The Wandering Blog

The Series So Far

In the previous articles, we introduced logic and discussed some basic concepts, including the informal test for invalidity:

1. Introduction
2. Soundness and Validity
3. Invalidity
4. The Test for Invalidity

In all of these presentations, we have looked at logic from an informal perspective.

Natural-language Arguments vs. Argument-forms

The next section of our Series introduces you to Formal Logic. Let’s start by looking at the following arguments.

1. If it is raining, then it is cloudy. It is raining. Therefore, it is cloudy.
2. If the trade gap narrows, the deficit will rise. The trade gap narrows. Therefore, the deficit will rise.
3. If you recycle your cans, you will feel warm and fuzzy. You recycle your cans. Therefore, you will feel warm and fuzzy.

These are all examples of natural-language arguments. Their content is different, but their form is the same. That is, they all have the same logical structure, which is:

• If A then B. A. Therefore, B.

This logical structure is called an argument-form, as opposed to a natural-language argument. A natural language argument has both content and form. An argument-form abstracts away the content, and preserves the logical structure. It has only form.

Try to figure out the logical argument-form for the following arguments:

1. You either have the coffee or the tea. You don’t have the tea. Therefore, you must have the coffee.
2. The prize is behind door number one or door number two. It isn’t behind door number two. Therefore, it must be behind door number one.
3. He meet us in the alley or on the corner. He doesn’t meet us on the corner. Therefore, he meets us in the alley.

The argument-form for these argument is:

• A or B. Not B. Therefore, A.

Here is a slightly more challenging set of arguments:

1. If you read this book, it opens your mind. Your mind is not open. Therefore, you did not read this book.
2. Provided that Bob likes potatoes, he will like your dinner. He doesn’t like your dinner. Therefore, he doesn’t like potatoes.
3. It rains only when it is cloudy. It is not cloudy. Therefore, it is not raining.

The argument-form of these arguments may be expressed in any of the following three ways:

• If A then B. Not B. Therefore, not A.
• Since A, B. Not B. Therefore, not A.
• A only when B. Not B. Therefore, not A.

All of these answers are equally correct. Logicians often prefer the first formulation, but these argument-forms are all equivalent, in the sense that they all express the same logical pattern.

Natural-language arguments are expressed in a naturally-occurring language (such as English, French, Ojibwe). Partly as a result of the richnesses of natural languages, ambiguities tend to crop up. Consider the following:

• I have a banana and an apple.
• Press the button and an alarm will sound.

In the first statement, the word “and” functions as a conjunction. That is, the statement is true if I have both a banana and an apple. In the second statement, the word “and” functions completely differently. In fact, it is difficult to know what its function is. Perhaps we are looking at a sign that contains a warning (or an enticement!): if the button is pressed, then an alarm will be sounded. This use of “and” is quite different from the first.

Logical Connectives

One of the virtues to which logicians aspire is clarity. In order to achieve a high level of clarity in argument-forms, we eschew natural-language, and instead use an artificial language. In this artificial language, we are only allowed to use the following words/phrases to express our argument-forms (called our “logical connectives”):

• And: I like marshmallows and I like cake.
• Or: You must take math or French to graduate.
• If … then: If this is a triangle, then it has three sides.
• Not: Your claim is not clear.
• If and only if: You may come to the party if and only if you are a member of the club.

Try to express the argument-forms of the following arguments, using only these words (use placeholders for the content, as above).

1. If the government rigs the elections, there will be riots. If there are riots, the government will fall. Therefore, if the government rigs the elections, the government will fall.
2. If Homer did not exist, it follows that the Odyssey was written by a committee or by a woman. But it was not written by a woman. So, it was written by a committee.
3. The players will go back to work if agreement is reached about their salaries. But this will be achieved only if some of them take early retirement. So the players will not go back if some of them do not retire early.

A properly expressed argument-form will have nothing other than placeholders and the words/phrases above, the words/phrases must be intact wholes (“only if” cannot stand by itself; you must have either “if” or “if and only if”), and a placeholder cannot implicitly contain any of the words/phrases above. By convention, we have been using “therefore” at the beginning of the conclusion.

In the previous article, we introduced a test for invalidity, and finished up with a short set of exercises. We’ll start here by going through those exercises, and then discuss some of the implications of the test for invalidity.

Invalidity Test Exercises

1. The winning ticket is number 500. Bill holds ticket number 501. Therefore, Bill does not hold the winning ticket.

2. Some people smoke cigars. Some people smoke pipes. Therefore, some people smoke cigars and pipes.

3. Anna does not believe there is a cat in the box. Therefore, Anna believes that there is no cat in the box.

Remember the test for invalidity: Imagine (1) the premisses are true and (2) the conclusion is false, and then (3) think of a scenario that might make that possible.

1. Imagine that the winning ticket is number 500, and Bill has 501. Also imagine that Bill does in fact hold the winning ticket. A possible explanation would be that Bill holds both tickets. The premise does not, after all, say that Bill holds only ticket number 501. The argument is therefore invalid.

2. Imagine that some people smoke cigars, some pipes, and further that it is not the case that some people smoke cigars and pipes. What could explain this is that we live in a world where the cigar smokers avoid pipes, and the pipe smokers avoid cigars; that is, nobody smokes both cigars and pipes. This makes it so that some people smoke cigars, some pipes, and nobody smokes both. The argument is therefore invalid.

3. Imagine that Anna does not believe there is a cat in the box, but she also does not believe there is no cat in the box. This is a little tricky, because it requires us to think about the nature of belief. Let’s look more closely at this example.

When we say you do not believe x, there are at least two possible meanings. The first is that you believe not-x. In other words, you would say “I do not believe that Tom is in the corner” if you believe that Tom is not in the corner. That is, you have a belief in not-x (Tom is not in the corner), which makes it impossible for you to believe x (Tom is in the corner).

The second possible meaning, when you say you do not believe x, is that you simply have no belief one way or the other regarding x. In other words, you would say “I do not believe that Tom is in the corner” if you meant that you do not hold the belief that Tom is in the corner. It isn’t that you believe that Tom is not it the corner; rather, you simply have no belief-state regarding Tom being in the corner.

Another way you could think about it is this: it is probably reasonable to say that you do not believe that I am six feet tall. This is because I have never talked about my height, so you have no reason to believe that I am six feet tall. It is not that you believe I am not six feet; it’s that you have no belief one way or the other.

Back to our exercise, which states, Anna does not believe there is a cat in the box. We might naturally think that this means Anna believes there is no cat in the box. But this interpretation is not the only possibility, as explained above. It could be that Anna simply does not have a belief one way or the other about a cat being in the box: perhaps she has never even thought about it. Looking at it this way, it would be possible for Anna not to believe that there is a cat in the box, and also for Anna not to believe that there is no cat in the box. The argument is therefore invalid.

Implications of the Test for Invalidity

The test for invalidity is a quick device that we apply to natural-language arguments, and in our case, that natural language in English. One of the great advantages of this test is that it gives us some insight into why invalid arguments are invalid.

But there are limitations to the test.

First, it only tells us when an argument is invalid, not when an argument is valid. If you can’t think up a possible situation where the premisses are true and the conclusion false, you might assume the argument is valid. Now you are in a bad spot: Is it really that the argument is valid, or is the argument invalid, but you just can’t come up with the situation that proves its invalidity?

Second, the test for invalidity doesn’t help us construct valid arguments. It only helps us spot invalid arguments.

Given these limitations, we have to move beyond the test for invalidity, and learn to look at an argument symbolically. We’ll explore what it means to look at arguments symbolically in the next article. To give you a bit of a preview: many natural-language arguments that, on the surface, look different from each other, actually share similar forms. It is these forms that we draw out when we look at an argument symbolically. Symbolic argument-forms are the core of the study of formal logic, and offer us a very powerful way to look at and to evaluate arguments in general. If this sounds intimidating, don’t worry. It will make sense when we go through it next time!

In the previous article, we introduced the concept of soundness. We said that a sound argument has two things: premisses that are true in the real world, and valid reasoning. We also defined validity: an argument is valid if and only if there is no possible situation where the premisses are true and the conclusion false. So far so good.

It turns out that there is a simple way to test if an argument is invalid. In this article, we are going to learn about this test and how to apply it.

Invalidity

A valid argument is one where there is no possible situation where the premisses are true and the conclusion is false. So, an invalid argument must be one where there is at least one possible situation where the premisses are true and the conclusion is false. Consider this example:

If we come upon a wild boar in the forest, then we are in trouble. We are in trouble. Therefore, we have come upon a wild boar in the forest.

This is an invalid argument, because we can think up a situation where the premisses are true and the conclusion false. Consider this line of thought. It seems reasonable to accept that if we come upon a wild boar in the forest, then we are in trouble. And further, let’s imagine that we are in trouble. Does it necessarily follow that we have come upon a wild boar in the forest? Take a moment and try to answer that question before reading further.

No, it does not necessarily follow that, if we accept the premisses, the conclusion follows. For one thing, it might be the case that although we are in trouble, we are in trouble for an entirely different reason than encountering a wild boar. Perhaps the reason we are in trouble is that we are on a magic island that has a monster that wants to kill us. It’s still true that if we came upon a wild boar in the forest, we would be in trouble; and it’s also true that we are in trouble. But according to our story, it’s not true that we have come upon a wild boar in the forest.

This gives us the outline of a method to test for the invalidity of an argument.

The Invalidity Test

In order to test an argument for invalidity, we must: (1) imagine that the premisses are true; (2) imagine that the conclusion is false; (3) come up with a coherent (non-contradictory) story that explains how (1) and (2) can be the case.

That’s it. Let’s look at another example. Suppose I’m charging money for people to hear me read this article, and I’m standing at the door, letting in people who pay me. And I say:

If you have paid your money, you can come into the room. Anna has not paid her money. Therefore, Anna can not come into the room.

Let’s use our method to show that this argument is invalid. (1) First, imagine that the premisses are true. You see signs in the hallway and on the door that say “if you have paid your money, you can come into the room.” And you know Anna has not paid. (2) Second, imagine that the conclusion is false. You see Anna come into the room.

(3) Come up with a story that explains how these things can all be the case. That is, can you explain how it could be that there are signs in the hallway saying you must pay, and you know Anna hasn’t paid, but she comes in anyway? There are many possibilities. Perhaps Anna is my sister, and I let family members in for free. That could explain it. Or, perhaps Anna is my co-lecturer, and has an important role to play in reading out examples.

In any case, the condition “if you have paid your money, you can come into the room” means that payers can gain entry; but it does not exclude other ways of getting into the room. If we wanted to make sure there was no other way to get into the room, we could say, “You can come into the room only if you have paid your money.” Now, Anna would have to pay to come in.

A few things to note about this method. First, remember that validity is independent of truth. We are testing the reasoning of the argument, not the truth of the premisses or conclusion. Second, the explanation you come up with can be wide-ranging. If you can find any explanation, no matter how outlandish, the argument is invalid.

Examples

Test yourself with these two examples, one fairly easy, the other one a bit more difficult.

1. We need to raise some money for our club. Having a bake sale would raise money. Therefore, we should have a bake sale.

2. Two shots were fired. Two bullets were found in the victim’s body. Therefore, two shots were fired at the victim.

Take a moment and use the invalidity test to show how these arguments are invalid. Remember the test: imagine (1) the premisses are true and (2) the conclusion is false, and then (3) explain how everything can work together.

1. Imagine that we are in a club that needs to raise money, and having a bake sale would certainly raise money. Also imagine that we should not have a bake sale. How can these claims go together? Under what scenario could these premisses be true, but the conclusion false? Here’s one possible answer. Perhaps there are better ways to raise money, such as having a rummage sale. So just because we need to raise money, and a bake sale would raise money, does not necessarily mean that we should have a bake sale.

2. Imagine that it’s true two shots were fired, and two bullets were found in the victim’s body. Also imagine that two shots were not fired at the victim. How can we explain this? Again, under what scenario could the premisses be true (two shots were fired, and two bullets were found in the victim), but the conclusion (two shots were fired at the victim), be false? Here’s one possible answer. Perhaps there were two shots, but these shots were only fired into the air. However, our victim was actually shot twice many years ago, and the surgeons were unable to remove the bullets. In other words, it’s possible that two shots were fired, and that there were two bullets in the victim, but that two shots were not fired at the victim. So the argument is invalid, because there are scenarios in which the premisses are true and the conclusion false.

This should give you a good idea how the test works. Here are a few examples to think about for next time, when we’ll discuss the implications of invalidity in greater depth.

1. The winning ticket is number 500. Bill holds ticket number 501. Therefore, Bill does not hold the winning ticket.

2. Some people smoke cigars. Some people smoke pipes. Therefore, some people smoke cigars and pipes.

3. Anna does not believe there is a cat in the box. Therefore, Anna believes that there is no cat in the box.

Sound Arguments

In the first post of this series, we described an argument as a form of discourse with some premisses and a conclusion. The question inevitably arises: what makes a good argument different from a bad argument? Logic has a very firm answer to this question:

A good argument is an argument that (1) has premisses that are actually true in the real world, and (2) has reasoning that is valid.

That’s it. And any argument that does not meet these two criteria is a bad argument. (In the terminology used by logicians, we do not talk about “good” and “bad” arguments, but “sound” and “unsound” arguments.)

Let’s look at the argument that we made reference to in our first post:

All humans are mortal. Socrates is human. Therefore, Socrates is mortal.

Is this a sound (good) argument? It is if we can say that (1) the premisses are actually true in the real world, and (2) the reasoning is valid.

Looking at (1), is it the case that the premisses are true, i.e. that all humans are mortal, and that Socrates is human? Yes, all humans are mortal. And yes, Socrates is human. So criterion (1) is met. Looking at (2), is it the case that the reasoning is valid? To know the answer to this question, we need to know what is meant by “valid”. You may think you know what that word means, but in the study of logic, the word valid has a very specific meaning.

Validity

To some extent, one can think of the study of logic as the study of validity. Logicians are very concerned with it. But what, then, is validity? The standard definition is:

An argument is valid if and only if there is no possible situation in which the premisses are true and the conclusion false.

In other words, for an argument to be valid, the conclusion cannot fail to be true if the premisses are true.

To illustrate what we mean, let’s return to our argument about Socrates. Is it possible for the premisses (all humans are mortal; Socrates is human) to be true, and for the conclusion (Socrates is mortal) to fail to be true? No, if those premisses are true, then the conclusion must be true.

You might ask yourself, why does this matter, since we already know the premisses and the conclusion of this argument are actually true in the real world? The answer to this is that the notion of validity allows us to judge whether a conclusion follows from the premisses regardless of the truth of those claims. And this helps us see the weakness in someone’s reasoning (and in our own reasoning!) very quickly. Take the following example:

If the moon is made of green cheese, it is tasty. The moon is made of green cheese. Therefore, it is tasty.

Of course, the moon is not actually made of green cheese. But if we are evaluating just the validity (not the soundness) of the argument, we need to ask ourselves: does the conclusion follow from the premisses? Yes it does. Even though the conclusion is false (the moon is tasty), the conclusion does follow from the premisses. So what we have here is a distinction between statements that are “true” on the one hand, and statements that “follow” from other statements on the other. A statement being true, and a statement following from other statements, are two different things.

Examples

Here are some additional examples:

1. If it’s raining, then it is cloudy. It’s raining. Therefore, it is cloudy.

2. Only citizens can vote. Anna is a citizen. Therefore, Anna can vote.

Consider whether these arguments are any good. Try to figure them out before you continue reading. And remember, for an argument to be good, or sound, the premisses must be true in the real world, and the conclusion cannot fail to be true if the premisses are true.

It turns out that the first example is unsound. The reasoning itself is valid, i.e. if the premisses are true, the conclusion must be true. But the first premiss is not true! In many cases, clouds are present when it’s raining. But sunshowers (and often rainbows) have rain with no clouds. Therefore, the argument is unsound. It is not a good argument.

The second example is also unsound. In this case, the argument is unsound because the reasoning is invalid. Even if Anna is a citizen in a country where only citizens can vote, Anna could be too young to vote, or could have lost her voting rights for some reason. So even if the premisses in this argument are true (only citizens can vote; Anna is a citizen), the conclusion (Anna can vote) does not necessarily follow. The reasoning is therefore not valid, making the argument unsound.

Summary

In this article, we have introduced the concepts of a sound argument and of valid reasoning. The main work of logic is to help you determine whether an argument is valid! So in the next instalment of this series, we’ll learn a simple test helps us spot invalid arguments.

One of the privileges of being human, one of the great distinctions between us and other animals, is the ability to reason. So the study of logic is nothing less than the study of what it means to be human. We employ some of the basic principles and practices of logic every day. But there is far more to it than most people realize. Understanding logic, and learning how to apply it to our own thinking and to how we interact with the world, is an eye-opening experience.

In this series, we will be discussing logic, its principle and its practices. In particular, we will be looking at what makes an argument a good argument, and examining the building-blocks which are available to us when constructing our own arguments. First, we’ll address a couple of basic questions: (a) what is logic, and (b) why study logic?

What is Logic?

Logic, broadly speaking, is the study of argument. An argument is a form of discourse that contains some premisses and a conclusion. It may also contain some reasoning that explains how the conclusion follows from the premisses. Here is an example (by tradition, one of the first examples when you study logic):

All humans are mortal. Socrates is human. Therefore, Socrates is mortal.

The first two sentences are the premisses (those statements which purportedly support the conclusion); and the third sentence is the conclusion (that statement which is purportedly supported by the premisses). The basic idea that underpins good reasoning is that if the premisses are true, then the conclusion cannot fail to be true.

Why Study Logic?

We all have an intuitive sense of what logic is, so you might wonder why you need to study it. Think about this, then. We all have an intuitive sense of how bodies move, and how reproduction works, and how people think. Does that mean that we understand physics, biology or psychology? That we should not study those subjects? In addition, because logic underpins reasoning, it underpins almost every other area of serious human endeavour (including physics, biology and psychology!).

People want and need to reason correctly, and the study of logic is what helps us distinguish good from bad reasoning. You might think you already know how to reason correctly, and in large part, you’re probably right. The problem is that we often end up in situations where is imperative that we reason correctly, but are unsure of how to do so. Or we are confronted with issues or controversies that are clouded by poor reasoning that most people, even very intelligent people, do not see through because they are not trained to see an argument to its logical end. In this series, we will study how to reason correctly in the general case, in a way that will be applicable to the specific cases as well.

In case you want to test your reasoning abilities, you might look at Monty Hall and Hemoccult Testing.

The Series

In this series, we will look at the following topics:

• natural-language arguments (for our purposes, arguments in English)
• argument-forms (the patterns underlying natural-language arguments)
• natural deduction (testing arguments for validity, constructing valid arguments)

The balance of our series will study natural deduction, since that is the topic most important to reasoning well.

Here is a short logic puzzle with which many people, including medical professionals, have a lot of difficulty.

A hemoccult test looks for blood in the stool, and is a method of detecting colon cancer. Here is some data we have about the test for the general population over 50 years of age:

The probability that any one person has colon cancer is 0.3 percent. If a person has colon cancer, the probability that the hemoccult test is positive is 50 percent. If a person does not have colon cancer, the probability that the test is positive is 3 percent.

Suppose that we have a person, 50 years of age, who has no symptoms, but has a positive hemoccult test. The question is: what is the probability that this person actually has colon cancer?

Now, you might think that you don’t need to be able to reason your way to the answer; after all, your doctor must know! Well, in a rather unofficial poll of 24 doctors, the most common answer was that the person who tests positive has a 50 percent chance of having colon cancer. What is your answer (try it out before reading further)?

Solving the Puzzle

The easiest way to think about what is going on, is to think of a group of people over 50. Let’s take 100,000 of them. Out of this group of 100,000 people, we know from the data that 0.3 percent of them will have colon cancer. So, that means 300 of them have colon cancer, and 99,700 of them do not have colon cancer.

Out of the 300 that have colon cancer, the data says that 50 percent of them will test positive. So 150 of them will test positive, and 150 of them will not.

Out of the 99,700 that do not have colon cancer, the data says that 3 percent will test positive. So 2,991 will test positive, and 96,709 will not.

Answer

From our group of 100,000 people, 150 tested positive (who have colon cancer), and 2,991 tested positive (who do not have colon cancer). Thus, we know that 3,141 people total tested positive. But since there only 150 of the people who tested positive actually have colon cancer, the probability of having colon cancer given that you tested positive is 150 in 3,141. In other words, a 4.78 percent chance.

The Monty Hall Paradox is a logical paradox that, for the most part, defies our intuition. It continues to generate controversy for this reason, even though the solution is definitive. We may state the paradox as follows:

Suppose you are on a game show, and you are given the choice of three doors: behind one of the doors is the prize, and behind the other two doors are goats. You choose a door, say No. 1; and the host, who knows what is behind each of the doors, opens another door, say No. 2, and reveals a goat. He then says to you, “Do you want to pick door No. 3 instead?” The question is, is it to your advantage to switch doors?

We assume, as part of the scenario, that there is an equal likelihood of each door having the prize at the outset; and that the host, who must open a door showing a goat, randomly chooses which door to open if both hide goats. Furthermore, the offer to switch must be made in every case. Furthermore, we define it to be to your advantage to switch doors if and only if there is a greater probability of your choosing the door with the prize when you switch.

Analysis #1:

There is a 1-in-3 chance of choosing the door that hides the prize at the outset. This much is given to us by the paradox itself. But the additional information from the game show host does not change this probability: there is still a 1-in-3 chance that our door hides the prize, and a 1-in-3 chance that the remaining door hides the prize. Therefore, there is no advantage in switching.

Analysis #2:

There is a 1-in-3 chance of choosing the door that hides the prize at the outset. But with the additional information from the game show host, we enjoy a slight boost in our chances: two doors remain, and so now there is a 1-in-2 chance that our door hides the prize. There is no advantage in switching, because the remaining door also has a 1-in-2 chance of hiding the prize.

Analysis #3

There is a 1-in-3 chance of choosing the door that hides the prize at the outset. But there is a 2-in-3 chance that the prize is behind the two doors we did not choose. When the game show host reveals that there is a goat behind one of those two doors we did not choose, there is still a 2-in-3 chance that the prize is behind the remaining door. Therefore, it is to our advantage to change (2-in-3 versus 1-in-3).

It turns out that Analysis #3 is correct: it is to your great advantage to switch. In order to understand why, it helps to think of more than three doors. Let’s imagine that there are one hundred doors. When we choose door No. 1, we readily understand that there is a 1-in-100 chance of our having chosen the door that hides the prize, and a 99-in-100 chance that we did not. Let’s now suppose that the game show host reveals that door Nos. 3–100 hide goats. This must mean that there is a 1-in-100 chance we are correct, and a 99-in-100 chance that the remaining door hides the prize. Thus, Analysis #3 must be the correct way to understand the paradox.

Computational Confirmation

In case you are still sceptical, we can easily write a short computer program to simulate the problem and calculate the results. It should be noted that in this version, for the sake of simplicity, the door that is revealed is always the next non-prizewinnig door (looped around from 3 to 1 if necessary). Here it is, written in Objective-C:

– (void)montyCalculation
{



  

srand((unsigned)time(NULL)); // Seed random number generator

NSInteger i=0; // Counter to do it 50000 times
NSInteger right=0; // Number of times right
NSInteger wrong=0; // Number of times wrong

while (i<50000)
{

    // Pick a prize door number between 1 and 3 (the others have goats).
    NSInteger prize = rand() % 3 + 1;

    // Make a choice between doors 1 and 3.
    NSInteger choice = rand() % 3 + 1;

    // Pick which door Monty reveals (next available door after prize that is != choice).
    NSInteger reveal = (prize % 3) + 1;
    if (reveal == choice) reveal = (reveal % 3) + 1;

    // Change doors (next available door after change that is != choice).
    NSInteger change = (reveal % 3) + 1;
    if (change == choice) change = (change % 3) + 1;

    prize == change ? right++ : wrong++;

    i++;
}

NSLog(@"Right: %d, Wrong %d, Percentage Right: %f",right,wrong,(double)right/(right+wrong)*100);




}

And the results, running through the loop 50000 times:

Right: 33418, Wrong 16582, Percentage Right: 66.836000
Right: 33378, Wrong 16622, Percentage Right: 66.756000
Right: 33596, Wrong 16404, Percentage Right: 67.192000
Right: 33382, Wrong 16618, Percentage Right: 66.764000

The results speak for themselves: it is to your advantage to switch.

Question: What’s the most unique, useful feature of Deductions?

Wandering Mango: We'd have to say that the most useful feature of Deductions is different from the most unique feature.  The most useful feature is that Deductions gives you immediate feedback on proofs: it lets you know whether you made a mistake by highlighting the mistake immediately.  The most unique feature is the hint engine.  Often, when you're working on proofs in logic, you can get stuck.  The hint engine gives you an idea how you might solve the problem.

Q: Why did you create Deductions?

WM: We created Deductions because the vast majority of natural deduction applications lack a modern look and feel, and one of our founders is a professor who works in the area of logic, and wanted a modern natural deduction program to use in class.  When you look at other programs on the market, most of them are antiquated. They are written for DOS, or Windows 95. Some are rudimentary Java applets. As you might expect, these programs tend to be awkward and primitive. We wrote Deduction to have an intuitive user interface, and to be a modern alternative.

Q: What is most interesting to you about developing software for the Mac platform?

WM: What is most interesting to us is the attention to design. From the Cocoa framework, to separation of the view and model, to the polished user interface that users expect, the Mac development environment is something really special.  It's true that Xcode has lagged behind its Windows equivalents in some important ways, but now it seems to have eclipsed the competition with Xcode 4.  We’ve worked on other platforms, but on the Mac we really feel like we’ve come to a place where people are very interested in what your program does and how your program does it. It is the interest in “how” that sets the platform apart.

Q: What features should a prospective buyer look into during a trial of your product?

WM: A prospective buyer should have a look at the video tutorials, and then if you know anything about natural deduction, run though some deductions to see how the program helps you learn.  Deductions is a great tool, and this will give you a good idea how and why.

Q: What are some interesting experiences you’ve had creating new versions of your software when OS is upgraded?

WM: An OS update always makes developers a little jittery, but so far everything has gone smoothly. I was able to plug directly into a number of Objective-C 2.0 features that came about it Leopard, and that made the development of Deductions a lot easier. I have no doubt that we’ll be incorporating some Snow Leopard technologies into Deductions too. I’m thinking about putting the hint engine into a separate thread using Grand Central Dispatch.

Q: What features would you like to add to your product that at this time seem improbable/impossible?

WM: I’d really like to improve the hint engine to the point that it gives advice several steps ahead: not just “derive a contradiction,” but “derive this particular contradiction in order to achieve this particular goal.” Now for really fascinating reasons (to me, anyway) that have to do with the computability of predicate logic, there are fixed limits on just how far advice can go. But I’d love to do the research necessary to design a hint engine that presses right up against those fixed limits. That just seems out of reach right now.

Q: What should a beginning logic student know before purchasing Deductions?

WM: If you are taking a class, you should know that formal logic is unlike mathematics in that the symbols and rules differ from textbook to textbook. Deductions works with over twelve textbooks out-of-the-box, and it is very configurable, so it can work with many more textbooks.  But it does not work with every one.  You can see the textbooks Deduction supports, complete with links to where you can get those textbooks, on our textbooks page.  Feel free to ask us if you're not sure.

Q: Can someone use Deductions even if not taking a logic course?

WM: Of course! Deductions has several video tutorials that you can follow, but if you really want to learn how to do proofs in formal logic, you’ll want to follow a textbook or set of lectures. Fortunately, Teller’s Modern Formal Logic Primer and Roy’s Accessible Introduction to Serious Mathematical Logic are freely available on the web.

Q: Can you us more about your company?

WM: We started Wandering Mango last year, and we now have two products: Deductions and Intuition.  We're pretty lucky in that the market for Deductions is well-defined (instructors and students who study formal logic in philosophy, mathematics, and computer science, and other people interested in formal logic).

Q: What’s with the name “Wandering Mango”?

WM: Well, Mac indie developers tend to have incongruous names that stick in your head: Rogue Amoeba, Flying Meat, Delicious Monster, and so on. Our name follows that tradition. A friend of ours decided that “Wandering Mango” sounded like a drink, and she created a recipe to celebrate our first app. It’s on our About page, and it’s really very tasty.