PHIL 452 · Modal Logic
We use possible worlds models to interpret the language of modal logic, which consist of a set of possible worlds and an accessibility relation on them.
What counts as valid with respect to a family of possible worlds models turns on what features we stipulate the accessibility relation to have.
We will use the expression \(\langle a, b \rangle\) to denote the ordered pair of \(a\) and \(b\).
Ordered pairs are governed by the principle:
\[ \langle a, b \rangle = \langle c, d\rangle \ \text{ iff } \ a = c \text{ and } b = d \] Ordered pairs are generally defined in terms of unordered sets:
\(\langle x, y \rangle\) is the set \(\{ \{a\}, \{a, b\}\}\).
\(A \times B = \{\langle x, y\rangle: x \in A \wedge y \in B\}\)
Given a set \(A\), \(R\) is a binary relation on \(A\) iff \(R \subseteq A \times A\).
\(R\) is a binary relation iff \(R\) is a binary relation on a set.
\(R\) is reflexive on \(A\) iff for every \(x \in A\), \(Rxx\).
Let \(A = \{a, b\}\) and \(R = \{\langle a, a \rangle, \langle b, b\rangle\}\). Then \(R\) is reflexive on \(W\).
\(R\) is irreflexive on \(A\) iff for every \(x \in A\), \(\neg Rxx\).
Let \(A = \{a, b\}\) and \(R = \{\langle a, b \rangle, \langle b, a\rangle \}\). Then \(R\) is irreflexive on \(W\).
\(R\) is non-reflexive on \(A\) iff for some \(x \in A\), \(\neg Rxx\).
Fix \(W\) as the set of English words. Check whether each relation below is reflexive, irreflexive, or non-reflexive:
\(R\) is symmetric on \(A\) iff for every \(x, y \in A\), if \(Rxy\), then \(Ryx\).
Let \(A = \{a, b\}\) and \(R = \{\langle a, b \rangle, \langle b, a\rangle\}\). Then \(R\) is symmetric on \(W\).
\(R\) is asymmetric on \(A\) iff for every \(x, y \in A\), if \(Rxy\), then \(\neg Ryx\).
\(R\) is antisymmetric on \(A\) iff for every \(x, y \in A\), if \(Rxy\) and \(Ryx\), then \(x = y\).
Let \(A = \{a, b\}\) and \(R = \{\langle a, a \rangle, \langle b, b\rangle\}\). Then \(R\) is antisymmetric on \(W\).
Fix \(W\) as the set of English words. Check whether each relation below is symmetric, asymmetric, or antisymmetric:
\(R\) is transitive on \(A\) iff for every \(x, y, z\in A\), if \(Rxy\) and \(Ryz\), then \(Rxz\).
Let \(A = \{a, b, c\}\) and \(R = \{\langle a, b \rangle, \langle b, c\rangle, \langle a, c\rangle\}\). Then \(R\) is transitive on \(W\).
\(R\) is intransitive on \(A\) iff for every \(x, y \in A\), if \(Rxy\) and \(Ryz\), then \(\neg Rxz\).
Let \(A = \{a, b, c\}\) and \(R = \{\langle a, b \rangle, \langle b, c\rangle\}\). Then \(R\) is intransitive on \(W\).
\(R\) is non-transitive on \(A\) iff for some \(x, y, z \in A\), \(Rxy\) and \(Ryz\), but \(\neg Rxz\).
Fix \(W\) as the set of English words. Check whether each relation below is transitive, intransitive, or non-transitive:
Synonymy is an equivalence relation on \(W\).
If \(R\) is an equivalence relation on a set \(A\), for each \(x \in A\), the equivalence class of \(x\), written \([x]_R\) is the set of elements of \(A\) that are \(R\)-equivalent to \(x\): \[ [x]_R := \{y\in A: Rxy\} \]
If \(A\) is the set of undergraduate students at USC and we let \(R\) be \(\{\langle x, y\rangle: x \ \text{is in the same year as} \ y\}\), then if \(a\) is a freshman at \(USC\), then \([a]_R\) is the set of freshmen at USC.
\(R\) is euclidean on \(A\) iff for every \(x, y, z\in A\), if \(Rxy\) and \(Rxz\), then \(Ryz\).
Symmetry and transitivity on a set \(A\) do not entail reflexivity on \(A\).
Let \(A = \{a, b\}\) and \(R = \{\langle a, a \rangle\}\).
Not every euclidean relation on \(A\) is an equivalence relation on \(A\).
Let \(A = \{a, b\}\) and \(R = \{\langle a, a \rangle\}\).