Changes

|}<br>

|}<br>

Thus, &alpha;₀ = &beta;₁₅ is a totally indiscriminate measure, one that accepts all propositions ''f'' : '''B'''² &rarr; '''B''', whereas &alpha;₁₅ and &beta;₀ are measures that value the constant propositions '''1''' : '''B'''² &rarr; '''B''' and '''0''' : '''B'''² &rarr; '''B''', respectively, above all others.

Thus, <math>\alpha_0 = \beta_{15}\!</math> is a totally indiscriminate measure, one that accepts all propositions <math>f : \mathbb{B}^2 \to \mathbb{B},</math> whereas <math>\alpha_{15}\!</math> and <math>\beta_0\!</math> are measures that value the constant propositions <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> and <math>0 : \mathbb{B}^2 \to \mathbb{B},</math> respectively, above all others.

Finally, in conformity with the use of the fiber notation to indicate sets of models, it is natural to use notations like:

Finally, in conformity with the use of the fiber notation to indicate sets of models, it is natural to use notations like:

: [| &alpha;_''i'' |] = (&alpha;_''i'')^(–1)(1),

: <math>[| \alpha_i |] = (\alpha_i)^{-1}(1),\!</math>

: [| &beta;_''i'' |] = (&beta;_''i'')^(–1)(1),

: <math>[| \beta_i |] = (\beta_i)^{-1}(1),\!</math>

: [| &Upsilon;_''p'' |] = (&Upsilon;_''p'')^(–1)(1),

: <math>[| \Upsilon_p |] = (\Upsilon_p)^{-1}(1),\!</math>

to denote sets of propositions that satisfy the umpires in question.

to denote sets of propositions that satisfy the umpires in question.

===Extending the Existential Interpretation to Quantificational Logic===

===Extending the Existential Interpretation to Quantificational Logic===

Previously I introduced a calculus for propositional logic, fixing its meaning according to what C.S. Peirce called the "existential interpretation".  As far as it concerns propositional calculus this interpretation settles the meanings that are associated with merely the most basic symbols and logical connectives.  Now we must extend and refine the existential interpretation to comprehend the analysis of "quantifications", that is, quantified propositions.  In doing so we recognize two additional aspects of logic that need to be developed, over and above the material of propositional logic.  At the formal extreme there is the aspect of higher order functional types, into which we have already ventured a little above.  At the level of the fundamental content of the available propositions we have to introduce a different interpretation for what we may call "elemental" or "singular" propositions.

Previously I introduced a calculus for propositional logic, fixing its meaning according to what C.S.&nbsp;Peirce called the ''existential interpretation''.  As far as it concerns propositional calculus this interpretation settles the meanings that are associated with merely the most basic symbols and logical connectives.  Now we must extend and refine the existential interpretation to comprehend the analysis of ''quantifications'', that is, quantified propositions.  In doing so we recognize two additional aspects of logic that need to be developed, over and above the material of propositional logic.  At the formal extreme there is the aspect of higher order functional types, into which we have already ventured a little above.  At the level of the fundamental content of the available propositions we have to introduce a different interpretation for what we may call ''elemental'' or ''singular'' propositions.

Let us return to the 2-dimensional case ''X''° = [''x'', ''y''].  In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers ''L''_''uv'' : ('''B'''² &rarr; '''B''') &rarr; '''B''' that have the following characters:

Let us return to the 2-dimensional case ''X''° = [''x'', ''y''].  In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers ''L''_''uv'' : ('''B'''² &rarr; '''B''') &rarr; '''B''' that have the following characters: