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What changes when a matrix holds formulas?

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A numeric C M stores finished answers: zero or one.

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A logical matrix, or L M, stores expressions that still need values for their variables.

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We will build those expression-valued cells, evaluate them under one consistent assignment,

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and learn to distinguish the symbolic matrix from its numeric result.

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Place X and not X in a column. Place Y and not Y in a row.

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Here not reverses a Boolean value. Multiplication in this logical construction

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means AND: both factors must be true for the product to be true.

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Their outer product gives four expressions. The top-left cell is X and Y.

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The top-right is X and not Y. The bottom row uses not X instead.

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We write M sub X Y equals this whole two-by-two logical matrix.

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For the positive assignment used here, set the original variables X and Y to one.

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Then not X and not Y become zero. Apply that same assignment to every expression.

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We are evaluating one symbolic object consistently, cell by cell.

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The top-left expression becomes one and one, which is one. The top-right becomes one and zero,

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which is zero. Notice that the written expressions stay different
even though we gave the original variables the same value.

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Both bottom expressions contain not X, now zero, so both evaluate to zero.

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The resulting numeric matrix is the AND operator in true-first order.

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The paper describes this relationship as taking the positive valuation of the logical matrix.

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To represent equivalence, start from the equivalence logical matrix shown here.

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Its diagonal entries test agreement. Its off-diagonal entries test exclusive-or.

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We changed the expressions before evaluating them;
valuation alone did not change AND into equivalence.

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Again set X and Y to one. What does the upper-right exclusive-or expression evaluate to?

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And what does an equivalence expression on the diagonal evaluate to?

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Work out those two cases before viewing the completed matrix.

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Exclusive-or is zero because the inputs agree. Equivalence is one for the same reason.

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So the diagonal entries become one and the other entries become zero.

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The original expressions determine which operator the valuation produces.

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You now have two useful layers: expressions while constructing a logical matrix,

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and bits after evaluating it. Next we will combine two complete logical matrices.

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Keeping their names and their entries separate will
make that larger construction much easier to read.