Between Burgess and Lewis – Part II: Semantics without Rational Monotonicity

Abstract The last 50 years of research has taught us that conditionals are non-monotonic in the antecedent. That is, they invalidate Antecedent Strengthening. Many accounts have been developed for such conditionals, starting with Stalnaker and Lewis. These accounts converge roughly to Burgess’ conditional logic $${{\,\mathrm{\textsf{B}}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mi>B</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> or the non-monotonic reasoning system $$\textbf{P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> . The latter two have Cautious Monotonicity as a weak replacement for Antecedent Strengthening. Lewis weakest conditional logic $${{\,\mathrm{\textsf{V}}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mi>V</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> or system $$\textbf{R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> are obtained by adding a stronger proxi for Antecedent Strengthening – the law of Rational Monotonicity. I argue that Rational Monotonicity is too much while Cautious Monotonicity is not enough monotonicity. I investigate two other monotonicity postulates, which are jointly weaker than Rational Monotonicity. This gives rise to three logics in between $${{\,\mathrm{\textsf{B}}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mi>B</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> and $${{\,\mathrm{\textsf{V}}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mi>V</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> and three other logics in between $${{\,\mathrm{\textsf{BN}}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mi>BN</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> and $${{\,\mathrm{\textsf{VN}}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mi>VN</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> obtained by strengthening $${{\,\mathrm{\textsf{B}}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mi>B</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> and $${{\,\mathrm{\textsf{V}}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mi>V</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> by a consistency axiom. In the first part of the paper, I proved soundness, completeness and decidability results for these logics, using set-selection semantics. In the present second part of the paper, I investigate five other semantics for the new logics: order semantics, a new closeness semantics, alternative ranking semantics inspired by Spohn, Huber and Raidl, and two new versions of similarity semantics inspired from Lewis and Burgess. Establishing truth preserving maps between these semantics, the soundness and completeness results from set-selection semantics can be transferred. Overall, the different new semantics provide new insights into why Rational Monotonicity fails.