<<PrSAT`

Carnap's Early Systems: (SuperDagger[𝔪]) and  (SuperStar[𝔪])

Here is the (SuperDagger[𝔪]) distribution, for the 2 prediate, 2 object case:

Symbolize[(Subscript[(SuperDagger[𝔪]), 2,2])];

(Subscript[(SuperDagger[𝔪]), 2,2])=PrSAT[{Pr[Ea∧Ga∧Eb∧Gb]==(1/16),Pr[Ea∧Ga∧Eb∧¬Gb]==(1/16),
Pr[Ea∧Ga∧¬Eb∧Gb]==(1/16),Pr[Ea∧Ga∧¬Eb∧¬Gb]==(1/16),
Pr[Ea∧¬Ga∧Eb∧Gb]==(1/16),Pr[Ea∧¬Ga∧Eb∧¬Gb]==(1/16),
Pr[Ea∧¬Ga∧¬Eb∧Gb]==(1/16),Pr[Ea∧¬Ga∧¬Eb∧¬Gb]==(1/16),
Pr[¬Ea∧Ga∧Eb∧Gb]==(1/16),Pr[¬Ea∧Ga∧Eb∧¬Gb]==(1/16),
Pr[¬Ea∧Ga∧¬Eb∧Gb]==(1/16),Pr[¬Ea∧Ga∧¬Eb∧¬Gb]==(1/16),
Pr[¬Ea∧¬Ga∧Eb∧Gb]==(1/16),Pr[¬Ea∧¬Ga∧Eb∧¬Gb]==(1/16),
Pr[¬Ea∧¬Ga∧¬Eb∧Gb]==(1/16),Pr[¬Ea∧¬Ga∧¬Eb∧¬Gb]==(1/16)}]

TruthTable[(Subscript[(SuperDagger[𝔪]), 2,2])]

Here are some salient Eacts about  (SuperDagger[𝔪]):

x_⊃y_:=¬x∨y;
x_≡y_:=(x⊃y)∧(y⊃x);

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)], (Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb∧Gb],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb|Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb|Ea≡Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb|Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb|Ea≡Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea∧¬Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ea∧¬Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga|Gb∧(Ea∧¬Eb)],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ea|Eb]>Pr[Ea|Eb∧(Ga∧¬Gb)]>Pr[Ea],(Subscript[(SuperDagger[𝔪]), 2,2])]

EvaluateProbability[Pr[Ga|Gb],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

In other words,  (SuperDagger[𝔪]) violates instantial relevance and analogy, but it leads to all Hempelian confirmatory instances for a universal generalization being confirmatory (probabilistically relevant) instances for a universal generalization.

Here is the (SuperStar[𝔪]) distribution:

Symbolize[(Subscript[(SuperStar[𝔪]), 2,2])];

(Subscript[(SuperStar[𝔪]), 2,2])=PrSAT[{Pr[Ea∧Ga∧Eb∧Gb]==(1/10),Pr[Ea∧Ga∧Eb∧¬Gb]==(1/20),
Pr[Ea∧Ga∧¬Eb∧Gb]==(1/20),Pr[Ea∧Ga∧¬Eb∧¬Gb]==(1/20),
Pr[Ea∧¬Ga∧Eb∧Gb]==(1/20),Pr[Ea∧¬Ga∧Eb∧¬Gb]==(1/10),
Pr[Ea∧¬Ga∧¬Eb∧Gb]==(1/20),Pr[Ea∧¬Ga∧¬Eb∧¬Gb]==(1/20),
Pr[¬Ea∧Ga∧Eb∧Gb]==(1/20),Pr[¬Ea∧Ga∧Eb∧¬Gb]==(1/20),
Pr[¬Ea∧Ga∧¬Eb∧Gb]==(1/10),Pr[¬Ea∧Ga∧¬Eb∧¬Gb]==(1/20),
Pr[¬Ea∧¬Ga∧Eb∧Gb]==(1/20),Pr[¬Ea∧¬Ga∧Eb∧¬Gb]==(1/20),
Pr[¬Ea∧¬Ga∧¬Eb∧Gb]==(1/20),Pr[¬Ea∧¬Ga∧¬Eb∧¬Gb]==(1/10)}]

TruthTable[(Subscript[(SuperStar[𝔪]), 2,2])]

Here are some salient Eacts about (SuperStar[𝔪]):

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)], (Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb∧Gb],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb|Ea∧Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb|Ea≡Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb|(Ea≡Ga)∧Eb],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb|Eb],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb|Ea∧Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb|Ea≡Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ea∧Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea∧¬Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea∧Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ea|Eb]>Pr[Ea|Eb∧(Ga∧¬Gb)]>Pr[Ea],(Subscript[(SuperStar[𝔪]), 2,2])]

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ea∧¬Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga|Gb∧(Ea∧¬Eb)],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga|Gb],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

In other words,  (SuperStar[𝔪]) satisfies instantial relevance and "analogy by similarity", but it also leads to all Hempelian confirmatory instances for a universal generalization being confirmatory (probabilistically relevant) instances for a universal generalization.

What about "Grue"?  Here, the 2-object, 3-predicate case is what we'd want to look at (E, O, G, and a, b, c).  It's pretty complex.  We need to be more clever now in specifying (Subscript[(SuperDagger[𝔪]), 2,3]) and (Subscript[(SuperStar[𝔪]), 2,3]).  We start with (Subscript[(SuperDagger[𝔪]), 2,3]).

Symbolize[(Subscript[(SuperDagger[𝔪]), 2,3])];

atoms={Ea, Eb, Oa, Ob, Ga, Gb};
tvs=Flatten[Outer[List,Sequence@@Table[{True,False},{6}]],5];
set[x_]:=Table[If[x[[i]]==True,atoms[[i]],¬atoms[[i]]],{i,1,Length[atoms]}];
sds=(And@@#&)/@(set/@tvs);
f[s_]:=Pr[s]==(1/(Length[sds]));
(Subscript[(SuperDagger[𝔪]), 2,3])=PrSAT[{f/@sds},BypassSearch->True];

x_≡y_:=(x⊃y)∧(y⊃x);

EvaluateProbability[Pr[(Ea⊃(Oa≡Ga))∧(Eb⊃(Ob≡Gb))],(Subscript[(SuperDagger[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb∧Gb],(Subscript[(SuperDagger[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Ob|Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Gb|Ea≡Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb|Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb|Ea≡Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea∧¬Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ea∧¬Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga|Gb∧(Ea∧¬Eb)],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga|Gb],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

Now, for (Subscript[(SuperStar[𝔪]), 2,3]), we must be a little more clever.  We have to compute the structure descriptions, and then the probabilities for them, etc.

perm[s_]:=s/.{Ea->Eb,Eb->Ea,Ga->Gb,Gb->Ga,Oa->Ob,Ob->Oa};
states=set/@tvs;

results={};
For[i=1,i≤Length[states],i++,a=states[[i]];temp={};AppendTo[temp,i];For[j=1,j≤Length[states],j++,b=states[[j]];If[Sort[a]==Sort[perm[b]],AppendTo[temp,j]]];AppendTo[results,temp]];
structIndices=Union/@Union[Sort/@results];

in[x_,S_]:=Or@@Table[x===S[[i]],{i,1,Length[S]}];
pr[sn_]:=(1/(36 Length[Flatten[Select[structIndices,in[sn,#1]&]]]));
f[n_]:=Pr[And@@states[[n]]]==pr[n];
(Subscript[(SuperStar[𝔪]), 2,3])=PrSAT[{f/@Range[64]},BypassSearch->True];

TruthTable[(Subscript[(SuperStar[𝔪]), 2,3])]

EvaluateProbability[Pr[(Ea⊃(Oa≡Ga))∧(Eb⊃(Ob≡Gb))],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[(Ea⊃(Oa≡Ga))∧(Eb⊃(Ob≡Gb))|Ea∧Oa∧Ga],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ea∧Oa∧Ga],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[(Ea⊃(Oa≡Ga))∧(Eb⊃(Ob≡Gb))|Ea∧Ga],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[(Ea⊃(Oa≡Ga))∧(Eb⊃(Ob≡Gb))|Ea∧Oa],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[(Ea⊃(Oa≡Ga))∧(Eb⊃(Ob≡Gb))|Ga∧Oa],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb⊃(Ob≡Gb)|Ea⊃(Oa≡Ga)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb⊃(Ob≡Gb)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb⊃(Ob≡Gb)|Ea∧(Oa∧Ga)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb∧(Ob≡Gb)|Ea∧(Oa∧Ga)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb∧(Ob≡Gb)|Ea∧(Oa≡Ga)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb∧(Ob≡Gb)|Ea∧(Oa≡Ga)∧(Oa∧Ob)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb∧(Ob≡Gb)|(Oa∧Ob)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb∧Gb|Ea∧(Oa≡Ga)∧(Oa∧Ob)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb∧Gb|Oa∧Ob],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb∧(Ob≡Gb)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb∧(Ob∧Gb)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb∧(Ob∧Gb)|Ea∧(Oa∧Ga)],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Gb|Ea≡Ga],(Subscript[(SuperStar[𝔪]), 2,3])]//N

Comparison with 2,2:

EvaluateProbability[Pr[Gb|Ea≡Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb|Ea≡Ga],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Gb],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Gb],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb≡Gb],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb≡Gb|Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb|Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,3])]//N

Here's a counterexample to Fine's L7:

EvaluateProbability[Pr[Eb≡Gb|Ea∧Ga],(Subscript[(SuperStar[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Eb≡Gb|Ea∧Ga],(Subscript[(SuperStar[𝔪]), 2,3])]//N

EvaluateProbability[Pr[Eb≡Gb|Ea≡Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea∧¬Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea∧Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ea∧¬Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|¬Ea],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[(Ea⊃Ga)∧(Eb⊃Gb)|Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga|Gb∧(Ea∧¬Eb)],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga|Gb],(Subscript[(SuperDagger[𝔪]), 2,2])]//N

EvaluateProbability[Pr[Ga],(Subscript[(SuperDagger[𝔪]), 2,2])]//N