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Among other things, these images make it visually obvious that the constraint on the three boolean variables <math>p, q, r\!</math> that is indicated by asserting either of the forms <math>\texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}</math> or <math>p \le q \le r</math> implies a constraint on the two boolean variables <math>p, r\!</math> that is indicated by either of the forms <math>\texttt{(} p \texttt{(} r \texttt{))}</math> or <math>p \le r,</math> but that it imposes additional constraints on these variables that are not captured by the illative conclusion.

Among other things, these images make it visually obvious that the constraint on the three boolean variables <math>p, q, r\!</math> that is indicated by asserting either of the forms <math>\texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}</math> or <math>p \le q \le r</math> implies a constraint on the two boolean variables <math>p, r\!</math> that is indicated by either of the forms <math>\texttt{(} p \texttt{(} r \texttt{))}</math> or <math>p \le r,</math> but that it imposes additional constraints on these variables that are not captured by the illative conclusion.

One way to view a proposition ''f'' : '''B'''^''k'' &rarr; '''B''' is to consider its ''fiber of truth'', ''f''^&minus;1(1) &sube; '''B'''^''k'', and to regard it as a ''k''-adic relation ''L'' &sube; '''B'''^''k''.

One way to view a proposition <math>f : \mathbb{B}^k \to \mathbb{B}</math> is to consider its ''fiber of truth'', <math>f^{-1}(1) \subseteq \mathbb{B}^k,</math> and to regard it as a <math>k\!</math>-adic relation <math>L \subseteq \mathbb{B}^k.</math>

By way of general definition, the ''fiber'' of a function ''f'' : ''X'' &rarr; ''Y'' at a given value ''y'' of its co-domain ''Y'' is the ''antecedent'' (pre-image or inverse image) of ''y'' under ''f''.  This is a subset, possibly empty, of the domain ''X'', notated as ''f''^&minus;1(''y'') &sube; ''X''.

By way of general definition, the ''fiber'' of a function <math>f : X \to Y</math> at a given value <math>y\!</math> of its co-domain <math>Y\!</math> is the ''antecedent'' (also known as the ''inverse image'' or ''pre-image'') of <math>y\!</math> under <math>f.\!</math> This is a subset, possibly empty, of the domain <math>X,\!</math> notated as <math>f^{-1}(y) \subseteq X.</math>

In particular, if ''f'' is a proposition ''f'' : ''X'' &rarr; '''B''', then we think of ''f''^&minus;1(''y'') as the subset of ''X'' that is ''indicated'' by the proposition ''f''.  Whenever we ''assert'' a proposition ''f'' : ''X'' &rarr; '''B''', we are saying that what it indicates is all that happens to be the case in the relevant universe of discourse ''X''.  Because the special case of the fiber of truth is used so often in logical contexts, we will sometimes use the notation <nowiki>[|</nowiki>''f''<nowiki>|]</nowiki> = ''f''^&minus;1(1).

In particular, if ''f'' is a proposition ''f'' : ''X'' &rarr; '''B''', then we think of ''f''^&minus;1(''y'') as the subset of ''X'' that is ''indicated'' by the proposition ''f''.  Whenever we ''assert'' a proposition ''f'' : ''X'' &rarr; '''B''', we are saying that what it indicates is all that happens to be the case in the relevant universe of discourse ''X''.  Because the special case of the fiber of truth is used so often in logical contexts, we will sometimes use the notation <nowiki>[|</nowiki>''f''<nowiki>|]</nowiki> = ''f''^&minus;1(1).