Logic and Computing

 

 

Tips on Learning

In her 2014 A Mind for Numbers, Barbara Oakley discusses the strategies that she and other accomplished instructors teach their students to prepare for STEM classes. These skills, however, could be applied to any challenging field. In fact, even chess players use these. If taken to heart, these techniques can help you succeed in acquiring new skills in challenging disciplines, including logic. As such, Oakley's work is invaluable for students who are starting their academic careers. Let's briefly review some of her recommendations.

In chapter 6, Oakley discusses the phenomenon of “choking”, the feeling of being overwhelmed during a test, making it so that none of the information that you need will come to mind. Many students report this happening to them when they have failed a test. Choking occurs when we have overloaded our working memory. To prevent this, we must take enough time to “chunk”, or integrate one or more concepts into a smoothly connected working thought pattern. In order to avoid this, you should give yourself several days to learn tough concepts and use retrieval practice during your study. That is to say, read slowly and, after each paragraph (or less), pause to say outloud what the main idea of what you just read is. You can also do this with homework problems. Don't just rush to finish them. Explain to yourself the reasoning behind your solution.

In chapter 11, Oakley discusses the role of working memory. To excel in your studies (and life), use memory tricks to become proficient at remembering important information. We’re good at remembering physical environments, like our homes. Whenever possible, “link” what you’re learning to physical locations and things. If you're learning physiology, perhaps the layers of the skin could be the layers of your house. If you're learning logic rules, you can make analogies to everyday household items, like we will later with the analogy of the lock and key. You can also carry pictures of flash cards of these connections with you when you workout, since working out regularly has been shown to improve memory. Take a glance before starting your training to prime your brain to think about those things in the background of your mind. It works!

Tips also found in chapter 11 include utilizing our predilection for music as an aid in our studies. Little songs that you make up might help you remember some important concepts or equations. The sillier the better. Oakley reminds us to not worry about being weird; some of the most brilliant thinkers were not very normal people. Also, sometimes a location can evoke a certain feeling. You can invoke certain memories by evoking this feeling. If you studied a lot in the library, visualize yourself in the library before starting a test.

Other tips:

• Make your own questions.
• Apply the concepts you’re learning to your life.
• Doodle visual metaphors for hard concepts.
• Review just before sleep and upon waking.

 

 

 

Level III: Sorites

For references, here are the two mains steps for working out sorites. First, make sure the argument is in standard form. This means that you must make sure that:

• All statements are standard-form categorical statements.
• Each term occurs twice.
• The predicate term of the conclusion appears in the first premise.
• Every statement up to the conclusion has a term in common with the statement immediately following.

Once this has been done, follow these steps:

1. Pair together two premises that have a term in common and derive an intermediate conclusion. This conclusion should have a term in common with one of the unused statements in the sorites.
2. Pair together these two statements and draw a conclusion from this second pair.
3. Repeat until all premises have been used.
4. Evaluate each individual syllogism.

Rule: If each individual syllogism is valid, the sorites is valid. If even one syllogism in the chain is invalid, the sorites is invalid.

For more information, check out the video below.

 

 

 

 

Storytime!

 

 

Computation today...

I'd like to add a caveat to this happy history of logic and computation. As if we need a reminder, the history of civilization isn't always in the "up" direction. Sometimes civilization takes a few steps back; sometimes it's several steps. Our experiments with computational devices are only just beginning, and there is no guarantee that there will be a happy ending. For example, in a recent study, researchers presented machine learning experts with 70 occupations and asked them about the likelihood that they will be automated. The researchers then used a probability model to project how susceptible to automation an additional 632 jobs were. The result: it is technically feasible to do about 47% of jobs with machines (Frey & Osborne 2013).

The jobs at risk of robotization are both low-skill and high-skill jobs. On the low-skill end, it is possible to fully automate most of the tasks performed by truckers, cashiers, and line cooks. It is conceivable that the automation of other low-skill jobs, like those of security guards and parcel delivery workers, could be accelerated due to the global COVID-19 pandemic.

High-skill workers won't be getting off the hook either, though. Many of the tasks performed by a paralegal could be automated. The same could be said for accountants.

Are these jobs doomed? Hold on a second... The researchers behind another study disputed Frey and Osborne’s methodology, arguing that it’s not the entire job that will be automated but separate tasks within the job role more generally. After reassessing, their result was that only about 38% could be done by machines...

It gets worse...¹

 

 

The Road Ahead

In the Storytime! above, we introduced some historical figures that will play a role in the development of logic that we will cover in this course, namely Alan Turing. Historically speaking, though, the most important group to discuss at this juncture is the Stoic school of Philosophy. It's true. In Aristotle's own time, there were criticisms to his approach to reasoning coming from a group of people that would sit at the stoa (steps) of the agora (marketplace). Stay tuned.

For now, we can admit that Aristotle's logic has some weaknesses. First off, once Boole introduced the concept of the hypothetical viewpoint, there emerges an unappealing relativity: some arguments are valid depending on whether or not you're taking the existential viewpoint or the hypothetical viewpoint. Not only is there something inelegant about this, it also makes it so that Aristotle's highly abstracted categorical statements (like "All S are P") are somehow "missing" information. How would we know if "All S are P" refers to actually existing things or not? There must be a way for expressing which viewpoint we are taking within our symbolized statements, but Aristotle's logic does not have a convention for this.

Second, Aristotle believed that all propositions could be forced into standard form for categorical statements. But as you might've noticed, this is really counterintuitive in some cases. For example, consider the sentence “All except truckers are happy with the new regulations”. To accurately convey the information in this expression, one must use two standard form categorical statements:

• All non-truckers are persons who are happy with the new regulations.
• No truckers are persons who are happy with the new regulations.

It goes without saying that this is extremely counterintuitive, inelegant, and cumbersome.

And so we take the next step in our introduction to the history of logic. We now take a look at the chief rival to Aristotle’s Categorical Logic: truth-functional logic.

 

 

FYI

Related Material—

• Video: TEDTalks, The wonderful and terrifying implications of computers that can learn | Jeremy Howard

• Reading: Nick Bostrom, SuperIntelligence (Answer to the 2009 EDGE QUESTION: “WHAT WILL CHANGE EVERYTHING?”)

 

Footnotes

1. The interested student can refer to literature on the potential of artificial intelligence becoming an existential risk to humans. The most famous example of this is the work of Nick Bostrom, particularly his Superintelligence: Paths, dangers, strategies.