axiomatic derivations

PHIL 452 · Modal Logic

Gabriel Uzquiano

axiomatic proofs

We aim for a characterization of sets of modal formulas valid in different classes of frames.

We begin with the system \(\textsf{K}\) which proves all and only formulas valid in all frames.

the minimal normal modal logic \(\textsf{K}\)

Axioms include:

• all substitution instances of axioms of propositional logic

• all substitution instances of axiom

  \[ \begin{array}{lll} \textsf{K} & & \Box (p \to q) \to (\Box p \to \Box q)\\ \end{array} \]

Two rules of inference:

• modus ponens

  \[ \varphi \to \psi, \varphi / \psi \tag{\textsf{MP}} \]

• necessitation

  \[ \varphi / \Box \varphi \tag{\textsf{RN}} \]

theorem

A proof in \(\textsf{K}\) is a finite sequence of modal formulas \((\chi_1, \dots , \chi_n)\) where for each \(m \leq n\), the formula \(\chi_m\) is either

• an axiom

• an outcome of an application of \(\textsf{MP}\) to two prior formulas

• an outcome of an application of \(\textsf{RN}\) to a prior formula

A theorem of \(\textsf{K}\) is formula \(\varphi\) is a formula that appears in the last line of a proof in \(\textsf{K}\). We write \(\vdash_{\textsf{K}} \varphi\) to indicate that \(\varphi\) is a theorem of \(\textsf{K}\).

example

\(\vdash_{\textsf{K}} \Box p \to \Box (q \to p)\)

Proof. \[ \begin{array}{lll} 1 & p \to (q \to p) & \textsf{A1}\\ 2 & \Box (p \to (q \to p)) & \textsf{RN}, 1\\ 3 & \Box (p \to (q \to p)) \to (\Box p \to \Box(q \to p)) & \textsf{K}[p/p, (q \to p)/q]\\ 4 & \Box p \to \Box(q \to p) & \textsf{MP} \ 2, 3\\ \end{array} \]

soundness

All theorems of \(\textsf{K}\) are valid in the class of all frames.

Proof.

• All substitution instances of tautologies are valid in the class of all frames.
• All substitution instances of \(\textsf{K}\) are valid in the class of all frames.
• \(\textsf{MP}\) preserves validity in the class of all frames.
• \(\textsf{RN}\) preserves validity in the class of all frames.

derived rules of inference

If \(\vdash_{\textsf{K}} \varphi_1, \dots, \vdash_{\textsf{K}} \varphi_n\) and \(\psi\) is derivable from \(\{\varphi_1, \dots, \varphi\}\) in propositional logic, then \(\vdash_{\textsf{K}} \psi\).

Proof. Given the Deduction Theorem for Propositional Logic, if \(\psi\) is derivable from \(\{ \varphi_1, \dots, \varphi_n\}\) in Propositional Logic, then \(\varphi_1 \to ( \varphi_2 \to \dots (\varphi_n \to \psi) \dots))\) is a theorem of Propositional Logic and thus one of \(\textsf{K}\). By repeated applications of \(\textsf{MP}\), \(\vdash_K \psi\).

\(\textsf{RK}\)

(\(\textsf{RK}\))  If \(\vdash_K \varphi \to \psi\), then \(\vdash_K \Box \varphi \to \Box \psi\)

Proof. We may transform a proof of \(\varphi \to \psi\) into a proof of \(\Box \varphi \to \Box \psi\).

\[\begin{array}{lllll} n & & \varphi \to \psi & \\ n+1 & & \Box (\varphi \to \psi) & \textsf{RN} \ n\\ n+2 & & \Box (\varphi \to \psi) \to (\Box \varphi \to \Box \psi) & \textsf{K}[\varphi/p, \psi/q]\\ n+3 & & \Box \varphi \to \Box \psi & \textsf{MP} \ n+1, n+2\\ \end{array}\]

\(\textsf{RK}_\Diamond\)

(\(\textsf{RK}\))  If \(\vdash_K \varphi \to \psi\), then \(\vdash_K \Diamond \varphi \to \Diamond \psi\)

Proof. We may transform a proof of \(\varphi \to \psi\) into a proof of \(\Diamond \varphi \to \Diamond \psi\).

\[ \begin{array}{lllll} n & & \varphi \to \psi & \\ n+1 & & \neg \psi \to \neg \varphi & \textsf{PL} \ n\\ n+2 & & \Box \neg \psi \to \Box \neg \varphi & \textsf{RK} \ n+1\\ n+4 & & \neg \Box \neg \varphi \to \neg \Box \neg \psi & \textsf{PL} \ n+4\\ n+5 & & \Diamond \varphi \to \Diamond \psi & \textsf{Def} \ \Diamond \ n+5 \end{array} \]

theorems of \(\textsf{K}\)

\(\vdash_K \Box (\varphi \to \psi) \to (\Diamond \varphi \to \Diamond \psi)\)

Proof. \[ \begin{array}{lllll} 1 & & (\varphi \to \psi) \to (\neg \psi \to \neg \varphi) & \textsf{PL} \\ 2 & & \Box (\varphi \to \psi) \to \Box(\neg \psi \to \neg \varphi) & \textsf{RK} \ 1 \\ 3 & & \Box (\neg \psi \to \neg \varphi) \to (\Box \neg \psi \to \Box \neg \varphi) & \textsf{K}[\neg \psi/p, \neg \varphi/q] \\ 4 & & (\Box \neg \psi \to \Box \neg \varphi) \to (\neg \Box \neg \varphi \to \neg \Box \neg \psi) & \textsf{PL}\\ 5 & & \Box (\varphi \to \psi) \to (\neg \Box \neg \varphi \to \neg \Box \neg \psi) & \textsf{PL} \ 2, 3, 4\\ 6 & & \Box (\varphi \to \psi) \to (\Diamond \varphi \to \Diamond \psi) & \textsf{Def} \ \Diamond \ 6\\ \end{array} \]

necessity commutes with conjunction (\(\Rightarrow\))

\(\vdash_K \Box (\varphi \wedge \psi) \to (\Box \varphi \wedge \Box \psi)\)

Proof.

\[\begin{array}{lllll} 1 & & (\varphi \wedge \psi) \to \varphi & \textsf{PL} \\ 2 & & \Box (\varphi \wedge \psi) \to \Box \varphi & \textsf{RK} \ 1 \\ 3 & & (\varphi \wedge \psi) \to \psi & \textsf{PL} \\ 4 & & \Box (\varphi \wedge \psi) \to \Box \psi & \textsf{RK} \ 3 \\ 5 & & \Box (\varphi \wedge \psi) \to (\Box \varphi \wedge \Box \psi) & \textsf{PL} \ 2, 4\\ \end{array}\]

necessity commutes with conjunction (\(\Leftarrow\))

\(\vdash_K (\Box \varphi \wedge \Box \psi) \to \Box (\varphi \wedge \psi)\)

Proof. \[ \begin{array}{lllll} 1 & & \varphi \to (\psi \to (\varphi \wedge \psi)) & \textsf{PL} \\ 2 & & \Box \varphi \to \Box (\psi \to (\varphi \wedge \psi)) & \textsf{RK} \ 1\\ 3 & & \Box (\psi \to (\varphi \wedge \psi)) \to (\Box \psi \to \Box (\varphi \wedge \psi)) & \textsf{K}[\psi/p, (\varphi \wedge \psi)/q]\\ 4 & & \Box \varphi \to (\Box \psi \to \Box (\varphi \wedge \psi)) & \textsf{PL} \ 2, 3\\ 5 & & (\Box \varphi \wedge \Box \psi) \to \Box (\varphi \wedge \psi) & \textsf{PL} \ 4\\ \end{array} \]

possibility commutes with disjunction (\(\Rightarrow\))

\(\vdash_K \Diamond (\varphi \vee \psi) \to (\Diamond \varphi \vee \Diamond \psi)\)

Proof. \[ \begin{array}{lllll} 1 & & (\neg \varphi \wedge \neg \psi) \to \neg (\varphi \vee \psi) & \textsf{PL} \\ 2 & & \Box (\neg \varphi \wedge \neg \psi) \to \Box \neg (\varphi \vee \psi) & \textsf{RK} \ 1 \\ 3 & & (\Box \neg \varphi \wedge \Box \neg \psi) \to \Box (\neg \varphi \wedge \neg \psi) & \textsf{Prop} \ 7.6 \\ 4 & & (\Box \neg \varphi \wedge \Box \neg \psi) \to \Box \neg (\varphi \vee \psi) & \textsf{PL} \ 2, 3 \\ 5 & & (\neg \Diamond \varphi \wedge \neg \Diamond \psi) \to \neg \Diamond (\varphi \vee \psi) & \textsf{Def} \ \Diamond, \textsf{PL} \ 4 \\ 6 & & \Diamond (\varphi \vee \psi) \to (\Diamond \varphi \vee \Diamond \psi) & \textsf{PL} \ 5 \end{array} \]

possibility commutes with disjunction (\(\Leftarrow\))

\(\vdash_K (\Diamond \varphi \vee \Diamond \psi) \to \Diamond (\varphi \vee \psi)\)

Proof. \[ \begin{array}{lllll} 1 & & \neg (\varphi \vee \psi) \to (\neg \varphi \wedge \neg \psi) & \textsf{PL} \\ 2 & & \Box \neg (\varphi \vee \psi) \to \Box (\neg \varphi \wedge \neg \psi) & \textsf{RK} \ 1 \\ 3 & & \Box (\neg \varphi \wedge \neg \psi) \to (\Box \neg \varphi \wedge \Box \neg \psi) & \textsf{Prop} \ 7.5 \\ 4 & & \Box \neg (\varphi \vee \psi) \to (\Box \neg \varphi \wedge \Box \neg \psi) & \textsf{PL} \ 2, 3 \\ 5 & & \neg \Diamond (\varphi \vee \psi) \to (\neg \Diamond \varphi \wedge \neg \Diamond \psi) & \textsf{Def} \ \Diamond, \textsf{PL} \ 4 \\ 6 & & (\Diamond \varphi \vee \Diamond \psi) \to \Diamond (\varphi \vee \psi) & \textsf{PL} \ 5 \end{array} \]

one more helpful theorem

\(\vdash_K (\Diamond \varphi \wedge \Box \psi) \to \Diamond (\varphi \wedge \psi)\)

Proof. \[ \begin{array}{lllll} 1 & & \neg (\varphi \wedge \psi) \to (\psi \to \neg \varphi) & \textsf{PL} \\ 2 & & \Box \neg (\varphi \wedge \psi) \to \Box (\psi \to \neg \varphi) & \textsf{RK} \ 1 \\ 3 & & \Box (\psi \to \neg \varphi) \to (\Box \psi \to \Box \neg \varphi) & \textsf{K}[\psi/p, \neg \varphi/q] \\ 4 & & \Box \neg (\varphi \wedge \psi) \to (\Box \psi \to \Box \neg \varphi) & \textsf{PL} \ 2, 3 \\ 5 & & (\Diamond \varphi \wedge \Box \psi) \to \Diamond (\varphi \wedge \psi) & \textsf{Def} \ \Diamond, \textsf{PL} \ 4 \end{array} \]