ADDING POSSIBILITIES

pvd

STARTING STATE
POSSIBILITY TO BE ADDED

CALCULATIONS

Starting state: A and B and C
Compositional truthmaker(s): {ABC}, Intensional truthmaker(s): {ABC}

Possibility added: -A or (-A & -B)
Compositional truthmaker(s): {-A-B, -A}, Intensional truthmaker(s): {-A}

Compositional possibility-adding:
{BC, C}: requirement-reduction method
{ABC, -ABC, -A-BC, -AC, -BC}: liberal possibility-addition method
{ABC, -ABC, -A-BC}: conservative possibility-addition method

Intensional possibility-adding:
{BC}: requirement-reduction method
{BC}: liberal possibility-addition method
{BC}: conservative possibility-addition method


INSTRUCTIONS

Enter the starting state and the possibility being added in the two boxes in the first section. For example, you can put A & B on the right and -A on the left and you will get B. The results of different ways of adding possibilities will be put below.

SYNTAX

Atomic letters: A to F, Legal connectives: &, and, or, |, ->, >, <->, iff, not, -, ~, formulas converted to sets of truthmakers via van Fraassen compositional construction of truthmakers. Feel free to just write out truthmakers with brackets of any sorts and commas, concatenation understood as conjunction. (e.g. AB or C-D gives the truth makers AB and C-D)

FIDNING TRUTHMAKERS

The compositional method uses the ideas from van Fraassen's work as well as Fine and Yablo. Indiviudal truthmakers are sets of literals: they can be thought of as the propositions corresponding to conjunction of its members, and the truthmaker of a sentence entails it. The compositional rules for determining the truthmakers (and falsemakers) of a sentence is propositional logic are as follows:
Atomic case:
$ \def\tm#1{ | #1 |^+} \def\fm#1{ | #1 |^-} \tm{x} = \{x\}$ atomic sentences are their own truthmakers
$\fm{x} = \{ \lnot x \}$ atomic sentences have their negations as falsemakers
Recursive rules:
$\tm{\lnot \phi} = \fm{\phi}$
$\fm{\lnot \psi} = \tm{\psi}$
$\tm{\phi \land \psi} = \{ ss' : s \in \tm{\phi}, s' \in \tm{\psi} \}$ $\fm{\phi \land \psi} = \fm{\phi} \cup \fm{\psi}$
$\tm{\phi \lor \psi} = \tm{\phi} \cup \tm{\psi}$
$\fm{\phi \lor \psi} = \{ ss' : s \in \fm{\phi}, s' \in \fm{\psi} \}$
(Where $ |\phi|^+ (|\phi|^-) $ is the set of truthmakers (falsemakers) of $\phi$, and $ss'$ is the truthmaker containing both $s$ and $s'$, in set theoretic terms the union of $s$ and $s'$. Here and above we describe truthmakers using concatenation. But since they are not order sensitive AB is the same as BA. Following Fine we can sometimes talk of them in mereological terms, so that -A is a part of -AB and -ABC is -A fused with BC.

The intensional method identifies the truthmakers with the prime implicants of the sentence. (The calculator uses the Quine-McClusky method to find these, borrowing this implementation.)

UPDATE RULES

Let $S$ be the set of truthmakers of the starting state. Let $P$ be the set of truthmakers of the new possibility.

The result of adding $P$ to $S$ on the requirement-reduction method is $\{s \uparrow p : s \in S , p \in P, \text{ and } \text{comp}(s,P) \}$ where $\text{comp}(s,P)$ iff there is a $p \in P$ that is logically compatible with $s$, and $s \uparrow p$ = the largest part of $s$ that is compatible with $p$.

The result of adding $P$ to $S$ on liberal possibility-adding method is $S \cup \{p \star s : s \in S , p \in P, \text{ and } \text{comp}(s,P) \}$ where $p \star s$ = the largest part of $p$ compatible with $s$ fused with the largest part of $s$ compatible with $p$. (In the hyperintensional case we add more explicit possibilities so as not to lose information about *what* is permitted exactly... to be described.)

The result of adding $P$ to $S$ on conservative possibility-adding method is $S \cup \{p \star s : s \in S , p \in P, \text{ and } \text{comp}(s,P) \}$ where $p \oplus s$ = the largest part of $p$ compatible with $s$ fused with $s$.


By Daniel Rothschild, using Python and Google App Engine. Methods used developed in collaboration with Steve Yablo, using ideas from his work as well as Kit Fine's.