jpg|500px]]
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| (44)
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| (&omicron;)
 
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| [[Image:Equational Inference Bar -- QED.jpg|500px]]
 
| [[Image:Equational Inference Bar -- QED.jpg|500px]]
 
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| (45)
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| (&pi;)
 
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Jon Awbrey
12,080

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