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 Material implication is problematic if I say it is Material implication is problematic if I say it is20151103 [PDF] [PERMANENT LINK]These are just some thoughts on how we teach logic to beginners. My experience is from introductory logic courses provided to philosophy students, I'm sure it's done differently in other contexts. The claim in the title is false. My saying there's something wrong with material implication doesn't make it so. But if the claim is understood to have the form of a material implication, then it is true if material implication is problematic—independently of what I say and why. This aspect of material implication, that its truth conditions are such as to ignore any connection between the antecedent and the consequent, has been regarded as a problem by some people since the dawn of modern symbolic logic around the turn of the last century. (How to understand the truth conditions for a conditional statement has been debated throughout the history of logic theories.) A quick recap: In truthfunctional logic, material implication is indicated by the symbol '→' (or sometimes by '⊃' or '⇒', but I will use the single arrow), and 'p → q' is read as 'if p then q.' The expression is true if and only if it is not the case that p is true and q is false. Hence If Russia won the race to the moon then The Avengers 3 premiers tonight would be true if interpreted as a material implication (since the antecedent is false), and If Donald Trump becomes president, then peace will reign on Earth is true, again if understood as a material implication, if peace will reign on Earth—even if Trump has nothing to do with it. But in their ordinary, nontechnical, senses these statements are not true under such conditions. For the longest time I used to think there was nothing, and could not be anything, wrong with material implication. After all, it has welldefined truth conditions, that are functions of the truth values of the sentences connected by the arrow, and what more could you ask of a truth functional connective? Truth conditions for material implication
But one day I realized that I had changed my mind. I was having some serious issues with material implication. The reason was one, and one only: teaching introductory logic courses. In introductory propositional logic, we teach the five logic connectives: negation ('¬'), conjunction ('∧'), disjunction ('∨'), implication ('→'), and equivalence ('↔'). It is usually necessary to repeat a few times to students that disjunction is 'inclusive.' That is to say, that 'p ∨ q' is true if both p and q are true—this is in contradiction with some everyday uses of 'or.' It may trip some students up a couple of times at first, but it's not a great hurdle. Negation, conjunction, and equivalence are usually not a problem. However, ask the students halfway through the course if p → ¬p is a contradiction, and around half, sometimes more, will say yes. Because 'if p then not p' really does look like a contradiction, even when you've had the truth conditions for material implication repeated to you several times. (It is not a contradiction, of course, but equivalent to ¬p.) For implication only do I have to resort to tables of translations to natural language and Venn diagrams to promote understanding. My problem with material implication is pedagogical, then. Now, many things are hard to learn in the beginning, and there isn't much you can do about that. And material implication is central in the single historically most important inference rule in logic: Modus Ponens. Modus Ponens
Modus Ponens is also a very natural inference rule. If it's Thursday, it's pea soup for lunch. It's Thursday. So, it's pea soup for lunch. What could be more obvious? That's part of the problem. You can understand and use Modus Ponens perfectly for every case you come across, and still be using a stronger version of 'if… then…' than material implication, and so still be thrown off by 'p → ¬p'. So, I have a suggestion. Let's not teach material implication, not at first. Let's only teach negation, conjunction, and disjunction. (You can do everything with these three.) But what about Modus Ponens? Let's teach Disjunctive Syllogism instead! Disjunctive Syllogism
It may not be as obvious as Modus Ponens, but once you've learned how to use it, there is little risk of misunderstanding. Then, when our beginning students have mastered these symbols and the inference rule, we can explain that p → q is equivalent to ¬p ∨ q, and that both Modus Ponens and its sibling Modus Tollens can be understood as special cases of the Disjunctive Syllogism. The arrow does not stand for 'if… then…' in any ordinary sense—it's a particular disjunction that satisfies only the truth functional part of the ordinary conditional. Or, more straightforwardly: the conditional is not a form in truthfunctional logic. Give students a chance to grasp this from the very beginning. Today is Sinead O'Connor vs The World Day20121003 [PDF] [PERMANENT LINK]This post began life on Facebook. Had I written it today I would have noted my sources, but I don't think any of this is hard to look up. 20 years ago today, Sinéad O'Connor tore up a picture of the pope on SNL in protest of the sexual abuse of children by catholic priests, and the church coverup. It ended her brief career in mainstream pop. People thought she was a lunatic. Catholic priests don't abuse children! Right. The next week's SNL host Joe Pesci said he "would have given her such a smack." Two years earlier O'Connor had refused to play at a New Jersey venue if they played the national anthem before she went on stage. She doesn't like national anthems, thinks they have nothing to do with music and are mostly war songs full of "nationalist tripe." Sounds about right to me, but the next night Frank Sinatra was on the same stage and said he would "like to kick her in the ass." So that was kind of a theme. O'Connor appeared on Bob Dylan's 30th anniversary concert two weeks after the SNL event. I watched this on Swedish tv. Kris Kristofferson announced her, clearly apprehensive of what would happen. When she got on stage the thousands in the audience roared. She smiled, walked up to the mike and said thank you. Then her face went blank as she realized that most of the people were booing and screaming. (Let's call it "Christian Rage.") She stood silent for a moment, waved to her band to be quiet, pulled out her ear plugs, and shouted an a capella version of Bob Marley's "War" over the noise. (The same she had performed at SNL.) After staring at the furious mob for a few more seconds, she walked off stage and collapsed into Kris Kristofferson's arms. We didn't have SNL so I had no idea what this was about, but I was stunned. Something real just happened on tv. One last story. O'Connor had never met Prince when she recorded his Nothing Compares 2 U, but afterward she was invited to his mansion. Prince, being a famous Jehovah's Witness, started to chastise her for swearing during interviews. (Her Twitter feed—by the way—is still one of the least family friendly.) When she told him to fuck off, he became, according to her, violent. She couldn't fight him, she said later, he packs a punch, but there was a lot of spitting. Apparently she wasn't applying for the Prince Babe Troupe. So, anyway, the SNL and Dylan concert episodes are easy enough to find on YouTube. I'll end with something from years before anything of great international significance happened to Sinéad O'Connor. Happy 20th anniversary, girl! Sinéad O'Connor  Troy (Pinkpop Festival 1988) The other solution to the base unit discrepancy20040701 [PDF] [PERMANENT LINK]This text originally appeared on a web forum dedicated to decimal time keeping systems, sometime in 2004. To my delight several of the users of that forum took my proposal seriously. 1 The Base unit mismatchIt seems generally accepted that because life on Earth tends to repeat in both daily and annual cycles, these two both need to be counted within a useful time keeping system. In effect, the system must be constructed so that there are two units, day and year, that correspond to the two astronomical cycles of Earth axis rotations and Earth solar revolutions, in such a way that day is a subunit of year. A decimal system is by definition a system where every unit is an exponent of 10. This means that every unit has a ratio to all of its sub and superunits that is also an exponent of 10. (E.g., 1 kilometer / 1 meter = 10^{3}, and 1 kilogram / 1 hectogram = 10^{1}.) Cyclic earth time dictates two units to us, that do not have a ratio that is compatible with a decimal system. There are approximately 365.25 days in a year, and that is not a power of 10. This is a fixed premise of the problem, and can't be changed. 2 The reason for a decimal systemWe prefer to use decimal systems for measurement because our number system is itself decimal. Powers of 10 are special in our decimal, positional number system in that they can be generated by simply inserting zeroes and/or moving the decimal point. If we use only decimal units for measurements as well, we use the same trivial procedure to translate between units and their sub or superunits. With nondecimal units, we need to make calculations for each translation, or look them up in a table. For mass, volume, and temperature, we are free to define our own units and subunits, and there is no reason to not make them decimal. As far as I know, time is the only case of measurement where we are given two units with a rational relationship by nature, as part of the premise. Because of this, efforts to make time keeping systems conform to our decimal number system and other decimal units will forever be imperfect. 3 The other solutionHowever, the very fact that we are free to define all other units as we please is the clue to a solution. If it is indeed the case that only in time measurements are we bound by a ratio in nature, then it would seem to make more sense to adapt our other unit systems to this ratio, than to accept a flawed and incompatible time keeping system. It's within our abilities to change the base of our number system and of all of our other units, while the number of days in a year is fixed. 4 A new baseWe then need to examine what new numerical base the ratio of days to years suggests to us. First of all, an exponentiation of a whole number is always a whole number, so 365.25 can't be the exponentiation of any whole number. The base of a number system is by definition the number of signs it uses, so it must be a positive integer, a whole number. We can't use 7.3 signs in the number system. It seems obvious then that to have any chance of succeeding, we need to look at 365 rather than 365.25. We'll get back later to the implications of this. We now notice that the square root of 365 is approximately 19.1. If it had been a whole number, that would have been our candidate for a new numerical base. Since 365 is such a cumbersome number however (can only be factored into 5 and 73, both primes), there seems to be no smaller number that we can use as our base. We are stuck with a new numerical base of 365. 5 Practical issues with having 365 as a numerical baseFirst of all, we will need 365 distinct signs to use in our new numerical system. This is actually a smaller problem than one might imagine. We can easily derive 365 signs from our existing number system, by combining the signs that we use to build the same numbers today. The signs for the numbers 12, 75, and 103 (for example), would then look something like this: , , These signs will all be easy to recognize and remember for someone familiar with our current system. I leave it as an exercise for the reader to come up with a convenient way to draw these signs without lifting the pencil from the paper. (You might want to revisit the Bridges of Konigsberg problem.) Another potential issue is how to build input devices for our trecentisexagintapentimal number system. Using an average decimal calculator as a model, the size of the keyboard would be about 10x30 centimeters, perhaps larger, since the trecentisexagintapentimal signs are somewhat more complex than their decimal counterparts. This is however also a problem to which a solution easily presents itself. Most Personal Digital Assistants today are equipped with a writing pad, for entering signs into the device using a pen. A similar contraption can be used for calculators to enter numerical signs, instead of having a button for each one. 6 Benefits of using 365 as a numerical baseIn these times when oil prices are increasing, computers are becoming faster and faster, average global temperatures are steadily rising, and people are getting fatter at an increasing rate, it is foresighted to switch even now to a numerical system that can express large numbers with fewer signs. In our current system, we need to employ three signs already at 100 units. With the trecentisexagintapentimal system, we can use two signs up to no less than 133,224 (expressed decimally) units! Writing space is an exhaustible resource, and using 365 as our base, we significantly push forward the inevitable date of depletion. Another benefit of the trecentisexagintapentimal system is that it happens to be compatible with our planet's year/day ratio. 7 Leap unitsWhat about that 0.25 leftover? Right. There are after all not 365 days in a year, but approximately 365.25. With a year of 365 days, we will "lose" one actual day every four years or so. Since we have had to use wholenumber units in our calendars (for the reason mentioned above), we have needed to round the number of days per year to the nearest whole number. However, this truncation leads to an unacceptable displacement of the calendar over the years. Hence, it's not possible to simply use the base 365 and leave it at that, as convenient as that would have been. Our astronomically based number system must compensate for the fact that there is not a whole number of Earth axis rotations in one solar revolution. Fortunately, there is a simple and sufficient algorithm to accomplish this. Append a leap sign, we call it "366," at the end of the regular number series, when one of the following criteria are fulfilled: Append a leap sign, "366," if and only if the next upcoming multiple of 365, divided by 365, is exactly divisible by 4, but not exactly divisible by 100, or is exactly 1 or the next upcoming multiple of 365, divided by 365, is exactly divisible by 100 and exactly divisible by 400. This algorithm can easily be built into all calculators, thermometers, scales, rulers, and so on, and will finally make them consistent with our time keeping units, which is good since mathematical consistency is the only thing that matters in this world. 8 Final wordsThe solution to the ageold problem of inconsistent base units between time and other measurements can be conveniently resolved by recognizing that nature only dictates one ratio, that of days to years, and that all other units, and the number system itself, use an arbitrary base. Hence, our universal numerical base should be derived from that ratio, to achieve complete and universal mathematical consistency between all cases of measurement and calculation. The change to a trecentisexagintapentimal system has the additional benefit of conserving writing space, a resource that will inevitably become scarcer as literacy rates increase. As the formal name of this system, taking into account the included leap unit feature, I suggest: Trecentisexagintapentimal+. Time zones are about space20040601 [PDF] [PERMANENT LINK]This post originally appeared on a web forum dedicated to decimal time keeping systems, sometime in 2004. The notion that we ought to abolish time zones because it's simpler if everyone uses the same time really was a thing at the time. I put it here mostly because I like the clock idea. In 1998 (Gregorian time) the Swiss watch company Swatch introduced "Internet Beat Time," in the presence of Internet guru and MIT Media Laboratory director Nicholas Negroponte. The suggested benefits with Internet Beat Time is that it divides the day into 1000 "beats," and hence is decimal, and that it did away with the annoying time zones, that are a hassle to keep track of. In the spirit of Internet comingtogetherness, we would all be able to use the same time, wherever on the globe we are. In a CNN article about Beat Time, an interviewed software developer said: "It's a clever idea. I mean we have developers in London and Geneva, and I guess it could come in handy if I wanted to call them at a certain time and not wake them up or something." So why have we used these silly and complicating time zones all this time, if it's so much simpler to use a single time for the whole globe? The truth is that there already is a single, universal time that is the same for the whole world. It's called UT or UTC, (Coordinated) Universal Time. Basically, it's Greenwhich Mean Time, the time on the zero meridian in England. But although some businesses, like banks, use it, there has been no movement to use it in our daily lives, as a replacement for our time zoneadjusted clocks. (Additionally, computers use UT. Any correctly configured computer can give you the correct Universal Time. No Swatch watch needed.) The reason is that cyclic earth time, that is calendars and clocks, are about space, not time. We live our lives according to astronomical, cyclical motions, the rotation of the earth around the sun, but especially the rotation of the earth around its own axis. The significance of the time of the day, is as an indicator of where we are in the earth's axis rotation. If I don't know where in that rotation a certain time is, knowing the time is of little use to me. I can tell if it's been five minutes since I last checked the clock, but I have no idea if it's night or day, if it's bedtime or lunchtime. The meaning of 12 o'clock is that I am now on the part of the globe that faces the sun. In the same manner, most calendars tell us where we are in earth's rotation around the sun. When I look at a calendar, I want to know if it's time to sow or reap, put on the winter tires or put the boat back in the water. This is decided by the earth's position in that rotation. It's just a coincidence that one of these rotations takes some particular amount of time. When we start to focus on these time units, instead of the astronomical events they were designed to keep track of, we reduce the system to one we can only use to tell relative time difference between two events. This is a useful function, but not as important as the system's main function, to tell whether it's night or day, summer or winter. So, I think it's fair to say that the software developer above is a little confused. With a universal clock, he would know what time it is in Geneva, but he would have no idea what that time means. It becomes an arbitrary number containing no information about night or day. This is a small problem between London and Geneva, so let's pretend he needs to communicate with Singapore instead. When it's 9 o'clock Beat Time in London, our friend knows that it's 9 o'clock in Singapore as well. But if he wants to avoid waking them up, as he says, knowing that doesn't help him. To do that, he must figure out what 9 o'clock means in Singapore. So, he needs to check the distance between London and Singapore, and see where that puts Singapore in relation to the earth's rotation around its axis. What he's doing is basically looking up Singapore's time zone. Time zones aren't artificial constructs. They are part of the nature of cyclic earth time, and some kind of time zones are unavoidable. To illustrate how earth time is all about space, we could construct a clock that shows the time, and shows it for all time zones at once, but that is a model of earth's axis rotation. We could build this using an ordinary clock mechanism, and just change the clock face and hands. First, we need to make the small hand move at half its normal speed, so it completes one lap around the clock face in one full, 24 hour day. Then, remove the big hand of the clock. Replace the small hand with a disc, picturing the earth seen from "above," that is with the north pole facing us. You could indicate on this picture where on the globe the large cities are situated. Paint a sun on the clock face where 12 would be, and there you have it. The clock is really a model of the earth rotation. Looking at the illustration below you can see that it becomes trivial to keep track of the time all over the planet. The shading to (roughly) indicate night time can be done on the clock's glass cover.
Figure 1: A clock is a model of the earth's rotation around its axis. It can also be seen as a map of a section of space always facing the sun, used as a reference system. 