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  • Number Theory
  • Probability
  • Everyday Math
  • Classical Mechanics
  • Electricity and Magnetism
  • Computer Science
  • Quantitative Finance

Take a guided, problem-solving based approach to learning Logic. These compilations provide unique perspectives and applications you won't find anywhere else.

What's inside

  • Introduction
  • Puzzles and Riddles
  • Multi-Level Thinking
  • The Rational Detective
  • Syllogisms and Sets
  • Logic Machines
  • Arithmetic With Logic Gates
  • Propositional Logic
  • First-Order Logic

Joy of Problem Solving

  • Intro to Problem Solving
  • Coin Rearrangements
  • Truth Tellers and Liars
  • Operator Searches
  • Matchstick Puzzles

Community Wiki

Browse through thousands of Logic wikis written by our community of experts.

  • Truth-Tellers and Liars
  • Cryptogram - Problem Solving
  • Solving Propositional Logic Word Problem
  • Mind Reading with Math
  • Information Compression
  • K-level thinking
  • Chess Puzzles
  • Arithmetic Puzzles - Operator Search
  • Arithmetic Puzzles - Fill in the Blanks
  • Elimination Grids
  • Grid Puzzles
  • Combinatorial Games - Definition
  • Combinatorial Games - Winning Positions
  • Tic Tac Toe
  • Sprague Grundy Theorem
  • Chess Puzzles - Reduced Games
  • Chess Puzzles - Opening Strategies
  • Chess Puzzles - Rook Strategies
  • Rook Polynomial
  • Game Theory
  • Nash Equilibrium
  • Zero-Sum Games
  • Prisoner's Dilemma
  • Braess' Paradox
  • Utility Functions
  • Cognitive Bias
  • Monty Hall Problem
  • Birthday Problem
  • Two-Envelope Paradox
  • Simpson's Paradox
  • Berkson's Paradox
  • Newcomb's Paradox
  • Benford's Law
  • Mathematics of Voting
  • Survivorship Bias
  • Russell's Paradox
  • Zeno's Paradox
  • Gabriel's Horn
  • Truth Tables
  • Proof by Contradiction
  • Mathematical Logic and Computability
  • Mathematical Logic and Computability II (continuation)
  • Propositional Logic Using Algebra
  • Venn Diagram
  • Predicate Logic

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Computers and the Internet

Course: computers and the internet   >   unit 2.

  • Circuits & Logic
  • Logic gates

Logic circuits

help with logic homework

Input AInput BOutput
00 fraction, like fraction, like decimal, like or
01 fraction, like fraction, like decimal, like or
10 fraction, like fraction, like decimal, like or
11 fraction, like fraction, like decimal, like or

Logic Homework help, solvers, FREE tutors, lessons

Put a stop to deadline pressure, and have your homework done by an expert.

Logic Homework Help | Professional Advice and Assistance

logic homework

You might need logic homework help for varied reasons. But, the major reason why many learners need help with logic homework is the lack of adequate time to complete the assignment. In most cases, this assignment takes a lot of time. And most learners need time to study for exams.

Logic is the study of informal and formal reasoning. Initially, logic was a part of philosophy. However, logic is now a mathematical discipline. It’s a core aspect of computer science and also a linguistic tool. Those studying logic at universities and colleges know what logic in computer science and other disciplines is all about.

Major Reasons to Seek Logic Homework Help

Logic exercises and assignments revolve around argumentative and critical thinking. As a branch in philosophy, logic is a formal study of arguments. Completing assignments in this discipline, therefore, involves defending ideas through logic thinking. You must be smart in presenting ideas and attacking your opponents.

Philosophers, lawyers, policymakers, writers, and editors are all interested in logic and critical thinking. The first philosopher to tackle logic seriously was Aristotle. He coined the word “syllogism”. This refers to a simple argument that lacks a conclusion and two premises yet it can be evaluated for validity and soundness. Essentially, not every syllogism is valid.

Most logic books and classes teach students to come up with valid arguments. They also teach them to consider logical fallacies or errors in thinking carefully. What’s more, students are exposed to different forms of arguments. Symbolic language is learned in logic classes and the best books on logic teach this as well. For instance, learners are taught about symbolic language. This involves arguments that are created with symbols and letters. They resemble different mathematical formulas. But, symbolic logic exercises are not easy.

Some professors require students to argue against all logic in their assignments. Others may require learners to discuss the science of logic extensively. These are not simple tasks. They require time and skills to research and present arguments and counter arguments properly. And because students have tight schedules and deadlines for multiple assignments, they end up seeking help with logic homework.

Using the Logic Homework 8 Inference Rules

Some students turn to experts for logic help because they don’t know how to use the inference rules of logic. In the science of logic, inference rule or the rule of inference is also called the transformation rule. It can be described as a logical form that comprises a function that takes the premises, analyzes the syntax, and then returns a conclusion.

For instance, the inference rule known as Modus ponens involves two premises with one forming the “If p then q” format while the other comes in the form of “p” and “q” as the conclusion. The validity of the rule depends on classical logic semantics and semantics of other non-classical logic. Essentially, if the premises are considered true based on interpretation, the conclusion is also considered true.

Completing homework assignments that require learners to use the 8 inference rules is not easy. It requires the learners to know how old is logic and what the 8 rules stipulate. Essentially, inference rules are templates that build valid arguments.

Inference rules are considered syntactical transform rules that are used to infer conclusions from the premises when creating arguments. These are the rules that people use to infer valid conclusions. These rules are mostly used in math assignments and visual logic puzzle exercises. They also help in forming good arguments in different disciplines and occasions.

Using Logic to Form Excellent Arguments

Logic largely entails studying arguments. People use logic to analyze arguments or reasoning. It’s through logic that experts determine whether an argument is valid or invalid. A good argument is a valid conclusion supported by premises or premises. And, certain rules should be followed to construct logical arguments to minimize errors.

So, how does logic help to form a good argument? Well, you must be familiar with logic rules to form a good argument. Logic means “reason” or “thought”. Therefore, using logic means reasoning or thinking well.

Therefore, a good argument comprises a logical connection between the presented assumptions and conclusion. Those pursuing geometry studies can compare this to writing geometrical proofs. A classic example is the ancient Greeks’ reasoning based on the logic rules.

It goes like this:

  • Socrates is a man
  • All men are mortal
  • Socrates, therefore, is mortal

This makes sense. But, some statements can appear logical when looked at. However, further examination proves that they don’t make sense.

For instance:

  • Aristotle and Socrates are different
  • Aristotle is, therefore, different from a man

Such an argument doesn’t make sense. It is a logical fallacy or an argument that concludes without following from the presented premises. A fallacy is also defined as faulty or invalid reasoning. It’s an argument that professes decisiveness of an issue yet it is not.

  • Logic proofs help in strengthening an argument. To provide these proofs, you must familiarize yourself with the subject that you want to argue about and stick to it. This will enable you to express your reasons for reaching the conclusions that you make. You also present the points that you wish to make in your argument. To be strong, an argument should have a minimum of three main points.
  • Practice making an argument from a different viewpoint when preparing for a debate or an argument. This will enable you to consider the counter-arguments that you’re likely to face. This is what most people term as being the devil’s advocate. It’s the practice that lawyers use to prepare for tough cases.
  • An argument that considers only one side of an issue can’t win. To come up with a good argument, you must switch from one side to another with ease. Your opponents should not lead or distract you from the topic.
  • Additionally, avoid getting emotional when presenting your argument. Avoiding getting emotional can boost logic in an argument. It enables you to remain calm even when your opponent provokes you. To avoid getting emotional, maintain the normal volume of your voice, avoid insults, and expect opposition. Learn to listen to your opponent without losing temper.

Practice Solving a Logic Puzzle

One of the best ways to enhance logical reasoning and arguing is by practicing. Practice constructing logical arguments or playing logic games. For instance, you can practice by arguing about the logic to never do homework. You can also read logic puzzle books before you start solving puzzles.

If you work on puzzles consistently, brain cells’ connections will be reinforced. New brain cells’ connections will also form. And, this will enhance your short-term memory. Memory is very important when it comes to completing puzzles. It enables a person to remember the sizes, shapes, and pieces. This enables you to visualize where different sizes and shapes of the puzzles fit.

What’s more, playing logic games help with focusing, reaction control, and attention. These are important skills when it comes to arguments. But, some logic games and puzzles are not easy. Nevertheless, they play a crucial role in enhancing the ability to make logical arguments with valid conclusions.

Get Professional Assistance

Completing homework in logic requires a vast knowledge of logic traditions. Mathematical notation, logic programming, and incompleteness theorems can also be involved. Essentially, there is so much that is involved when completing logic assignments.

Students must also ensure that their work is not laced with plagiarized content. They must also meet the submission deadlines set by their educators. These are some of the major challenges that prompt learners to turn to experts that require them to simply say, ‘do my logic homework’.

You should get assistance with your homework if:

  • You’re confused by multiple writing assignments in logic subjects
  • You don’t have adequate time to acquire knowledge on the subjects separately and complete the assignments
  • You have more important things to do instead of spending all your time researching, learning, and writing assignments

You can approach professional homework helpers if faced with these and other problems. Your professor, guardians, or friends might not always be available to help with homework or show you how to solve a logic puzzle. However, online logic homework helpers are available 24/7. Contact them if you need help with any logic task.

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How To Find Solutions For Your Logic Homework Problems

Logic homework is difficult, so it’s not surprising if you struggle with it. You’re not alone, though! Many people find it hard to do. And that’s exactly why we’ve written this handy guide to finding solutions for your logic homework problems with ease.

Use your textbook

Your first port of call when you need assistance should always be your textbook, so try looking for the solutions to your logic homework problems there. Even if you don’t find the exact answers, you should find very similar examples with their corresponding answers. And, you can use those to work out the answers for your assignment questions. If you don’t have a decent textbook, it really is worth getting one, as you can consult it every time you get stuck with the work. In addition, you can use it when you study for tests and exams.

Use a decent study guide

Another excellent place to look for assistance is in a good study guide. Like your textbook, it may have similar examples with their corresponding answers. It may even have the exact questions and answers you’re looking for! You can buy a study guide for yourself in reputable bookshop, or even online. You’ll also find that you use your study guide for more than just assignments. Normally, study guides explain work very well, so you can use it to study the theory, too.

Ask your friends

You can also try asking your friends for solutions to your logic homework, as they may have already done the assignments, or even found the solutions somewhere else. Just remember that your friends may not be willing to let you copy their work if they’ve slaved over it themselves.

Look online

A really great place to look for assignment answers is on the Internet, so try searching there. You can look on homework websites, as they often have answers for assignments. In addition, there are many websites dedicated to the study of logic, so you may find some decent answers on those sites, too. Also, don’t forget to check some of the live help websites, as there are so many great people who are willing to give their time to help students with their schoolwork.

Do the work yourself

Even though there are so many excellent places to get solutions for your logic homework, there’s no substitute for actually doing the work yourself. You’ll never learn to do it if you don’t try, so attempt the questions on your own first. Then you can start looking for the answers to check your work.

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Sat / act prep online guides and tips, the 5 best homework help websites (free and paid).

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Other High School , General Education

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Listen: we know homework isn’t fun, but it is a good way to reinforce the ideas and concepts you’ve learned in class. But what if you’re really struggling with your homework assignments?

If you’ve looked online for a little extra help with your take-home assignments, you’ve probably stumbled across websites claiming to provide the homework help and answers students need to succeed . But can homework help sites really make a difference? And if so, which are the best homework help websites you can use? 

Below, we answer these questions and more about homework help websites–free and paid. We’ll go over: 

  • The basics of homework help websites
  • The cost of homework help websites 
  • The five best homework websites out there 
  • The pros and cons of using these websites for homework help 
  • The line between “learning” and “cheating” when using online homework help 
  • Tips for getting the most out of a homework help website

So let’s get started! 

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The Basics About Homework Help Websites–Free and Paid

Homework help websites are designed to help you complete your homework assignments, plain and simple. 

What Makes a Homework Help Site Worth Using

Most of the best sites allow users to ask questions and then provide an answer (or multiple possible answers) and explanation in seconds. In some instances, you can even send a photo of a particular assignment or problem instead of typing the whole thing out! 

Homework help sites also offer more than just help answering homework questions. Common services provided are Q&A with experts, educational videos, lectures, practice tests and quizzes, learning modules, math solving tools, and proofreading help. Homework help sites can also provide textbook solutions (i.e. answers to problems in tons of different textbooks your school might be using), one-on-one tutoring, and peer-to-peer platforms that allow you to discuss subjects you’re learning about with your fellow students. 

And best of all, nearly all of them offer their services 24/7, including tutoring! 

What You Should Should Look Out For

When it comes to homework help, there are lots–and we mean lots –of scam sites out there willing to prey on desperate students. Before you sign up for any service, make sure you read reviews to ensure you’re working with a legitimate company. 

A word to the wise: the more a company advertises help that veers into the territory of cheating, the more likely it is to be a scam. The best homework help websites are going to help you learn the concepts you’ll need to successfully complete your homework on your own. (We’ll go over the difference between “homework help” and “cheating” a little later!) 

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You don't need a golden piggy bank to use homework help websites. Some provide low or no cost help for students like you!

How Expensive Are the Best Homework Help Websites?

First of all, just because a homework help site costs money doesn’t mean it’s a good service. Likewise, just because a homework help website is free doesn’t mean the help isn’t high quality. To find the best websites, you have to take a close look at the quality and types of information they provide! 

When it comes to paid homework help services, the prices vary pretty widely depending on the amount of services you want to subscribe to. Subscriptions can cost anywhere from $2 to $150 dollars per month, with the most expensive services offering several hours of one-on-one tutoring with a subject expert per month.

The 5 Best Homework Help Websites 

So, what is the best homework help website you can use? The answer is that it depends on what you need help with. 

The best homework help websites are the ones that are reliable and help you learn the material. They don’t just provide answers to homework questions–they actually help you learn the material. 

That’s why we’ve broken down our favorite websites into categories based on who they’re best for . For instance, the best website for people struggling with math might not work for someone who needs a little extra help with science, and vice versa. 

Keep reading to find the best homework help website for you! 

Best Free Homework Help Site: Khan Academy

  • Price: Free!
  • Best for: Practicing tough material 

Not only is Khan Academy free, but it’s full of information and can be personalized to suit your needs. When you set up your account , you choose which courses you need to study, and Khan Academy sets up a personal dashboard of instructional videos, practice exercises, and quizzes –with both correct and incorrect answer explanations–so you can learn at your own pace. 

As an added bonus, it covers more course topics than many other homework help sites, including several AP classes.

Runner Up: Brainly.com offers a free service that allows you to type in questions and get answers and explanations from experts. The downside is that you’re limited to two answers per question and have to watch ads. 

Best Paid Homework Help Site: Chegg

  • Price: $14.95 to $19.95 per month
  • Best for: 24/7 homework assistance  

This service has three main parts . The first is Chegg Study, which includes textbook solutions, Q&A with subject experts, flashcards, video explanations, a math solver, and writing help. The resources are thorough, and reviewers state that Chegg answers homework questions quickly and accurately no matter when you submit them.  

Chegg also offers textbook rentals for students who need access to textbooks outside of their classroom. Finally, Chegg offers Internship and Career Advice for students who are preparing to graduate and may need a little extra help with the transition out of high school. 

Another great feature Chegg provides is a selection of free articles geared towards helping with general life skills, like coping with stress and saving money. Chegg’s learning modules are comprehensive, and they feature solutions to the problems in tons of different textbooks in a wide variety of subjects. 

Runner Up: Bartleby offers basically the same services as Chegg for $14.99 per month. The reason it didn’t rank as the best is based on customer reviews that say user questions aren’t answered quite as quickly on this site as on Chegg. Otherwise, this is also a solid choice!

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Best Site for Math Homework Help: Photomath

  • Price: Free (or $59.99 per year for premium services) 
  • Best for: Explaining solutions to math problems

This site allows you to t ake a picture of a math problem, and instantly pulls up a step-by-step solution, as well as a detailed explanation of the concept. Photomath also includes animated videos that break down mathematical concepts to help you better understand and remember them. 

The basic service is free, but for an additional fee you can get extra study tools and learn additional strategies for solving common math problems.

Runner Up: KhanAcademy offers in-depth tutorials that cover complex math topics for free, but you won’t get the same tailored help (and answers!) that Photomath offers. 

Best Site for English Homework Help: Princeton Review Academic Tutoring

  • Price: $40 to $153 per month, depending on how many hours of tutoring you want 
  • Best for: Comprehensive and personalized reading and writing help 

While sites like Grammarly and Sparknotes help you by either proofreading what you write via an algorithm or providing book summaries, Princeton Review’s tutors provide in-depth help with vocabulary, literature, essay writing and development, proofreading, and reading comprehension. And unlike other services, you’ll have the chance to work with a real person to get help. 

The best part is that you can get on-demand English (and ESL) tutoring from experts 24/7. That means you can get help whenever you need it, even if you’re pulling an all-nighter! 

This is by far the most expensive homework site on this list, so you’ll need to really think about what you need out of a homework help website before you commit. One added benefit is that the subscription covers over 80 other subjects, including AP classes, which can make it a good value if you need lots of help!  

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Best Site for STEM Homework Help: Studypool

  • Best for: Science homework help
  • Price: Varies; you’ll pay for each question you submit

When it comes to science homework help, there aren’t a ton of great resources out there. The best of the bunch is Studypool, and while it has great reviews, there are some downsides as well. 

Let’s start with the good stuff. Studypool offers an interesting twist on the homework help formula. After you create a free account, you can submit your homework help questions, and tutors will submit bids to answer your questions. You’ll be able to select the tutor–and price point–that works for you, then you’ll pay to have your homework question answered. You can also pay a small fee to access notes, lectures, and other documents that top tutors have uploaded. 

The downside to Studypool is that the pricing is not transparent . There’s no way to plan for how much your homework help will cost, especially if you have lots of questions! Additionally, it’s not clear how tutors are selected, so you’ll need to be cautious when you choose who you’d like to answer your homework questions.  

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What Are the Pros and Cons of Using Homework Help Sites?

Homework help websites can be a great resource if you’re struggling in a subject, or even if you just want to make sure that you’re really learning and understanding topics and ideas that you’re interested in. But, there are some possible drawbacks if you don’t use these sites responsibly. 

We’ll go over the good–and the not-so-good–aspects of getting online homework help below. 

3 Pros of Using Homework Help Websites 

First, let’s take a look at the benefits. 

#1: Better Grades Beyond Homework

This is a big one! Getting outside help with your studies can improve your understanding of concepts that you’re learning, which translates into better grades when you take tests or write essays. 

Remember: homework is designed to help reinforce the concepts you learned in class. If you just get easy answers without learning the material behind the problems, you may not have the tools you need to be successful on your class exams…or even standardized tests you’ll need to take for college. 

#2: Convenience

One of the main reasons that online homework help is appealing is because it’s flexible and convenient. You don’t have to go to a specific tutoring center while they’re open or stay after school to speak with your teacher. Instead, you can access helpful resources wherever you can access the internet, whenever you need them.

This is especially true if you tend to study at off hours because of your extracurriculars, work schedule, or family obligations. Sites that offer 24/7 tutoring can give you the extra help you need if you can’t access the free resources that are available at your school. 

#3: Variety

Not everyone learns the same way. Maybe you’re more of a visual learner, but your teacher mostly does lectures. Or maybe you learn best by listening and taking notes, but you’re expected to learn something just from reading the textbook . 

One of the best things about online homework help is that it comes in a variety of forms. The best homework help sites offer resources for all types of learners, including videos, practice activities, and even one-on-one discussions with real-life experts. 

This variety can also be a good thing if you just don’t really resonate with the way a concept is being explained (looking at you, math textbooks!).

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Not so fast. There are cons to homework help websites, too. Get to know them below!

3 Cons of Using Homework Help Websites 

Now, let’s take a look at the drawbacks of online homework help. 

#1: Unreliable Info

This can be a real problem. In addition to all the really good homework help sites, there are a whole lot of disreputable or unreliable sites out there. The fact of the matter is that some homework help sites don’t necessarily hire people who are experts in the subjects they’re talking about. In those cases, you may not be getting the accurate, up-to-date, and thorough information you need.

Additionally, even the great sites may not be able to answer all of your homework questions. This is especially true if the site uses an algorithm or chatbot to help students…or if you’re enrolled in an advanced or college-level course. In these cases, working with your teacher or school-provided tutors are probably your best option. 

#2: No Clarification

This depends on the service you use, of course. But the majority of them provide free or low-cost help through pre-recorded videos. Watching videos or reading info online can definitely help you with your homework… but you can’t ask questions or get immediate feedback if you need it .

#3: Potential For Scamming 

Like we mentioned earlier, there are a lot of homework help websites out there, and lots of them are scams. The review comments we read covered everything from outdated or wrong information, to misleading claims about the help provided, to not allowing people to cancel their service after signing up. 

No matter which site you choose to use, make sure you research and read reviews before you sign up–especially if it’s a paid service! 

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When Does “Help” Become “Cheating”?

Admittedly, whether using homework help websites constitutes cheating is a bit of a grey area. For instance, is it “help” when a friend reads your essay for history class and corrects your grammar, or is it “cheating”? The truth is, not everyone agrees on when “help” crosses the line into “cheating .” When in doubt, it can be a good idea to check with your teacher to see what they think about a particular type of help you want to get. 

That said, a general rule of thumb to keep in mind is to make sure that the assignment you turn in for credit is authentically yours . It needs to demonstrate your own thoughts and your own current abilities. Remember: the point of every homework assignment is to 1) help you learn something, and 2) show what you’ve learned. 

So if a service answers questions or writes essays for you, there’s a good chance using it constitutes cheating. 

Here’s an example that might help clarify the difference for you. Brainstorming essay ideas with others or looking online for inspiration is “help” as long as you write the essay yourself. Having someone read it and give you feedback about what you need to change is also help, provided you’re the one that makes the changes later. 

But copying all or part of an essay you find online or having someone write (or rewrite) the whole thing for you would be “cheating.” The same is true for other subjects. Ultimately, if you’re not generating your own work or your own answers, it’s probably cheating.

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5 Tips for Finding the Best Homework Help Websites for You

Now that you know some of our favorite homework help websites, free and paid, you can start doing some additional research on your own to decide which services might work best for you! Here are some top tips for choosing a homework help website. 

Tip 1: Decide How You Learn Best 

Before you decide which site or sites you’re going to use for homework help, y ou should figure out what kind of learning style works for you the most. Are you a visual learner? Then choose a site that uses lots of videos to help explain concepts. If you know you learn best by actually doing tasks, choose a site that provides lots of practice exercises.

Tip 2: Determine Which Subjects You Need Help With

Just because a homework help site is good overall doesn’t mean that it’s equally good for every subject. If you only need help in math, choose a site that specializes in that area. But if history is where you’re struggling, a site that specializes in math won’t be much help. So make sure to choose a site that you know provides high-quality help in the areas you need it most. 

Tip 3: Decide How Much One-On-One Help You Need 

This is really about cost-effectiveness. If you learn well on your own by reading and watching videos, a free site like Khan Academy is a good choice. But if you need actual tutoring, or to be able to ask questions and get personalized answers from experts, a paid site that provides that kind of service may be a better option.

Tip 4: Set a Budget

If you decide you want to go with a paid homework help website, set a budget first . The prices for sites vary wildly, and the cost to use them can add up quick. 

Tip 5: Read the Reviews

Finally, it’s always a good idea to read actual reviews written by the people using these homework sites. You’ll learn the good, the bad, and the ugly of what the users’ experiences have been. This is especially true if you intend to subscribe to a paid service. You’ll want to make sure that users think it’s worth the price overall!

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What’s Next?

If you want to get good grades on your homework, it’s a good idea to learn how to tackle it strategically. Our expert tips will help you get the most out of each assignment…and boost your grades in the process.

Doing well on homework assignments is just one part of getting good grades. We’ll teach you everything you need to know about getting great grades in high school in this article.

Of course, test grades can make or break your GPA, too. Here are 17 expert tips that’ll help you get the most out of your study prep before you take an exam.

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Ashley Sufflé Robinson has a Ph.D. in 19th Century English Literature. As a content writer for PrepScholar, Ashley is passionate about giving college-bound students the in-depth information they need to get into the school of their dreams.

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Frequently asked questions

  
  
     
  
  
  
  
     Chapters 5 & 3 & 4  
  "All wolverines are ferocious"
      translates into syllogistic logic as:

      {1}  all W is F
      {2}  W is F
      {3}  w is F
      {4}  all F is W

  What is your answer, Harry? _

  TO ANSWER: Click the answer, type its number,  
  or highlight it (using spacebar or arrow keys)  
  and hit ENTER.
INFO: Display logic rules. GETANS: Have LogiCola get you the next answer. DOPROB: Have LogiCola finish the whole problem for you. REPEAT: Do the previous problem again with scoring disabled.
    
     
     
     
      no P is B  
      some C is B  
  ∴ some C is not P  
      a is F  
      a is G  
  ∴ some F is G  
is B  
  no is   
        
      no * is *  
      some C is B  
  ∴ some C* is not   
      no * is *  
      some C is not *  
  ∴ some C* is P*  
      a is F  
      a is G  
  ∴ some F* is G*  
      No pure water is burnable.  
      Some Cuyahoga River
         water is burnable.  
  ∴ Some Cuyahoga River
         water is not pure water.  
      no * is *  Valid  
      some C is B  
  ∴ some C* is not   
      some A is B  
      all B is C  
  ∴ ??
      some A is B  
      all B* is C  
  ∴ ??
      some A is B  Valid  
      all B* is C  
  ∴ some A* is C*  
      no P is B  
      some C is B  
  ∴ some C is not P  

  Whatever is good in itself ought to be desired. But  
  whatever ought to be desired is capable of being  
  desired. So only pleasure is good in itself, since  
  only pleasure is capable of being desired.  
      all G* is O  Valid  
      all O* is C  
      all C* is P  
  ∴ all G is P*  
  =  
  =  
  =  
  =  
  =  
Not A  
Both A and B  
Either A or B  
If A then B  
A if and only if B  
  =  
  =  
(∼A ∨ B)  
∼(A ∨ B)  
  =  
  =  
(A ⊃ (B • C))  
((A ⊃ B) • C)  
  =  
  =  
(B ⊃ A)  
(A ⊃ B)  
  =  
  =  
(A ⊃ B)  
(B ⊃ A)  
  =  
  =  
  =  
(A ⊃ B)  
(∼A ⊃ ∼B)  
(A ≡ B)  
 
0  0        0   
0  1        0   
1  0        0   
1  1        1   
 
0  0        0    
0  1        1    
1  0        1    
1  1        1    
 
0  0        1    
0  1        1    
1  0        0    
1  1        1    
 
0  0        1    
0  1        0    
1  0        0    
1  1        1    
 
0     1 
1     0 
 
0
1
 
0  0
0  1
1  0
1  1
 
0  0  0
0  0  1
0  1  0
0  1  1
1  0  0
1  0  1
1  1  0
1  1  1
 
0  0  0                      1     
0  0  1                      1     
0  1  0                      0     
0  1  1                      1     
1  0  0                      0     
1  0  1                      1     
1  1  0                      0     
1  1  1                      1     
      It's in my left hand or my right hand.  
      It's not in my right hand.  
  ∴ It's in my left hand.  
      (L ∨ R)  
      ∼R  
  ∴ L  
 
0  0         0         1        0
0  1         1         0        0
1  0         1         1        1
1  1         1         0        1
 
0  0         1       0       0
0  1         1       1       0
1  0         0       0       1
1  1         1       1       1


⇐ Invalid (we get "110")  


      (Lº ∨ Rº)  ≠  1  Valid  
      ∼Rº  =  1
  ∴ Lº  =  0
      ∼(A • B¹)  =  1  
  ∴ ∼B¹  =  0
      ∼(Aº • B¹)  =  1  Invalid  
  ∴ ∼B¹  =  0
      If our country will be weak,
         there will be war.
      Our country will not be weak.  
  ∴ There will not be war.  
      (Kº ⊃ R¹)  =  1  Invalid  
      ∼Kº  =  1
  ∴ ∼R¹  =  0
  Knowledge is good in itself only if it  
  can be desired for its own sake. So  
  knowledge is good in itself, since it  
  can be desired for its own sake.  
      (Gº ⊃ C¹)  =  1  Invalid  
      C¹  =  1
  ∴ Gº  =  0

  

P, Q


  
∼P, ∼Q


  
P, ∼Q
      I'm not in Paris and I'm not in Quebec.  
  ∴ I'm not in Paris.
  ∴ I'm not in Quebec.
  
∼P, ∼Q
      You're not both in Paris and also in Quebec.  
  ∴ ???
  ∴ ???
  
nil
  
∼Q
  
∼P
  
Q
  
P
  
Q
  
∼P
      You're not both in Paris and also in Quebec.  
      You're in Paris.
  ∴ You're not in Quebec.
  
∼Q
      You're not both in Paris and also in Quebec.  
      You're not in Paris.
  ∴ You're in Quebec.
  
Q
  
∼B
      If-then.  
      Affirm first.
  ∴ Affirm second.  
  
Q
      If-then.  
      Deny second.
  ∴ Deny first.
  
∼P
      If you're a dog, you're an animal.  
      You're a dog.
  ∴ You're an animal.  
  
A
      If you're a dog, you're an animal.  
      You're not an animal.
  ∴ You're not a dog.
  
∼D
      If you're a dog, you're an animal.  
      You're not a dog.
  ∴ You're not an animal.  
  
∼A
      If you're a dog, you're an animal.  
      You're an animal.
  ∴ You're a dog.
  
D
  
A
  
(C ∨ D)
  
∼(A • B)
    1    (S ⊃ ∼C)
    2    (S ∨ ∼A)
     [∴ ∼(A • C)

  What's next, Harry? _

  TO ANSWER: Type a derived line ("A"), assumption
  ("ASM: A"), or "REFUTE" (to finish an invalid argument).  
  Click a wff (or use PgUp-PgDn keys) to copy it to the
  answer space. "Info" reviews the rules.
  *  1    (S ⊃ ∼C)
  *  2    (S ∨ ∼A)
     [∴ ∼(A • C)
  *  3 ┌ asm: (A • C)
      4 │ ∴ A  {from 3}
      5 │ ∴ C  {from 3}
      6 │ ∴ ∼S  {from 1 and 5}
      7 └ ∴ ∼A  {from 2 and 6}
      8 ∴ ∼(A • C)  {from 3; 4 contradicts 7}  
  




  *  1    (A ⊃ B)    Valid
       [∴ (∼A ∨ B)
  *  2 ┌ asm: ∼(∼A ∨ B)
      3 │ ∴ A  {from 2}
      4 │ ∴ ∼B  {from 2}
      5 └ ∴ B  {from 1 and 3}
      6 ∴ (∼A ∨ B)  {from 2; 4 contradicts 5}  
      1    (Aº ⊃ B¹)  =  1  
       [∴ (B¹ ⊃ Aº)  =  0
  *  2    asm: ∼(B ⊃ A)
      3    ∴ B  {from 2}
      4    ∴ ∼A  {from 2}  
  *  1    (B ∨ A)    Valid
      2    (B ⊃ A)
     [∴ ∼(A ⊃ ∼A)
  *  3 ┌ asm: (A ⊃ ∼A)
      4 │┌ asm: B  {break 1}
      5 ││ ∴ A  {from 2 and 4}
      6 │└ ∴ ∼A  {from 3 and 5}
      7 │ ∴ ∼B  {from 4; 5 contradicts 6}
      8 │ ∴ A  {from 1 and 7}
      9 └ ∴ ∼A  {from 3 and 8}
    10 ∴ ∼(A ⊃ ∼A)  {from 3; 8 contradicts 9}  
      1    (A¹ ∨ ∼(B ⊃ C¹))  =  1  
      2    (Dº ⊃ (A¹ ⊃ B))  =  1  
       [∴ (C¹ ⊃ ∼(Dº ∨ A¹))  =  0
  *  3    asm: ∼(C ⊃ ∼(D ∨ A))
      4    ∴ C  {from 3}
      5    ∴ (D ∨ A)  {from 3}
      6      asm: A  {break 1}
      7        asm:  ∴ ∼D  {break 2}  
  =   Romeo is Italian.
  =   x is Italian.
  =   All are Italian.
For all x, x is Italian.
  =   Some are Italian.
For some x, x is Italian.  
  =   (x)Ix  
  =   ∼(x)Ix  
  =   (∃x)Ix  
  =   ∼(∃x)Ix  
constant  
  *  1    (∃x)(Fx • Gx)    Valid
       [∴ (∃x)Fx
  *  2 ┌ asm: ∼(∃x)Fx
      3 │ ∴ (x)∼Fx  {from 2}
      4 │ ∴ (Fa • Ga)  {from 1}
      5 │ ∴ ∼Fa  {from 3}
      6 └ ∴ Fa  {from 4}
      7 ∴ (∃x)Fx  {from 2; 5 contradicts 6}  
is true if and only if
are true.
is true if and only if  
is true.
      1    (x)(Fx ⊃ Gx)  =  1
  *  2    (∃x)Fx  =  1  
       [∴ (x)Gx  =  0
  *  3    asm: ∼(x)Gx
  *  4    ∴ (∃x)∼Gx  {from 3}
      5    ∴ ∼Ga  {from 4}
      6    ∴ Fb  {from 2}
  *  7    ∴ (Fa ⊃ Ga)  {from 1}
  *  8    ∴ (Fb ⊃ Gb)  {from 1}
      9    ∴ ∼Fa  {from 5 and 7}  
    10    ∴ Gb  {from 6 and 8}
        1    ((x)Fx ⊃ (x)Gx)  =  1
         [∴ (Fa ⊃ Ga)  =  0
    *  2    asm: ∼(Fa ⊃ Ga)
        3    ∴ Fa  {from 2}
        4    ∴ ∼Ga  {from 2}
        5      asm: ∼(x)Fx  {break 1}  
   **  6      ∴ (∃x)∼Fx  {from 5}
   **  7      ∴ ∼Fb  {from 6}
  *  1    (∃x)(∃y)∼x=y    Valid
       [∴ ∼(∃x)(y)y=x
  *  2 ┌ asm: (∃x)(y)y=x
  *  3 │ ∴ (∃y)∼a=y  {from 1}
      4 │ ∴ ∼a=b  {from 3}
      5 │ ∴ (y)y=c  {from 2}
      6 │ ∴ a=c  {from 5}
      7 │ ∴ b=c  {from 5}
      8 └ ∴ a=b  {from 6 and 7}
      9 ∴ ∼(∃x)(y)y=x  {from 2; 4 contradicts 8}  
      (x)(∃y)Lxy  
  ∴ (∃y)(x)Lxy  
      Everyone loves someone or other.
  ∴ There is someone that everyone loves.  
      (x)(∃y)Lxy  
  ∴ (∃y)Lay
  ∴ Lab
  ∴ (∃y)Lby
  ∴ Lbc
  ∴ (∃y)Lcy  ...  
      Everyone loves someone or other.  
  ∴ a loves someone or other.
  ∴ a loves b.
  ∴ b loves someone or other.
  ∴ b loves c.
  ∴ c loves someone or other.  ...
      (x)(∃y)Lxy  =  1  
  ∴ (∃y)(x)Lxy  =  0  



      1    (x)(y)Lxy    Valid
       [∴ (∃x)Lax
  *  2 ┌ asm: ∼(∃x)Lax
      3 │ ∴ (x)∼Lax  {from 2}
      4 │ ∴ ∼Laa  {from 3}
      5 │ ∴ (y)Lay  {from 1}
      6 └ ∴ Laa  {from 5}
      7 ∴ (∃x)Lax  {from 2; 4 contradicts 6}  
= =
= =
= =
=
=
string of W's  
      1    ☐(A • B)    Valid
       [∴ ☐A
  *  2 ┌ asm: ∼☐A
  *  3 │ ∴ ◊∼A  {from 2}
      4 │ W ∴ ∼A  {from 3}
      5 │ W ∴ (A • B)  {from 1}
      6 └ W ∴ A  {from 5}
      7 ∴ ☐A  {from 2; 4 contradicts 6}  
      1    ☐(A ⊃ B)  =  1
  *  2    ◊A  =  1  
       [∴ ☐B  =  0
  *  3    asm: ∼☐B
  *  4    ∴ ◊∼B  {from 3}
      5    W ∴ A  {from 2}
      6    WW ∴ ∼B  {from 4}
  *  7    W ∴ (A ⊃ B)  {from 1}
  *  8    WW ∴ (A ⊃ B)  {from 1}
      9    W ∴ B  {from 5 and 7}  
    10    WW ∴ ∼A  {from 6 and 8}  
Drop boxes :  If we have a "☐A" line, we can derive a "A" line. We can use any world prefixes (the same or different) in either line.
    1    ◊◊P
    2    ◊Q
      ...
  11    W ∴ ◊P  {from 1}
  12    WW ∴ P  {from 11}
  13    WWW ∴ Q  {from 2}  
  

#  ⇒  W
W  ⇒  WW
#  ⇒  WWW
      1    ☐A
       [∴ ☐☐A
  *  2 ┌ asm: ∼☐☐A
  *  3 │ ∴ ◊∼☐A  {from 2}
  *  4 │ W ∴ ∼☐A  {from 3}
  *  5 │ W ∴ ◊∼A  {from 4}
      6 │ WW ∴ ∼A  {from 5}
      7 └ WW ∴ A  {from 1}
      8 ∴ ☐☐A  {from 2; 6 contradicts 7}  


      1    (x)◊Ax  =  1
       [∴ ◊(x)Ax  =  0
  *  2    asm: ∼◊(x)Ax
      3    ∴ ☐∼(x)Ax  {from 2}
  *  4    ∴ ◊Aa  {from 1}
      5    W ∴ Aa  {from 4}
  *  6    W ∴ ∼(x)Ax  {from 3}
  *  7    W ∴ (∃x)∼Ax  {from 6}  
      8    W ∴ ∼Ab  {from 7}
  *  9    ∴ ◊Ab  {from 1}
    10    WW ∴ Ab  {from 9}
  * 11    WW ∴ ∼(x)Ax  {from 3}
  * 12    WW ∴ (∃x)∼Ax  {from 6}  
      (x)◊Ax  =  1  
  ∴ ◊(x)Ax  =  0  
      It's possible for anyone to be above average.
  ∴ It's possible for everyone to be above average.  
  
  You're doing A  

A
Au
  
Do A


A
  =   Don't do A.
• )  =   Do A and B.
• )  =   Don't both do A and do B.
Don't combine doing A with doing B.
• ∼ )  =   Don't combine doing A with not doing B.  
Don't do A without doing B.
  =   You're doing A and you're doing B.  
)  =   You're doing A, but do B.
• )  =   Do A and B.
  =   If you're doing A, then you're doing B.  
)  =   If you (in fact) are doing A, then do B.
⊃ B)  =   Do A, only if you (in fact) are doing B.
)  =   If you (in fact) are doing A, then don't do B.  
• ∼ )  =   Don't combine doing A with not doing B.  
  =   X, do (or be) A.
Let it be the case that X does A.
Would that X do A.
y  =   X, do A to Y.
Let it be the case that X does A to Y.  
Would that X do A to Y.
  =   Let everyone do A.  
  =   Let someone do A.
)  =   Let everyone who (in fact) is doing A do B.  
)  =   Let someone who (in fact) is doing A do B.  
• B )  =   Let someone both do A and do B.
  =   It ought to be that A.
Act A is obligatory (required, a duty).  
  =   It would be all right that A.
Act A is right (permissible, OK).
  =  O∼
• )
)
• )
y
y
)
• Rx)
• R )
      If the cocoa is boiling, remove  
         it from the heat.
      The cocoa is boiling.  
  ∴ Remove it from the heat.  
      (B ⊃ )  Valid  
      B
  ∴
      Don't combine accelerating  
         with braking.
      You're accelerating.  
  ∴ Don't brake.  
      ∼( • )  Invalid  
      A
  ∴ ∼
      ∼( • )  
      A
  ∴ ∼
,      =  0  
  =  1
)
  If you're accelerating,  
then don't brake.
• )
  Don't combine accelerating  
with braking.
)
If you believe that
  you ought to do A,  
then do A.
• ∼ )
  Don't combine believing  
that you ought to do A
with not doing A.
  *  1    (Bº ⊃ º)  =  1  
       [∴ ∼( ¹ • ∼ º)  =  0
  *  2    asm: ( • ∼ )
      3    ∴    {from 2}
      4    ∴ ∼   {from 2}
      5    ∴ ∼B  {from 1 and 4}  
, ∼   
  ⇒  R∼  
  ∼R   ⇒  O∼   
  ⇒  D ∴ ,  
  use a string of D's  
  ⇒  D ∴ ,  
  use a blank or any string of D's  

  ———  
∴  ◊A
" is the corresponding  
indicative wff.
      1    O(x)(Ax ⊃ B )    Valid
      2    Ag
       [∴ OB
  *  3 ┌ asm: ∼OB
  *  4 │ ∴ R∼B   {from 3}
      5 │ D ∴ ∼B   {from 4}
      6 │ D ∴ (x)(Ax ⊃ B )  {from 1}
  *  7 │ D ∴ (Ag ⊃ B )  {from 6}
      8 │ D ∴ ∼Ag  {from 5 and 7}
      9 └ ∴ ∼Ag  {from 8 by indicative transfer rule}  
    10 ∴ OB   {from 3; 2 contradicts 9}  
  =   You believe that A is true.
  =   You don't believe that A is true.
  =   You believe that A is false.
  =   You take no position on A.
  =   If you believe A, then you don't believe B.  
  =   You believe that you ought to do A.
  =   You believe that everyone ought to do A.
  =   Everyone believes that you ought to do A.  
:A  =   Believe that A is true.
:A  =   Don't believe that A is true.
:∼A  =   Believe that A is false.
:A • ∼ :∼A)  =   Take no position on A.
:B)  =   If you believe A, then don't believe B.  
:OA   =   Believe that you ought to do A.
:(x)OA   =   Believe that everyone ought to do A.
:OA   =   Let everyone believe that you ought to do A.  
:B)  =   If you (in fact) believe A, then don't believe B.  
:A • :B)  =   Don't combine believing A with believing B.
  =   You accept (endorse, assent to) "Let A be done."  
You will that A be done.
)  =   You desire that some who kill repent.
• Rx)  =   You desire that some kill who repent.
• R )  =   You desire that some both kill and repent.  

  u:A  =  You believe A.  
  u:   =  You will A.

   :A  =  Believe A.  
   :   =  Will A.
⊃ ∼A )
=
If you believe that it's wrong for
you to do A, then don't do A.
:O∼A • :A )
=
  Don't combine believing that it's wrong for  
you to do A with acting to do A.
:A  =   A is evident to you.
It's obligatory (rationally required) that you believe A.  
Insofar as intellectual considerations are concerned,
  you ought to believe A.
:A  =   A would be reasonable for you to believe.
It's all right (rationally permissible) that you believe A.  
Insofar as intellectual considerations are concerned,
  it would be all right for you to believe A.
⊃ O∼A )
=
If you believe that it's wrong for you to
do A, then you ought not to do A.
:O∼A • :A )
=
  You ought not to combine believing that it's  
wrong for you to do A with acting to do A.
:A • :∼A)
Don't combine believing A
with believing not-A.
:(A • B) • ∼ :A)
  Don't combine believing A-and-B  
with not believing A.
:A  ⇒  u ∴ ∼A,  
  use a string of u's  
:A  ⇒  u ∴ A,  
  use any string of u's  
       [∴ ∼( :A • :∼A)    Valid
  *  1 ┌ asm: ( :A • :∼A)
      2 │ ∴  :A  {from 1}
      3 │ ∴  :∼A  {from 1}
      4 │ u ∴ A  {from 2}
      5 └ u ∴ ∼A  {from 3}
      6 ∴ ∼( :A • :∼A)  {from 1; 4 contradicts 5}  
       [∴ O∼( :O∼A • :A )    Valid
  *  1 ┌ asm: ∼O∼( :O∼A • :A )
  *  2 │ ∴ R( :O∼A • :A )  {from 1}
  *  3 │ D ∴ ( :O∼A • :A )  {from 2}
      4 │ D ∴  :O∼A   {from 3}
      5 │ D ∴  :A   {from 3}
      6 │ Du ∴ O∼A   {from 4}
      7 │ Du ∴ A   {from 5}
      8 └ Du ∴ ∼A   {from 6}
      9 ∴ O∼( :O∼A • :A )  {from 1; 7 contradicts 8}  
      N percent of A's are B's.
      X is an A.
      This is all we know about the matter.  
  ∴ It's N percent probable that X is a B.
      N percent of examined A's are B's.
      A large and varied group of A's has been examined.  
  ∴ Probably roughly N percent of all A's are B's.
      Most of the things true of X are also true of Y.  
      X is A.
      This is all we know about the matter.
  ∴ Probably Y is A.
      A occured more than once.
      B is the only additional factor that
         occured if and only if A occured.
  ∴ Probably B caused A, or A caused B.  
      A occurred in some case.
      B didn't occur in the same case.  
  ∴ A doesn't necessarily cause B.
      A occurred in the first case but not the second.
      The cases are otherwise identical, except that B also  
         occurred in the first case but not in the second.
  ∴ Probably B is (or is part of) the cause of A, or A is (or  
         is part of) the cause of B.
      A changes in a certain way if and only if B  
         also changes in a certain way.
  ∴ Probably B caused A, or A caused B, or
         some C caused both A and B.
         
    *    
         
  *       
         
       *  
 *       
    *    
       * 
 *     * 
         
 *     * 
 *     * 
    *    
 *     * 
 *     * 
 *     * 
 *     * 
  *       
         
       *  
 *     * 
         
 *     * 
 *       
    *    
       * 
 
   lower   
 
 
   2 3 4 5 6 7 8 9 10 J Q K A   
 
 
   higher   
 
 3       
 ♥      

       ♥ 
      3 
 4       
 ♥    ♥ 
         
 ♥    ♥ 
       4 
 5       
 ♥    ♥ 

 ♥    ♥ 
       5 
 6        
♥  ♥
♥♥
♥  ♥
       6 
 7       
♥ ♥
♥♥♥
♥ ♥
      7 
 8        
♥  ♥
♥♥♥♥
♥  ♥
       8 
 9       
♥♥♥
♥♥♥
♥♥♥
      9 
 10      
♥♥♥♥
♥  ♥
♥♥♥♥
     10 
 J        
 ♥      
         
       ♥ 
       J 
 Q        
 ♥      
         
       ♥ 
       Q 
 K        
 ♥      
         
       ♥ 
       K 
 A        
 

 
       A 
 3       
 ♦      

       ♦ 
      3 
 4       
 ♦    ♦ 
         
 ♦    ♦ 
       4 
 5       
 ♦    ♦ 

 ♦    ♦ 
       5 
 6        
♦  ♦
♦♦
♦  ♦
       6 
 7       
♦ ♦
♦♦♦
♦ ♦
      7 
 8        
♦  ♦
♦♦♦♦
♦  ♦
       8 
 9       
♦♦♦
♦♦♦
♦♦♦
      9 
 10      
♦♦♦♦
♦  ♦
♦♦♦♦
     10 
 J        
 ♦      
         
       ♦ 
       J 
 Q        
 ♦      
         
       ♦ 
       Q 
 K        
 ♦      
         
       ♦ 
       K 
 A        
 

 
       A 
 3       
 ♣      

       ♣ 
      3 
 4       
 ♣    ♣ 
         
 ♣    ♣ 
       4 
 5       
 ♣    ♣ 

 ♣    ♣ 
       5 
 6        
♣  ♣
♣♣
♣  ♣
       6 
 7       
♣ ♣
♣♣♣
♣ ♣
      7 
 8        
♣  ♣
♣♣♣♣
♣  ♣
       8 
 9       
♣♣♣
♣♣♣
♣♣♣
      9 
 10      
♣♣♣♣
♣  ♣
♣♣♣♣
     10 
 J        
 ♣      
         
       ♣ 
       J 
 Q        
 ♣      
         
       ♣ 
       Q 
 K        
 ♣      
         
       ♣ 
       K 
 A        
 

 
       A 
 3       
 ♠      

       ♠ 
      3 
 4       
 ♠    ♠ 
         
 ♠    ♠ 
       4 
 5       
 ♠    ♠ 

 ♠    ♠ 
       5 
 6        
♠  ♠
♠♠
♠  ♠
       6 
 7       
♠ ♠
♠♠♠
♠ ♠
      7 
 8        
♠  ♠
♠♠♠♠
♠  ♠
       8 
 9       
♠♠♠
♠♠♠
♠♠♠
      9 
 10      
♠♠♠♠
♠  ♠
♠♠♠♠
     10 
 J        
 ♠      
         
       ♠ 
       J 
 Q        
 ♠      
         
       ♠ 
       Q 
 K        
 ♠      
         
       ♠ 
       K 
 A        
 

 
       A 
hearts  ♥
   diamonds  ♦   
clubs  ♣
spades  ♠
   2 3 4 5 6 7 8 9 10 J Q K A   
   2 3 4 5 6 7 8 9 10 J Q K A   
   2 3 4 5 6 7 8 9 10 J Q K A   
   2 3 4 5 6 7 8 9 10 J Q K A   
 4       
 ♥    ♥ 
         
 ♥    ♥ 
       4 

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Philosophy

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Try to solve other problems with the mechanisms you’ve learned. Remember that practice makes perfect. It’s an old saying, but it works every time, so it’s better to spend half an hour honing your new skill than to forget it almost instantly.

Of course, these methods don’t work for every single individual. If you have other approaches that suit you more than they do anyone else, just follow them and trust yourself.

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Bill Belichick used a different kind of X’s and O’s during his first encounter with now-girlfriend Jordon Hudson three years ago.

The ex-Patriots coach, 72, whose relationship with the 24-year-old Hudson was first revealed last week , autographed a textbook belonging to the former cheerleader during a February 2021 flight from the Boston area to Florida, according to images published Tuesday by TMZ .

“Jordon, Thanks for giving me a course on logic! Safe travels!” the message inside “Deductive Logic” read above Belichick’s signature.

Belichick appeared to include the numbers of which Super Bowls the Patriots won under his reign: 36, 38, 39, 49, 51, and 53 with the word “championships” to conclude it.

Jordon Hudson is dating former Patriots coach Bill Belichick.

TMZ previously reported how Belichick and Hudson forged a connection over a philosophy project she had been working on.

Their friendship turned romantic following Belichick’s breakup with longtime girlfriend Linda Holliday.

Page Six exclusively reported in September 2023 that Belichick and Holliday, a TV presenter, had split after more than a decade together.

Bill Belichick apparently met Jordon Hudson on a flight in 2021.

The exes were said to have “been involved in a drawn-out, back-and-forth breakup for nearly a year.”

Months after Belichick’s split with Holliday was reported, the six-time Super Bowl-winning coach of the Patriots parted ways with the team after 24 seasons.

He returned to Gillette Stadium last week to support his former quarterback, Tom Brady, who was inducted into the Patriots Hall of Fame and had his No. 12 jersey retired.

Bill Belichick attends Tom Brady's Patriots Hall of Fame ceremony in June 2024.

Hudson was also in attendance and posted a photo on her Instagram Story from the festivities.

Belichick won six championships alongside Brady in New England before the future Hall-of-Fame quarterback took his talents to Tampa Bay after the 2019 season.

Brady won his seventh Super Bowl as a member of the Buccaneers in 2021.

Bill Belichick signed Jordon Hudson's book.

The quarterback, who retired in 2023 after 23 seasons, praised Belichick in his Patriots Hall of Fame speech over their storied legacy.

“To coach Belichick, thank you for your tireless commitment to develop and push me to be my very best,”  Brady said , according to  Pro Football Talk . “It wasn’t me, it wasn’t you, it was us.”

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Jordon Hudson is dating former Patriots coach Bill Belichick.

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    Bill Belichick used a different kind of X's and O's during his first encounter with now-girlfriend Jordon Hudson three years ago. The ex-Patriots coach, 72, whose relationship with the 24-year ...