This lesson introduces logic as a language and as a way to represent statements. Another lesson builds on this one, exploring arguments, truth/validity, deduction/induction & logical fallacies.

1. Introduction, Constants, Variables

Intro to these lessons

We use the word logic and terms derived from that same Greek word λόγος in a variety of ways. You've probably said that someone was "being illogical" because they were too emotional or didn't agree with you. You might have thought of logic as a way of proving things or reasoning better. I plan to dig a little deeper and talk about logic as one basic way to approach our use of language. For starters, that means I'd like to distinguish between logic as a way of talking about thinking, including arguments, fallacies & truth, and logic as a way of talking about language and what goes into using language, into formulating thoughts.

I don't mean to propose this division as a hard and exceptionless rule, but however you plan to use logic, as long as you're communicating with other humans, your basic starting point will be logic as a way to analyze language. Since the systems we use to communicate analysis are themselves languages, it follows that we consider logic as a language. This short series will introduce logic as a language. More specifically, I will present logic as a formal language, not because it's more polite, but because it focuses on form instead of meaning.

Before getting to the basics, consider what I will and will not cover in this series. You will get to:

  • consider logic as a language,
  • see how to translate into that language, and
  • view symbolic logic as a backbone model for the fuller statements made in everyday language.

On the other hand, you will not:

  • spend time analyzing good or bad arguments (syllogisms, informal and formal fallacies),
  • discuss inductive vs. deductive reasoning,
  • consider admittedly important questions about logic, truth and reality, or
  • apply the language of logic to any particular field.

If those are the topics you're after, check out the logic of arguments. That course looks into premises & conclusions, form & content and truth & validity.

Constants & Variables

I threw the terms "logic" and "language" at you. I know those are abstract terms, but think about learning a new language, perhaps one spoken in some foreign country. If you reflect for a moment on the components of that language, you'll have to take on new sounds (phonology), new parts of words (morphology), the specific words used in that language (lexis), the meaning of those words (semantics), and rules for constructing well-formed phrases & sentences (syntax).

One way to simplify all this is by trying to find some core concept. We might imagine that speakers use language to convey things to listeners. For example, when I tell you about a 'house', you might picture an image of a house. So the word 'house' acts as a symbol for that image.

Here we took "words" as our core concept and said that words are symbols. It looks like words aren't just symbols, they're fixed symbols with specific definitions. When you parsed the word 'house', you related the word to a single concept. The word 'house', like other words, acts like a constant symbol or a constant. One key feature of these constants like house, dog and blue is that they have semantic value - they are meaningful.

These meaning-filled constants can relate to one another in a variety of ways. Two broad and basic relationships are intension and extension. Intension refers to what a word connotes - other words commonly associated with that word or other words that are properties of that word. The image of a red house connotes 'house', 'red' and so on. Extension is what a word denotes - its definition or examples. For example, 'house' extends to any image of a house.

Of course, we don't only speak in single words. We also make statements, like 'the house is red' or 'fish can swim'. Those statements are also meaningful. So we could consider those as symbols, too. So far it follows that those statements work like constants. Now we're taking on so many symbols - all kinds of words, all kinds of sentences. It would be hard to cOUnt all of them, let alone analyze them.

What if we could strip language down to the symbols? That's the aim of symbolic logic. We want a set of symbols that helps us evaluate structure without getting tethered in meaning and use. We may still want constants, but we'll definitely need symbols that aren't so fixed. Since their meanings wouldn't be fixed, these symbols would vary. So they're called variables. To make it easy on ourselves, we'll use very simple symbols like x for variables. We can use these variables for words - x can mean 'house'. However, the variable is not limited to that value. It might mean 'dog'. It might also represent an entire statement like 'the house is red', or 'I'm speaking English'.

Variable symbols will help us strip down our language and take a good look at the structure. For variables, I can use symbols like x, y and z. For statements with some truth value, it's common to use p and q. The variable p might be filled by 'fish can swim' or 'Gillsworth is a fish'. These are all conventions.

In concrete cases, you will want to use constants. It makes sense to use a constnat symbol that helps you keep track of your logic. An example is e when you're exclusively reading e as 'the English language'. In this case, you're using that symbol e as an individual constant, a name with a specific interpretation.

That's your background for the upcoming lessons and videos. In the next lesson, you'll get to see the key role subjects and predicates play in logic. I'll also use variables and constants to take a look at the logic behind some normal language sentences using formal logic.

2. Subjects & Variables

Review

The first lesson in this series presented logic as a language. Specifically, logic is a formal language that can help us think about human language and individual languages. Now that we've seen why it's useful to have a logic that's a formal system with constant symbols and variable symbols, I want to explore the parts of statements and the constants and variables used to represent those parts of statements in logic.

Basics of Subjects & Predicates

We'll start by dividing all statements we can make into subjects and predicates. So far, I've presented words and statements as symbols. We then used even simpler alphabetic symbols to stand for words and statements, like x and p. When our symbols name something or focus on a topic, they are subjects. All of the individual constants we've used so far, like h for 'house' and e for the English language, can be used as subjects.

Subjects are part of a larger frame. That frame includes the predicate, which relates information about the subject. If you went through my intro to the verb and its arguments, particularly the first two videos on arguments and valency, you got a linguistic perspective on subjects and predicates. (You saw that verbs take arguments, and the verb and its arguments form the core of a sentence frame.) But you don't need to be a linguist to get a feel for the difference between subjects and predicates. Nor do you need to take it all the way back to Aristotle's heavy discussions on the topic. Just consider the two terms in relation. The rough idea is that a subject is a term that a predicate says something about, and a predicate is a term that says something about a subject.

If a predicate says something about a subject, it makes sense that a predicate would not include that subject. Some examples of predicates are '...is red', '...is a language' and '...sees'. Let's simplify these predicates by using constant symbols. Unlike variables - x, y, z - and individual constants like "h" and "e", the symbols for predicates are capital letters. Using capital letters for predicate constants is a convention in logic. So the predicates end up looking like this:

predicate symbol predicate translation
L is a language
R is red
S sees

Our first two predicates say something about a single subject. For example, we could say that 'English is a language'. We could write 'English is a language' in logic like this: Le. That third predicate takes multiple arguments. We could say 'John sees Mary' or 'Jill sees the movie', and we'll write the sentence in symbolic logic like this: Sjm. Notice that the predicate comes first: L 'is-language' + e 'English'; S 'sees' + j 'John' + m 'Mary'. The predicate comes before its subject whether it's the predicate of a single individual (Le) or the predicate of multiple individuals (Sjm).

Notice that we used constants for our predicates, which holds true for all predicates in basic logic. Let's scrap our individual constants (j, m and e) for a moment and focus on using variables with our predicates. Instead of speaking specifically about English as a language, we might want to find out whether or not something is a language. In this case, we'll use x to stand for that 'something': Lx 'x is-language'.

I mentioned it in the last lesson, but it's worth repeating for clarity's sake - that variable x can be filled by many values, but it doesn't stand for any specific value. A fancy way of saying this is "a variable ranges over the domain of discourse", where the domain of discourse (universe of discourse) is just the set of everything that the variable can represent.

When it comes to L 'is a language', you can think of the combination of predicate and variable Lx as '__blank__ is a language'. And Rx as '___blank____ is running'. And what about Sxy, where S is 'sees'? You guessed it: '___blank___ sees ___blank____'. More precisely, Sxy is '___blank___ sees ____some other blank___'. If you have strong analytic skills, you can always think in terms of the variables instead: 'x is a language', 'x is running' and 'x sees y'.

With variables in mind, let's look at the valency of our predicates. This refers to the number of terms that a predicate says something about. The predicates L 'is a language' and R 'is running' both say something about one term, which we can symbolize with the variable x, so they have a valence of 1. The predicate S 'sees', on the other hand, leaves two slots open for us, one for x and another for y. It has a valence of 2. But let's not stop there! The deceptively short predicate 'gave' has a place for a giver x, a thing given y and a "givee" or recipient z. The result is Gxyz 'x gives y to z', and its valence is 3!

If you're familiar with mathematics or computer science, you'll see similarities between the logical subject and predicate and the concepts of function and argument, or subroutines and parameters. The specialized applications of those terms have been refined from and contributed to the development of logic.

Let's back up for a moment and reconsider something I mentioned in the first section. Whole logical statements may be represented just by the symbols p and q. This gives you options - you won't always need to drill down to the level of individual predicates. The big picture may even be a better option if you're comparing or evaluating statements as a whole. The statement p might be 'English is a language' or 'somebody saw something'.

Now that you have a bit of a handle on subjects and predicates in logic, here are some ordinary language statements to translate into logic. I have a simple hint as you use your logic here - don't overthink!

  1. Socrates is a gadfly.
  2. George went on vacation.
  3. The fuzzy dog took the bone from my hand.
  4. Well, somebody said something to someone!
  5. Those women are such good people.

So, you've seen the symbols that stand for words and statements, you understand constants and variables, and now you know something about subjects and predicates. But this is all amateur stuff. It's time move on and look at quantifiers and bound variables, which will really tighten up your logical statements.

3. Quantifiers & Bound Variables

Review

In the last two sections you learned to think about the structure of language using logic. I introduced symbols for constants and variables. I also broke statements into subjects and predicates. Now we turn to look at some logical symbols that will give our sentences a bit more structure.

Universal Quantifier 'All'

We use quantity to refer to the number of instances of a thing ("how many"). Ordinary language is fairly general in its treatment of quantitiy - 'house' (for 1 house) versus 'houses' (for 2 or more houses). Logic also treats quantity generally. We'll look at three basic quantities: none, some (1 or more) and all. Each of these quantities is a symbol, so we will have three symbols called quantifiers.

First, there's the quantifier 'all'. When we used predicates with variables, we ended up with sentences like Px 'x is a person'. This statement tells us very little about x. Let's use x as a quantifier - (x)Px 'for any x, x is a person'. Let's try the same thing with the sentence Sxy 'x saw y' - (x)Sxy 'for any x, x saw y'. It's one step further to (x)(y)Sxy 'for any x, for any y, x saw y'. When I use 'any' with these variables x and y, I'm really referring to all the things x and y can represent.

Let's throw the universal quantifier ∀ 'all' into the mix. We get the sentence (∀x)Px 'for all x, x is a person'. In ordinary language, you can read that as 'all of them are people' or just 'all people'. Let's look at another one: (∀x)Sjx 'for all x, Jill saw x'. 'Jill saw everything' serves as an adequate translation. If we abandoned these quantifiers and just left the predicate, we end up with '__blank__ is a person' (Px) and 'Jill saw __blank___' (Sjx).

Existential Quantifier 'Some'

Now we move on to the existential quantifier. It acts like the universal quantifier 'all' you just learned about, but it means 'some', or more specifically 'there exists at least one'. Return to our variable x in the phrase Px 'x is a person'. Let's say we want to talk about 'some person', in other words 'for some x, where x is a person'. We need to use an existential quantifier: (∃x)Px. We'll also need to use it for the sentence (∃x)Sjx 'Jill saw something'. Remember that (∀x)Sjx 'Jill saw everything' used the universal quantifier, while 'Jill saw something' requires the quantifier ∃ on x.

Negative Quantifier 'No'

Our final quantifier is the symbol 'no'. Take a statement p. This p might represent 'John saw the movie', 'my house is red' or any other statement. Te negation of this statement p is 'not p', which has the same form as 'John did not see the movie', 'my house is not red', or any negative statement. The negative statement 'not-p' is written ~p in logic.

Take another statement, this time with an individual constant and a predicate. As you saw in the last lesson, you can write 'John saw the movie' this way: Sjm. Now turn this same statement into a negative: ~Sjm 'John didn't see the movie'. The same goes for 'my house isn't red': ~Rh. This is how to negate a predicate, even a predicate with variables: ~Px 'x isn't a person' or ~Sxy 'x didn't see y'.

Don't forget that you learned to quantify your variables with ∀ 'all' and ∃ 'some'. Examples include 'some person', which I wrote as (∃x)Px 'there exists at least one x, where x is a person' and 'all people', which I wrote as (∀x)Px 'for all x, x is a person".

We can actually negate both of these quantifiers. If we negate ∃x as ~∃x, what do you think it means? 'Not for some x'. So (~∃x)Px symbolizes 'not for some x, x is a person', which makes explicit what we mean when we say 'no people' or 'no person'. If we negate the universal quantifier instead, what does ~∀x mean? 'Not all x'. So (~∀x)Px means 'not all x, where x is a person'.

What is the difference between (~∀x)Px and (~∃x)Px? If there is not some x, then there is no x whatsoever ('no people'). If there is not all x, we are still left with some x ('not all people'). Take another example for clarity: (∃x)Tx 'something is a turtle' versus (∀x)Tx 'everything is a turtle'. If I negate the existential quantifier, I get 'nothing is a turtle'. If I negate the universal one, I get 'not everything is a turtle'. If I negate either predicate, the result is simply 'not a turtle': (∃x)~Tx 'some x, x is not a turtle' and (∀x)~Tx 'all x, x is not a turtle'.

Before you take quantifiers and run with them, consider how we have used variables with quantifiers. Instead of translating 'all languages' as "∀l" (with an individual constant l for 'language'), we say (∀x)Lx 'for all x, x is a language'. The same applies to existential quantifiers. The logic for 'some fish' is (∃x)Fx 'given some x, x is a fish' instead of just "∃f". This is how to use predicates and variables to lay out logical sentences.

Logical statement Translation
Fx ___ is a fish.
(∃x)Fx Something is a fish.
(~∃x)Fx Nothing is a fish.
(∀x)Fx Everything is a fish.
(~∀x)Fx Not everything is a fish.
(∃x)~Fx Something is not a fish.
(∀x)~Fx Everything is not a fish.

The Scope of a Quantifier

Let's render more general terms in logic. When you come across words like 'someone', 'somewhere' or 'something', the existential quantifier comes into play: 'someone' is (∃x)Hx 'for some x, x is human'; 'somewhere' is (∃x)Px 'for some x, x is a place'; 'something' is (∃x) 'for some x'. With that in mind, what are 'everybody' and 'everything' in logic? Take a moment to think. If your answers use the universal quantifier ((∀x)Hx and so on), congratulations! If not, don't despair.

You are ready to gain some perspective on the scope of a quantifier. Variables of a predicate are bound by a quantifier to their left (the nearest applicable quantifier, to be precise). x is bound by a quantifier on x, y by a quantifier on y, z by a quantifier on z. In the proposition (∀x)(Sxy) 'for all x, x sees y', x is bound by the universal quantifier but y is free. On the other hand, both x and y are bound in (∀x)(∃y)Sxy 'for all x, there exists some y, where x sees y'. Notice that the predicate "x sees y" falls within the scope of these quantifiers. The variables x and y are also bound in (x)Rx 'for x, x is red' and (y)Sxy 'for y, x sees y'. In that second example, y is bound but x is free.

Why would you want to bind variables? Picture it this way: the variable is like a blank space. Sjx has a free x, so it translates to 'John saw __blank___'. If you bind the variable, you can form a sentence in a language: (∃x)Sjx 'John saw something'. If you fail to bind the variable, you leave the blank space, and fail to represent a real sentence in logic. Binding requires more work, but is crucial to logical thinking and modeling ordinary language with logic.

Let's get to some translations before we bring this lesson to a close. Do your best to translate the following statements into logic.

  1. All those things fell down.
  2. Jim said nothing.
  3. All of them are human.
  4. Jim didn't watch TV.
  5. Something moved something else.

And there you have it. Next time we'll take a look at logical operators to add some flesh to the bones of your logic skills.

4. Logical Connectives

Review

By now, you have a feel for variables & constants, predicates & subjects, the quantifiers 'no', 'some' & 'all', and taking advantage bound variables and scope to translate everyday phrases into logic. It's time to move on to logical operators or connectives.

Connectives AND, OR & XOR

We can build our sentences out a bit more. To do that, let's start with the logical connectives 'and' and 'or'.

We can join two symbols with 'and', called the conjunction. The conjunction can come between two statements, like 'She saw Bob' (p) & 'the house is red' (q) or 'this is a fish' (p) & 'fish can swim' (q). These statements are joined with the conjunction - p · q. The conjunction may also fall between two variables, like x · y.

The conjunction has less obvious applications. If we want to say that 'Bob is a good person', we can break that sentence into Hb · Gb 'Bob is human AND Bob is good'. Using the same reasoning, we can render 'some good person' as (∃x)Hx·Gx 'for some x, x is human AND x is good'. As with other operators, different texts use different symbols for 'and': & · ^. Stick with one and learn others as needed.

Also, let me mention that the meaning of this logical 'and' is very strong. It asserts both things being joined, so 'it's vanilla and chocolate' means that it is BOTH vanilla AND chocolate. The conjunction is not the weak 'and' of 'you can choose between vanilla and chocolate', where you actually mean 'or'.

We can also join sentences with the disjunction 'or'. The disjunction might come between two statements, p ∨ q 'p or q', like this is vanilla OR this is chocolate'. It also applies to variables, like x ∨ y 'x or y'. This 'or' is called inclusive OR, which means presenting the first thing (x), the second thing (y) or both of them (x · y). For example, the statement (∃x) Vx ∨ Cx 'for some x, x is vanilla or x is chocolate' means that something is vanilla, chocolate, or both vanilla & chocolate. If you want to rule out the option 'both of them', you have to use an exclusive XOR - (∃x) Vx ⊕ Cx 'for some x, x is vanilla XOR x is chocolate' for either vanilla or chocolate, but NOT chocolate+vanilla.

These ANDs and ORs bring up an earlier topic, namely the scope of the quantifier. When two predicates fall in a quantfier's scope, the quantifier applies to both of them - (∀x) Hx · Sx 'for all x, x is human and x can swim' just means that everybody can swim! In the same way, (∃x) Dx ∨ Cx 'for some x, x is dirty or x is cluttered' translates 'something is dirty or cluttered', where you mean that this thing may be dirty, cluttered, or both dirty and cluttered. Last, (∃x) Dx ⊕ Cx 'for some x, x is dirty or is cluttered' with an exclusive OR, which means 'something is either dirty or cluttered'.

What happens when you negate two symbols joined by 'and' or 'or', like 'not chocolate or vanilla'? A set of rules known as De Morgan's laws will help you deal with these cases. If you want to say ~Cx · ~Vx 'NOT chocolate and vanilla', you can represent the phrase as ~(Cx ∨ Vx) 'not (chocolate or vanilla)'. On the flipside, when you want to say ~Cx ∨ ~Vx 'NOT chocolate or vanilla', this comes out as ~(Cx · Vx) 'not (vanilla and chocolate)'. In logic, NOT (chocolate OR vanilla) is treated the same as NOT chocolate AND NOT vanilla, while NOT (chocolate AND vanilla) is treated like NOT chocolate OR NOT vanilla.

Before we move on to the rest of the connectives, here are a few phrases to translate. Use 'and' or 'or' - that's EITHER 'and' OR 'or' - for each statement.

  1. every good dog
  2. Bill saw the movie or he went running
  3. not friends and enemies
  4. either something ugly or something beautiful

Connectives IF...THEN, ONLY IF

On to some new operators. You're probably already familiar with the equals symbol (=). As you expect, this indicates a logical equality, like (∃x)(∃y)(x = y) 'some x is equal to some y'. Nothing new here, so let's move on.

Consider sentences in which you say that IF one thing is true, THEN something else follows. This 'if...then' conditional operator is very important in logic. We use it to structure even very basic statements where it's hidden in ordinary language. Take the statement 'English is a language' - the logic for this simple sentence is actually 'for all x, if X is English then x is a language'. We can translate it as (∀x) Ex ⊃ Lx, where the horseshoe symbol ⊃ is the 'if...then' conditional. Take another sentence, this time 'everybody swims', which is (∀x) Hx ⊃ Sx 'for all x, if x is human then x swims' in logic.

There are other popular symbols for this conditional, like the left-to-right arrow ->. Like the versions of 'and' and 'not', they're different ways of writing the same symbol, so their logical value is the same - (∃x) Dx -> Lx and (∃x) Dx ⊃ Lx both translate 'some dogs are loyal'; (∀x)(Hx -> Mx) and (∀x)(Hx ⊃ Mx) both mean 'humans are mortal'.

There's another type of conditional called a biconditional. The meaning here is 'if and only if', which is quite different from the plain conditional. Let's take the sentence 'only water is refreshing', meaning 'for all x, x is refreshing if and only if x is water' - (∀x)(Rx ≡ Wx). Ratchet this up one level to 'it's only edible if it's good food', where we mean (∀x)(Ex ≡ (Gx · Fx)) 'for all x, x is edible if and only if x is good and x is food'.

Notice that this "if any only if" is really strong. What's refreshing has to be water, and what's water has to be refreshing. Likewise, what's good food has to be edible, and what's edible has to be good food.

Next, when you're drawing a conclusion in language, you often use the word 'therefore'. This 'therefore' indicates that whatever follows is implied by whatever came before. You can show this in logic with a triple dot before your conclusion. Say you propose a simple line of reasoning like (1) 'Everybody knows something', (2) 'John's somebody', (3) 'So he knows something'. You're probably getting used to the translations into logic by now, so you can breeze over the logic of these three statements - (1) 'for all x, for some y, if x is human then x knows y', (2) 'John is human', (3) ∴ (THEREFORE) 'for some y, John knows y'.

That's just about all the basic stuff you need to begin uncovering the logic behind everyday language. Of course, there are other logics that the expand the use of formal logic and give it a variety of applications. To offer but one example, modal logic allows you to incorporate possibility and necessity into the formal structure. Examples of modal verbs in English are 'can', 'may' and 'must'. Modal logic adds symbols for 'it is necessary' and 'it is possible' to our logic toolbox, which allows us to express modals. Notice that when we say something like 'can' or 'it is possible', that leaves a lot of room to discuss the probability of that possibility, which is one of the topics I'll tackle in the next video.

We've come a long way in these lessons, but I'm fairly sure you're already getting a feel for how this kind of logic gets down to the structure of thoughts expressed in language. Let's do a few translations.

  1. Everyone goes somewhere on vacation.
  2. Nobody goes everywhere on vacation.
  3. It's only exciting if it's dangerous!
  4. Jill ate everything that's not chocolate and not vanilla.
  5. Steve taught everyone, but not everyone listened to him.

We'll wrap up these lessons next time with a simple discussion of how to represent members of a set and to express the language of probability in logic. Until then, keep practicing!

5. Language of Sets & Probability

Welcome to more logic! I plan to make this the last installment in my brief introduction to logic as a way of looking at language. If you haven't gone through the previous videos one through four, please take the time to do so. Go ahead, I'll wait for you.

Before getting in to the new topics, I want to offer a brief note on translating into logic. I realize that I did not take the time to break translations into a step-by-step process. Let's slow the pace a bit and work on parsing ordinary sentences in logical language, then translating that logical language into symbolic logic.

First, make sure you understand the logic of the sentence you're analyzing. If your sentence is 'English is a tough language', break it down. You're actually saying that 'English is tough' and that 'English is a language'. Similarly, 'a good person' is someone who is both good and a person.

Then, break the translation into stages. Reword the sentence in logical language. 'English is a tough language' becomes 'English is tough AND English is a language'. Once you've structured it clearly, start putting the sentence into logic. Figure out how many variables you have and where they fall. 'For all x, if x is English then x is tough and x is a language' is a good example: (∀x)(Ex ⊃ (Tx · Lx)).

As you go through this process, look for buzzwords. You've learned logical operators for 'if...then', 'if and only if', 'and', 'or' and so on. Figure out where to place those operators. Also, pick out universal and existential terms - 'always' and 'anytime' mean 'for all times', 'somewhere' is 'for some place', and the like. So the logic of 'everybody's talking to everybody else' is more transparent once it's reworded, 'all persons are talking to all (other) persons'. It's just a step from there to 'for all x, for all y, if x is a person AND y is a person, then x is talking to y'. The only thing left for you to do at that point is to map the symbolic logic onto the structure: (∀x)(∀y)((Px · Py) ⊃ Txy)

Sets, Members & Subsets

On to the new stuff. I mentioned the domain of discourse when introducing variables. It is the set of all things a variable could represent. That brings up two concepts not covered yet - a set and members of a set.

In logic, a set is just a grouping of things that belong together. There is a set of colors of paint, a set of ingredients in ice cream, a set of people whose company you enjoy. The things inside each set are its members. Members of the set of ingredients in ice cream include eggs, cream and sugar. If something a is an element of a set A, it's represented this way: a ∈ A. For example, eggs ∈ Ice Cream Ingredients. We can also say that some b is NOT an element of set A: b ∉ A. For example, mutton ∉ Ice Cream Ingredients.

Consider that some members of Ice Cream Ingredients are also members of the set of Dairy Products. It follows that the two sets intersect each other - Ice Cream Ingredients ∩ Dairy Products. That's the intersection of the two sets, which includes cream (but not eggs and not sugar).

You can also speak of the union of the two sets. This is the same concept underpinning logical OR. The union of Ice Cream Ingredients and Dairy Products includes every member of both sets: Ice Cream Ingredients ∪ Dairy Products.

Let's get a little more detailed. There aren't simply sets, because sets may have subsets. For instance, Ice Cream Ingredients are a subset of Foods: I ⊆ F. If we flip the statement around, we can call Foods the superset of Ice Cream Ingredients: F ⊇ I. Of course, what we really mean is that Ice Cream Ingredients and Foods have different members, so it's more precise to call Ice Cream Ingredients a proper subset of the set Foods: I ⊂ F. Foods are then a proper superset of Ice Cream Ingredients: F ⊃ I.

Of course, two sets can also coincide. The set of Ice Cream Ingredients may be exactly the same as the set of ingredients for something else, in which case every member of one would be a member of the other: I = A.

The Basic Language of Probability

Let's switch gears and focus on probability. I'm not going to look at solving problems in probability. I will simply introduce the logic of probability.

Keep the concepts you just learned for sets in mind here. The probability of a single event A is represented as probability of A, written P(A). A separate event B has the probability of B, or P(B). Now think back to sets. Taking both the probability of A OR probability of B (and that's a logical OR), we have the probability of the union of A and B, or P(A ∪ B). And you can bring back the concept of the intersection, too - here it's the probability of both A and B being true: P(A ∩ B).

If you think that the probability of event A depends on event B, you'll talk about the probability of A on B, written P(A|B). Likewise, if the probability of B depends on A, then that's the probability of B on A P(B|A).

This is all well and good for the language of probability, but if you want to calculate probabilities, you'll need more information than the symbols I've given you here.

Brief Conclusion & Farewell

Those are all the topics I have to cover in this quick run through the basics of logic. Of course, logic covers so many subjects and has so many uses that I haven't even touched on in these lessons. Logic has applications in many fields within mathematics, philosophy, computer science and more. But crucial to those studies is logical validity. Instead, this course has simply used logic to understand sentences in language. I hope this short introduction to formal logic will help you take a structured look at language, and provide you with a good jumping-off point for learning more.

Answers to the Exercises

Subjects & Predicates

Hints. If you're struggling, try these hints:

  1. socrates isGadfly
  2. george Vacationed
  3. dog Took bone, from hand
  4. x Said y, to z
  5. women areGood

Answers. Original phrase on the left, logical translation to the right of the arrow:

  1. Socrates is a gadfly. -> Gs
  2. George went on vacation. -> Vg
  3. The fuzzy dog took the bone from my hand. -> Tdbh
  4. Well, somebody said something to someone! -> Sxyz
  5. Those women are such good people. -> Gw

Quantifiers & Scope

Hints. If you're struggling, try these hints:

  1. for all x, x Fell down
  2. for no x, jim Saw x / for all x, jim did not See x
  3. for all x, x is Human
  4. jim did not Watch tv
  5. for some x, for some y, x Moved y

Answers. Original phrase on the left, logical translation to the right of the arrow:

  1. Everything fell down. -> (∀x)(Fx)
  2. Jim saw nothing. -> (~∃x)Sjx / (x∀)Sjx
  3. All of them are human. -> (x)Hx
  4. Jim didn't watch TV. -> ~Wjt
  5. Something moved something else. -> (∃x)(∃y)Mxy

Connectives/Operators

AND/OR/XOR hints. If you're struggling, try these hints:

  1. for all x, x is Good AND x is a Dog
  2. bill Saw the movie OR bill went Running
  3. NOT (friends OR enemies)
  4. for some x, x is Ugly XOR x is Beautiful

AND/OR/XOR answers. Original phrase on the left, logical translation to the right of the arrow:

  1. (∀x)(Gx·Dx)
  2. Sbm ∨ Rb
  3. ~(f ∨ e)
  4. (∃x)(Ux ⊕ Bx)

IF...THEN hints. If you're struggling, try these hints. They are a little less obvious than previous hints:

  1. if all x are human then x vacations (or: if all x is human and some y is a place then x vacations at y)
  2. for no x, for all y, if x is human and y is a place then x vacations at y (or: for all x, for all y, if x is human and y is a place then x does not vacation at y
  3. all x are exciting only-if they are dangerous
  4. for all x, if it's not the case that (x is chocolate or x is vanilla), then jill is eating x. Reread the bit on De Morgan's laws.
  5. if x is human then Steve taught x and x did not listen to Steve

IF...THEN answers. Original phrase on the left, logical translation to the right of the arrow:

  1. (∀x)(Hx⊃Vx) or (∀x)(∃y)((Hx·Py)⊃Vxy). Congratulations if you went for the second one!
  2. (~∃x)(∀y)((Hx·Py)⊃Vxy) or (∀x)(∀y)((Hx·Hy)⊃~Vxy). Both mean the same thing.
  3. (∀x)(Ex ≡ Dx)
  4. (∃x)(~(Cx v Vx) ⊃ Esx)
  5. (∀x)(Hx ⊃ (Tsx · ~Lxs)

More resources & about this author

This guide's author has written a number of introductory books on linguistics and language learning, including Native Grammar: How Languages Work and The IPA for Language Learning.