Intro to Logic: What is Logic?

by Luke Muehlhauser on May 30, 2009 in Intro to Logic

Welcome to my course Intro to Logic (index). Here, we learn the basic skills of good thinking and their benefits in real life.

Last time, I summarized 16 techniques of critical thinking. Critical thinking is a broad set of skills for questioning the meaning of truth claims and whether there are good reasons to believe them. It is also an attempt to minimize your own bias in all circumstances.

Critical thinking is a core part of good thinking. Other parts of good thinking are studied in rationality theory. One hugely important part of good thinking, and one that is agreed on by most modern philosophers, is the main topic of this course: logic.

What is logic?

Logic is the study of correct thinking about arguments. The aim of logic is to place our thinking, wherever possible, into a formal structure called an argument, which can then be assessed as a good argument or a bad argument based on the rules of logic.

In a way, logic is like math. It is possible to prove things 100% in math and logic, but not in science. Experts disagree on the foundations of logic, but the vast majority of modern thinkers agree on the laws of logic. If nothing else, I tend to think of math and logic basically as an invented structure for which we all choose to play by the same rules. That is, whenever we each refer to “2″ or “all” or “square root,” we’ve agreed in advance exactly what those words mean. For example, when I say “3″, this means that I do not also mean “2″ at the same time and in the same way.

Here are the three basic rules of logic we all agree to:

  1. The law of identity: p is equal to p at the same time in the same way. Thus: Einstein is Einstein.
  2. The law of non-contradiction: If p is true, then not-p can’t be true in the same way at the same time. Thus, if the statement “Einstein is alive” is true, then the statement “Einstein is not alive” can’t be true at the same time in the same way.
  3. The law of the excluded middle: Either p or not-p must be true. There is no in-between. If the statement “Einstein is not alive” is false, then the statement “Einstein is alive” must be true.

So you’re thinking, “Well, duh!” These three things are self-evidently true by what we mean when we use words. If somebody disagrees with one of these laws, then they’re just using words in a different way than everybody else, and that’s going to cause confusion.

But here’s the amazing part. Everything else in (propositional) logic follows from these three laws. If you agree to play by these three laws, then you must accept all the complicated laws of logic that have been discovered in the past several millennia. In the same way, if you agree about the most basic mathematical rules, then you must also accept all the mathematical theorems that have been proved since Euler, for example Fermat’s Last Theorem.

Anyway… since logic is about analyzing arguments, we must first ask “What is an argument?”

What is an argument?

An argument presents premises which supposedly support the truth of the conclusion. For example, here is an argument for the mortality of Socrates:

  1. All men are mortal.
  2. Socrates is a man.
  3. Therefore, Socrates is mortal.

The first two lines are premises, and the third is the conclusion. As it happens, this is a valid deductive argument, so if you agree with the premises then you must accept the conclusion. We’ll talk about that later.

But not all arguments are presented in such a clear form, with numbered premises and a conclusion. Sometimes an argument must be extracted from several pages of scattered premises. Other times, an argument is squeezed into a very compact sentence, like this one:

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.

This is an argument, but to analyze it, we must unpack it into something like this:

  1. Languages die.
  2. The work of Aeschylus is in a language.
  3. So the work of Aeschylus will eventually die.
  4. Mathematical ideas do not die.
  5. The work of Archimedes was in mathematical ideas.
  6. So the work of Archimedes will not die.
  7. So the work of Archimedes will live longer than the work of Aeschylus.
  8. Therefore, Archimedes will be remembered when Aeschylus is forgotten.

But even this is not a valid argument, as we will see. To make it valid would require several additional premises. Not only that, but we might question the truth of some of the premises.

The use of arguments

Philosophers use arguments to support their conclusions about the existence of God, the nature of existence, moral values, political systems, when and how we can trust our senses, and so on. These arguments are published in academic books and articles, and then analyzed and revised by other philosophers.

But you will find arguments everywhere in everyday life, too. When deciding to what to do for the evening, your friend will say, “Let’s go to Joe’s tonight – they’ve got 2-for-1 drink specials from 5-8 and I think Chad’s band is doing a set there tonight.” That’s an argument. An argument like that may not be worth picking apart, but sometimes it might be. For example, you might want to argue that one of his premises is false (“No, Chad’s band is playing tomorrow night”), or that there are better reasons for doing something else.

On more important matters, politicians use arguments all the time (and they are almost always invalid). Once we finish covering the basics of logic, you might want to grab a newspaper, identify an argument, and analyze it. It’s good practice.

Next, we’ll learn about the building blocks of the argument: propositions.

But first, I’d like to mention that Christian blogger Brian Auten has started his own logic course called Logic Primer. It will be interesting to see if we disagree about anything. Supposedly, logic is religion-neutral. (Although, I think it ravages religious belief. Perhaps Brian thinks it destroys atheism. We’ll see.)

(Also see the post index to this Intro to Logic series.)

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{ 15 comments… read them below or add one }

Dace May 30, 2009 at 3:05 pm

Is everyone new to logic aware that p stands for a proposition? Good stuff, anyway.

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Lorkas May 30, 2009 at 6:38 pm

Should it be “assessed” instead of “accessed” in the first paragraph under “What is logic?”?

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lukeprog May 30, 2009 at 8:29 pm

Lorkas,

Yes! Thank you!

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Chuck May 30, 2009 at 9:37 pm

George Bush has hair.

Bears have hair.
Therefore, George Bush is a bear!

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Chuck May 30, 2009 at 9:43 pm

But seriously, I think many scientists would disagree that math is an invented structure. Rather, it is discovered.

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Josh May 30, 2009 at 9:50 pm

Chuck: But seriously, I think many scientists would disagree that math is an invented structure. Rather, it is discovered.

And I would have to say that they are wrong.  Well, sorta.  Once you lay down your axioms, then every provable statement is already there, just waiting to be “discovered”.  But the axioms are the important part, and they are just invented.

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lukeprog May 30, 2009 at 9:51 pm

Chuck,

Clearly, you are in need of a post on the logical fallacy of affirming the consequent. :)

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lukeprog May 30, 2009 at 10:01 pm

Josh: Once you lay down your axioms, then every provable statement is already there, just waiting to be “discovered”. But the axioms are the important part, and they are just invented.

Yeah. I would say that logic, math, and other axiomatic systems are both invented and discovered. The axioms are invented by convention of what we mean when we use words like “is” and “2″ and “sum.” If somebody uses “is” in such a way that the law of identity is broken, then he is simply speaking a different language than I am. In the same way, if somebody says that 2+2=5, then he appears to be using different definitions for “2″ or “+” or “=” or “5″, and he is speaking a different language than I am.

But after the axioms are laid down, the necessary consequences of those axioms await discovery, and can be quite surprising.

Quite a separate matter is the empirical discovery that many physical systems appear to adhere to certain mathematical rules, such that deductive discoveries in mathematics can actually lead to inductive discoveries in science. See:

Wigner – The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960)

Putnam – What is Mathematical Truth? (1975)

Hamming – The Unreasonable Effectiveness of Mathematics (1980)

Sarukkai – Revisiting the Unreasonable Effectiveness of Mathematics (2005)

Tegmark – Our Mathematical Universe (2007)

Carrier – Our Mathematical Universe (2007)

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Lorkas May 31, 2009 at 5:42 am

2+2=5
(for large values of 2 and small values of 5)

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Democritus June 1, 2009 at 5:58 am

You might want to say that those are the laws of “binary logic”. There are other forms of (non-classical) logic, you know – though they are definitely not as widely used as binary logic; of course, one could argue that not all arguments can be defined in terms of binary logic, but that argument would be binary logic in itself. Cool stuff. :-)

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Pete June 1, 2009 at 4:01 pm

“Everything else in logic follows from these three laws.”

You mean everything else in propositional logic, right?

For predicate logic, you need more axioms (not to speak of modal logic, higher-order logic, etc. ).

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lukeprog June 1, 2009 at 8:29 pm

Pete, I fixed the propositional logic bit.

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D.R. Page August 23, 2010 at 8:16 pm

I think this article on logic has a lot of flaws. It doesn’t distinguish between philosophical logic and mathematical logic. As a theoretical computer scientist we refer to logic in foundations as mathematics (i.e. not as math but, is maths).

Another thing I need to point out is many of the great theorems in foundations maths show that one cannot prove everything 100%. These include theorems in regards to completeness, consistency, and decidability. The three main notions in logical systems. I could easily say it is impossible to attain all three by definition of theorems such as Godel’s Incompleteness Theorem and the Halting Problem. I must also mention that it is possible to attain mathematical certainty of proof in science.

Logic is something falled under Formal sciences in Mathematics, Computer Science and Statistics. Formal sciences are a part of science and should not be distinguished from the umbrella of Science. One needs to clear up that there are two main branches of science. Formal Sciences (as discussed above) and Natural/Empirical Sciences (Biology, Chemistry, Physics,…). Just my 2 cents on the subject because, I think this discussion has a lot of holes and assumptions made all over the place since as a scientist I would definitely not consider a lot of these things correct.

For example: It is possible to build a Turing Machine to show 1+1 can equal 6. As a scientist I assume only for number systems transitive to the natural numbers that 1+1 = 2 because, that is an axiom due to the nature that arithmetic is incomplete and undecidable.

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lukeprog August 23, 2010 at 10:06 pm

Thanks for your feedback.

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Paul Baird January 25, 2011 at 5:05 am

What if Einstein is undergoing an NDE. Is he alive or dead ? More importantly, during an NDE, where is Einstein ?

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