Logic Articles

Cheat Sheet / Updated 02-14-2022

Logic is more than a science, it’s a language, and if you’re going to use the language of logic, you need to know the grammar, which includes operators, identities, equivalences, and quantifiers for both sentential and quantifier logic. And, if you’re studying the subject, exam tips can come in handy.

Article / Updated 12-22-2021

With all the restrictions placed upon it, you may think that logic is too narrow to be of much use. But this narrowness is logic's great strength. Logic is like a laser — a tool whose best use is not illumination, but rather focus. A laser may not provide light for your home, but, like logic, its great power resides in its precision. The following sections describe just a few areas in which logic is commonly used. Pick a number (math) Mathematics is tailor made to use logic in all its power. In fact, logic is one of the three theoretical legs that math stands on. (The other two are set theory and number theory, if you're wondering.) Logic and math work so well together because they're both independent from reality and because they're tools that are used to help people make sense of the world. For example, reality may contain three apples or four bananas, but the ideas of three and four are abstractions, even though they're abstractions that most people take for granted. Math is made completely of such abstractions. When these abstractions get complicated — at the level of algebra, calculus, and beyond — logic can be called on to help bring order to their complexities. Mathematical ideas, such as number, sum, fraction, and so on, are clearly defined without exceptions. That's why statements about these ideas are much easier to verify than a statement about reality, such as "people are generally good at heart" or even "all ravens are black." Fly me to the moon (science) Science uses logic to a great advantage. Like math, science uses abstractions to make sense of reality and then applies logic to these abstractions. The sciences attempt to understand reality by: Reducing reality to a set of abstractions, called a model Working within this model to reach a conclusion Applying this conclusion back to reality again Logic is instrumental during the second step, and the conclusions that science attains are, not surprisingly, logical conclusions. This process is most successful when a good correlation exists between the model and reality and when the model lends itself well to the type of calculations that logic handles comfortably. The areas of science that rely most heavily on logic and math are the quantifiable sciences, such as physics, engineering, and chemistry. The qualitative sciences — biology, physiology, and medicine — use logic but with a bit less certainty. Finally, the social sciences — such as psychology, sociology, and economics — are the sciences whose models bear the least direct correlation to reality, which means they tend to rely less on pure logic. Switch on or off (computer science) Medicine used to be called the youngest science, but now that title has been handed over to computer science. A huge part of the success of the computer revolution rests firmly on logic. Every action your computer completes happens because of a complex structure of logical instructions. At the hardware level — the physical structure of the machine — logic is instrumental in the design of complex circuits that make the computer possible. And, at the software level — the programs that make computers useful — computer languages based on logic provide for the endless versatility that sets the computer apart from all other machines. Tell it to the judge (law) As with mathematics, laws exist primarily as sets of definitions: contracts, torts, felonies, intent to cause bodily harm, and so on. These concepts all come into being on paper and then are applied to specific cases and interpreted in the courts. A legal definition provides the basis for a legal argument, which is similar to a logical argument. For example, to demonstrate copyright infringement, a plaintiff may need to show that the defendant published a certain quantity of material under his own name, for monetary or other compensation, when this writing was protected by a preexisting copyright. These criteria are similar to the premises in a logical argument: If the premises are found to be true, the conclusion — that the defendant has committed copyright infringement — must also be true. Find the meaning of life (philosophy) Logic had its birth in philosophy and is often still taught as an offshoot of philosophy rather than math. Aristotle invented logic as a method for comprehending the underlying structure of reason, which he saw as the motor that propelled human attempts to understand the universe in the widest possible terms. As with science, philosophy relies on models of reality to help provide explanations for what we see. Because the models are rarely mathematical, however, philosophy tends to lean more toward rhetorical logic than mathematical logic.

Article / Updated 05-11-2021

Because deduction rhymes with reduction, you can easily remember that in deduction, you start with a set of possibilities and reduce it until a smaller subset remains. For example, a murder mystery is an exercise in deduction. Typically, the detective begins with a set of possible suspects — for example, the butler, the maid, the business partner, and the widow. By the end of the story, he or she has reduced this set to only one person — for example, "The victim died in the bathtub but was moved to the bed. But, neither woman could have lifted the body, nor could the butler with his war wound. Therefore, the business partner must have committed the crime." Induction begins with the same two letters as the word increase, which can help you remember that in induction, you start with a limited number of observations and increase that number by generalizing. For example, suppose you spend the weekend in a small town and the first five people you meet are friendly, so you inductively conclude the following: "Everybody here is so nice." In other words, you started with a small set of examples and you increased it to include a larger set. Logic allows you to reason deductively with confidence. In fact, it's tailor-made for sifting through a body of factual statements (premises), ruling out plausible but inaccurate statements (invalid conclusions), and getting to the truth (valid conclusions). For this reason, logic and deduction are intimately connected. Deduction works especially well in math, where the objects of study are clearly defined and where little or no gray area exists. For example, each of the counting numbers is either even or odd. So, if you want to prove that a number is odd, you can do so by ruling out that the number is divisible by 2. On the other hand, as apparently useful as induction is, it's logically flawed. Meeting five friendly people — or 10 or 10,000 — is no guarantee that the next one you meet won't be nasty. Meeting 10,000 people doesn't even guarantee that most people in the town are friendly — you may have just met all the nice ones. Logic, however, is more than just a good strong hunch that a conclusion is correct. The definition of logical validity demands that if your premises are true, the conclusion is also true. Because induction falls short of this standard, it's considered the great white elephant of both science and philosophy: It looks like it may work, but in the end it just takes up a lot of space in the living room.

Article / Updated 05-11-2021

When people say "let's be logical" about a given situation or problem, they usually mean "let's follow these steps:" Figure out what we know to be true. Spend some time thinking about it. Determine the best course of action. In logical terms, this three-step process involves building a logical argument. An argument contains a set of premises at the beginning and a conclusion at the end. In many cases, the premises and the conclusion will be linked by a series of intermediate steps. In the following sections, these steps are discussed in the order that you're likely to encounter them. Generating premises The premises are the facts of the matter: the statements that you know (or strongly believe) to be true. In many situations, writing down a set of premises is a great first step to problem solving. For example, suppose you're a school board member trying to decide whether to endorse the construction of a new school that would open in September. Everyone is very excited about the project, but you make some phone calls and piece together your facts, or premises. Premises: The funds for the project won't be available until March. The construction company won't begin work until they receive payment. The entire project will take at least eight months to complete. So far, you only have a set of premises. But when you put them together, you're closer to the final product — your logical argument. In the next section, you'll discover how to combine the premises together. Bridging the gap with intermediate steps Sometimes an argument is just a set of premises followed by a conclusion. In many cases, however, an argument also includes intermediate steps that show how the premises lead incrementally to that conclusion. Using the school construction example from the previous section, you may want to spell things out like this: According to the premises, we won't be able to pay the construction company until March, so they won't be done until at least eight months later, which is November. But, school begins in September. Therefore. . . The word therefore indicates a conclusion and is the beginning of the final step. Forming a conclusion The conclusion is the outcome of your argument. If you've written the intermediate steps in a clear progression, the conclusion should be fairly obvious. For the school construction example, here it is: Conclusion: The building won't be complete before school begins. If the conclusion isn't obvious or doesn't make sense, something may be wrong with your argument. In some cases, an argument may not be valid. In others, you may have missing premises that you'll need to add. Deciding if the argument is valid After you've built an argument, you need to be able to decide if it's valid, which is to say if it's a good argument. To test an argument's validity, assume that all of the premises are true and then see if the conclusion follows automatically from them. If the conclusion automatically follows, you know it's a valid argument. If not, the argument is invalid. Understanding enthymemes The school construction example argument may seem valid, but you also may have a few doubts. For example, if another source of funding became available, the construction company may start earlier and perhaps finish by September. Thus, the argument has a hidden premise called an enthymeme (pronounced EN-thi-meem), as follows: There is no other source of funds for the project. Logical arguments about real-world situations (in contrast to mathematical or scientific arguments) almost always have enthymemes. So, the clearer you become about the enthymemes hidden in an argument, the better chance you have of making sure your argument is valid.

Article / Updated 03-26-2016

Quantifier logic encompasses the rules of sentential logic and expands upon them so that you can write whole statements with logic symbols. Those symbols come into play when you work with identities, or interchangeable constants. The rules of identity are shown here: And, when talking about identities, you can quantify statements, using the rules in the following table:

Article / Updated 03-26-2016

Taking an exam in logic calls for a clear head and a clear plan. The tips in the following list can help you approach a logic exam with the best chance to prove your proficiency: Start by glancing over the whole exam to get a feel for what is covered. Warm up with an easy problem first. Fill in truth tables column by column. If you know you made a mistake, say so — you may get partial credit. If time is short, finish the tedious stuff. Check — and double-check — your work.

Article / Updated 03-26-2016

Working with sentential logic means working with a language designed to express logical arguments with precision and clarity. To make use of this language of logic, you need to know what operators to use, the input-output tables for those operators, and the implication rules. This table introduces sentential logic operators: The next tables offer input–output tables for sentential logic operators: Logic helps you reach conclusions, which you do with the help of implication rules for sentential logic:

Article / Updated 03-26-2016

In any logic system, you compare statements to prove or disprove their validity. With sentential logic, you use the following equivalence rules to make those comparisons: