 The basic insight underlying Bolzano’s method of idea-variation is quite simple (WL II, 77 ff.). Let us take as our first example \(S_1\) the proposition [Kant is a German philosopher]. (In order to simplify matters linguistically, we do not adhere to Bolzano’s formulation ‘\(A\) has \(b\)’ but will allow also formulations of the kind ‘\(A\) is (a) \(B\)’. Moreover, we will take in what follows words like ‘German’, ‘French’, ‘European’, ‘American’ etc. in the sense of ‘born in Germany’, ‘born in France’, ‘born in Europe’, ‘born in America’ etc.) The extra-logical ideas that are parts of \(S_1\) are the ideas [Kant], [German], and [philosopher]; we consider now one or more of them to be variable in the sense that we think that they are replaced in \(S_1\) by other ideas fitting to the former ones (i.e., belonging to the same “category”). In this way the idea [Kant] can be “varied” in \(S_1\) and replaced, e.g., by [Hegel]; in other words, we can substitute [Hegel] for [Kant] in \(S_1\). The variation in question is a kind of replacement or substitution. It results in a “new” (or better: in another) proposition, viz. in the true proposition [Hegel is a German philosopher]; we will say that [Hegel is a German philosopher] is the [Hegel]/[Kant]-variant of \(S_1\). A false [Kant]-variant of \(S_1\) is its [Sartre]/[Kant]-variant [Sartre is a German philosopher]. Similarly, [Kant is an European philosopher] is a true and [Kant is an American philosopher] is a false [German]-variant of \(S_1\). The operation of replacement (or substitution) can also be performed on two or more parts of a proposition simultaneously: Replacing [Kant] and [philosopher] in \(S_1\) simultaneously by, e.g., [Gauss] and [mathematician], results in the true ([Gauss], [mathematician]/[Kant], [philosopher])-variant [Gauss is a German mathematician]. A false ([Kant], [philosopher])-variant of \(S_1\) is, e.g., [Sartre is a German musician]. We can also replace all extra-logical parts of \(S_1\) simultaneously: A true ([Kant], [German], [philosopher])-variant of \(S_1\) is [Mozart is an Austrian composer], whereas [Sartre is a Greek mathematician] is a false one. Following Bolzano, we will here use this generalized operation of simultaneous replacement (or variation). Given an arbitrary proposition \(s\) and two sequences \(i_1, i_2 ,\ldots ,i_n\) and \(j_1, j_2 ,\ldots ,j_n\) of ideas, a proposition \(s'\) is uniquely determined by this operation; due to this operation, for each \(k\) \((1 \le k \le n)\), the idea \(i_k\) is replaced in \(s\) uniformly (i.e., wherever it “occurs” in \(s)\) by one and the same corresponding idea \(j_k\). The resulting proposition \(s'\) is the \((j_1, j_2 ,\ldots ,j_n /i_1, i_2 ,\ldots ,i_n)\)-variant of \(s\), or — briefly put — the \(j/i\)-variant of \(s\). (Thereby we take \(i = \langle i_1, i_2 ,\ldots ,i_n\rangle\) and \(j = \langle j_1, j_2 ,\ldots ,j_n\rangle\). Moreover, we are using here ‘\(s\)’, ‘\(s'\)’, ‘\(s_1\)’, ‘\(s_2\)’,… as variables for propositions, ‘\(i_1\)’, ‘\(i_2\)’,…, ‘\(j_1\)’, ‘\(j_2\)’,… as variables for ideas, and ‘\(i\)’ and ‘\(j\)’ as variables for sequences of ideas of the same length.) The close relationship between a variant of a proposition and a substitution instance of a sentence is quite obvious. In order for the \(j/i\)-variant of an arbitrary proposition \(s\) to be uniquely determined and to fulfill certain criteria of adequacy, however, several restrictions are required: (i) Each of the \(i_k\)s \((1 \le k \le n)\) must be simple or at least “relatively” simple (in the sense that in each particular context under consideration they are not further analyzed into parts but “taken” to be simple); (ii) each of the \(i_k\)s is an extra-logical idea; (iii) the \(i_k\)s are pair-wise distinct; moreover (iv), in order to keep the result of the replacement operation “well-formed”, i.e., a genuine proposition, we must require that each \(j_k\) “fits” the corresponding \(i_k\), i.e., is of the same semantic category; and finally (v), we must also require that at least one of the ideas \(i_k\) must be contained in \(s\) as one of its parts so that the operation of replacement is never performed vacuously.