Modal Logic (Lecture)

I followed this course during my 3rd Semester of the Bachelor at the LMU Munich. The course has been held by Dr. Marra at the mcmp. Here I collect the notes from the material available online. I had an oral exam on the 22nd May 2024 and got 1.3; the mark counts in the B.A. Philosophy (LMU).

1. Intro and Lecture

• Modal: a mode of truth like: necessity, knowledge, belief, obligation, present
  • Alethic: $\square \phi$ means “it is necessary that $\phi$“.
    • Counterfactual: $\phi \square \to \psi$ means “if $\phi$ were the case, then $\psi$ would be the case”.
  • Epistemic: $K_{\phi }$ means “the agent knows $\phi$”.
    • $K_i\phi$ means “agent $i$ knows that $\phi$”.
    • $L_i\phi$ means “the agent $i$ cannot rule out that $\phi$“.
  • Deontic: $O_{\phi }$ means “it is obligatory that $\phi$“.
  • Temporal: $G\phi$ means ” it will always be the case that $\phi$”.
    • $H_{\phi }$ means ” it was always the case that $\phi$”.
• wff. are generated by the usual (finitary) recursion with $\neg ,\wedge ,\square$
  • $◊\phi :=\neg \square \neg \phi$

2. Truth

• Kripke Frames: $\mathcal{F}:=(W,R)$ for $W=\{w_1,w_2,...\}$ non empty, $R\subseteq W\times W$ a binary relation on $W$.
  • $R$, the accessibility relation, is interpreted in different ways:
    • Deontic Logic: Ideality
    • Temporal Logic: Immediate Predecessor
  • $R$ can be:
    • Reflective, Symmetric, Transitive
    • Serial: ${\forall }_x{\exists }_{y}xRy$
    • Conversely well-founded: there is no inf. ascending chain $x_{0}R{x}_1\wedge x_{1}R{x}_2\wedge x_{2}R{x}_3...$
    • Euclidian: ${\forall }_{x,y,z}((xRy\wedge xRz)\to yRz)$
• Kripke Models: $\mathcal{M}:=(W,R,V)$ s.t. $\mathcal{F}=(W,R)$ is K.fr., $V:AT\to \mathcal{P}(W)$ for $AT$ the set of atomic sentences mapped to the set of worlds where it holds. Write $w\in V(p)$ means ”$p$ is true at $w$“.
• Kripke Semantics:
  • $\mathcal{M},w\models p$ iff $w\in V(p)$
  • $\mathcal{M},w\models \neg \phi$ iff not $\mathcal{M},w\models \phi$
  • $\mathcal{M},w\models \phi \wedge \psi$ iff $\mathcal{M},w\models \phi$ and $\mathcal{M},w\models \psi$
  • $\mathcal{M},w\models \square \phi$ iff for all $v$: if $wRv$ then $\mathcal{M},v\models \phi$
    • those are local, only affected by accessible worlds.
    • those are intensional, $\square \phi$ at $w$ depends on other things
  • $\mathcal{M},w\models ◊\phi$ iff for some $v$: $wRv$ and $\mathcal{M},v\models \phi$
  • For convention: $V(\perp )=\varnothing$ and $V(\top )=W$
  • $w$ is an end point if $\neg {\exists }_{v}vRw$

3. Expressive Power and Invariance

• Bisimulation: denoted with $E$, is a relation between $W_1$ and $W_2$ of two Kripke Models for fixed $w\in W_1,v\in W_2$, s.t.:
  • ${\forall }_{p\in AT}wEv\Rightarrow (w\in V_1(p)\iff v\in V_2(p))$ (Atomic Harmony)
  • $wEv\wedge w{R}_1{w}^{\prime }\Rightarrow {\exists }_{v^{\prime }\in {\mathcal{M}}_2}v{R}_2{v}^{\prime }\wedge w^{\prime }E{v}^{\prime }$ (ZIG)
  • $wEv\wedge v{R}_2{v}^{\prime }\Rightarrow {\exists }_{w^{\prime }\in {\mathcal{M}}_1}w{R}_1{w}^{\prime }\wedge w^{\prime }E{v}^{\prime }$ (ZAG)
• Pointed Model: denoted with $\mathcal{M},w$, is a Kripke model $\mathcal{M}$ with a designated world $w\in W$.
• Bisimilar: ${\mathcal{M}}_1,w$ and ${\mathcal{M}}_2,v$ pointed models are bisimilar if there is a bisimulation $E$ s.t. $wEv$, write it ${\mathcal{M}}_1,w\underline{\leftrightarrow }{\mathcal{M}}_2,v$
  • This is a weaker but similar notion of isomorphism:
    • whenever it is possible to make a transition in one model, it is possible to make a matching transition in the other.
• Invariance Lemma if ${\mathcal{M}}_1,w\underline{\leftrightarrow }{\mathcal{M}}_2,v$ then for all $\phi$ we have ${\mathcal{M}}_1,w\models \phi \iff {\mathcal{M}}_2,v\models \phi$, i.e. ${\mathcal{M}}_1,w\equiv {\mathcal{M}}_2,v$.
  • Undefinable properties can differ in bisimilar models, those are:
    • “having a predecessor”
    • “uniqueness”: there is exactly one world s.t. $\phi$.
  • Prove undefinability:
    • if we claim yes: write down a definition in the modal language
    • if we claim no: give two bisimular models, one with the property and one without.
• Unraveling and Contraction, one can expand a Kripke model adding an infinite tree, one can also contract it, both operations require bisimulations.

4. Frame Correspondence

• We have seen four notions of validity:
  1. Truth at a world in a model $\mathcal{M},w\models \phi$
  2. Validity in a model $\mathcal{M}\models \phi$ iff $\mathcal{M},w\models \phi$ for all $w\in W$
  3. Validity on a frame $\mathcal{F}\models \phi$ iff $\mathcal{M}\models \phi$ for all models $\mathcal{M}=(\mathcal{F},V)$
  4. Validity in a class of frames $\mathcal{K}\models \phi$ iff $\mathcal{F}\models \phi$ for all $\mathcal{F}\in \mathcal{K}$.
     • Validity in the class of all frames is denoted with $\models \phi$
     • Distributive of $\square$ and $◊$ on $\wedge$ and $\vee$ are valid on the class of all frames.
• Frame Correspondence: $\mathcal{F}$ has $P$ iff $\mathcal{F}\models \phi$, then $\phi$ corresponds to $P$.
  • Examples:
    • $\square \phi \to \phi$ corr. to refl.
    • $\phi \to \square ◊\phi$ corr. to sym.
    • $\square \phi \to \square \square \phi$ corr. to trans.
    • $\square \phi \to ◊\phi$ corr. to seriality
    • $◊\phi \to \square ◊\phi$ corr. to euclideaness
  • Not all properties (first order expressible) of frames corr. to modal formulae, not all modal formulae correspond to properties (first order expressible) of frames.
    • the Löb Axiom: $\square (\square \phi \to \phi )\to \square \phi )$ corr. to trans and converse well-foundness, the latter is not expressible in first order logic.

5. Finite Model Property

• Is there a $\phi$ that corr. to finiteness of a model? No, if $\phi$ is true in a model, there is a finite model in which it is true too.
  • $|\mathcal{M}|<2^n$ for $n$ the number of connectives and operators in $\phi$.
    • since the language is finitary, $n<\omega$.
    • There are finitely many models $\mathcal{M}$ s.t. $|\mathcal{M}|<2^n$.
• Finite Model Property: for $\mathcal{M},w$ and $\phi$, there is a finite model ${\mathcal{M}}^{\prime },w^{\prime }$ s.t. $\mathcal{M},w\models \phi \iff {\mathcal{M}}^{\prime },w^{\prime }\models \phi$, furthermore it holds $|{\mathcal{M}}^{\prime }|<2^n$ for $n$ the number of connectives and operators in $\phi$. - It corresponds to compactness in Model Theory (Lecture) (not in its last part).
• Subformula Closed Sets: let $K=\{{\phi }_1,...,{\phi }_n\}$ is subformula closed if each subformula of formulae in the set is in the set.
• Method of Filtration: ${\mathcal{M}}^{\prime }:=(W^{{\mathcal{M}}^{\prime }},R^{\prime },V^{\prime })$ is a filtration of $\mathcal{M}:=(W^{\mathcal{M}},R,V)$ by the subformula set $K$ if
  1. $W^{{\mathcal{M}}^{\prime }}=\{[w]:w\in W^{\mathcal{M}}\}$ where for $w\in W^{\mathcal{M}}$, $[w]:=\{\phi \in K:\mathcal{M},w\models \phi \}$
  2. $[w]R^{\prime }[v]\iff {\forall }_{◊\phi \in K}\phi \in [v]\to ◊\phi \in [w]$
     • when $◊\phi \notin K$, then trivially $◊\phi \notin [w]$, hence such worlds are connected from each world.
  3. $[w]\in V^{\prime }(p)\iff p\in [w]$
• Finite Model Property via Filtrations: for $\mathcal{M}$, $\phi$ and ${\mathcal{M}}^{\prime }$ the filtration of $\mathcal{M}$ on the smallest subformula-closed set $K$ containing $\phi$, then:
  1. ${\forall }_{w\in \mathcal{M}}\mathcal{M},w\models \phi \leftrightarrow {\mathcal{M}}^{\prime },[w]\models \phi$
  2. $|{\mathcal{M}}^{\prime }|<2^{|K|-1}$ ($|K|-1$ is the number of subformulae of $\phi$)
  • hence we derive ${\forall }_{\psi \in K}{\forall }_{[w]\in {\mathcal{M}}^{\prime }}{\mathcal{M}}^{\prime },[w]\models \psi \iff \psi \in [w]$

6. Consequence Relations

• we defined "$\models$" in Lecture 2 already (Kripke Semantics), also consider
  • $\Gamma {\models }_{\mathcal{K}}\phi$ iff for all frames $\mathcal{F}\in \mathcal{K}$, for all models $\mathcal{M}$ on $\mathcal{F}$, for all $w\in W$, we have: if $\mathcal{M},w\models \psi$ for all $\psi \in \Gamma$, then $\mathcal{M},w\models \phi$.
• "$\psi ⊢\phi$" if there is a Hilbert-style proof from $\psi$ to $\phi$.
  • each passage can either be: (i) tautology, (ii) appl. of a rule of inf., (iii) instance of an axiom of $\Lambda$.
• $\Lambda$
  • $\Lambda$ is a normal modal logic, i.e. a set of formulae s.t.:
    • Contains
      • (K) $\square (\phi \to \psi )\leftrightarrow (\square \phi \to \square \psi )$
      • (Dual) $◊\phi \leftrightarrow \neg \square \neg \phi$
    • Closed under:
      • Modus Ponens
      • if $\phi \in \Lambda$ then $\square \phi \in \Lambda$
  • ${\Lambda }_K$ is the smallest $\Lambda$, one can also add further axioms.
  • ${⊢}_{\Lambda }\phi$ iff $\phi \in \Lambda$
  • $\Gamma {⊢}_{\Lambda }\phi$ iff exists $\{{\phi }_1,...,{\phi }_n\}\subseteq \Gamma$ s.t. ${⊢}_{\Lambda }({\psi }_1\wedge ...\wedge {\psi }_n)\to \phi$
    • ${⊢}_K:={⊢}_{{\Lambda }_K}$
    • (Inheritance) if $(\phi \to \psi )\in \Lambda$, then $(\square \phi \to \square \psi )\in \Lambda$
  • Further Axioms:
    • (T) $\square \phi \to \phi$
    • (4) $\square \phi \to \square \square \phi$
    • (B) $\phi \to \square ◊\phi$
    • (D) $\square \phi \to ◊\phi$
    • (5) $◊\phi \to \square ◊\phi$
  • Further Normal Logics:
    • ${\Lambda }_{KT}:=T:={\Lambda }_K+(T)$
    • ${\Lambda }_{KT4}:=S4:={\Lambda }_{KT}+(4)$
    • ${\Lambda }_{KT4B}:=S5:={\Lambda }_{KT4}+(B)={\Lambda }_{KT}+(5)$

7. Soundness and Completeness I

• for any $\Gamma$
  • if $\Gamma {⊢}_{\Lambda }\psi$, then $\Gamma {\models }_{\mathcal{K}}\psi$ (Soundness)
  • if $\Gamma {\models }_{\mathcal{K}}\psi$, then $\Gamma {⊢}_{\Lambda }\psi$ (Completeness)
  • if ${\models }_{\mathcal{K}}\psi$, then ${⊢}_{\Lambda }\psi$ (Weak Completeness)
• for $\Gamma$ cons. set of formulae, there exists $\Delta$ maximally consistent set s.t. $\Gamma \subseteq \Delta$. (Lindenbaum)

8. Completeness II

• Note: Completeness $\iff$ every $\Lambda$-consistent set has a model in $\mathcal{K}$.
• Completeness proof plan:
  1. claim: every consistent set of formulae has a model
  2. By Lindenbaum we can extend it to a maximally consistent set
     • for $\Gamma$, let $\Delta$ be MCS s.t. $\Gamma \subseteq \Delta$
  3. by truth lemma there is a canonical model that makes all those formaulae true.
     • ${\mathcal{M}}^C=(W^C,R^C,V^C)$ for $W^C:=\{\Gamma :\Gamma$ is MCS$\}$, $\Gamma R^c\Delta \iff {\forall }_{\phi \in \Delta }◊\phi \in \Gamma \iff {\forall }_{\square \phi \in \Gamma }\phi \in \Delta$ and $\Gamma \in V^C(p)\iff p\in \Gamma$.
     • Truth Lemma: $\phi \in \Gamma$ iff ${\mathcal{M}}^C,\Gamma \models \phi$.
     • Existence Lemma: $◊\phi \in \Gamma$ iff there is a MCS $\Delta$ with $\phi \in \Delta$ and $\Gamma R^C\Delta$.
  • $\phi$ is canonical for a class of frames $\mathcal{K}$ if ${\Lambda }_{K+\phi }^C\in \mathcal{K}$
  • if $\phi$ is canonical for a class of frames $\mathcal{K}$, then the logic ${\Lambda }_{K+\phi }$ is strongly complete.

9. Temporal Logic

• ${\mathcal{L}}_T=\mathcal{L}+\{G,H\}$
  • $G\phi$ means “it will always be the case that $\phi$”
  • $H\phi$ means “it was always the case that $\phi$”
  • $F\phi :=\neg G\neg \phi$ reads as “it will at some time be the case that $\phi$”
  • $P\phi :=\neg H\neg \phi$ reads as “it was at some time the case that $\phi$”
• Temporal Frame $\mathcal{F}:=(T,<)$, works as intuitively should
• Semantics:
  • $\mathcal{M},t\models p$ iff $t\in V(p)$
  • $\mathcal{M},t\models \neg \phi$ iff $\mathcal{M},t/\models \phi$
  • $\mathcal{M},t\models \phi \wedge \psi$ iff $\mathcal{M},t\models \phi$ and $\mathcal{M},t\models \psi$
  • $\mathcal{M},t\models G\phi$ iff for all $s$ if $s<t$ then $\mathcal{M},s\models \phi$
  • $\mathcal{M},t\models H\phi$ iff for all $s$ if $s>t$ then $\mathcal{M},s\models \phi$
• $NEXT\phi$ is not definable (proof with bisimulation)
• Basic Temporal Logi (BTL) is the smallest set of formulae s.t.:
  • Contains all prop. tautologies
  • All instances of:
    • $G(\phi \to \psi )\to (G\phi \to G\psi )$ (Distribution)
    • $H(\phi \to \psi )\to (H\phi \to H\psi )$
    • $F\phi \leftrightarrow \neg G\neg \phi$ (Dual)
    • $P\phi \leftrightarrow \neg H\neg \phi$
    • $\phi \to GP\phi$ (Converse)
    • $\phi \to HF\phi$
  • Is closed under
    • Modus Ponens
    • Temp. Gener. if $\phi \in BTL$, then $H\phi \in BTL$ and $G\phi \in BTL$.
• Further Restrictions: "$<$" is assumed to be:
  • Transitive; which corresponds to $G\phi \to GG\phi$ and $H\phi \to HH\phi$
  • Irreflexivity: no corresponding modal formula

10. Deontic Logic I

• It determines “obligation” & “permission”, and is related to “good”
• There is no explicit agency “it is obligatory that”
• it adds to classical logic the operator $O\phi$ “it ought that $\phi$”
• Deontic Frames are just like serial Modal Frames, i.e. s.t. ${\forall }_w{\exists }_v(wRv)$, the models and semantics are also the same. It also is sound and (strongly) complete.
• $SDL$, Standard Deontic Logic, is ${\Lambda }_{KD}$, namely, a normal modal logic with:
  • (KD) $O(\phi \to \psi )\to (O\phi \to O\psi )$
  • (Dual) $P\phi \leftrightarrow \neg O\neg \phi$
  • (D) $O\phi \to P\phi$
• It follows that:
  • if $(\phi \to \psi )\in SDL$ then $(O\phi \to O\psi )\in SDL$ (Monotonicity)
  • $\neg (O\phi \wedge O\neg \phi )$ ($\iff O\phi \to P\phi$) (No Deontic Conflicts)
  • Distributivity of $\wedge$ and $\vee$ on $O$
  • $\neg O\perp$
  • $(\phi \wedge (\phi \to O\psi ))\to O\psi$ (Factual Detachment)
  • $(O\phi \wedge O(\phi \to \psi ))\to O\psi$ (Deontic Detachment)
• The Gentle Murder Puzzle, consider the following passages:
  1. Smith ought not to kill John
  2. If Smith kills John, then Smith ought to kill John gently
  3. Smith kills John
  • We can formalise this as:
    • 1. $O(\neg k)$, 2. $O(k\to g)$, 3. $k$
      • Not independent: in this formalisation $O(k\to g)$ follows from $O(\neg k)$, how?
    • 1. $O(\neg k)$, 2. $k\to Og$, 3. $k$
      • Inconsistent: $O(k)$ follows from 2. and 3.
  • the $O$ operator describes how things are in the ideal world, worlds where “kill John gently” are sub-ideal worlds that are not always supported by $SDL$.
  • A solution is to consider proposition of the form “if $p$, then it ought to be that $q$” with a non-material conditional, s.t. Factual Detachment holds.
    • hence 2. becomes: $k\Rightarrow Og$, this still gives rise to problem, hence:
  • Dyadic Deontic Logic:
    • has the classical logic with $O({\phi }_2/{\phi }_1)$
    • frames are $(W,⪯)$ where $⪯$ represents “betterness”, hence we formalise the puzzle as:
      • 1. $O(\neg k/\perp )$, 2. $O(g/k)$, 3. $k$

11. Deontic Logic II

• Paper: mining validity of reasoning by cases in deontic logic.
• Miner Puzzle: runs as follows:
  1. 10 miners either all in shaft A or all in shaft B
  2. Flood is approaching, we can wither block shaft A or shaft B, consequences are:
     1. if we block the shaft with all miners, all survive
     2. if we block the shaft without miners, all die
     3. if we block none of the shaft, one dies
  • To get to the solution, we may consider the following reasoning:
    1. The miners are in A or they are in B
    2. If they are in A, then we ought to block A
    3. If they are in B, then we ought to block B
    • by cases it follows: we ought to block A or we ought to block B
  • Reactions could be:
    • perhaps that is the right answer
    • we should reject one of the assumptions of the reasoning
    • we should reject the inference method: inference by cases
      • this is the path taken in the paper.
• Non-Theorems of $SDL$:
  • $O\phi \to \phi$
  • $O(O\phi \to \phi )$
• Extension: $SDL+$, we add the axiom $O(O\phi \to \phi )$
  • the correspondence of the axiom is secondary reflexivity ${\forall }_w{\forall }_v(wRv\to vRv)$

12. STIT Logic

• Seeing To It That: an agent $i$ sees to it that $\phi$, i.e. an action of $i$ can guarantee that $\phi$.
  • $Ab{l}_i\phi$ reads: “agent $i$ is able to see to it that $\phi$”
• this logic requires branching time frames, both acting and not acting must be considered. No probabilities or intentions are considered.
  • Branching Time Frames $⟨T,<⟩$
    • $T$ is a set of moments
    • $<$ is a tree-like ordering on $T$, i.e. s.t.:
      • $<$ irreflexive
      • $<$ transitive
      • $<$ s.t. ${\forall }_{x,yz}((y<x\wedge z<x)\to (y<z\vee z<y\vee \le y))$ (Past-Linearity)
    • History = maximal chain
    • $H_m$ = set of histories passing through the moment $m$
• STIT Logic: is ${\mathcal{L}}_T$ (temporal logic) with $[i-stit]\phi$ “the agent $i$ sees to it that $\phi$“.
  • Frames: $\mathcal{F}=(T,<,Agent,Choice)$
    • $T$ a set of moments
    • $<$ irrefl., trans., past-lin.
    • $Agent$ a set of agents
    • $Choice$ is a function mapping each agent $i$ and moment $m$ into a disj. partition of $H_m$
    • Write $Choic{e}_i^m(h)$ to indicate the set of histories $h$ belonging to the same equivalence class of $h$.
      • No choice between undivided histories (different choice bring to different histories)
      • Independence of agents:
  • Models & Semantics: $\mathcal{M},m/h\models [i-stit]\phi$ iff for all $h^{\prime }\in Choic{e}_i^m(h),\mathcal{M},m/h^{\prime }\models \phi$.
    • $[i-stit]$ is an $S5$ operator
    • Ability: $Ab{l}_i\phi :=◊[i-stit]\phi$
      • $Ab{l}_i\phi \to \phi$, $\phi \to Ab{l}_i\phi$, $Ab{l}_i\top$, distributives and inverse for $\wedge$

13. Epistemic Logic

• To the classical language we add $K\phi$, the agent knows $\phi$
• Epistemic Model: $(W,R_K,V)$ where $R_K$ is an eq. relation (minimally refl.) and reads as “epistemic equivalence”.
  • $\mathcal{M},w\models K\phi$ iff for all $v$: if $w{R}_{K}v$ then $\mathcal{M},v\models \phi$,
    • its dual is $L\phi$ “the agent considers possible that $\phi$”
    • it’s Syntax is just like $S5$.
• Multi-Agent Epistemic Logic
  • Models are $(W,Agents,\{R_K{\}}_i,V)$, $\mathcal{M},w\models K_i\phi$ iff all $v$: if $w{R}_{K_i}v$ then $\mathcal{M},v\models \phi$.
  • shared knowledge can be of $n$th-order up to common ($n=\omega$), requires an infinite formula, i.e. not expressible
  • Distributed knowledge (if they all share information) is not expressible, not with individual knowledge modalities.
• Positive Introspection $K\phi \to KK\phi$
  • Argument by Hintikka:
    1. we know that if $\{K\alpha ,\neg K\neg \beta \}$ is consistent then $\{K\alpha ,\beta \}$ is consistent.
    2. assume that pos.intr. fails, i.e. $\{K\phi ,\neg KK\phi \}$ is consistent
    3. then, for $\alpha =\phi$ and $\beta =\neg K\phi$, we get that $\{K\phi ,\neg K\phi \}$ is consistent

14. Dynamic Epistemic Logic

• Knowledge is not static, you change it constantly.
• Public Announcement Logic.
  • it handles the hard information, i.e. trustworthy, true and public.
  • ${\mathcal{L}}_{PAL}$ adds the operator $[!\alpha ]\phi$ for $\alpha \in {\mathcal{L}}_{EP}$, the epistemic language.
  • for $\mathcal{M}=(W,Agents,\{R_{K_i}\},V)$ a mutli-agent-Kripke model, then ${\mathcal{M}}_{|\phi }=(W_{|\phi },Agents,\{R_{K_{i_{|\phi }}}\},V_{|\phi })$ defined intuitively.
    • ${\mathcal{M}}_{|\phi }$ is the model restricted to the worlds where $\phi$ holds.
    • $\mathcal{M},w\models [!\alpha ]\phi$ iff $\mathcal{M},w\models \neg \alpha$ or ${\mathcal{M}}_{|\alpha },w\models \phi$.
    • $\mathcal{M},w\models ⟨!\alpha ⟩\phi$ iff $\mathcal{M},w\models \alpha$ and ${\mathcal{M}}_{|\alpha },w\models \phi$.
• Announcements don’t change facts on the world but change our knowledge on them
  • i.e. truth of $\phi$ may well change.
    • $[!\alpha ](\phi \wedge \psi )\leftrightarrow ([!\alpha ]\phi \wedge [!\alpha ]\psi )$
    • $[!\alpha ]\neg \phi \leftrightarrow (\alpha \to \neg [!\alpha ]\phi )$
    • $[!\alpha ]K_i\phi \leftrightarrow (\alpha \to K_i(\alpha \to [!\alpha ]\phi ))$
  • but truth of $p$ cannot.
    • $[!\alpha ]p\leftrightarrow (\alpha \to p)$
• for each $\phi \in {\mathcal{L}}_{PAL}$ exist ${\phi }^{\prime }\in {\mathcal{L}}_{EP}$ s.t. $\mathcal{M},w\models \phi$ iff $\mathcal{M},w\models {\phi }^{\prime }$ (Reduction Theorem)
  • you can get to this ${\phi }^{\prime }$ by the four rules above
• Then $PAL$ is normal modal logic $S5$ together with the four reduction axioms above.
• Moore-sentences, i.e. of the form $p\wedge \neg K_{i}p$, are false if announced: $[!(p\wedge \neg K_{i}p)]\neg (p\wedge \neg K_{i}p)$
  • such statements were in fact true before being pronounced and false the moment afterwards. There are in fact truths that cannot be known, the discussion of Gettier examples and JTB comes afterwards.

Exercises

2. Truth

2.1.

Draw a Kripke Model $\mathcal{M}$ s.t. $\mathcal{M},w/\models (p\to \square q)\leftrightarrow \square (p\to q)$. I need a world $w$ where one of the two formulae is true and the other false. I need to have $p\to q$ true in all worlds connected, though I can have, for instance, $\neg p$ in $w$ and $q$ true in all connected worlds. Consider for instance $w⟶v$ for $v\in V(p)$ while $w\notin V(p)$ and $v,w\in V(q)$, then we have $\mathcal{M},w\models \neg (p\to \square q)$ since $\mathcal{M},w\models \neg p$ and $\mathcal{M},w\models \square q$ and also $\mathcal{M},w\models \square (p\to q)$ since $\mathcal{M},v\models p\to q$ and $v$ is the only world connected to $w$.

2.2.

Follow a similar principle with more complex computations.