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# Deducive Logic Puzzle to Formal Logic

 Posts: 2,742 Add as FriendChallenge to a DebateSend a Message 2/9/2016 1:48:21 PMPosted: 3 years agoBelow is a logic puzzle that I decided to try to solve using formal logic in order to test my understanding of it. If anyone knows formal logic, how would you write the clues and the necessary axioms to solve this puzzle? I've done so, but I'd like to see how others would go about this...----------------------------------------------Esmeralda's Shape-Shifting Studio has a loyal following of people who want to learn to change themselves into various animals, objects, and famous personages. During their very first workshop, five people (Charlie, Juliette, Morgan, Omar, and Tess), each of whom had signed up for a different number of lessons (4, 12, 16, 20, or 24), each transformed him- or herself into a different animal (beetle, crab, marmoset, penguin, or weasel), a different object (cushion, mailbox, pencil sharpener, rope, or toaster), and a different celebrity (Bob Dylan, George Clooney, Jennifer Lopez, Oprah Winfrey, or Stephen King). Despite their initial success, transmogrification is by no means a simple process, and one that should not be tried on one's own; in fact, Esmeralda recently had to make a house call to assist a student who was stuck in the form of a coffee mug and unable to change back to his natural form! From the information provided, determine the number of lessons that each person is taking, as well as the animal, object, and celebrity whose form each student took.1. The person who transmogrified him- or herself into a marmoset, a toaster, and Stephen King signed up for fewer lessons than the one who changed into a beetle.2. Charlie (who didn't change into a crab) turned himself into a very convincing rope. The one who shifted into the form of both a cushion and a weasel isn't the student who changed into the shape of Jennifer Lopez.3. Juliette is taking exactly 8 fewer lessons than the penguin impersonator. The one who turned into George Clooney signed up for neither 16 nor 20 lessons.4. The person who shifted into both a pencil sharpener and Oprah Winfrey is taking the most lessons.5. Omar is taking fewer lessons than Tess but more lessons than the one who transmogrified into a penguin, a mailbox, and Bob Dylan.
 Posts: 792 Add as FriendChallenge to a DebateSend a Message 2/9/2016 5:41:25 PMPosted: 3 years agoAt 2/9/2016 1:48:21 PM, Chaosism wrote:Below is a logic puzzle that I decided to try to solve using formal logic in order to test my understanding of it. If anyone knows formal logic, how would you write the clues and the necessary axioms to solve this puzzle? I've done so, but I'd like to see how others would go about this...----------------------------------------------Esmeralda's Shape-Shifting Studio has a loyal following of people who want to learn to change themselves into various animals, objects, and famous personages. During their very first workshop, five people (Charlie, Juliette, Morgan, Omar, and Tess), each of whom had signed up for a different number of lessons (4, 12, 16, 20, or 24), each transformed him- or herself into a different animal (beetle, crab, marmoset, penguin, or weasel), a different object (cushion, mailbox, pencil sharpener, rope, or toaster), and a different celebrity (Bob Dylan, George Clooney, Jennifer Lopez, Oprah Winfrey, or Stephen King). Despite their initial success, transmogrification is by no means a simple process, and one that should not be tried on one's own; in fact, Esmeralda recently had to make a house call to assist a student who was stuck in the form of a coffee mug and unable to change back to his natural form! From the information provided, determine the number of lessons that each person is taking, as well as the animal, object, and celebrity whose form each student took.1. The person who transmogrified him- or herself into a marmoset, a toaster, and Stephen King signed up for fewer lessons than the one who changed into a beetle.2. Charlie (who didn't change into a crab) turned himself into a very convincing rope. The one who shifted into the form of both a cushion and a weasel isn't the student who changed into the shape of Jennifer Lopez.3. Juliette is taking exactly 8 fewer lessons than the penguin impersonator. The one who turned into George Clooney signed up for neither 16 nor 20 lessons.4. The person who shifted into both a pencil sharpener and Oprah Winfrey is taking the most lessons.5. Omar is taking fewer lessons than Tess but more lessons than the one who transmogrified into a penguin, a mailbox, and Bob Dylan.We know that the one who transmogrified into a penguin is not taking 24 lessons (because he or she also transmogrified into Bob Dylan, and the one who transmogrified into Oprah Winfrey is taking the most lessons), 16 lessons (because Juliette is taking 8 fewer lessons than the penguin impersonator, and none of the options are 8 fewer than 16), 4 lessons (for the same reason as 16) or 20 lessons (because Omar is taking more lessons than the penguin impersonator but cannot be taking 24 lessons because Tess is taking more than him); therefore, the person who transmogrified into a penguin is taking 12 lessons.Because Juliette is taking 8 fewer lessons than the penguin impersonator, she is taking 4 lessons.Because the person who transmogrified into George Clooney is not taking 16 or 20 lessons, the Bob Dylan impersonator is taking 12 lessons, and the Oprah Winfrey impersonator is taking 24 lessons, she is taking 4 lessons and is Juliette.Because Tess is taking more lessons than Omar, who is taking more lessons than the penguin impersonator, Tess is taking either 20 or 24 lessons. If she is taking 24 lessons, than Omar is taking either 16 or 20 lessons. If she is taking 20 lessons, than Omar is taking 16 lessons.Charlie turned into a rope, and the penguin impersonator turned into a mailbox, Charlie is not the penguin impersonator and is taking 16, 20, or 24 lessons. Since Juliette, Tess, Omar, and Charlie are not the penguin impersonator, Morgan is.Since the person who transmogrified into a marmoset, a toaster, and Stephen King is not taking the most lessons, he is either Charlie, Omar, or Tess. Whoever it is, he or she is taking fewer lessons than the beetle impersonator, which is either Charlie, Omar, or Tess.Tess is either the beetle, the crab, the weasel, or the marmoset.Omar is either the marmoset, the crab, the beetle, or the weasel.Charlie is either the marmoset or the beetle.Juliette is either the crab or the weasel.Charlie is not the marmoset because he is the rope and the marmoset is the toaster, so he is the beetle. Because the beetle is taking more lessons than marmoset, the person who is taking the most lessons is a pencil sharpener, and Charlie is a rope, the following can be concluded:Tess is taking 24 lessons, is Oprah Winfrey, and is the pencil sharpener.Omar is taking 16 lessons, is the marmoset, is the toaster, and is Stephen King.Because Tess is the pencil sharpener, she is not the cushion and is not the weasel.Juliette is the weasel and the cushion, and Tess is the crab.Charlie is taking 20 lessons and is Jennifer Lopez.The answerTess- 24 lessons, Oprah Winfrey, pencil sharpener, and crab.Charlie- 20 lessons, Jennifer Lopez, rope, and beetle.Omar- 16 lessons, Stephen King, toaster, and marmoset.Morgan- 12 lessons, Bob Dylan, mailbox, and penguin.Juliette- 4 lessons, George Clooney, cushion, and weasel.Please tell me if there are any gaps in my logic.
 Posts: 430 Add as FriendChallenge to a DebateSend a Message 2/9/2016 6:04:15 PMPosted: 3 years agoAt 2/9/2016 5:41:25 PM, Bob13 wrote:At 2/9/2016 1:48:21 PM, Chaosism wrote:Below is a logic puzzle that I decided to try to solve using formal logic in order to test my understanding of it. If anyone knows formal logic, how would you write the clues and the necessary axioms to solve this puzzle? I've done so, but I'd like to see how others would go about this...----------------------------------------------Esmeralda's Shape-Shifting Studio has a loyal following of people who want to learn to change themselves into various animals, objects, and famous personages. During their very first workshop, five people (Charlie, Juliette, Morgan, Omar, and Tess), each of whom had signed up for a different number of lessons (4, 12, 16, 20, or 24), each transformed him- or herself into a different animal (beetle, crab, marmoset, penguin, or weasel), a different object (cushion, mailbox, pencil sharpener, rope, or toaster), and a different celebrity (Bob Dylan, George Clooney, Jennifer Lopez, Oprah Winfrey, or Stephen King). Despite their initial success, transmogrification is by no means a simple process, and one that should not be tried on one's own; in fact, Esmeralda recently had to make a house call to assist a student who was stuck in the form of a coffee mug and unable to change back to his natural form! From the information provided, determine the number of lessons that each person is taking, as well as the animal, object, and celebrity whose form each student took.1. The person who transmogrified him- or herself into a marmoset, a toaster, and Stephen King signed up for fewer lessons than the one who changed into a beetle.2. Charlie (who didn't change into a crab) turned himself into a very convincing rope. The one who shifted into the form of both a cushion and a weasel isn't the student who changed into the shape of Jennifer Lopez.3. Juliette is taking exactly 8 fewer lessons than the penguin impersonator. The one who turned into George Clooney signed up for neither 16 nor 20 lessons.4. The person who shifted into both a pencil sharpener and Oprah Winfrey is taking the most lessons.5. Omar is taking fewer lessons than Tess but more lessons than the one who transmogrified into a penguin, a mailbox, and Bob Dylan.We know that the one who transmogrified into a penguin is not taking 24 lessons (because he or she also transmogrified into Bob Dylan, and the one who transmogrified into Oprah Winfrey is taking the most lessons), 16 lessons (because Juliette is taking 8 fewer lessons than the penguin impersonator, and none of the options are 8 fewer than 16), 4 lessons (for the same reason as 16) or 20 lessons (because Omar is taking more lessons than the penguin impersonator but cannot be taking 24 lessons because Tess is taking more than him); therefore, the person who transmogrified into a penguin is taking 12 lessons.Because Juliette is taking 8 fewer lessons than the penguin impersonator, she is taking 4 lessons.Because the person who transmogrified into George Clooney is not taking 16 or 20 lessons, the Bob Dylan impersonator is taking 12 lessons, and the Oprah Winfrey impersonator is taking 24 lessons, she is taking 4 lessons and is Juliette.Because Tess is taking more lessons than Omar, who is taking more lessons than the penguin impersonator, Tess is taking either 20 or 24 lessons. If she is taking 24 lessons, than Omar is taking either 16 or 20 lessons. If she is taking 20 lessons, than Omar is taking 16 lessons.Charlie turned into a rope, and the penguin impersonator turned into a mailbox, Charlie is not the penguin impersonator and is taking 16, 20, or 24 lessons. Since Juliette, Tess, Omar, and Charlie are not the penguin impersonator, Morgan is.Since the person who transmogrified into a marmoset, a toaster, and Stephen King is not taking the most lessons, he is either Charlie, Omar, or Tess. Whoever it is, he or she is taking fewer lessons than the beetle impersonator, which is either Charlie, Omar, or Tess.Tess is either the beetle, the crab, the weasel, or the marmoset.Omar is either the marmoset, the crab, the beetle, or the weasel.Charlie is either the marmoset or the beetle.Juliette is either the crab or the weasel.Charlie is not the marmoset because he is the rope and the marmoset is the toaster, so he is the beetle. Because the beetle is taking more lessons than marmoset, the person who is taking the most lessons is a pencil sharpener, and Charlie is a rope, the following can be concluded:Tess is taking 24 lessons, is Oprah Winfrey, and is the pencil sharpener.Omar is taking 16 lessons, is the marmoset, is the toaster, and is Stephen King.Because Tess is the pencil sharpener, she is not the cushion and is not the weasel.Juliette is the weasel and the cushion, and Tess is the crab.Charlie is taking 20 lessons and is Jennifer Lopez.The answerTess- 24 lessons, Oprah Winfrey, pencil sharpener, and crab.Charlie- 20 lessons, Jennifer Lopez, rope, and beetle.Omar- 16 lessons, Stephen King, toaster, and marmoset.Morgan- 12 lessons, Bob Dylan, mailbox, and penguin.Juliette- 4 lessons, George Clooney, cushion, and weasel.Please tell me if there are any gaps in my logic.Is what I got. Could both be wrong!http://i.imgur.com...
 Posts: 2,742 Add as FriendChallenge to a DebateSend a Message 2/9/2016 6:56:07 PMPosted: 3 years agoAt 2/9/2016 5:41:25 PM, Bob13 wrote:The answerTess- 24 lessons, Oprah Winfrey, pencil sharpener, and crab.Charlie- 20 lessons, Jennifer Lopez, rope, and beetle.Omar- 16 lessons, Stephen King, toaster, and marmoset.Morgan- 12 lessons, Bob Dylan, mailbox, and penguin.Juliette- 4 lessons, George Clooney, cushion, and weasel.Please tell me if there are any gaps in my logic.That appears to be all correct. Well done!However...I was trying to refer to formal logic as in...1. The person who transmogrified him- or herself into a marmoset, a toaster, and Stephen King signed up for fewer lessons than the one who changed into a beetle.Translated, this could be something like:1. (Ex)(Txm&Txt&Txk&(Ay)(Tyb>Fxy))b :: beetlek :: Stephen Kingm :: marmosett :: toasterT :: ...transmogrified into...F :: ...took fewer lessons than...Of course, there are a lot of variables, so I used double-character subjects; instead of "m" I used "am", for example ("animal/marmoset").
 Posts: 2,742 Add as FriendChallenge to a DebateSend a Message 2/9/2016 6:57:08 PMPosted: 3 years agoAt 2/9/2016 6:04:15 PM, Heterodox wrote:The answerTess- 24 lessons, Oprah Winfrey, pencil sharpener, and crab.Charlie- 20 lessons, Jennifer Lopez, rope, and beetle.Omar- 16 lessons, Stephen King, toaster, and marmoset.Morgan- 12 lessons, Bob Dylan, mailbox, and penguin.Juliette- 4 lessons, George Clooney, cushion, and weasel.Please tell me if there are any gaps in my logic.Is what I got. Could both be wrong!http://i.imgur.com...Nope. Victory is yours!! But, see my previous post, here...
 Posts: 792 Add as FriendChallenge to a DebateSend a Message 2/9/2016 7:00:07 PMPosted: 3 years agoAt 2/9/2016 6:56:07 PM, Chaosism wrote:At 2/9/2016 5:41:25 PM, Bob13 wrote:The answerTess- 24 lessons, Oprah Winfrey, pencil sharpener, and crab.Charlie- 20 lessons, Jennifer Lopez, rope, and beetle.Omar- 16 lessons, Stephen King, toaster, and marmoset.Morgan- 12 lessons, Bob Dylan, mailbox, and penguin.Juliette- 4 lessons, George Clooney, cushion, and weasel.Please tell me if there are any gaps in my logic.That appears to be all correct. Well done!However...I was trying to refer to formal logic as in...1. The person who transmogrified him- or herself into a marmoset, a toaster, and Stephen King signed up for fewer lessons than the one who changed into a beetle.Translated, this could be something like:1. (Ex)(Txm&Txt&Txk&(Ay)(Tyb>Fxy))b :: beetlek :: Stephen Kingm :: marmosett :: toasterT :: ...transmogrified into...F :: ...took fewer lessons than...Of course, there are a lot of variables, so I used double-character subjects; instead of "m" I used "am", for example ("animal/marmoset").I don't know how formal logic works.
 Posts: 2,742 Add as FriendChallenge to a DebateSend a Message 2/9/2016 7:03:56 PMPosted: 3 years agoAt 2/9/2016 7:00:07 PM, Bob13 wrote:At 2/9/2016 6:56:07 PM, Chaosism wrote:At 2/9/2016 5:41:25 PM, Bob13 wrote:The answerTess- 24 lessons, Oprah Winfrey, pencil sharpener, and crab.Charlie- 20 lessons, Jennifer Lopez, rope, and beetle.Omar- 16 lessons, Stephen King, toaster, and marmoset.Morgan- 12 lessons, Bob Dylan, mailbox, and penguin.Juliette- 4 lessons, George Clooney, cushion, and weasel.Please tell me if there are any gaps in my logic.That appears to be all correct. Well done!However...I was trying to refer to formal logic as in...1. The person who transmogrified him- or herself into a marmoset, a toaster, and Stephen King signed up for fewer lessons than the one who changed into a beetle.Translated, this could be something like:1. (Ex)(Txm&Txt&Txk&(Ay)(Tyb>Fxy))b :: beetlek :: Stephen Kingm :: marmosett :: toasterT :: ...transmogrified into...F :: ...took fewer lessons than...Of course, there are a lot of variables, so I used double-character subjects; instead of "m" I used "am", for example ("animal/marmoset").I don't know how formal logic works.It's fun if your nerdy enough, I suppose. :) Here's where I learned it, in case you're interested at all: https://www.youtube.com...
 Posts: 2,742 Add as FriendChallenge to a DebateSend a Message 2/9/2016 7:04:29 PMPosted: 3 years agoAt 2/9/2016 7:03:56 PM, Chaosism wrote:It's fun if *your nerdy enough, I suppose. :) ...*you're
 Posts: 430 Add as FriendChallenge to a DebateSend a Message 2/9/2016 10:19:40 PMPosted: 3 years agoAt 2/9/2016 6:57:08 PM, Chaosism wrote:At 2/9/2016 6:04:15 PM, Heterodox wrote:The answerTess- 24 lessons, Oprah Winfrey, pencil sharpener, and crab.Charlie- 20 lessons, Jennifer Lopez, rope, and beetle.Omar- 16 lessons, Stephen King, toaster, and marmoset.Morgan- 12 lessons, Bob Dylan, mailbox, and penguin.Juliette- 4 lessons, George Clooney, cushion, and weasel.Please tell me if there are any gaps in my logic.Is what I got. Could both be wrong!http://i.imgur.com... /UvGTJPb.pngNope. Victory is yours!! But, see my previous post, here...I was just here for the logic puzzle, the way I use to do them as a kid. No clue about what you are asking about.
 Posts: 2,742 Add as FriendChallenge to a DebateSend a Message 2/10/2016 1:25:18 PMPosted: 3 years agoAt 2/9/2016 10:19:40 PM, Heterodox wrote:At 2/9/2016 6:57:08 PM, Chaosism wrote:At 2/9/2016 6:04:15 PM, Heterodox wrote:The answerTess- 24 lessons, Oprah Winfrey, pencil sharpener, and crab.Charlie- 20 lessons, Jennifer Lopez, rope, and beetle.Omar- 16 lessons, Stephen King, toaster, and marmoset.Morgan- 12 lessons, Bob Dylan, mailbox, and penguin.Juliette- 4 lessons, George Clooney, cushion, and weasel.Please tell me if there are any gaps in my logic.Is what I got. Could both be wrong!http://i.imgur.com... /UvGTJPb.pngNope. Victory is yours!! But, see my previous post, here...I was just here for the logic puzzle, the way I use to do them as a kid. No clue about what you are asking about.No problem. I have a book of them, I can post another one at a later time, if you'd like.To see what I'm talking about (if interested), check out the link I provided in my previous post in this thread.
 Posts: 430 Add as FriendChallenge to a DebateSend a Message 2/11/2016 2:32:50 PMPosted: 3 years agoAt 2/10/2016 1:25:18 PM, Chaosism wrote:At 2/9/2016 10:19:40 PM, Heterodox wrote:At 2/9/2016 6:57:08 PM, Chaosism wrote:At 2/9/2016 6:04:15 PM, Heterodox wrote:The answerTess- 24 lessons, Oprah Winfrey, pencil sharpener, and crab.Charlie- 20 lessons, Jennifer Lopez, rope, and beetle.Omar- 16 lessons, Stephen King, toaster, and marmoset.Morgan- 12 lessons, Bob Dylan, mailbox, and penguin.Juliette- 4 lessons, George Clooney, cushion, and weasel.Please tell me if there are any gaps in my logic.Is what I got. Could both be wrong!http://i.imgur.com... /UvGTJPb.pngNope. Victory is yours!! But, see my previous post, here...I was just here for the logic puzzle, the way I use to do them as a kid. No clue about what you are asking about.No problem. I have a book of them, I can post another one at a later time, if you'd like.To see what I'm talking about (if interested), check out the link I provided in my previous post in this thread.I vaguely remember it, from the mensa/logic books/magazines I use to get when I was a kid. Never really looked into it. Not really interested in another, thanks though. Just did that one for the nostalgia (haven't done one since I was a kid).
 Posts: 2,762 Add as FriendChallenge to a DebateSend a Message 2/11/2016 2:57:03 PMPosted: 3 years agoI did formal logic at school last semester, but I'm not sure I want to use natural deduction to take on a puzzle of this size :OI think it is well established that the only reason aliens come to earth is to slice up cows and examine inside peoples' bottoms. Unless you are a cow or suffer haemerrhoids I don't think there is anything to worry about from aliens. - keithprosser Don't be a stat cynic: http://www.debate.org... Response to conservative views on deforestation: http://www.debate.org... Topics I'd like to debate (not debating ATM): http://tinyurl.com...
 Posts: 3,981 Add as FriendChallenge to a DebateSend a Message 2/13/2016 12:11:01 PMPosted: 3 years agoanalyzing ..."Life calls the tune, we dance." John Galsworthy
 Posts: 2,742 Add as FriendChallenge to a DebateSend a Message 2/15/2016 5:23:45 PMPosted: 3 years agoHere's my setup...Operative Symbols (modified to accommodate the limited DDO format)~ :: Negation& :: Conjunctionv :: Disjunction> :: Implication<> :: Equivalence!= :: Not EqualRepresentation for subjectsI had to use double-characters due to the high quantity of subjects:People:pa=Charlie, pb=Juliette, pc=Morgan, pd=Omar, pe=TessAnimals:aa=Beetle, ab=Crab, ac=Marmoset, ad=Penguin, ae=WeaselObjects:oa=Cushion, ob=Mailbox, oc=Pencil Sharpener, od=Rope, oe=ToasterCelebrities:ca=Bob Dylan, cb=George Clooney, cc=Jennifer Lopez, cd=Oprah Winfrey, ce=Stephen KingRepresentation for predicatesC = "... transmogrified into the celebrity-form of ..."L = "... took a number of lessons equal to ..."N = "... transmogrified into the animal-form of a(n) ..."O = "... transmogrified into the object-form of a(n) ..."Q = "... took exactly eight fewer lessons than ..."M = "... took more lessens than ..."Expression of cluesP1) (Epx)(Npxac&Opxoe&Cpxce&(Apy)(Npyaa>Mpypx))"The person who transmogrified him- or herself into a marmoset, a toaster, and Stephen King signed up for fewer lessons than the one who changed into a beetle."P2) ~Npaab&Opaod&(Epx)(Opxoa&Npxaev~Cpxcc)"Charlie (who didn't change into a crab) turned himself into a very convincing rope. The one who shifted into the form of both a cushion and a weasel isn't the student who changed into the shape of Jennifer Lopez."P3) (Apx)(Npxad<>Qpbpx)&(Apy)(Cpycb>~(LpylcvLpyld))"Juliette is taking exactly 8 fewer lessons than the penguin impersonator. The one who turned into George Clooney signed up for neither 16 nor 20 lessons."P4) (Apx)((Opxoc<>Lpxle)&(Cpxcd<>Lpxle))"The person who shifted into both a pencil sharpener and Oprah Winfrey is taking the most lessons."P5) Mpepd&(Epx)(Apxad&Opxob&Cpxca&Mpdpx)"Omar is taking fewer lessons than Tess but more lessons than the one who transmogrified into a penguin, a mailbox, and Bob Dylan."AxiomsA1) (pa!=pb)&(pa!=pc)&(pa!=pd)&(pa!=pe)&(pb!=pc)&(pb!=pd)&(pb!=pe)&(pc!=pd)&(pc!=pe)&(pd!=pe)A2) Lq&(la!=lb)&(la!=lc)&(la!=ld)&(la!=le)&(lb!=lc)&(lb!=ld)&(lb!=le)&(lc!=ld)&(lc!=le)&(ld!=le)A3) Nq&(aa!=ab)&(aa!=ac)&(aa!=ad)&(aa!=ae)&(ab!=ac)&(ab!=ad)&(ab!=ae)&(ac!=ad)&(ac!=ae)&(ad!=ae)A4) Oq&(oa!=ob)&(oa!=oc)&(oa!=od)&(oa!=oe)&(ob!=oc)&(ob!=od)&(ob!=oe)&(oc!=od)&(oc!=oe)&(od!=oe)A5) Cq&(ca!=cb)&(ca!=cc)&(ca!=cd)&(ca!=ce)&(cb!=cc)&(cb!=cd)&(cb!=ce)&(cc!=cd)&(cc!=ce)&(cd!=ce)A6) (Apx)(LpxlavLpxlbvLpxlcvLpxldvLpxle)A7) (Apx)(NpxaavNpxabvNpxacvNpxadvNpxae)A8) (Apx)(OpxoavOpxobvOpxocvOpxodvOpxoe)A9) (Apx)(CpxcavCpxcbvCpxccvCpxcdvCpxce)A10) (AF)(Azx)(FpazxvFpbzxvFpczxvFpdzxvFpezx)A11) (AF)(Apx)(Azx)(Fpxzx>(Apy)((Fq&(px!=py))>~Fpyzx))A12) AF)(Apx)(Azx)(Fpxzx>(Azy)((Fq&(zx!=zy))>~Fpxzy))A13) (AF)(Azx)(Epx)(Fq>Fpxzx)A14) (Apx)(Apy)(Qpxpy>(Mpypx&((Lpxla&Lpylb)v(Lpxlb&Lpyld)v(Lpxlc&Lpyle))))A15) (Apx)(Apy)(Mpxpy>(~Lpxla&~Lpyle))A16) (Apx)(Apy)(Apz)((Mpxpy&Mpypz)>(~Lpzld&~Lpxlb&Mpxpz))A17) (Apw)(Apx)(Apy)(Apz)((Mpwpx&Mpxpy&Mpypz)>(~Lpzlc&~Lpwlc&Mpzpw))A18) (Apx)(Apy)((Lpxlc&Lpyld)>Mpypx)A19) (Apx)(Apy)((px!=py)>(Mpxpy<>~Mpypx))Notes:1) A2-A5: the generic "Xq" statement is there as an generic qualifier to regulate Universal Instantiations in axioms A11,A12, & A13.2) The italicized "F" represents a generic, unspecific predicate for the purposes of Universal Instantiation (Second Order Logic).3) A15-A17: these express the impossibilities of the "M" predicate in relation to the maximum and minimum "L" values....in case anyone at all is interested. ~awaits passing tumbleweeds~