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What are proofs? (informal)

The following submission from an anonymous engineer to the January, 1902 edition of Popular Mechanics caught my eye. Seems like somethingevery Boy/Girl Scout and Architect should know.

HOW TO USE THE WATCH AS A COMPASS: Very few people are aware of the fact that in a watch they are alwaysprovided with a compass, with which, when the sun is shining, the cardinal points can be determined. All one has to do is to point thehour hand to the sun and south is exactly half way between the hour and the figure 12 on the watch. This may seem strange to the averagereader, but it is easily explained. While the sun is passing over 180 degrees (east to west) the hour hand of the watch passes over 360degrees (from 6 o'clock to 6 o'clock). Therefore the angular movement of the sun in one hour corresponds to the angular movement of the hourhand in half an hour; hence, if we point the hour hand toward the sun the line from the point midway between the hour hand and 12 o'clock tothe pivot of the hands will point to the south.Engineer.

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They give an argument of correctness; is that really a proof? Well, there are some ambiguities: Do I hold the watch vertically,or, in the plane of the sun's arc? Certainly I can't hold it up-side down, even though this isn't explicitly stated.Furthermore, the correctness of the reasoning relies on some unstated assumptions. E.g. , the sun is at its highest (northernmost) point of its transit at noon.Is this actually true? Does it depend on the time of year?I'm not exactly sure (and will have to sit down and scratch my head and draw pictures of orbits, to convince myself).Certainly there are at least a couple of caveats: even beyond account for Daylight Savings Time,the solar-time and clock-time only align at time-zone boundaries, and they drift up to an hour apart,before the next boundary rectifies the difference. Is this presuming I'm in the northern hemisphere?What if I'm on the equator?

To be fair, the intent of this anecdote was to give enough evidence to convince you,not necessarily to be a complete, stand-alone self-contained proof. But in writing out a careful proof, one is forced to consider all thepoints just made; being forced to understand these can lead you to better understandthe procedure yourself. But be careful to distinguish between something which sounds reasonable,and something that you're certain of.

An argument by form

How can we tell true proofs from false ones? What, exactly, are the rules of a proof?These are the questions which will occupy us.

Proofs are argument by form. We'll illustrate this with three parallel examples ofa particular proof form called syllogism .

1 All people are mortal. Premise
2 Socrates is a person. Premise
3 Therefore, Socrates is mortal. Syllogism, lines 1,2
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1 All [substitution ciphers] are [vulnerable to brute-force attacks]. Premise
2 The [Julius Caesar cipher] is a [substitution cipher]. Premise
3 Therefore, the [Julius Caesar cipher] is[vulnerable to brute-force attacks]. Syllogism, lines 1,2

Note that you don't need to know anything about cryptography to know that the conclusion follows from the two premises.(Are the premises indeed true? That's a different question.)

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Questions & Answers

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Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
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