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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Homework 5" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## 5.1" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "Rules:\n", | ||
| "1. If the unicorn is mythical, then it is immortal.\n", | ||
| "2. If it is not mythical, then it is a mortal mammal. \n", | ||
| "3. If the unicorn is either immortal or a mammal, then it is horned. \n", | ||
| "4. If the unicorn is horned, then it is magical. \n", | ||
| "\n", | ||
| "You cannot prove the unicorn is mythical, but you can prove both that it is magical and that it is horned.\n", | ||
| "We can't prove the unicorn is mythical because neither the first rule nor the second is a biconditional. So we have no way to conclude anything about whether the unicorn is mythical or not, since none of the other rules involve the unicorn's mythicality at all.\n", | ||
| "\n", | ||
| "To start proving the unicorn is horned and magical, either the unicorn is mythical or it is not mythical.\n", | ||
| "\n", | ||
| "Suppose the unicorn is mythical.\n", | ||
| "\n", | ||
| "Since it is mythical, it is immortal. \n", | ||
| "\n", | ||
| "Since it is immortal, it is either immortal or a mammal.\n", | ||
| "\n", | ||
| "Since it is either immortal or a mammal, it is horned.\n", | ||
| "\n", | ||
| "Since it is horned, then it is magical.\n", | ||
| "\n", | ||
| "Therefore, if the unicorn is mythical, then it is horned and magical.\n", | ||
| "\n", | ||
| "\n", | ||
| "Alternatively, suppose the unicorn is not mythical.\n", | ||
| "\n", | ||
| "Since it is not mythical, it is a mortal and a mammal.\n", | ||
| "\n", | ||
| "Since it is a mortal and a mammal, it is a mammal.\n", | ||
| "\n", | ||
| "Since it is a mammal, it is either immortal or a mammal.\n", | ||
| "\n", | ||
| "Since it is either immortal or a mammal, then it is horned.\n", | ||
| "\n", | ||
| "Since it is horned, then it is magical.\n", | ||
| "\n", | ||
| "Therefore, if the unicorn is not mythical, then it is horned and magical.\n", | ||
| "\n", | ||
| "Since the unicorn is either mythical or not mythical, and either way the unicorn is horned and magical, the unicorn is horned and magical." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### More formally: " | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "1. mythical(unicorn) => immortal(unicorn)\n", | ||
| "2. !mythical(unicorn) => mammal(unicorn) ^ mortal(unicorn)\n", | ||
| "3. immortal(unicorn) v mammal(unicorn) => horned(unicorn)\n", | ||
| "4. horned(unicorn) => magical(unicorn)\n", | ||
| "_______________________________________\n", | ||
| "5. mythical(unicorn) v !mythical(unicorn)\n", | ||
| " 1. Suppose mythical(unicorn)\n", | ||
| " _________________________________\n", | ||
| " 2. immortal(unicorn) | (conditional elimination/modus ponens, from statements 1 and A)\n", | ||
| " 3. immortal(unicorn) v mammal(unicorn) | (disjunctive introduction, statement B)\n", | ||
| " 4. horned(unicorn) | (conditional elimination/modus ponens, from statements 3 and C)\n", | ||
| " 5. magical(unicorn) | (conditional elimination/modus ponens, from statements 4 and D)\n", | ||
| " 6. magical(unicorn) ^ horned(unicorn) | (conjunctive introduction, statements D and E)\n", | ||
| "6. mythical(unicorn) => magical(unicorn) ^ horned(unicorn) | (conditional introduction, statements A-F)\n", | ||
| " 7. Suppose !mythical(unicorn)\n", | ||
| " _________________________________\n", | ||
| " 8. mortal(unicorn) ^ mammal(unicorn) | (conditional elimination/modus ponens, from statements 2 and G)\n", | ||
| " 9. mammal(unicorn) | (conjunction elimination, statement H)\n", | ||
| " 10. immortal(unicorn) v mammal(unicorn) | (disjunctive introduction, statement I)\n", | ||
| " 11. horned(unicorn) | (conditional elimination/modus ponens, from statements 3 and J)\n", | ||
| " 12. magical(unicorn) | (conditional elimination/modus ponens, from statements 4 and K)\n", | ||
| " 13. magical(unicorn) ^ horned(unicorn) | (conjunctive introduction, statements K and L)\n", | ||
| "7. !mythical(unicorn) => magical(unicorn) ^ horned(unicorn) | (conditional introduction, statements G-M)\n", | ||
| "8. magical(unicorn) ^ horned(unicorn) | (disjunction elimination, statements 5, 6, and 7)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### Prolog" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "I think this program would be pretty difficult to program in Prolog. The main issue would be the second rule, where we conclude the unicorn is a mortal mammal as the result of the unicorn *not* being mythical. Rule one would be very easy; and while it might require multiple statements to state that the unicorn is a mortal mammal, that aspect of rule two would be doable. The final rules would be very easy. But the *not* aspect of rule two would complicate things considerably, I think, maybe even to the point of making it impossible. Making a negatory statement in Prolog is not simple." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## 5.2" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| " crossword(V1,V2,V3,H1,H2,H3) :- word(V1, _,A,_,B,_,C,_), \n", | ||
| "\t\t\t\tword(V2, _,D,_,E,_,F,_),\n", | ||
| "\t\t\t\tword(V3, _,G,_,H,_,I,_),\n", | ||
| "\t\t\t\tword(H1, _,A,_,D,_,G,_),\n", | ||
| "\t\t\t\tword(H2, _,B,_,E,_,H,_),\n", | ||
| "\t\t\t\tword(H3, _,C,_,F,_,I,_),\n", | ||
| "\t\t\t\tV1 \\== V2,\n", | ||
| "\t\t\t\tV1 \\== V3,\n", | ||
| "\t\t\t\tV1 \\== H1,\n", | ||
| "\t\t\t\tV1 \\== H2,\n", | ||
| "\t\t\t\tV1 \\== H3,\n", | ||
| "\t\t\t\tV2 \\== V3,\n", | ||
| "\t\t\t\tV2 \\== H1,\n", | ||
| "\t\t\t\tV2 \\== H2,\n", | ||
| "\t\t\t\tV2 \\== H3,\n", | ||
| "\t\t\t\tV3 \\== H1,\n", | ||
| "\t\t\t\tV3 \\== H2,\n", | ||
| "\t\t\t\tV3 \\== H3,\n", | ||
| "\t\t\t\tH1 \\== H2,\n", | ||
| "\t\t\t\tH1 \\== H3,\n", | ||
| "\t\t\t\tH2 \\== H3." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Python 3.7 (tensorflow)", | ||
| "language": "python", | ||
| "name": "tensorflow" | ||
| }, | ||
| "language_info": { | ||
| "codemirror_mode": { | ||
| "name": "ipython", | ||
| "version": 3 | ||
| }, | ||
| "file_extension": ".py", | ||
| "mimetype": "text/x-python", | ||
| "name": "python", | ||
| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython3", | ||
| "version": "3.7.7" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 4 | ||
| } |