![\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260125114132-6976016c0d5f6.png\")
Ancient number systems compared with Leibniz\u2019s binary, made by CounterComplex \u53e4\u4ee3\u8bb0\u6570\u7cfb\u7edf\u4e0e\u83b1\u5e03\u5c3c\u8328\u4e8c\u8fdb\u5236\u7684\u5bf9\u6bd4\u56fe，\u7531 CounterComplex \u521b\u4f5c<\/p>\n\u6ce8：\u672c\u6587\u4e3a \u201c\u903b\u8f91\u6f14\u8fdb\u4e0e\u5e94\u7528\u201d \u76f8\u5173\u5408\u8f91\u3002 \u82f1\u6587\u5f15\u6587，\u673a\u7ffb\u672a\u6821\u3002 \u4e2d\u6587\u5f15\u6587，\u7565\u4f5c\u91cd\u6392\u3002 \u56fe\u7247\u6e05\u6670\u5ea6\u53d7\u5f15\u6587\u539f\u56fe\u6240\u9650\u3002 \u5982\u6709\u5185\u5bb9\u5f02\u5e38，\u8bf7\u770b\u539f\u6587\u3002<\/p>\n
June 17, 2017 by Nexperia<\/p>\n
Digital logic as we know it today required collaboration from thousands of people over thousands of years. \u5982\u4eca\u6211\u4eec\u6240\u719f\u77e5\u7684\u6570\u5b57\u903b\u8f91，\u662f\u6570\u5343\u5e74\u6765\u65e0\u6570\u4eba\u534f\u4f5c\u7814\u7a76\u7684\u6210\u679c\u3002<\/p>\n
Digital logic as we know it today required collaboration from thousands of people over thousands of years. \u5982\u4eca\u6211\u4eec\u6240\u719f\u77e5\u7684\u6570\u5b57\u903b\u8f91，\u662f\u6570\u5343\u5e74\u6765\u65e0\u6570\u4eba\u534f\u4f5c\u7814\u7a76\u7684\u6210\u679c\u3002<\/p>\n
Before electronic devices were ever created, mathematicians and philosophers were unknowingly creating the language of binary and laying the foundation for modern computing devices. These concepts were later applied to similar communication systems like Morse Code and the binary that we\u2019re familiar with today. Mathematicians like George Boole and Charles Babbage applied these concepts to mechanical devices to automate them. While computing has a rich history, the logic gates that allow computers to function get less attention. \u5728\u7535\u5b50\u8bbe\u5907\u8bde\u751f\u4e4b\u524d，\u6570\u5b66\u5bb6\u4e0e\u54f2\u5b66\u5bb6\u4fbf\u5728\u65e0\u610f\u95f4\u521b\u7acb\u4e86\u4e8c\u8fdb\u5236\u7684\u8868\u8fbe\u4f53\u7cfb，\u4e3a\u73b0\u4ee3\u8ba1\u7b97\u8bbe\u5907\u7684\u8bde\u751f\u5960\u5b9a\u4e86\u57fa\u7840\u3002\u8fd9\u4e9b\u7406\u8bba\u6982\u5ff5\u540e\u6765\u88ab\u5e94\u7528\u4e8e\u83ab\u5c14\u65af\u7535\u7801\u7b49\u540c\u7c7b\u901a\u4fe1\u7cfb\u7edf，\u4e5f\u6210\u4e3a\u4e86\u6211\u4eec\u5982\u4eca\u6240\u719f\u77e5\u7684\u4e8c\u8fdb\u5236\u7684\u96cf\u5f62\u3002\u4e54\u6cbb\u00b7\u5e03\u5c14\u3001\u67e5\u5c14\u65af\u00b7\u5df4\u8d1d\u5947\u7b49\u6570\u5b66\u5bb6\u5c06\u8fd9\u4e9b\u6982\u5ff5\u5e94\u7528\u4e8e\u673a\u68b0\u8bbe\u5907，\u5b9e\u73b0\u4e86\u8bbe\u5907\u7684\u81ea\u52a8\u5316\u8fd0\u884c\u3002\u5c3d\u7ba1\u8ba1\u7b97\u673a\u9886\u57df\u6709\u7740\u60a0\u4e45\u4e14\u4e30\u5bcc\u7684\u53d1\u5c55\u5386\u53f2，\u4f46\u652f\u6491\u8ba1\u7b97\u673a\u5b9e\u73b0\u5404\u9879\u529f\u80fd\u7684\u903b\u8f91\u95e8，\u5374\u5e76\u672a\u5f97\u5230\u540c\u7b49\u7a0b\u5ea6\u7684\u5173\u6ce8\u3002<\/p>\n
In the 10th century in ancient China, a scholar named Shao Yong rearranged the hexagrams from the I Ching into a format similar to binary. Some scholars even suggest that the I Ching was Gottfried Leibniz\u2019s inspiration for binary code. \u516c\u5143 10 \u4e16\u7eaa\u7684\u4e2d\u56fd\u53e4\u4ee3，\u5b66\u8005\u90b5\u96cd\u5c06\u300a\u6613\u7ecf\u300b\u4e2d\u7684\u5366\u8c61\u91cd\u65b0\u6392\u5217，\u5f62\u6210\u4e86\u7c7b\u4e8c\u8fdb\u5236\u7684\u6392\u5e03\u5f62\u5f0f\u3002\u90e8\u5206\u5b66\u8005\u751a\u81f3\u8ba4\u4e3a，\u300a\u6613\u7ecf\u300b\u6b63\u662f\u6208\u7279\u5f17\u91cc\u5fb7\u00b7\u83b1\u5e03\u5c3c\u8328\u521b\u9020\u4e8c\u8fdb\u5236\u7f16\u7801\u7684\u7075\u611f\u6765\u6e90\u3002<\/p>\n
Ancient number systems compared with Leibniz\u2019s binary, made by CounterComplex \u53e4\u4ee3\u8bb0\u6570\u7cfb\u7edf\u4e0e\u83b1\u5e03\u5c3c\u8328\u4e8c\u8fdb\u5236\u7684\u5bf9\u6bd4\u56fe，\u7531 CounterComplex \u521b\u4f5c<\/p>\n
There were also truth tables in ancient Greece with YES and NO functioning like ON and OFF in the case of transistors. Leibniz most likely found inspiration from Greek mathematicians because their works were the most accessible for scholars in Europe during the 17th and 18th centuries. \u53e4\u5e0c\u814a\u65f6\u671f\u4fbf\u5df2\u51fa\u73b0\u771f\u503c\u8868，\u5176\u4e2d\u7684\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\u903b\u8f91，\u4e0e\u6676\u4f53\u7ba1\u4e2d\u7684\u201c\u5bfc\u901a\u201d\u548c\u201c\u5173\u65ad\u201d\u72b6\u6001\u5177\u6709\u76f8\u901a\u7684\u4f5c\u7528\u539f\u7406\u3002\u83b1\u5e03\u5c3c\u8328\u7684\u7814\u7a76\u7075\u611f\u5927\u6982\u7387\u6765\u6e90\u4e8e\u53e4\u5e0c\u814a\u6570\u5b66\u5bb6，\u56e0\u4e3a\u5728 17 \u81f3 18 \u4e16\u7eaa，\u53e4\u5e0c\u814a\u6570\u5b66\u5bb6\u7684\u8457\u4f5c\u662f\u6b27\u6d32\u5b66\u8005\u6700\u6613\u83b7\u53d6\u7684\u7814\u7a76\u8d44\u6599\u3002<\/p>\n
There was also the small, remote Polynesian island of Mangareva which used a binary numeral system to help with mental arithmetic. Many ancient civilizations had numeric systems with similarities to binary as we know it today. Whether Leibniz was inspired by one of these other ancient systems will remain one of history\u2019s mysteries. The study of ancient mathematics and binary is very interesting, but it also helps to temper one\u2019s expectations and be prepared for misleading headlines and titles. \u5730\u5904\u6ce2\u5229\u5c3c\u897f\u4e9a\u7684\u504f\u8fdc\u5c0f\u5c9b\u8292\u963f\u96f7\u74e6，\u4e5f\u66fe\u4f7f\u7528\u4e00\u5957\u4e8c\u8fdb\u5236\u8bb0\u6570\u7cfb\u7edf\u8f85\u52a9\u5fc3\u7b97\u3002\u8bf8\u591a\u53e4\u4ee3\u6587\u660e\u6240\u4f7f\u7528\u7684\u8bb0\u6570\u65b9\u5f0f，\u90fd\u4e0e\u5982\u4eca\u6211\u4eec\u719f\u77e5\u7684\u4e8c\u8fdb\u5236\u5b58\u5728\u76f8\u4f3c\u6027\u3002\u83b1\u5e03\u5c3c\u8328\u662f\u5426\u66fe\u53d7\u8fd9\u4e9b\u53e4\u4ee3\u8bb0\u6570\u4f53\u7cfb\u7684\u542f\u53d1，\u8fd9\u4e00\u95ee\u9898\u81f3\u4eca\u4ecd\u662f\u5386\u53f2\u672a\u89e3\u4e4b\u8c1c\u3002\u5bf9\u53e4\u4ee3\u6570\u5b66\u4e0e\u4e8c\u8fdb\u5236\u7684\u7814\u7a76\u517c\u5177\u8da3\u5473\u6027，\u540c\u65f6\u4e5f\u80fd\u8ba9\u7814\u7a76\u8005\u4fdd\u6301\u7406\u6027\u7684\u7814\u7a76\u89c6\u89d2，\u907f\u514d\u88ab\u8bef\u5bfc\u6027\u7684\u6807\u9898\u4e0e\u4fe1\u606f\u6240\u5f71\u54cd\u3002<\/p>\n
Although Gottfried Wilhelm Leibniz is credited with inventing binary as we know it today, the first logic based devices were created by Charles Babbage. His first Difference Engine was a mechanical calculator designed to tabulate polynomial equations and print the results on a roll of paper. \u5c3d\u7ba1\u6208\u7279\u5f17\u91cc\u5fb7\u00b7\u5a01\u5ec9\u00b7\u83b1\u5e03\u5c3c\u8328\u88ab\u516c\u8ba4\u4e3a\u73b0\u4ee3\u4e8c\u8fdb\u5236\u7684\u53d1\u660e\u8005，\u4f46\u9996\u6b3e\u57fa\u4e8e\u903b\u8f91\u539f\u7406\u7684\u8bbe\u5907\u7531\u67e5\u5c14\u65af\u00b7\u5df4\u8d1d\u5947\u7814\u5236\u3002\u4ed6\u8bbe\u8ba1\u7684\u7b2c\u4e00\u53f0\u5dee\u5206\u673a\u662f\u4e00\u6b3e\u673a\u68b0\u8ba1\u7b97\u5668，\u53ef\u5b8c\u6210\u591a\u9879\u5f0f\u65b9\u7a0b\u7684\u5236\u8868\u8fd0\u7b97，\u5e76\u5c06\u8fd0\u7b97\u7ed3\u679c\u6253\u5370\u5728\u7eb8\u5377\u4e0a\u3002<\/p>\n
In Babbage\u2019s design, mechanical gears served the function of logic gates. He designed a second Difference Engine, also referred to as the Analytical Engine, but could never obtain funding to make it. His designs were later noticed in the 1980s and a fully functional model was finally created to his specifications. \u5728\u5df4\u8d1d\u5947\u7684\u8bbe\u8ba1\u4e2d，\u673a\u68b0\u9f7f\u8f6e\u627f\u62c5\u4e86\u903b\u8f91\u95e8\u7684\u529f\u80fd\u3002\u4ed6\u8fd8\u8bbe\u8ba1\u4e86\u7b2c\u4e8c\u53f0\u5dee\u5206\u673a，\u4e5f\u88ab\u79f0\u4e3a\u5206\u6790\u673a，\u4f46\u59cb\u7ec8\u672a\u80fd\u83b7\u5f97\u7814\u53d1\u8d44\u91d1\u3002\u76f4\u5230 20 \u4e16\u7eaa 80 \u5e74\u4ee3，\u4ed6\u7684\u8bbe\u8ba1\u65b9\u6848\u624d\u53d7\u5230\u5173\u6ce8，\u4eba\u4eec\u6700\u7ec8\u4f9d\u7167\u5176\u8bbe\u8ba1\u89c4\u683c，\u5236\u9020\u51fa\u4e86\u4e00\u53f0\u53ef\u6b63\u5e38\u8fd0\u884c\u7684\u6837\u673a\u3002<\/p>\n
The first common logic gates were electromagnetic relays, which were basically ON-OFF switches. The electromechanical relay was invented by Joseph Henry in 1835, but the brilliance of his invention was not realized until later when his relays were used in the telegraph. After the invention of the rotary dial in 1890, more complex relays with 10 positions were developed. \u9996\u6b3e\u88ab\u5e7f\u6cdb\u5e94\u7528\u7684\u903b\u8f91\u95e8\u662f\u7535\u78c1\u7ee7\u7535\u5668，\u5176\u672c\u8d28\u4e3a\u901a\u65ad\u5f00\u5173\u3002\u7535\u78c1\u7ee7\u7535\u5668\u7531\u7ea6\u745f\u592b\u00b7\u4ea8\u5229\u4e8e 1835 \u5e74\u53d1\u660e，\u4f46\u8fd9\u4e00\u53d1\u660e\u7684\u4ef7\u503c\u5728\u5f53\u65f6\u5e76\u672a\u88ab\u53d1\u6398，\u76f4\u5230\u540e\u6765\u88ab\u5e94\u7528\u4e8e\u7535\u62a5\u7cfb\u7edf，\u5176\u4f18\u52bf\u624d\u5f97\u4ee5\u663e\u73b0\u30021890 \u5e74\u65cb\u8f6c\u62e8\u53f7\u76d8\u53d1\u660e\u540e，\u53ef\u5b9e\u73b0 10 \u6863\u4f4d\u63a7\u5236\u7684\u66f4\u590d\u6742\u7ee7\u7535\u5668\u88ab\u7814\u5236\u51fa\u6765\u3002<\/p>\n
In 1898, Nikola Tesla filed a patent for an invention he called \u201cteleautomation\u201d, which he demonstrated with a radio-controlled miniature boat, the first remote control device ever made. The remote control system contained a method for decoding Hertzian waves. This system would toggle actions based on different signals, functioning as an AND gate. 1898 \u5e74，\u5c3c\u53e4\u62c9\u00b7\u7279\u65af\u62c9\u4e3a\u5176\u53d1\u660e\u7684\u201c\u8fdc\u7a0b\u81ea\u52a8\u5316\u6280\u672f\u201d\u7533\u8bf7\u4e86\u4e13\u5229，\u5e76\u901a\u8fc7\u4e00\u8258\u65e0\u7ebf\u7535\u9065\u63a7\u5fae\u578b\u8239\u5b8c\u6210\u4e86\u6280\u672f\u6f14\u793a，\u8fd9\u8258\u8239\u4e5f\u662f\u4eba\u7c7b\u53f2\u4e0a\u9996\u6b3e\u9065\u63a7\u8bbe\u5907\u3002\u8be5\u9065\u63a7\u7cfb\u7edf\u642d\u8f7d\u4e86\u8d6b\u5179\u6ce2\u89e3\u7801\u6280\u672f，\u53ef\u6839\u636e\u4e0d\u540c\u7684\u4fe1\u53f7\u89e6\u53d1\u4e0d\u540c\u7684\u52a8\u4f5c，\u5176\u5de5\u4f5c\u539f\u7406\u4e0e\u4e0e\u95e8\u4e00\u81f4\u3002<\/p>\n
The first digital, programmable computing devices\u2019 logic circuitry laid in vacuum tubes. This includes famous computers like ENIAC, Colossus, and Alan Turing\u2019s Pilot Ace. These early digital computers were enormous and had thousands of vacuum tubes. In order to make computing devices accessible to the public, the logic circuitry would need to shrink. \u9996\u6b3e\u6570\u5b57\u53ef\u7f16\u7a0b\u8ba1\u7b97\u8bbe\u5907\u7684\u903b\u8f91\u7535\u8def\u4ee5\u771f\u7a7a\u7ba1\u4e3a\u6838\u5fc3\u5143\u4ef6，\u7ecf\u5178\u7684 ENIAC \u8ba1\u7b97\u673a\u3001\u5de8\u4eba\u8ba1\u7b97\u673a\u4ee5\u53ca\u827e\u4f26\u00b7\u56fe\u7075\u8bbe\u8ba1\u7684\u9886\u822a\u8005 ACE \u8ba1\u7b97\u673a\u5747\u5c5e\u6b64\u7c7b\u3002\u8fd9\u4e9b\u65e9\u671f\u6570\u5b57\u8ba1\u7b97\u673a\u4f53\u79ef\u5e9e\u5927，\u5185\u90e8\u642d\u8f7d\u4e86\u6570\u5343\u6839\u771f\u7a7a\u7ba1\u3002\u82e5\u8981\u8ba9\u8ba1\u7b97\u8bbe\u5907\u8d70\u5411\u6c11\u7528，\u5176\u903b\u8f91\u7535\u8def\u5fc5\u987b\u5b9e\u73b0\u5fae\u578b\u5316\u3002<\/p>\n
![\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260125114132-6976016c5005b.png\")
ENIAC occupied 1600 square feet of space. A modern smartphone has about 12,000 times as much computing power. Courtesy of the US Army ENIAC \u8ba1\u7b97\u673a\u7684\u5360\u5730\u9762\u79ef\u8fbe 1600 \u5e73\u65b9\u82f1\u5c3a，\u800c\u4e00\u53f0\u73b0\u4ee3\u667a\u80fd\u624b\u673a\u7684\u8fd0\u7b97\u80fd\u529b\u7ea6\u4e3a\u5176 12000 \u500d\u3002\u56fe\u7247\u7531\u7f8e\u56fd\u9646\u519b\u63d0\u4f9b<\/p>\n
This needed decrease in size came in the 1960s with resistor-transistor logic (RTL) and transistor-transistor logic (TTL). RTLs were the first logic devices to be incorporated into integrated circuits and were later used in the Apollo guidance computer. The first TTL integrated circuits were developed in 1963 by Sylvania and were popularized by TI\u2019s 7400 series. Since then, modern logic circuitry has been shrinking these concepts and adding additional Boolean gates. 20 \u4e16\u7eaa 60 \u5e74\u4ee3，\u7535\u963b-\u6676\u4f53\u7ba1\u903b\u8f91\u7535\u8def（RTL）\u4e0e\u6676\u4f53\u7ba1-\u6676\u4f53\u7ba1\u903b\u8f91\u7535\u8def（TTL)\u7684\u51fa\u73b0，\u5b9e\u73b0\u4e86\u903b\u8f91\u7535\u8def\u7684\u5fae\u578b\u5316\u9700\u6c42\u3002\u7535\u963b-\u6676\u4f53\u7ba1\u903b\u8f91\u7535\u8def\u662f\u9996\u6b3e\u88ab\u96c6\u6210\u5230\u96c6\u6210\u7535\u8def\u4e2d\u7684\u903b\u8f91\u5668\u4ef6，\u540e\u7eed\u8fd8\u88ab\u5e94\u7528\u4e8e\u963f\u6ce2\u7f57\u767b\u6708\u8ba1\u5212\u7684\u5236\u5bfc\u8ba1\u7b97\u673a\u30021963 \u5e74，\u897f\u5c4b\u7535\u6c14\u7814\u5236\u51fa\u9996\u6b3e\u6676\u4f53\u7ba1-\u6676\u4f53\u7ba1\u903b\u8f91\u96c6\u6210\u7535\u8def，\u5fb7\u5dde\u4eea\u5668\u63a8\u51fa\u7684 7400 \u7cfb\u5217\u5219\u8ba9\u8be5\u7c7b\u578b\u7535\u8def\u5b9e\u73b0\u4e86\u666e\u53ca\u3002\u81ea\u6b64，\u73b0\u4ee3\u903b\u8f91\u7535\u8def\u5728\u8fd9\u4e9b\u7406\u8bba\u57fa\u7840\u4e0a\u4e0d\u65ad\u5411\u5fae\u578b\u5316\u53d1\u5c55，\u540c\u65f6\u65b0\u589e\u4e86\u591a\u79cd\u5e03\u5c14\u903b\u8f91\u95e8\u7c7b\u578b\u3002<\/p>\n
Centuries of innovation have led to tiny and powerful logic devices like FPGAs, PLCs, and Pico Gates, which allow tiny devices to have computing power and control functionality that dwarfs the massive computers of old. These days, there are tiny logic packages for just about every function imaginable, whether it comes in an array or a discrete package. \u5386\u7ecf\u6570\u767e\u5e74\u7684\u6280\u672f\u9769\u65b0，\u73b0\u573a\u53ef\u7f16\u7a0b\u95e8\u9635\u5217（FPGA）\u3001\u53ef\u7f16\u7a0b\u903b\u8f91\u63a7\u5236\u5668（PLC）\u4ee5\u53ca\u5fae\u578b\u903b\u8f91\u95e8（Pico Gates）\u7b49\u5fae\u578b\u5316\u3001\u9ad8\u6027\u80fd\u903b\u8f91\u5668\u4ef6\u5e94\u8fd0\u800c\u751f\u3002\u8fd9\u4e9b\u5668\u4ef6\u8ba9\u5c0f\u578b\u8bbe\u5907\u62e5\u6709\u4e86\u8fdc\u8d85\u65e9\u671f\u5927\u578b\u8ba1\u7b97\u673a\u7684\u8fd0\u7b97\u4e0e\u63a7\u5236\u80fd\u529b\u3002\u5982\u4eca，\u5404\u7c7b\u529f\u80fd\u7684\u5fae\u578b\u903b\u8f91\u5668\u4ef6\u5df2\u5b9e\u73b0\u5168\u8986\u76d6，\u5f62\u5f0f\u4e0a\u65e2\u5305\u62ec\u9635\u5217\u5f0f，\u4e5f\u5305\u62ec\u5206\u7acb\u5f0f\u3002<\/p>\n
SQC Consulting, Mallow, County Cork, Ireland Gerard O\u2019Regan First Online: 10 August 2017<\/p>\n
Syllogistic logic \/ \u4e09\u6bb5\u8bba\u903b\u8f91<\/p>\n<\/li>\n
Fallacies \/ \u8c2c\u8bef<\/p>\n<\/li>\n
Paradoxes \/ \u6096\u8bba<\/p>\n<\/li>\n
Stoic logic \/ \u65af\u591a\u845b\u903b\u8f91<\/p>\n<\/li>\n
Boole\u2019s symbolic logic \/ \u5e03\u5c14\u7b26\u53f7\u903b\u8f91<\/p>\n<\/li>\n
Digital computing \/ \u6570\u5b57\u8ba1\u7b97<\/p>\n<\/li>\n
Propositional logic \/ \u547d\u9898\u903b\u8f91<\/p>\n<\/li>\n
Predicate logic \/ \u8c13\u8bcd\u903b\u8f91<\/p>\n<\/li>\n
Universal and existential quantifiers \/ \u5168\u79f0\u91cf\u8bcd\u4e0e\u5b58\u5728\u91cf\u8bcd<\/p>\n<\/li>\n<\/ul>\n
Logic is concerned with reasoning and with establishing the validity of arguments. It allows conclusions to be deduced from premises according to logical rules, and the logical argument establishes the truth of the conclusion provided that the premises are true. \u903b\u8f91\u7814\u7a76\u63a8\u7406\u53ca\u8bba\u8bc1\u6709\u6548\u6027\u7684\u5efa\u7acb\u3002\u5b83\u5141\u8bb8\u6839\u636e\u903b\u8f91\u89c4\u5219\u4ece\u524d\u63d0\u63a8\u5bfc\u51fa\u7ed3\u8bba，\u4e14\u82e5\u524d\u63d0\u4e3a\u771f，\u903b\u8f91\u8bba\u8bc1\u53ef\u786e\u7acb\u7ed3\u8bba\u7684\u771f\u5b9e\u6027\u3002<\/p>\n
The origins of logic are with the Greeks who were interested in the nature of truth. The sophists (e.g. Protagoras and Gorgias) were teachers of rhetoric, who taught their pupils techniques in winning an argument and convincing an audience. Plato explores the nature of truth in some of his dialogues, and he is critical of the position of the sophists who argue that there is no absolute truth, and that truth instead is always relative to some frame of reference. The classic sophist position is stated by Protagoras \u201cMan is the measure of all things: of things which are, that they are, and of things which are not, that they are not\u201d. In other words, what is true for you is true for you, and what is true for me is true for me. \u903b\u8f91\u7684\u8d77\u6e90\u53ef\u8ffd\u6eaf\u81f3\u5bf9\u771f\u7406\u672c\u8d28\u611f\u5174\u8da3\u7684\u53e4\u5e0c\u814a\u4eba\u3002\u667a\u8005\u5b66\u6d3e（\u5982\u666e\u7f57\u6cf0\u6208\u62c9\u548c\u9ad8\u5c14\u5409\u4e9a）\u662f\u4fee\u8f9e\u5b66\u6559\u5e08，\u4ed6\u4eec\u5411\u5b66\u751f\u4f20\u6388\u8d62\u5f97\u8fa9\u8bba\u548c\u8bf4\u670d\u542c\u4f17\u7684\u6280\u5de7\u3002\u67cf\u62c9\u56fe\u5728\u5176\u90e8\u5206\u5bf9\u8bdd\u5f55\u4e2d\u63a2\u8ba8\u4e86\u771f\u7406\u7684\u672c\u8d28，\u5e76\u5bf9\u667a\u8005\u5b66\u6d3e\u7684\u89c2\u70b9\u63d0\u51fa\u6279\u5224\u2014\u2014\u667a\u8005\u5b66\u6d3e\u8ba4\u4e3a\u4e0d\u5b58\u5728\u7edd\u5bf9\u771f\u7406，\u771f\u7406\u59cb\u7ec8\u76f8\u5bf9\u4e8e\u67d0\u4e00\u53c2\u7167\u6846\u67b6\u800c\u8a00\u3002\u666e\u7f57\u6cf0\u6208\u62c9\u63d0\u51fa\u4e86\u7ecf\u5178\u7684\u667a\u8005\u5b66\u6d3e\u4e3b\u5f20：\u201c\u4eba\u662f\u4e07\u7269\u7684\u5c3a\u5ea6：\u662f\u5b58\u5728\u8005\u5b58\u5728\u7684\u5c3a\u5ea6，\u4e5f\u662f\u4e0d\u5b58\u5728\u8005\u4e0d\u5b58\u5728\u7684\u5c3a\u5ea6\u3002\u201d \u6362\u53e5\u8bdd\u8bf4，\u5bf9\u4f60\u800c\u8a00\u4e3a\u771f\u7684\u4e1c\u897f\u5bf9\u4f60\u6765\u8bf4\u5c31\u662f\u771f\u7684，\u5bf9\u6211\u800c\u8a00\u4e3a\u771f\u7684\u4e1c\u897f\u5bf9\u6211\u6765\u8bf4\u5c31\u662f\u771f\u7684\u3002<\/p>\n
Socrates had a reputation for demolishing an opponent\u2019s position, and the Socratic enquiry consisted of questions and answers in which the opponent would be led to a conclusion incompatible with his original position. The approach was similar to a reductio ad absurdum argument, although Socrates was a moral philosopher who did no theoretical work on logic. \u82cf\u683c\u62c9\u5e95\u4ee5\u63a8\u7ffb\u5bf9\u624b\u7684\u89c2\u70b9\u800c\u95fb\u540d，\u82cf\u683c\u62c9\u5e95\u95ee\u7b54\u6cd5\u901a\u8fc7\u4e00\u8fde\u4e32\u63d0\u95ee\u4e0e\u56de\u7b54，\u5f15\u5bfc\u5bf9\u624b\u5f97\u51fa\u4e0e\u5176\u539f\u59cb\u7acb\u573a\u76f8\u77db\u76fe\u7684\u7ed3\u8bba\u3002\u8fd9\u79cd\u65b9\u6cd5\u7c7b\u4f3c\u4e8e\u5f52\u8c2c\u6cd5\u8bba\u8bc1，\u5c3d\u7ba1\u82cf\u683c\u62c9\u5e95\u662f\u4e00\u4f4d\u9053\u5fb7\u54f2\u5b66\u5bb6，\u5e76\u672a\u5728\u903b\u8f91\u9886\u57df\u5f00\u5c55\u7406\u8bba\u7814\u7a76\u3002<\/p>\n
Aristotle did important work on logic, and he developed a system of logic, syllogistic logic, that remained in use up to the nineteenth century. Syllogistic logic is a \u201cterm-logic\u201d, with letters used to stand for the individual terms. A syllogism consists of two premises and a conclusion, where the conclusion is a valid deduction from the two premises. Aristotle also did some early work on modal logic and was the founder of the field. \u4e9a\u91cc\u58eb\u591a\u5fb7\u5728\u903b\u8f91\u9886\u57df\u505a\u51fa\u4e86\u91cd\u8981\u8d21\u732e，\u4ed6\u53d1\u5c55\u4e86\u4e00\u5957\u540d\u4e3a\u4e09\u6bb5\u8bba\u903b\u8f91\u7684\u4f53\u7cfb，\u8be5\u4f53\u7cfb\u4e00\u76f4\u6cbf\u7528\u81f3 19 \u4e16\u7eaa\u3002\u4e09\u6bb5\u8bba\u903b\u8f91\u662f\u4e00\u79cd\u201c\u8bcd\u9879\u903b\u8f91\u201d，\u7528\u5b57\u6bcd\u4ee3\u8868\u5404\u4e2a\u8bcd\u9879\u3002\u4e00\u4e2a\u4e09\u6bb5\u8bba\u7531\u4e24\u4e2a\u524d\u63d0\u548c\u4e00\u4e2a\u7ed3\u8bba\u7ec4\u6210，\u5176\u4e2d\u7ed3\u8bba\u662f\u4ece\u4e24\u4e2a\u524d\u63d0\u4e2d\u6709\u6548\u63a8\u5bfc\u5f97\u51fa\u7684\u3002\u4e9a\u91cc\u58eb\u591a\u5fb7\u8fd8\u5bf9\u6a21\u6001\u903b\u8f91\u8fdb\u884c\u4e86\u65e9\u671f\u7814\u7a76，\u662f\u8be5\u9886\u57df\u7684\u5960\u57fa\u4eba\u3002<\/p>\n
The Stoics developed an early form of propositional logic, where the assertibles (propositions) have a truth-value such that at any time they are either true or false. The assertibles may be simple or non-simple, and various connectives such as conjunctions, disjunctions and implication are used in forming more complex assertibles. \u65af\u591a\u845b\u5b66\u6d3e\u53d1\u5c55\u4e86\u65e9\u671f\u7684\u547d\u9898\u903b\u8f91，\u5176\u4e2d\u53ef\u65ad\u8a00\u53e5（\u547d\u9898）\u5177\u6709\u771f\u503c，\u5373\u5728\u4efb\u4f55\u65f6\u523b\u8981\u4e48\u4e3a\u771f，\u8981\u4e48\u4e3a\u5047\u3002\u53ef\u65ad\u8a00\u53e5\u53ef\u5206\u4e3a\u7b80\u5355\u53e5\u548c\u975e\u7b80\u5355\u53e5，\u4eba\u4eec\u4f1a\u4f7f\u7528\u5408\u53d6\u3001\u6790\u53d6\u548c\u8574\u542b\u7b49\u591a\u79cd\u8054\u7ed3\u8bcd\u6765\u6784\u6210\u66f4\u590d\u6742\u7684\u53ef\u65ad\u8a00\u53e5\u3002<\/p>\n
George Boole developed his symbolic logic in the mid-1800s, and it later formed the foundation for digital computing. Boole argued that logic should be considered as a separate branch of mathematics, rather than a part of philosophy. He argued that there are mathematical laws to express the operation of reasoning in the human mind, and he showed how Aristotle\u2019s syllogistic logic could be reduced to a set of algebraic equations. \u4e54\u6cbb\u00b7\u5e03\u5c14\u4e8e 19 \u4e16\u7eaa\u4e2d\u53f6\u521b\u7acb\u4e86\u7b26\u53f7\u903b\u8f91，\u8be5\u903b\u8f91\u540e\u6765\u6210\u4e3a\u6570\u5b57\u8ba1\u7b97\u7684\u57fa\u7840\u3002\u5e03\u5c14\u8ba4\u4e3a，\u903b\u8f91\u5e94\u88ab\u89c6\u4e3a\u6570\u5b66\u7684\u4e00\u4e2a\u72ec\u7acb\u5206\u652f，\u800c\u975e\u54f2\u5b66\u7684\u4e00\u90e8\u5206\u3002\u4ed6\u63d0\u51fa，\u5b58\u5728\u6570\u5b66\u5b9a\u5f8b\u53ef\u8868\u8fbe\u4eba\u7c7b\u601d\u7ef4\u4e2d\u7684\u63a8\u7406\u8fc7\u7a0b，\u5e76\u5c55\u793a\u4e86\u5982\u4f55\u5c06\u4e9a\u91cc\u58eb\u591a\u5fb7\u7684\u4e09\u6bb5\u8bba\u903b\u8f91\u7b80\u5316\u4e3a\u4e00\u7ec4\u4ee3\u6570\u65b9\u7a0b\u3002<\/p>\n
Logic plays a key role in reasoning and deduction in mathematics, but it is considered a separate discipline to mathematics. There were attempts in the early twentieth century to show that all mathematics can be derived from formal logic, and that the formal system of mathematics would be complete, with all the truths of mathematics provable in the system (see Chap. 13 of [1]). However, this program failed when the Austrian logician, Kurt G\u00f6del, showed that that there are truths in the formal system of arithmetic that cannot be proved within the system (i.e. first-order arithmetic is incomplete). \u903b\u8f91\u5728\u6570\u5b66\u7684\u63a8\u7406\u548c\u6f14\u7ece\u4e2d\u8d77\u7740\u5173\u952e\u4f5c\u7528，\u4f46\u5b83\u88ab\u89c6\u4e3a\u4e0e\u6570\u5b66\u72ec\u7acb\u7684\u5b66\u79d1\u300220 \u4e16\u7eaa\u521d，\u6709\u4eba\u8bd5\u56fe\u8bc1\u660e\u6240\u6709\u6570\u5b66\u90fd\u53ef\u4ece\u5f62\u5f0f\u903b\u8f91\u63a8\u5bfc\u800c\u6765，\u4e14\u6570\u5b66\u7684\u5f62\u5f0f\u7cfb\u7edf\u662f\u5b8c\u5907\u7684\u2014\u2014\u6240\u6709\u6570\u5b66\u771f\u7406\u90fd\u80fd\u5728\u8be5\u7cfb\u7edf\u4e2d\u5f97\u5230\u8bc1\u660e（\u53c2\u89c1\u6587\u732e [1] \u7684\u7b2c 13 \u7ae0）\u3002\u7136\u800c，\u5965\u5730\u5229\u903b\u8f91\u5b66\u5bb6\u5e93\u5c14\u7279\u00b7\u54e5\u5fb7\u5c14\u8bc1\u660e\u4e86\u7b97\u672f\u5f62\u5f0f\u7cfb\u7edf\u4e2d\u5b58\u5728\u65e0\u6cd5\u5728\u7cfb\u7edf\u5185\u8bc1\u660e\u7684\u771f\u7406（\u5373\u4e00\u9636\u7b97\u672f\u662f\u4e0d\u5b8c\u5907\u7684），\u8fd9\u4e00\u8ba1\u5212\u968f\u4e4b\u5931\u8d25\u3002<\/p>\n
Early work on logic was done by Aristotle in the fourth century B.C. in the Organon [2]. Aristotle regarded logic as a useful tool of enquiry into any subject, and he developed syllogistic logic. This is a form of reasoning in which a conclusion is drawn from two premises, where each premise is in a subject\u2013predicate form. A common or middle term is present in each of the two premises but not in the conclusion. For example: \u516c\u5143\u524d 4 \u4e16\u7eaa，\u4e9a\u91cc\u58eb\u591a\u5fb7\u5728\u300a\u5de5\u5177\u8bba\u300b[2] \u4e2d\u5f00\u5c55\u4e86\u65e9\u671f\u903b\u8f91\u7814\u7a76\u3002\u4e9a\u91cc\u58eb\u591a\u5fb7\u5c06\u903b\u8f91\u89c6\u4e3a\u63a2\u7a76\u4efb\u4f55\u5b66\u79d1\u7684\u6709\u7528\u5de5\u5177，\u5e76\u53d1\u5c55\u4e86\u4e09\u6bb5\u8bba\u903b\u8f91\u3002\u8fd9\u662f\u4e00\u79cd\u63a8\u7406\u5f62\u5f0f，\u7ed3\u8bba\u4ece\u4e24\u4e2a\u524d\u63d0\u4e2d\u5f97\u51fa，\u6bcf\u4e2a\u524d\u63d0\u5747\u4e3a\u4e3b\u8c13\u7ed3\u6784\u3002\u4e24\u4e2a\u524d\u63d0\u4e2d\u5b58\u5728\u4e00\u4e2a\u5171\u540c\u7684\u8bcd\u9879（\u4e2d\u9879），\u4f46\u8be5\u8bcd\u9879\u4e0d\u5728\u7ed3\u8bba\u4e2d\u51fa\u73b0\u3002\u4f8b\u5982：<\/p>\n
<\/p>\n All\u00a0Greeks\u00a0are\u00a0mortal,\u00a0<\/p>\n \u6240\u6709\u5e0c\u814a\u4eba\u90fd\u662f\u4f1a\u6b7b\u7684<\/p>\n Socrates\u00a0is\u00a0a\u00a0Greek,\u00a0<\/p>\n \u82cf\u683c\u62c9\u5e95\u662f\u5e0c\u814a\u4eba<\/p>\n Therefore\u00a0Socrates\u00a0is\u00a0mortal,\u00a0\u56e0\u6b64\u82cf\u683c\u62c9\u5e95\u662f\u4f1a\u6b7b\u7684<\/p>\n \\\\begin{array}{l} \\\\text{All Greeks are mortal, \\\\qquad\\\\quad\u6240\u6709\u5e0c\u814a\u4eba\u90fd\u662f\u4f1a\u6b7b\u7684} \\\\\\\\ \\\\text{Socrates is a Greek, \\\\quad\\\\quad\\\\qquad\u82cf\u683c\u62c9\u5e95\u662f\u5e0c\u814a\u4eba} \\\\\\\\ \\\\hline \\\\text{Therefore Socrates is mortal, \u56e0\u6b64\u82cf\u683c\u62c9\u5e95\u662f\u4f1a\u6b7b\u7684} \\\\end{array} <\/p>\n <\/span><\/span><\/span><\/span><\/span>All\u00a0Greeks\u00a0are\u00a0mortal,\u00a0<\/span><\/span><\/span>\u6240\u6709\u5e0c\u814a\u4eba\u90fd\u662f\u4f1a\u6b7b\u7684<\/span><\/span><\/span><\/span><\/span>Socrates\u00a0is\u00a0a\u00a0Greek,\u00a0<\/span><\/span><\/span><\/span>\u82cf\u683c\u62c9\u5e95\u662f\u5e0c\u814a\u4eba<\/span><\/span><\/span><\/span><\/span>Therefore\u00a0Socrates\u00a0is\u00a0mortal,\u00a0<\/span>\u56e0\u6b64\u82cf\u683c\u62c9\u5e95\u662f\u4f1a\u6b7b\u7684<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n The common (or middle) term in this example is \u201cGreek\u201d. It occurs in both premises but not in the conclusion. The above argument is valid, and Aristotle studied and classified the various types of syllogistic arguments to determine those that were valid or invalid. Each premise contains a subject and a predicate, and the middle term may act as subject or a predicate. Each premise is a positive or negative affirmation, and an affirmation may be universal or particular. The universal and particular affirmations and negatives are described in Table 5.1. \u8be5\u4f8b\u5b50\u4e2d\u7684\u5171\u540c（\u4e2d）\u9879\u662f\u201c\u5e0c\u814a\u4eba\u201d，\u5b83\u5728\u4e24\u4e2a\u524d\u63d0\u4e2d\u90fd\u51fa\u73b0，\u4f46\u672a\u5728\u7ed3\u8bba\u4e2d\u51fa\u73b0\u3002\u4e0a\u8ff0\u8bba\u8bc1\u662f\u6709\u6548\u7684，\u4e9a\u91cc\u58eb\u591a\u5fb7\u5bf9\u5404\u79cd\u7c7b\u578b\u7684\u4e09\u6bb5\u8bba\u8bba\u8bc1\u8fdb\u884c\u4e86\u7814\u7a76\u548c\u5206\u7c7b，\u4ee5\u786e\u5b9a\u5176\u6709\u6548\u6027\u3002\u6bcf\u4e2a\u524d\u63d0\u90fd\u5305\u542b\u4e00\u4e2a\u4e3b\u9879\u548c\u4e00\u4e2a\u8c13\u9879，\u4e2d\u9879\u65e2\u53ef\u4ee5\u4f5c\u4e3a\u4e3b\u9879，\u4e5f\u53ef\u4ee5\u4f5c\u4e3a\u8c13\u9879\u3002\u6bcf\u4e2a\u524d\u63d0\u8981\u4e48\u662f\u80af\u5b9a\u9648\u8ff0，\u8981\u4e48\u662f\u5426\u5b9a\u9648\u8ff0，\u4e14\u9648\u8ff0\u8981\u4e48\u662f\u5168\u79f0\u7684，\u8981\u4e48\u662f\u7279\u79f0\u7684\u3002\u5168\u79f0\u4e0e\u7279\u79f0\u7684\u80af\u5b9a\u9648\u8ff0\u548c\u5426\u5b9a\u9648\u8ff0\u5982\u8868 5.1 \u6240\u793a\u3002<\/p>\n Table 5.1 Types of syllogistic premises \u8868 5.1 \u4e09\u6bb5\u8bba\u524d\u63d0\u7684\u7c7b\u578b<\/p>\n This leads to four basic forms of syllogistic arguments (Table 5.2) where the middle is the subject of both premises; the predicate of both premises; and the subject of one premise and the predicate of the other premise. \u8fd9\u4ea7\u751f\u4e86\u56db\u79cd\u57fa\u672c\u7684\u4e09\u6bb5\u8bba\u8bba\u8bc1\u5f62\u5f0f（\u8868 5.2），\u5206\u522b\u5bf9\u5e94：\u4e2d\u9879\u662f\u4e24\u4e2a\u524d\u63d0\u7684\u4e3b\u9879；\u4e2d\u9879\u662f\u4e24\u4e2a\u524d\u63d0\u7684\u8c13\u9879；\u4e2d\u9879\u662f\u4e00\u4e2a\u524d\u63d0\u7684\u4e3b\u9879\u4e14\u662f\u53e6\u4e00\u4e2a\u524d\u63d0\u7684\u8c13\u9879\u3002<\/p>\n Table 5.2 Forms of syllogistic premises \u8868 5.2 \u4e09\u6bb5\u8bba\u524d\u63d0\u7684\u5f62\u5f0f<\/p>\n There are four types of premises (A, E, I, O) and therefore sixteen sets of premise pairs for each of the forms above. However, only some of these premise pairs will yield a valid conclusion. Aristotle went through every possible premise pair to determine whether a valid argument may be derived. The syllogistic argument above is of form (iv) and is valid: \u524d\u63d0\u6709\u56db\u79cd\u7c7b\u578b（A\u3001E\u3001I\u3001O），\u56e0\u6b64\u6bcf\u79cd\u5f62\u5f0f\u4e0b\u6709 16 \u7ec4\u524d\u63d0\u5bf9\u3002\u7136\u800c，\u53ea\u6709\u90e8\u5206\u524d\u63d0\u5bf9\u80fd\u5f97\u51fa\u6709\u6548\u7ed3\u8bba\u3002\u4e9a\u91cc\u58eb\u591a\u5fb7\u9010\u4e00\u5206\u6790\u4e86\u6240\u6709\u53ef\u80fd\u7684\u524d\u63d0\u5bf9，\u4ee5\u786e\u5b9a\u662f\u5426\u80fd\u63a8\u5bfc\u51fa\u6709\u6548\u8bba\u8bc1\u3002\u4e0a\u8ff0\u4e09\u6bb5\u8bba\u8bba\u8bc1\u5c5e\u4e8e\u5f62\u5f0f（iv），\u4e14\u662f\u6709\u6548\u7684：<\/p>\n G A M Syllogistic logic is a \u201cterm-logic\u201d with letters used to stand for the individual terms. Syllogistic logic was the first attempt at a science of logic, and it remained in use up to the nineteenth century. There are many limitations to what it may express, and on its suitability as a representation of how the mind works. \u4e09\u6bb5\u8bba\u903b\u8f91\u662f\u4e00\u79cd\u201c\u8bcd\u9879\u903b\u8f91\u201d，\u7528\u5b57\u6bcd\u4ee3\u8868\u5404\u4e2a\u8bcd\u9879\u3002\u5b83\u662f\u903b\u8f91\u79d1\u5b66\u5316\u7684\u9996\u6b21\u5c1d\u8bd5，\u4e00\u76f4\u6cbf\u7528\u81f3 19 \u4e16\u7eaa\u3002\u4f46\u5b83\u5728\u8868\u8fbe\u80fd\u529b\u4e0a\u5b58\u5728\u8bf8\u591a\u5c40\u9650，\u4f5c\u4e3a\u4eba\u7c7b\u601d\u7ef4\u8fd0\u4f5c\u65b9\u5f0f\u7684\u8868\u5f81\u4e5f\u4e0d\u591f\u8d34\u5207\u3002<\/p>\n A paradox is a statement that apparently contradicts itself, and it presents a situation that appears to defy logic. Some logical paradoxes have a solution, whereas others are contradictions or invalid arguments. There are many examples of paradoxes, and they often arise due to self-reference in which one or more statements refer to each other. We discuss several paradoxes such as the liar paradox and the sorites paradox, which were invented by Eubulides of Miletus, and the barber paradox, which was introduced by Russell to explain the contradictions in na\u00efve set theory. \u6096\u8bba\u662f\u4e00\u79cd\u8868\u9762\u4e0a\u81ea\u76f8\u77db\u76fe\u7684\u9648\u8ff0，\u5448\u73b0\u51fa\u770b\u4f3c\u8fdd\u80cc\u903b\u8f91\u7684\u60c5\u5883\u3002\u6709\u4e9b\u903b\u8f91\u6096\u8bba\u5b58\u5728\u89e3\u51b3\u65b9\u6848，\u800c\u53e6\u4e00\u4e9b\u5219\u662f\u77db\u76fe\u6216\u65e0\u6548\u8bba\u8bc1\u3002\u6096\u8bba\u6709\u8bb8\u591a\u4f8b\u5b50，\u5176\u4ea7\u751f\u5f80\u5f80\u6e90\u4e8e\u81ea\u6307\u2014\u2014\u5373\u4e00\u4e2a\u6216\u591a\u4e2a\u9648\u8ff0\u76f8\u4e92\u6307\u4ee3\u3002\u6211\u4eec\u5c06\u8ba8\u8bba\u51e0\u4e2a\u6096\u8bba，\u4f8b\u5982\u7c73\u5229\u90fd\u7684\u6b27\u5e03\u91cc\u5fb7\u63d0\u51fa\u7684\u8bf4\u8c0e\u8005\u6096\u8bba\u548c\u8c37\u5806\u6096\u8bba，\u4ee5\u53ca\u7f57\u7d20\u4e3a\u89e3\u91ca\u6734\u7d20\u96c6\u5408\u8bba\u4e2d\u7684\u77db\u76fe\u800c\u5f15\u5165\u7684\u7406\u53d1\u5e08\u6096\u8bba\u3002<\/p>\n An example of the liar paradox is the statement \u201cEverything that I say is false\u201d, which is made by the liar. This looks like a normal sentence, but it is also saying something about itself as a sentence. If the statement is true, then the statement must be false, since the meaning of the sentence is that every statement (including the current statement) made by the liar is false. If the current statement is false, then the statement that everything that I say is false is false, and so this must be a true statement. \u8bf4\u8c0e\u8005\u6096\u8bba\u7684\u4e00\u4e2a\u4f8b\u5b50\u662f\u8bf4\u8c0e\u8005\u6240\u8bf4\u7684\u201c\u6211\u6240\u8bf4\u7684\u4e00\u5207\u90fd\u662f\u5047\u7684\u201d\u3002\u8fd9\u53e5\u8bdd\u770b\u4f3c\u666e\u901a，\u4f46\u5b83\u540c\u65f6\u4e5f\u5728\u5bf9\u81ea\u8eab\u4f5c\u4e3a\u4e00\u4e2a\u53e5\u5b50\u8fdb\u884c\u9648\u8ff0\u3002\u5982\u679c\u8be5\u9648\u8ff0\u4e3a\u771f，\u90a3\u4e48\u7531\u4e8e\u53e5\u5b50\u7684\u542b\u4e49\u662f\u8bf4\u8c0e\u8005\u6240\u8bf4\u7684\u6bcf\u4e00\u53e5\u8bdd（\u5305\u62ec\u5f53\u524d\u8fd9\u53e5\u8bdd）\u90fd\u662f\u5047\u7684，\u56e0\u6b64\u8be5\u9648\u8ff0\u5fc5\u987b\u4e3a\u5047；\u5982\u679c\u5f53\u524d\u9648\u8ff0\u4e3a\u5047，\u90a3\u4e48\u201c\u6211\u6240\u8bf4\u7684\u4e00\u5207\u90fd\u662f\u5047\u7684\u201d\u8fd9\u4e00\u8bf4\u6cd5\u5c31\u662f\u5047\u7684，\u56e0\u6b64\u5b83\u53c8\u5fc5\u987b\u662f\u4e00\u4e2a\u771f\u9648\u8ff0\u3002<\/p>\n The Epimenides paradox is a variant of the liar paradox. Epimenides was a Cretan who allegedly stated \u201cAll Cretans are liars\u201d. If the statement is true, then since Epimenides is Cretan, he must be a liar, and so the statement is false and we have a contradiction. However, if we assume that the statement is false and that Epimenides is lying about all Cretan being liars, then we may deduce (without contradiction) that there is at least one Cretan who is truthful. So in this case, the paradox can be avoided. \u57c3\u5e87\u7c73\u5c3c\u5f97\u65af\u6096\u8bba\u662f\u8bf4\u8c0e\u8005\u6096\u8bba\u7684\u4e00\u4e2a\u53d8\u4f53\u3002\u57c3\u5e87\u7c73\u5c3c\u5f97\u65af\u662f\u514b\u91cc\u7279\u4eba，\u636e\u79f0\u4ed6\u66fe\u8bf4\u8fc7\u201c\u6240\u6709\u514b\u91cc\u7279\u4eba\u90fd\u662f\u9a97\u5b50\u201d\u3002\u5982\u679c\u8be5\u9648\u8ff0\u4e3a\u771f，\u90a3\u4e48\u7531\u4e8e\u57c3\u5e87\u7c73\u5c3c\u5f97\u65af\u662f\u514b\u91cc\u7279\u4eba，\u4ed6\u5fc5\u7136\u662f\u9a97\u5b50，\u56e0\u6b64\u8be5\u9648\u8ff0\u4e3a\u5047，\u4ea7\u751f\u77db\u76fe；\u7136\u800c，\u5982\u679c\u6211\u4eec\u5047\u8bbe\u8be5\u9648\u8ff0\u4e3a\u5047，\u5373\u57c3\u5e87\u7c73\u5c3c\u5f97\u65af\u5173\u4e8e\u201c\u6240\u6709\u514b\u91cc\u7279\u4eba\u90fd\u662f\u9a97\u5b50\u201d\u7684\u8bf4\u6cd5\u662f\u8c0e\u8a00，\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5（\u65e0\u77db\u76fe\u5730）\u63a8\u5bfc\u51fa\u81f3\u5c11\u6709\u4e00\u4e2a\u514b\u91cc\u7279\u4eba\u662f\u8bda\u5b9e\u7684\u3002\u56e0\u6b64，\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b，\u6096\u8bba\u53ef\u4ee5\u88ab\u907f\u514d\u3002<\/p>\n The sorites paradox (paradox of the heap) involves a heap of sand in which grains are individually removed. It is assumed that removing a single grain of sand does not turn a heap into a non-heap, and the paradox is to consider what happens after when the process is repeated often enough. Is a single remaining grain a heap? When does it change from being a heap to a non-heap? This paradox may be avoided by specifying a fixed boundary of the number of grains of sand required to form a heap, or to define a heap as a collection of multiple grains (\u22652 grains). Then, any collection of grains of sand less than this boundary is not a heap. \u8c37\u5806\u6096\u8bba（\u5806\u6096\u8bba）\u6d89\u53ca\u4e00\u5806\u6c99\u5b50，\u5176\u4e2d\u6bcf\u6b21\u53ea\u79fb\u9664\u4e00\u7c92\u6c99\u3002\u5047\u8bbe\u79fb\u9664\u4e00\u7c92\u6c99\u4e0d\u4f1a\u4f7f\u4e00\u5806\u6c99\u5b50\u53d8\u6210\u975e\u5806，\u90a3\u4e48\u6096\u8bba\u5c31\u5728\u4e8e：\u5f53\u8fd9\u4e2a\u8fc7\u7a0b\u91cd\u590d\u8db3\u591f\u591a\u6b21\u540e\u4f1a\u53d1\u751f\u4ec0\u4e48？\u4ec5\u5269\u4e00\u7c92\u6c99\u65f6\u8fd8\u662f\u5806\u5417？\u5b83\u4f55\u65f6\u4ece\u5806\u53d8\u6210\u975e\u5806？\u89e3\u51b3\u8fd9\u4e00\u6096\u8bba\u7684\u65b9\u6cd5\u53ef\u4ee5\u662f\u8bbe\u5b9a\u6784\u6210\u5806\u6240\u9700\u7684\u6c99\u5b50\u7c92\u6570\u7684\u56fa\u5b9a\u754c\u9650，\u6216\u5c06\u5806\u5b9a\u4e49\u4e3a\u591a\u7c92\u6c99\u5b50\u7684\u96c6\u5408（\u22652 \u7c92）\u3002\u8fd9\u6837\u4e00\u6765，\u4efb\u4f55\u5c11\u4e8e\u8be5\u754c\u9650\u7684\u6c99\u5b50\u96c6\u5408\u90fd\u4e0d\u662f\u5806\u3002<\/p>\n The barber paradox is a variant of Russell\u2019s paradox (a contradiction in na\u00efve set theory), which was discussed in Chap. 4. In a village, there is a barber who shaves everyone who does not shave himself, and no one else. Who shaves the barber? The answer to this question results in a contradiction, as the barber cannot shave himself, since he shaves only those who do not shave themselves. Further, as the barber does not shave himself then he falls into the group of people who would be shaved by the barber (himself). Therefore, we conclude that there is no such barber. \u7406\u53d1\u5e08\u6096\u8bba\u662f\u7f57\u7d20\u6096\u8bba（\u6734\u7d20\u96c6\u5408\u8bba\u4e2d\u7684\u4e00\u4e2a\u77db\u76fe）\u7684\u53d8\u4f53，\u7f57\u7d20\u6096\u8bba\u5728\u7b2c 4 \u7ae0\u5df2\u8ba8\u8bba\u8fc7\u3002\u5728\u4e00\u4e2a\u6751\u5e84\u91cc，\u6709\u4e00\u4f4d\u7406\u53d1\u5e08，\u4ed6\u7ed9\u6240\u6709\u4e0d\u81ea\u5df1\u522e\u80e1\u5b50\u7684\u4eba\u522e\u80e1\u5b50，\u4e14\u53ea\u7ed9\u8fd9\u4e9b\u4eba\u522e\u80e1\u5b50\u3002\u90a3\u4e48\u8c01\u7ed9\u7406\u53d1\u5e08\u522e\u80e1\u5b50？\u8fd9\u4e2a\u95ee\u9898\u7684\u7b54\u6848\u4f1a\u4ea7\u751f\u77db\u76fe：\u56e0\u4e3a\u7406\u53d1\u5e08\u53ea\u7ed9\u4e0d\u81ea\u5df1\u522e\u80e1\u5b50\u7684\u4eba\u522e\u80e1\u5b50，\u6240\u4ee5\u4ed6\u4e0d\u80fd\u7ed9\u81ea\u5df1\u522e\u80e1\u5b50；\u4f46\u53e6\u4e00\u65b9\u9762，\u7531\u4e8e\u7406\u53d1\u5e08\u4e0d\u81ea\u5df1\u522e\u80e1\u5b50，\u4ed6\u53c8\u5c5e\u4e8e\u4f1a\u88ab\u7406\u53d1\u5e08（\u4ed6\u81ea\u5df1）\u522e\u80e1\u5b50\u7684\u4eba\u7fa4\u3002\u56e0\u6b64，\u6211\u4eec\u5f97\u51fa\u7ed3\u8bba：\u4e0d\u5b58\u5728\u8fd9\u6837\u7684\u7406\u53d1\u5e08\u3002<\/p>\n Table 5.3 Fallacies in arguments \u8868 5.3 \u8bba\u8bc1\u4e2d\u7684\u8c2c\u8bef<\/p>\n The purpose of a debate is to convince an audience of the correctness of your position and to challenge and undermine your opponent\u2019s position. Often, the arguments made are factual, but occasionally individuals skilled in rhetoric and persuasion will introduce bad arguments as a way to persuade the audience. Aristotle studied and classified bad arguments (known as fallacies), and these include fallacies such as the ad hominem argument; the appeal to authority argument; and the straw man argument. The fallacies are described in more detail in Table 5.3. \u8fa9\u8bba\u7684\u76ee\u7684\u662f\u8ba9\u542c\u4f17\u76f8\u4fe1\u4f60\u7684\u7acb\u573a\u662f\u6b63\u786e\u7684，\u5e76\u8d28\u7591\u548c\u524a\u5f31\u5bf9\u624b\u7684\u7acb\u573a\u3002\u901a\u5e38，\u8fa9\u8bba\u4e2d\u63d0\u51fa\u7684\u8bba\u8bc1\u662f\u57fa\u4e8e\u4e8b\u5b9e\u7684，\u4f46\u6709\u65f6\u64c5\u957f\u4fee\u8f9e\u548c\u8bf4\u670d\u6280\u5de7\u7684\u4eba\u4f1a\u4f7f\u7528\u7cdf\u7cd5\u7684\u8bba\u8bc1（\u5373\u8c2c\u8bef）\u6765\u529d\u8bf4\u542c\u4f17\u3002\u4e9a\u91cc\u58eb\u591a\u5fb7\u5bf9\u8fd9\u4e9b\u7cdf\u7cd5\u7684\u8bba\u8bc1\u8fdb\u884c\u4e86\u7814\u7a76\u548c\u5206\u7c7b，\u5305\u62ec\u4eba\u8eab\u653b\u51fb\u8c2c\u8bef\u3001\u8bc9\u8bf8\u6743\u5a01\u8c2c\u8bef\u548c\u7a3b\u8349\u4eba\u8c2c\u8bef\u7b49\u3002\u8868 5.3 \u5bf9\u8fd9\u4e9b\u8c2c\u8bef\u8fdb\u884c\u4e86\u66f4\u8be6\u7ec6\u7684\u63cf\u8ff0\u3002<\/p>\n The Stoic school\u00b9 was founded in the Hellenistic period by Zeno of Citium (in Cyprus) in the late fourth\/early third century B.C. (Fig. 5.1). The school presented its philosophy as a way of life, and it emphasized ethics as the main focus of human knowledge. The Stoics stressed the importance of living a good life in harmony with nature. \u65af\u591a\u845b\u5b66\u6d3e\u00b9 \u7531\u585e\u6d66\u8def\u65af\u7684\u5b63\u8482\u6602\u7684\u829d\u8bfa\u5728\u516c\u5143\u524d 4 \u4e16\u7eaa\u672b\u81f3\u516c\u5143\u524d 3 \u4e16\u7eaa\u521d\u7684\u5e0c\u814a\u5316\u65f6\u671f\u521b\u7acb（\u56fe 5.1）\u3002\u8be5\u5b66\u6d3e\u5c06\u5176\u54f2\u5b66\u89c6\u4e3a\u4e00\u79cd\u751f\u6d3b\u65b9\u5f0f，\u5f3a\u8c03\u4f26\u7406\u662f\u4eba\u7c7b\u77e5\u8bc6\u7684\u6838\u5fc3\u3002\u65af\u591a\u845b\u5b66\u6d3e\u6ce8\u91cd\u4e0e\u81ea\u7136\u548c\u8c10\u5171\u5904\u7684\u7f8e\u597d\u751f\u6d3b\u3002<\/p>\n \u00b9 The origin of the word Stoic is from the Stoa Poikile (\u03a3\u03c4\u03bf\u03ac \u03a0\u03bf\u03b9\u03ba\u03af\u03bb\u03b7), which was a covered walkway in the Agora of Athens. Zeno taught his philosophy in a public space at this location, and his followers became known as Stoics. \u00b9 \u201cStoic\u201d（\u65af\u591a\u845b）\u4e00\u8bcd\u6e90\u4e8e\u201cStoa Poikile\u201d（\u5f69\u7ed8\u67f1\u5eca），\u8fd9\u662f\u96c5\u5178\u5e7f\u573a\u4e0a\u7684\u4e00\u6761\u6709\u9876\u8d70\u5eca\u3002\u829d\u8bfa\u5728\u6b64\u516c\u5171\u7a7a\u95f4\u4f20\u6388\u5176\u54f2\u5b66\u601d\u60f3，\u5176\u8ffd\u968f\u8005\u56e0\u6b64\u88ab\u79f0\u4e3a\u201c\u65af\u591a\u845b\u5b66\u6d3e\u201d\u3002<\/p>\n The Stoics recognized the importance of reason and logic, and Chrysippus, the head of the Stoics in the third century B.C., developed an early version of propositional logic. This was a system of deduction in which the smallest unanalyzed expressions are assertibles (Stoic equivalent of propositions). The assertibles have a truth-value such that at any moment of time they are either true or false. True assertibles are viewed as facts in the Stoic system of logic, and false assertibles are defined as the contradictories of true ones. \u65af\u591a\u845b\u5b66\u6d3e\u8ba4\u8bc6\u5230\u7406\u6027\u548c\u903b\u8f91\u7684\u91cd\u8981\u6027，\u516c\u5143\u524d 3 \u4e16\u7eaa\u65af\u591a\u845b\u5b66\u6d3e\u7684\u9886\u8896\u514b\u5415\u897f\u6ce2\u53d1\u5c55\u4e86\u65e9\u671f\u7684\u547d\u9898\u903b\u8f91\u3002\u8fd9\u662f\u4e00\u79cd\u6f14\u7ece\u7cfb\u7edf，\u5176\u4e2d\u6700\u5c0f\u7684\u672a\u5206\u6790\u8868\u8fbe\u5f0f\u662f\u53ef\u65ad\u8a00\u53e5（\u65af\u591a\u845b\u5b66\u6d3e\u5bf9\u547d\u9898\u7684\u79f0\u547c）\u3002\u53ef\u65ad\u8a00\u53e5\u5177\u6709\u771f\u503c，\u5373\u5728\u4efb\u4f55\u65f6\u523b\u8981\u4e48\u4e3a\u771f，\u8981\u4e48\u4e3a\u5047\u3002\u5728\u65af\u591a\u845b\u903b\u8f91\u4f53\u7cfb\u4e2d，\u771f\u7684\u53ef\u65ad\u8a00\u53e5\u88ab\u89c6\u4e3a\u4e8b\u5b9e，\u5047\u7684\u53ef\u65ad\u8a00\u53e5\u5219\u88ab\u5b9a\u4e49\u4e3a\u771f\u53ef\u65ad\u8a00\u53e5\u7684\u77db\u76fe\u547d\u9898\u3002<\/p>\n Truth is temporal, and assertions may change their truth-value over time. The assertibles may be simple or non-simple (more than one assertible), and there may be present tense, past tense and future tense assertibles. Chrysippus distinguished between simple and compound propositions, and he introduced a set of logical connectives for conjunction, disjunction and implication that are used to form non-simple assertibles from existing assertibles. \u771f\u7406\u5177\u6709\u65f6\u95f4\u6027，\u65ad\u8a00\u7684\u771f\u503c\u4f1a\u968f\u65f6\u95f4\u53d8\u5316\u3002\u53ef\u65ad\u8a00\u53e5\u53ef\u5206\u4e3a\u7b80\u5355\u53e5\u548c\u975e\u7b80\u5355\u53e5（\u5305\u542b\u591a\u4e2a\u53ef\u65ad\u8a00\u53e5），\u4e14\u6709\u73b0\u5728\u65f6\u3001\u8fc7\u53bb\u65f6\u548c\u5c06\u6765\u65f6\u4e4b\u5206\u3002\u514b\u5415\u897f\u6ce2\u533a\u5206\u4e86\u7b80\u5355\u547d\u9898\u548c\u590d\u5408\u547d\u9898，\u5e76\u5f15\u5165\u4e86\u4e00\u7ec4\u903b\u8f91\u8054\u7ed3\u8bcd（\u5408\u53d6\u3001\u6790\u53d6\u548c\u8574\u542b），\u7528\u4e8e\u4ece\u73b0\u6709\u53ef\u65ad\u8a00\u53e5\u6784\u5efa\u975e\u7b80\u5355\u53ef\u65ad\u8a00\u53e5\u3002<\/p>\n The conjunction connective is of the form \u201cboth \u2026 and \u2026\u201d, and it has two conjuncts. The disjunction connective is of the form \u201ceither \u2026 or \u2026 or \u2026\u201d, and it consists of two or more disjuncts. Conditionals are formed from the connective \u201cif \u2026, \u2026\u201d, and they consist of an antecedent and a consequence. \u5408\u53d6\u8054\u7ed3\u8bcd\u7684\u5f62\u5f0f\u4e3a\u201c\u65e2\u2026\u2026\u53c8\u2026\u2026\u201d，\u5305\u542b\u4e24\u4e2a\u5408\u53d6\u652f；\u6790\u53d6\u8054\u7ed3\u8bcd\u7684\u5f62\u5f0f\u4e3a\u201c\u8981\u4e48\u2026\u2026\u8981\u4e48\u2026\u2026\u8981\u4e48\u2026\u2026\u201d，\u5305\u542b\u4e24\u4e2a\u6216\u591a\u4e2a\u6790\u53d6\u652f；\u6761\u4ef6\u53e5\u7531\u8054\u7ed3\u8bcd\u201c\u5982\u679c\u2026\u2026\u90a3\u4e48\u2026\u2026\u201d\u6784\u6210，\u5305\u542b\u524d\u4ef6\u548c\u540e\u4ef6\u3002<\/p>\n His deductive system included various logical argument forms such as modus ponens and modus tollens. His propositional logic differed from syllogistic logic, in that the Stoic logic was based on propositions (or statements) as distinct from Aristotle\u2019s term-logic. However, he could express the universal affirmation in syllogistic logic (e.g. All As are B) by rephrasing it as a conditional statement that if something is A then it is B. \u4ed6\u7684\u6f14\u7ece\u7cfb\u7edf\u5305\u542b\u591a\u79cd\u903b\u8f91\u8bba\u8bc1\u5f62\u5f0f，\u5982\u80af\u5b9a\u524d\u4ef6\u5f0f\u548c\u5426\u5b9a\u540e\u4ef6\u5f0f\u3002\u4ed6\u7684\u547d\u9898\u903b\u8f91\u4e0e\u4e09\u6bb5\u8bba\u903b\u8f91\u4e0d\u540c：\u65af\u591a\u845b\u903b\u8f91\u4ee5\u547d\u9898（\u6216\u9648\u8ff0）\u4e3a\u57fa\u7840，\u800c\u4e9a\u91cc\u58eb\u591a\u5fb7\u7684\u903b\u8f91\u662f\u8bcd\u9879\u903b\u8f91\u3002\u4e0d\u8fc7，\u4ed6\u53ef\u4ee5\u5c06\u4e09\u6bb5\u8bba\u903b\u8f91\u4e2d\u7684\u5168\u79f0\u80af\u5b9a\u547d\u9898（\u5982\u201c\u6240\u6709 A \u90fd\u662f B\u201d）\u91cd\u65b0\u8868\u8ff0\u4e3a\u6761\u4ef6\u53e5\u201c\u5982\u679c\u67d0\u7269\u662f A，\u90a3\u4e48\u5b83\u662f B\u201d\u3002<\/p>\n Chrysippus\u2019s propositional logic did not replace Aristotle\u2019s syllogistic logic, and syllogistic logic remained in use up to the mid-nineteenth century. George Boole developed his symbolic logic in the mid-1800s, and his logic later formed the foundation for digital computing. Boole\u2019s symbolic logic is discussed in the next section. \u514b\u5415\u897f\u6ce2\u7684\u547d\u9898\u903b\u8f91\u5e76\u672a\u53d6\u4ee3\u4e9a\u91cc\u58eb\u591a\u5fb7\u7684\u4e09\u6bb5\u8bba\u903b\u8f91，\u4e09\u6bb5\u8bba\u903b\u8f91\u4e00\u76f4\u6cbf\u7528\u81f3 19 \u4e16\u7eaa\u4e2d\u53f6\u3002\u4e54\u6cbb\u00b7\u5e03\u5c14\u4e8e 19 \u4e16\u7eaa\u4e2d\u53f6\u521b\u7acb\u4e86\u7b26\u53f7\u903b\u8f91，\u8be5\u903b\u8f91\u540e\u6765\u6210\u4e3a\u6570\u5b57\u8ba1\u7b97\u7684\u57fa\u7840\u3002\u4e0b\u4e00\u8282\u5c06\u8ba8\u8bba\u5e03\u5c14\u7684\u7b26\u53f7\u903b\u8f91\u3002<\/p>\n George Boole (Fig. 5.2) was born in Lincoln, England, in 1815. His father (a cobbler who was interested in mathematics and optical instruments) taught him mathematics and showed him how to make optical instruments. Boole inherited his father\u2019s interest in knowledge, and he was self-taught in mathematics and Greek. He taught at various schools near Lincoln, and he developed his mathematical knowledge by working his way through Newton\u2019s Principia, as well as applying himself to the work of mathematicians such as Laplace and Lagrange. \u4e54\u6cbb\u00b7\u5e03\u5c14（\u56fe 5.2）\u4e8e 1815 \u5e74\u51fa\u751f\u5728\u82f1\u56fd\u6797\u80af\u90e1\u3002\u4ed6\u7684\u7236\u4eb2\u662f\u4e00\u540d\u978b\u5320，\u5bf9\u6570\u5b66\u548c\u5149\u5b66\u4eea\u5668\u611f\u5174\u8da3，\u6559\u4ed6\u6570\u5b66\u5e76\u5411\u4ed6\u5c55\u793a\u5982\u4f55\u5236\u4f5c\u5149\u5b66\u4eea\u5668\u3002\u5e03\u5c14\u7ee7\u627f\u4e86\u7236\u4eb2\u5bf9\u77e5\u8bc6\u7684\u70ed\u7231，\u81ea\u5b66\u4e86\u6570\u5b66\u548c\u5e0c\u814a\u8bed\u3002\u4ed6\u5728\u6797\u80af\u90e1\u9644\u8fd1\u7684\u591a\u6240\u5b66\u6821\u4efb\u6559，\u901a\u8fc7\u7814\u8bfb\u725b\u987f\u7684\u300a\u81ea\u7136\u54f2\u5b66\u7684\u6570\u5b66\u539f\u7406\u300b\u4ee5\u53ca\u62c9\u666e\u62c9\u65af\u3001\u62c9\u683c\u6717\u65e5\u7b49\u6570\u5b66\u5bb6\u7684\u8457\u4f5c，\u4e0d\u65ad\u63d0\u5347\u81ea\u5df1\u7684\u6570\u5b66\u6c34\u5e73\u3002<\/p>\n George Boole<\/p>\n He published regular papers from his early twenties, and these included contributions to probability theory, differential equations and finite differences. He developed his symbolic algebra, which is the foundation for modern computing, and he is considered (along with Babbage) to be one of the grandfathers of computing. His work was theoretical, and he never actually built a computer or calculating machine. However, Boole\u2019s symbolic logic was the perfect mathematical model for switching theory, and for the design of digital circuits. \u4ed6\u4ece\u4e8c\u5341\u51fa\u5934\u5c31\u5f00\u59cb\u5b9a\u671f\u53d1\u8868\u8bba\u6587，\u5185\u5bb9\u6d89\u53ca\u6982\u7387\u8bba\u3001\u5fae\u5206\u65b9\u7a0b\u548c\u6709\u9650\u5dee\u5206\u7b49\u9886\u57df\u3002\u4ed6\u521b\u7acb\u7684\u7b26\u53f7\u4ee3\u6570\u6210\u4e3a\u73b0\u4ee3\u8ba1\u7b97\u7684\u57fa\u7840，\u4ed6\u4e5f\u56e0\u6b64\u4e0e\u5df4\u8d1d\u5947\u4e00\u540c\u88ab\u89c6\u4e3a\u201c\u8ba1\u7b97\u4e4b\u7236\u201d\u4e4b\u4e00\u3002\u4ed6\u7684\u7814\u7a76\u5c5e\u4e8e\u7406\u8bba\u5c42\u9762，\u4ece\u672a\u5b9e\u9645\u5236\u9020\u8fc7\u8ba1\u7b97\u673a\u6216\u8ba1\u7b97\u5668，\u4f46\u5e03\u5c14\u7b26\u53f7\u903b\u8f91\u4e3a\u5f00\u5173\u7406\u8bba\u548c\u6570\u5b57\u7535\u8def\u8bbe\u8ba1\u63d0\u4f9b\u4e86\u5b8c\u7f8e\u7684\u6570\u5b66\u6a21\u578b\u3002<\/p>\n Boole became interested in formulating a calculus of reasoning, and he published a pamphlet titled \u201cMathematical Analysis of Logic\u201d in 1847 [3]. This short book developed novel ideas on a logical method, and he argued that logic should be considered as a separate branch of mathematics, rather than a part of philosophy. He argued that there are mathematical laws to express the operation of reasoning in the human mind, and he showed how Aristotle\u2019s syllogistic logic could be reduced to a set of algebraic equations. He corresponded regularly on logic with Augustus De Morgan.\u00b2 \u5e03\u5c14\u5bf9\u6784\u5efa\u63a8\u7406\u6f14\u7b97\u4ea7\u751f\u4e86\u5174\u8da3，\u5e76\u4e8e 1847 \u5e74\u51fa\u7248\u4e86\u9898\u4e3a\u300a\u903b\u8f91\u7684\u6570\u5b66\u5206\u6790\u300b\u7684\u5c0f\u518c\u5b50 [3]\u3002\u8fd9\u672c\u77ed\u4e66\u63d0\u51fa\u4e86\u5173\u4e8e\u903b\u8f91\u65b9\u6cd5\u7684\u65b0\u89c2\u70b9，\u4ed6\u4e3b\u5f20\u903b\u8f91\u5e94\u88ab\u89c6\u4e3a\u6570\u5b66\u7684\u4e00\u4e2a\u72ec\u7acb\u5206\u652f，\u800c\u975e\u54f2\u5b66\u7684\u4e00\u90e8\u5206\u3002\u4ed6\u8ba4\u4e3a\u5b58\u5728\u6570\u5b66\u5b9a\u5f8b\u53ef\u8868\u8fbe\u4eba\u7c7b\u601d\u7ef4\u4e2d\u7684\u63a8\u7406\u8fc7\u7a0b，\u5e76\u5c55\u793a\u4e86\u5982\u4f55\u5c06\u4e9a\u91cc\u58eb\u591a\u5fb7\u7684\u4e09\u6bb5\u8bba\u903b\u8f91\u7b80\u5316\u4e3a\u4e00\u7ec4\u4ee3\u6570\u65b9\u7a0b\u3002\u4ed6\u8fd8\u4e0e\u5965\u53e4\u65af\u5854\u65af\u00b7\u5fb7\u6469\u6839\u00b2 \u5c31\u903b\u8f91\u95ee\u9898\u5b9a\u671f\u901a\u4fe1\u3002<\/p>\n \u00b2 De Morgan was a nineteenth British mathematician based at University College London. De Morgan\u2019s laws in Set Theory and Logic state that: <\/p>\n (<\/p>\n A<\/p>\n \u222a<\/p>\n B<\/p>\n )<\/p>\n c<\/p>\n =<\/p>\n A<\/p>\n c<\/p>\n \u2229<\/p>\n B<\/p>\n c<\/p>\n (A \\\\cup B)^c = A^c \\\\cap B^c<\/p>\n <\/span><\/span>(<\/span>A<\/span><\/span>\u222a<\/span><\/span><\/span><\/span>B<\/span>)<\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2229<\/span><\/span><\/span><\/span>B<\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and <\/p>\n \u00ac<\/p>\n (<\/p>\n A<\/p>\n \u2228<\/p>\n B<\/p>\n )<\/p>\n \u2261<\/p>\n \u00ac<\/p>\n A<\/p>\n \u2227<\/p>\n \u00ac<\/p>\n B<\/p>\n \\\\neg(A \\\\vee B) \\\\equiv \\\\neg A \\\\land \\\\neg B<\/p>\n <\/span><\/span>\u00ac<\/span>(<\/span>A<\/span><\/span>\u2228<\/span><\/span><\/span><\/span>B<\/span>)<\/span><\/span>\u2261<\/span><\/span><\/span><\/span>\u00ac<\/span>A<\/span><\/span>\u2227<\/span><\/span><\/span><\/span>\u00ac<\/span>B<\/span><\/span><\/span><\/span><\/span> \u00b2 \u5fb7\u6469\u6839\u662f 19 \u4e16\u7eaa\u82f1\u56fd\u6570\u5b66\u5bb6，\u4efb\u804c\u4e8e\u4f26\u6566\u5927\u5b66\u5b66\u9662\u3002\u96c6\u5408\u8bba\u548c\u903b\u8f91\u4e2d\u7684\u5fb7\u6469\u6839\u5b9a\u5f8b\u6307\u51fa：<\/p>\n (<\/p>\n A<\/p>\n \u222a<\/p>\n B<\/p>\n )<\/p>\n c<\/p>\n =<\/p>\n A<\/p>\n c<\/p>\n \u2229<\/p>\n B<\/p>\n c<\/p>\n (A \\\\cup B)^c = A^c \\\\cap B^c<\/p>\n <\/span><\/span>(<\/span>A<\/span><\/span>\u222a<\/span><\/span><\/span><\/span>B<\/span>)<\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2229<\/span><\/span><\/span><\/span>B<\/span><\/span>c<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e14 <\/p>\n \u00ac<\/p>\n (<\/p>\n A<\/p>\n \u2228<\/p>\n B<\/p>\n )<\/p>\n \u2261<\/p>\n \u00ac<\/p>\n A<\/p>\n \u2227<\/p>\n \u00ac<\/p>\n B<\/p>\n \\\\neg(A \\\\vee B) \\\\equiv \\\\neg A \\\\land \\\\neg B<\/p>\n <\/span><\/span>\u00ac<\/span>(<\/span>A<\/span><\/span>\u2228<\/span><\/span><\/span><\/span>B<\/span>)<\/span><\/span>\u2261<\/span><\/span><\/span><\/span>\u00ac<\/span>A<\/span><\/span>\u2227<\/span><\/span><\/span><\/span>\u00ac<\/span>B<\/span><\/span><\/span><\/span><\/span><\/p>\n He introduced two quantities \u201c0\u201d and \u201c1\u201d with the quantity 1 used to represent the universe of thinkable objects (i.e. the universal set), and the quantity 0 represents the absence of any objects (i.e. the empty set). He then employed symbols such as <\/p>\n x<\/p>\n x<\/p>\n <\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>, \n
Type \u7c7b\u578bSymbol \u7b26\u53f7Example \u793a\u4f8b<\/tr>\n \n\n Universal affirmative \u5168\u79f0\u80af\u5b9a<\/td>\n G A M<\/td>\n All Greeks are mortal \u6240\u6709\u5e0c\u814a\u4eba\u90fd\u662f\u4f1a\u6b7b\u7684<\/td>\n<\/tr>\n \n Universal negative \u5168\u79f0\u5426\u5b9a<\/td>\n G E M<\/td>\n No Greek is mortal \u6ca1\u6709\u5e0c\u814a\u4eba\u662f\u4f1a\u6b7b\u7684<\/td>\n<\/tr>\n \n Particular affirmative \u7279\u79f0\u80af\u5b9a<\/td>\n G I M<\/td>\n Some Greek is mortal \u6709\u4e9b\u5e0c\u814a\u4eba\u662f\u4f1a\u6b7b\u7684<\/td>\n<\/tr>\n \n Particular negative \u7279\u79f0\u5426\u5b9a<\/td>\n G O M<\/td>\n Some Greek is not mortal \u6709\u4e9b\u5e0c\u814a\u4eba\u4e0d\u662f\u4f1a\u6b7b\u7684<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n
Form (i) \u5f62\u5f0f（i）Form (ii) \u5f62\u5f0f（ii）Form (iii) \u5f62\u5f0f（iii）Form (iv) \u5f62\u5f0f（iv）<\/tr>\n \n\n Premise 1 \u524d\u63d0 1<\/td>\n M P<\/td>\n P M<\/td>\n P M<\/td>\n M P<\/td>\n<\/tr>\n \n Premise 2 \u524d\u63d0 2<\/td>\n M S<\/td>\n S M<\/td>\n M S<\/td>\n S M<\/td>\n<\/tr>\n \n Conclusion \u7ed3\u8bba<\/td>\n S P<\/td>\n S P<\/td>\n S P<\/td>\n S P<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n S I G
\n——-
\n S I M<\/p>\n3 Paradoxes and Fallacies<\/h3>\n
3 \u6096\u8bba\u4e0e\u8c2c\u8bef<\/h3>\n
\n
Fallacy \u8c2c\u8bef\u7c7b\u578bDescription\/Example \u63cf\u8ff0\/\u793a\u4f8b<\/tr>\n \n\n Hasty\/Accident generalization \u8f7b\u7387\/\u5076\u7136\u6982\u62ec<\/td>\n This is a bad argument that involves a generalization that disregards exceptions \u4e00\u79cd\u7cdf\u7cd5\u7684\u8bba\u8bc1，\u5176\u6982\u62ec\u8fc7\u7a0b\u5ffd\u7565\u4e86\u4f8b\u5916\u60c5\u51b5<\/td>\n<\/tr>\n \n Slippery slope \u6ed1\u5761\u8c2c\u8bef<\/td>\n This argument outlines a chain reaction leading to a highly undesirable situation that will occur if a certain situation is allowed. The claim is that even if one step is taken onto the slippery slope, then we will fall all the way down to the bottom \u8be5\u8bba\u8bc1\u63cf\u7ed8\u4e86\u4e00\u4e2a\u8fde\u9501\u53cd\u5e94，\u58f0\u79f0\u82e5\u5141\u8bb8\u67d0\u4e00\u60c5\u51b5\u53d1\u751f，\u5c06\u5bfc\u81f4\u6781\u5177\u5371\u5bb3\u6027\u7684\u7ed3\u679c\u3002\u5176\u4e3b\u5f20\u662f，\u4e00\u65e6\u8e0f\u4e0a\u201c\u6ed1\u5761\u201d\u7684\u7b2c\u4e00\u6b65，\u5c31\u4f1a\u4e00\u8def\u6ed1\u5230\u8c37\u5e95<\/td>\n<\/tr>\n \n Against the person Ad Hominem \u4eba\u8eab\u653b\u51fb\u8c2c\u8bef<\/td>\n The focus of this argument is to attack the person rather than the argument that the person has made \u8bba\u8bc1\u7684\u7126\u70b9\u662f\u653b\u51fb\u63d0\u51fa\u89c2\u70b9\u7684\u4eba，\u800c\u975e\u5176\u63d0\u51fa\u7684\u8bba\u8bc1\u672c\u8eab<\/td>\n<\/tr>\n \n Appeal to people Ad Populum \u8bc9\u8bf8\u516c\u4f17\u8c2c\u8bef<\/td>\n This argument involves an appeal to popular belief to support an argument, with a claim that the majority of the population supports this argument. However, popular opinion is not always correct \u501f\u52a9\u666e\u904d\u7684\u4fe1\u5ff5\u6765\u652f\u6301\u8bba\u8bc1，\u58f0\u79f0\u5927\u591a\u6570\u4eba\u90fd\u8ba4\u540c\u8be5\u89c2\u70b9\u3002\u4f46\u516c\u4f17\u610f\u89c1\u5e76\u975e\u603b\u662f\u6b63\u786e\u7684<\/td>\n<\/tr>\n \n Appeal to authority (Ad Verecundiam) \u8bc9\u8bf8\u6743\u5a01\u8c2c\u8bef<\/td>\n This argument is when an appeal is made to an authoritative figure to support an argument, and where the authority is not an expert in this area \u8bd5\u56fe\u501f\u52a9\u6743\u5a01\u4eba\u7269\u7684\u58f0\u671b\u652f\u6301\u8bba\u8bc1，\u4f46\u8be5\u6743\u5a01\u5728\u76f8\u5173\u9886\u57df\u5e76\u975e\u4e13\u5bb6<\/td>\n<\/tr>\n \n Appeal to pity (Ad Misericordiam) \u8bc9\u8bf8\u601c\u60af\u8c2c\u8bef<\/td>\n This is where the arguer tries to get people to accept a conclusion by making them feel sorry for someone \u8bba\u8bc1\u8005\u8bd5\u56fe\u901a\u8fc7\u8ba9\u4eba\u4eec\u540c\u60c5\u67d0\u4eba\u6765\u4f7f\u5176\u63a5\u53d7\u7ed3\u8bba<\/td>\n<\/tr>\n \n Appeal to ignorance \u8bc9\u8bf8\u65e0\u77e5\u8c2c\u8bef<\/td>\n The arguer makes the case that there is no conclusive evidence on the issue at hand and that therefore his conclusion should be accepted \u8bba\u8bc1\u8005\u58f0\u79f0\u5f53\u524d\u8bae\u9898\u4e0d\u5b58\u5728\u786e\u51ff\u8bc1\u636e，\u56e0\u6b64\u5e94\u63a5\u53d7\u5176\u7ed3\u8bba<\/td>\n<\/tr>\n \n Straw man argument \u7a3b\u8349\u4eba\u8c2c\u8bef<\/td>\n The arguer sets up a version of an opponent\u2019s position of his argument and defeats this watered down version of his opponent\u2019s position \u8bba\u8bc1\u8005\u5148\u6b6a\u66f2\u5bf9\u624b\u7684\u7acb\u573a，\u6784\u5efa\u4e00\u4e2a\u5f31\u5316\u7248\u7684\u89c2\u70b9，\u7136\u540e\u53cd\u9a73\u8fd9\u4e2a\u865a\u6784\u7684\u7acb\u573a<\/td>\n<\/tr>\n \n Begging the question \u5faa\u73af\u8bba\u8bc1\u8c2c\u8bef<\/td>\n This is a circular argument where the arguer relies on a premise that says the same thing as the conclusion and without providing any real evidence for the conclusion \u8bba\u8bc1\u8fc7\u7a0b\u662f\u5faa\u73af\u7684，\u8bba\u8bc1\u8005\u4f9d\u8d56\u7684\u524d\u63d0\u4e0e\u7ed3\u8bba\u8868\u8ff0\u7684\u662f\u540c\u4e00\u56de\u4e8b，\u4e14\u672a\u4e3a\u7ed3\u8bba\u63d0\u4f9b\u4efb\u4f55\u771f\u5b9e\u8bc1\u636e<\/td>\n<\/tr>\n \n Red herring \u8f6c\u79fb\u8bdd\u9898\u8c2c\u8bef<\/td>\n The arguer goes off on a tangent that has nothing to do with the argument in question \u8bba\u8bc1\u8005\u504f\u79bb\u4e3b\u9898，\u5f15\u5165\u4e0e\u5f53\u524d\u8bba\u8bc1\u65e0\u5173\u7684\u5185\u5bb9<\/td>\n<\/tr>\n \n False dichotomy \u865a\u5047\u4e8c\u5206\u6cd5\u8c2c\u8bef<\/td>\n The arguer presents the case that there are only two possible outcomes (often there are more). One of the possible outcomes is then eliminated leading to the desired outcome. The argument suggests that there is only one outcome \u8bba\u8bc1\u8005\u58f0\u79f0\u53ea\u6709\u4e24\u79cd\u53ef\u80fd\u7684\u7ed3\u679c（\u901a\u5e38\u5b9e\u9645\u5b58\u5728\u66f4\u591a\u7ed3\u679c），\u7136\u540e\u6392\u9664\u5176\u4e2d\u4e00\u79cd\u7ed3\u679c，\u4ece\u800c\u5f97\u51fa\u81ea\u5df1\u60f3\u8981\u7684\u7ed3\u8bba，\u6697\u793a\u8be5\u7ed3\u679c\u662f\u552f\u4e00\u53ef\u80fd\u7684<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 4 Stoic Logic<\/h3>\n
4 \u65af\u591a\u845b\u903b\u8f91<\/h3>\n
![\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260125114132-6976016c6f902.png\")
Zeno of Citium<\/p>\n5 Boole\u2019s Symbolic Logic<\/h3>\n
5 \u5e03\u5c14\u7b26\u53f7\u903b\u8f91<\/h3>\n
![\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260125114132-6976016c8dbfd.png\")
<\/p>\n