By Kaufman R. M.

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This text discusses a few tools of describing and pertaining to mathematical items and of continuously and unambiguously signaling the logical constitution of mathematical arguments.

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**Example text**

Since the reﬂection symmetries about the selected axes are perceived, features entailed by those symmetries may also be perceived. 9. It is perceived as symmetrical about its vertical and horizontal axes. But it would not look symmetrical about the vertical axis unless its upper angles looked equal and its lower angles looked equal. It would not look symmetrical about the horizontal unless the angles on the left looked equal and the angles on the right looked equal. So perceiving these symmetries entails perceiving every pair of adjacent angles as equal.

12 The argument is that there may be nothing in reality answering to a concept (no reference or semantic value), in which case a general thought that issues from the concept will not be true. So in order to know that it is true, one must know that the concept has a reference, and that knowledge must have an evidential basis independent of the concept. As an example, consider Priestley’s concept of phlogiston. g. ‘‘x is combustible; ∴ x contains phlogiston’’. But nothing simultaneously satisﬁes all those inference forms; that is, whatever real thing (substance kind) we take as the reference of ‘‘phlogiston’’, not all of those inference forms will be truth preserving.

Perceptual state] 4. Anything perceived as perfectly square appears symmetric about its diagonals. [Category speciﬁcation for squares] 5. ∴ Figure a appears symmetric about its diagonals. [3, 4] 6. ∴ x is symmetrical about its diagonals. [2, 3, 5] 7. ∴ The parts of x either side of a diagonal are congruent. [6, by concepts for symmetry and congruence] 8. If x is a perfect square, the parts of x either side of a diagonal are congruent. [7, discharging the assumption] 9. ∴ The parts of any perfect square either side of a diagonal are congruent.

### A. F. Lavriks truncated equations by Kaufman R. M.

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