Galactic Travel. System S5 lets us go from ÈA in any world W1 to A in any world W2. System T We need a ticket from W1 to W2.
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1 Galactic Travel System S5 lets us go from ÈA in any world W1 to A in any world W2. Weaker systems require a suitable travel ticket between the worlds. We get travel tickets when you drop diamonds and we use them when dropping boxes. System T We need a ticket from W1 to W2. System S4 Like T, but we also can use a series of tickets. System B Like T, but a ticket also works backwards. LogiCola KG Pages
2 1 È(N Ä ÈN) * 2 ÇN [ Á ÈN * 3 1 asm: ÀÈN * 4 2 Á ÇÀN {from 3} * 5 2 W Á N {from 2} 6 2 WW Á ÀN {from 4} * 7 2 W Á (N Ä ÈN) {from 1} 8 2 W Á ÈN {from 5 and 7} 9 3 WW Á N {from 8} 10 Á ÈN {from 3; 6 contradicts 9} # W # WW T or any other system Need S5 In dropping È (steps 8 to 9), we need to use a series of tickets (and one backwards), from W to WW. LogiCola KG Pages
3 1 ÈA [ Á ÈÈA * 2 1 asm: ÀÈÈA * 3 2 Á ÇÀÈA {from 2} * 4 2 W Á ÀÈA {from 3} * 5 2 W Á ÇÀA {from 4} 6 2 WW Á ÀA {from 5} 7 3 WW Á A {from 1} 8 Á ÈÈA {from 2; 6 contradicts 7} # W W WW Need S4 or S5 In dropping È (from 1 to 7), we need to use a series of tickets, from # to WW. LogiCola KG Pages
4 1 A [ Á ÈÇA * 2 1 asm: ÀÈÇA * 3 2 Á ÇÀÇA {from 2} * 4 2 W Á ÀÇA {from 3} 5 2 W Á ÈÀA {from 4} 6 3 Á ÀA {from 5} 7 Á ÈÇA {from 2; 1 contradicts 6} # W Need B or S5 In dropping È (from 5 to 6), we need to use a ticket backwards. LogiCola KG Pages
5 * 1 ÇA [ Á ÈÇA * 2 1 asm: ÀÈÇA * 3 2 Á ÇÀÇA {from 2} * 4 2 W Á ÀÇA {from 3} 5 2 W Á ÈÀA {from 4} 6 2 WW Á A {from 1} 7 3 WW Á ÀA {from 5} 8 Á ÈÇA {from 2; 6 contradicts 7} # W # WW Need S5 In dropping È (steps 5 to 7), we need to use a series of tickets (and one backwards), from W to WW. LogiCola KG Pages
6 s Quantified Modal Translations Ç(Æx)Ax Ç(x)Ax (x)çax It s possible for someone to be above average. It s possible for everyone to be above average. It s possible for anyone to be above average. ÈFx (Fx  ÇÀFx) F is a necessary (essential) property of x. x is necessarily F. F is a contingent (accidental) property of x. x is F but could have lacked F. LogiCola J (QM & QT) Pages
7 This ambiguous sentence could have either of two meanings: All bachelors are necessarily unmarried. Simple necessity (x)(bx Ä ÈUx) All bachelors are inherently unmarriable in no possible world would anyone marry them. This attributes a necessity property to the thing (de re). Conditional necessity È(x)(Bx Ä Ux) It s necessarily true that all bachelors are unmarried. (The meaning of bachelor makes this true.) This attributes necessity to the proposition (de dicto). LogiCola J (QM & QT) Pages
8 Ç(Æx)Ax Ç(x)Ax (x)çax It s possible for someone to be above average. It s possible for everyone to be above average. It s possible for anyone to be above average. ÈFx (Fx  ÇÀFx) F is a necessary (essential) property of x. x is necessarily F. F is a contingent (accidental) property of x. x is F but could have lacked F. All bachelors are necessarily unmarried (x)(bx Ä ÈUx) or È(x)(Bx Ä Ux) LogiCola J (QM & QT) Pages
9 1 È(x)xx Valid [ Á (x)èxx * 2 1 asm: À(x)Èxx * 3 2 Á (Æx)ÀÈxx {from 2} * 4 2 Á ÀÈaa {from 3} * 5 2 Á ÇÀaa {from 4} 6 2 W Á Àaa {from 5} 7 2 W Á (x)xx {from 1} 8 3 W Á aa {from 7} 9 Á (x)èxx {from 2; 6 contradicts 8} 1. Reverse squiggles (modal and quantificational). 2. Drop weak operators (modal and quantificational). 3. Lastly, drop strong operators (modal and quantificational). LogiCola KQ Pages
10 All bachelors are necessarily unmarried. You re a bachelor. Á You re unmarried is logically necessary. 1 (x)(bx Ä ÈUx) Valid 2 Bu [ Á ÈUu 3 1 asm: ÀÈUu * 4 2 Á (Bu Ä ÈUu) {from 1} 5 3 Á ÈUu {from 4 and 2} 6 Á ÈUu {from 3; 3 contradicts 5} (While this is valid, premise 1 is false.) 1 È(x)(Bx Ä Ux) Invalid 2 Bu Bu, Uu [ Á ÈUu * 3 asm: ÀÈUu W ÀBu, ÀUu * 4 Á ÇÀUu {from 3} 5 W Á ÀUu {from 4} 6 W Á (x)(bx Ä Ux) {from 1} 7 Á (x)(bx Ä Ux) {from 1} * 8 W Á (Bu Ä Uu) {from 6} * 9 Á (Bu Ä Uu) {from 7} 10 W Á ÀBu {from 5 and 8} 11 Á Uu {from 2 and 9} LogiCola KQ Pages
11 It s possible for anyone to be above average. Á It s possible for everyone to be above average.aa (x)çax Á Ç(x)Ax This lead into an endless loop. Using ingenuity, we can devise a refutation: W WW a, b Aa, ÀAb Ab, ÀAa LogiCola KQ Pages
12 Problem: Our system doesn t recognize the ambiguity here: The number I m thinking of is necessarily odd ÈOn I m thinking of just one number, and it has the necessary property of being odd. (Perhaps true!) (Æx)((Tx  À(Æy)(Àxy  Ty))  ÈOx) This is necessary: I m thinking of just one number and it is odd. (False!) È(Æx)((Tx  À(Æy)(Àxy  Ty))  Ox) Solution: Analyze the using Russell s theory of descriptions. Pages
13 Problem: Our system makes this valid which it isn t! en Èee Á Èen 8 is the number I m thinking of. It s necessary that 8 is 8. Á It s necessary that 8 is the number I m thinking of. Solution: Analyze the using Russell s theory of descriptions. Pages
14 Problem: Our system makes every entity a necessary being! Here s a proof that in every possible world there exists a being who is Gensler: [ Á È(Æx)xg Valid??? * 1 1 asm: ÀÈ(Æx)xg * 2 2 Á ÇÀ(Æx)xg {from 1} * 3 2 W Á À(Æx)xg {from 2} 4 2 W Á (x)àxg {from 3} 5 2 W Á Àgg {from 4}??? 6 3 W Á gg {self-identity rule} 7 Á È(Æx)xg {from 1; 5 contradicts 6} Solution: Reject the step from 4 to 5. Move to a free logic one that is free of the assumption that individual constants like g always refer to existing beings. Pages
15 Free logic reformulates the rules for dropping (Æx) and (x) : (Æx)Fx Fa, (Æx)xa, use a new constant (x)fx, (Æx)xa Fa, use any constant Some existing being is F. Á a is F. Á a is an existing being. Every existing being is F. a is an existing being. Á a is F. Pages
16 Our sophisticated system uses free logic rules (free of the assumption that constants like g refer to existing beings) for dropping (x) & (Æx) : (x)fx, (Æx)xa Fa, use any constant (Æx)Fx Fa, (Æx)xa, use a new constant Every existing being is F. a is an existing being. Á a is F. Some existing being is F. Á a is F. Á a is an existing being. [ Á È(Æx)xg Valid??? * 1 1 asm: ÀÈ(Æx)xg * 2 2 Á ÇÀ(Æx)xg {from 1} * 3 2 W Á À(Æx)xg {from 2} 4 2 W Á (x)àxg {from 3} 5 2 W Á Àgg {from 4}??? 6 3 W Á gg {self-identity rule} 7 Á È(Æx)xg {from 1; 5 contradicts 6} Without free logic, every entity (e.g. Gensler) is a necessary being that exists in every possible world. Free logic rejects step 5 in the proof to the left. Now what entities exist can vary from world to world. Pages
17 Our sophisticated system always uses Russell s theory of descriptions to analyze definite descriptions (terms of the form the so and so ). This invalidates the following invalid argument: en Èee Á Èen 8 is the number I m thinking of. It s necessary that 8 is 8. Á It s necessary that 8 is the number I m thinking of. This also enables us to express the two senses of The number I m thinking of is necessarily odd (which in naïve QMC is just ÈOn): This is necessary: I m thinking of just one number and it is odd. (False!) È(Æx)((Tx  À(Æy)(Àxy  Ty))  Ox) I m thinking of just one number, and it has the necessary property of being odd. (Perhaps true!) (Æx)((Tx  À(Æy)(Àxy  Ty))  ÈOx) Pages
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