(This is going to be a long one, but bear with me. Once we make it through this, I get to start talking about the cool stuff.)
If you’ve been doing the questions for Parts II and III you may have run into some interesting quirks with symbolizing one of the arguments from Part I. Specifically:
- P1) All men are mortal.
- P2) Socrates is a man.
- C) Socrates is mortal.
Now, with a little bit of reflection, it’s clear that this argument ought to be valid. However, if we try to symbolize it using what you’ve already learned, we end up with something like:
- P1) R
- P2) S
- C) M
Which, it’s pretty clear, is not going to be a valid argument. The reason it doesn’t end up working out is because the basic logical language we’ve learned (known as Sentential or Propositional Logic) has no way of formalizing the concept of “all”.
That’s why there’s a second logical language that we’ll be talking about called First-Order Logic (or Predicate Logic). It is a strict extension of Propositional Logic, which is to say that everything you’ve already learned will still apply. (Well… you won’t be doing truth tables any more, but you won’t really miss them, I promise). In our extension we introduce two new concepts to our language: “some” and “all”. Because these describe quantities, they are called “quantifiers”.
Before we learn about quantifiers, however, we need to discuss something called predicates. While they do have a fancy formal definition, I’d like to leave that for a future discussion and just cover them in an intuitive sense.
A predicate is like an adjective, in that it describes something about a subject. For example, in the sentence “a baseball is round”, the predicate “is round” describes a particular subject (in this case “a baseball”). Because we have separated the two things out, we can then go on to apply the predicate to other round things: “the Earth is round”, “circles are round” and “soccer balls are round” will, instead of each being their own sentence, be represented as a single predicate applied to three different round subjects.
Symbolically, we tend to denote predicates by uppercase letters, and subjects as lowercase letters. So in the examples above, we could let represent “is round” and then our four sentences become and (where represent “baseball”, “Earth”, “circles” and “soccer balls” respectively).
Typically however, we reserve letters from the beginning of the alphabet (,etc…) when we talk about particular subjects (called constants), and letters from the end of the alphabet (,etc…) when we talk about generic subjects (called variables).
4.1.2. Universal Quantification
So now that we have predicates, we can apply our first quantifier to it: the universal quantifier: . This symbol indicates that every subject has a particular property. In order to use it in a sentence, we must bind it to a variable such as . Once we’ve done this, we can create our sentence: or “everything is round”.
Now, this isn’t particularly helpful, since any useful predicate will be true of some things and not of others. That’s why we’re able to quantify over not just predicates, but other sentences. So, a more useful sentence might be which could be read as “anything which is a circle is also round”. In fact, we can even place a quantifier in front of a quantified sentence to end up with something like (which is a redundant sentence, but one that makes my point). Typically, we will abbreviate as .
In fact, this is the general form that most universal quantifications will take. For all if has some sort of property then has some other sort of property. To give another example, to say “all men are mortal” we would translate that as “all things that are men are mortal” or .
4.1.3. Existential Quantification
The other type of quantifier we can apply to a sentence is called an existential quantifier and is used to denote that some (or at least one) subject has a particular property. The symbol to denote this is and is used in a sentence exactly the same way as a universal quantifier: (read as “something is round”).
Again, this might not seem entirely useful, because it tells us nothing about what sorts of things the predicate applies to (in this case, what sorts of things are round?). But we can do another trick with them to narrow what we’re looking at by using conjunction. Take the predicate to mean “ is a plate$ then we have or “some plates are round”.
Relations are similar to predicates, except that they use more than one variable. For example, if you wish to say “James is the father of Harry”, we could take the predicate to describe a relationship between James () and Harry (). Symbolically we would write this as .
We aren’t just limited to two-place relations either. The predicate could represent the sentence “the priest married the husband and wife”. Examples with bigger and bigger relations do become more and more contrived, but technically speaking, there’s nothing to stop us.
We can also quantify over just part of a sentence. For example would say “someone is Harry’s father” and would say “James is the father of someone”.
Additionally, for two-place relations, they are sometimes represented with something called “infix notation”. Here, instead of saying we would write for whatever relation we’re dealing with. I prefer not to use this notation as it doesn’t work for relations with three or more subjects. With one notable exception:
We have the option of adding a special relation to our language, called identity (). It is a two-place relation written with infix notation that indicates that two things are the same. If we know that these two things are the same, then we can substitute one for the other in sentences. So the following sentence is valid (ie: always true) for any predicate :
Or, stated plainly, if two things are the same then any property that holds for one will also hold for the other.
I prefer to deal with first-order logic that makes use of identity, and so that is what I shall do. You may ask why we would choose not to do so, and the answer to that is basically just that it can make proving things about first-order logic a bit trickier. But that’s outside of the scope we’re dealing with here today.
Identity also has the following properties:
Reflexivity tells us that everything is equal to itself. Symmetry tells us that it doesn’t matter which way we order our terms. Transitivity tells us that if one thing is equal to another, and the second thing is equal to some third, then the first and third things are also equal. These are the three properties that give us something called an equivalence relation, but we’ll get to what that is when we talk about set theory.
With all of that under our belts, let’s look back at the original question. The argument:
- P1) All men are mortal.
- P2) Socrates is a man.
- C) Socrates is mortal.
Can now easily be symbolized as:
Easy enough, right? However this still doesn’t give us a way to tell whether the argument is valid. We can’t use a truth table, because there’s simply no way to express quantification in a truth table. (Well technically we could, but only if we had a finite domain and even then it’s a pain in the ass…)
So we are still in need of a way to tell whether arguments in first-order logic are valid. Which brings us to…
4.2. Fitch-Style Calculus
There are a number of different ways to determine validity in first-order logic. I’m going to be talking about one called Fitch-Style Calculus, but there are many more including Natural Deduction (which I won’t be talking about) and Sequent Calculus and Semantic Tableaux (which I probably will).
I personally like Fitch because I find it very intuitive. A basic Fitch proof involves taking your premises and deriving a conclusion. We no longer deal with truth values, but instead we consider all of our assumptions to be true, but in a particular context. Just as an argument has two parts: premises and conclusions; so do Fitch proofs. These two parts are delimited by a giant version of the symbol. Premises (or assumptions) go on top of the symbol and conclusions go underneath. Additionally, each new line in your derivation must also be justified by a rule, which we’ll get to.
The most important concept in such derivations is that of the sub-proof. Basically a sub-proof is a brief argument-within-an-argument with its own assumptions which can be used within the sub-proof, but not in the outside proof as a whole. We will also make use of the term “scope” which basically refers to how many sub-proofs deep we are in a current argument. The rules (mentioned above) include ways to move sentences between different scopes.
I know this all sounds confusing, but we’re going to look at these rules and see whether they make sense. Before we do, I want to show you an example of a (admittedly complicated) proof, so that you can see what it looks like. You can check it out here. Take particular notice of the sub-proofs (and the sub-proofs within sub-proofs) to get an idea of how this sort of thing gets laid out. Also notice that each line is justified either as an assumption (appearing above the horizontal line of the symbol and abbreviated “Ass.” (try not to laugh)) or justified by a rule and the lines which the rule is making use of. Finally, notice that the last line is outside of any scope lines. We’ll see what this means later on.
But onto the rules! Once again, there are two different kinds: rules of inference and rewriting rules.
4.2.1. Fitch-Style Rules of Inference
Rules of inference are used to either introduce or eliminate a connective or quantifier from a sentence, or move sentences between scopes. They may require one or several lines earlier in the proof in order to make use of them. In the examples below, the line with the rule’s name next to it represents an application of that rule. Additionally, for these rules we will be using Greek letters to represent sentences which may not necessarily be atomic. This is because these rules are general forms which can be applied to very complex sentences. Thus, the symbol may stand for , , or even something as complicated as .
Each logical symbol in our language will have both an introduction rule (how to get that symbol to appear in a sentence) and an elimination rule (how to get it out). Thus, we have two rules each for plus one more, for a total of 19 rules.
The first rule of inference is kind of a silly one. Quite frankly, it almost always gets left out of written proofs (in fact, it’s been left out of my example proof above). Basically it states that any line of a proof can be repeated (reiterated) at any line within the same scope or deeper. In a proof, it looks something like:
Where basically you have some sentence and later on in the proof you repeat it. Notice that we are justifying the second appearance of it with the name of the rule: “Reit.” This is important to do in every line of your proof. Normally (as seen in my example proof) we also need to justify what lines the rule is being derived from. Because we have no line numbers here, they are omitted. There is a second (more useful, but still generally omitted) version of this rule which allows you to move a sentence into a sub-proof:
Notice however that although we can move sentences deeper into sub-proofs, we can’t take a sentence in a sub-proof and move it back outside of the sub-proof. In order to get things out of sub-proofs, we need different rules, which we’ll see shortly.
220.127.116.11. Conjunction Introduction and Elimination
These two rules are fairly straightforward. First, and-introduction:
And-elimination is simply the same thing, backwards:
If we have a conjunction, then we can derive either of the two terms. Put another way, if we have then we clearly must have and also .
18.104.22.168. Implication-Introduction and Elimination
Implication-introduction is the first time we will see the use of a sub-proof in one of our rules. It has the form:
This deserves a bit of explanation, but is pretty easy to understand once you see what’s going on. In our sub-proof, we are assuming . Now, it doesn’t actually matter whether is true or not, but for the sake of argument we are assuming that it is. Once we’ve done so, if we can somehow further derive then we are able to end the sub-proof (also called discharging our assumption) then we can conclude , which is just the formal way of saying “if then $psi$”, or “if we assume then we can conclude $\psi$”.
Implication elimination is identical to the Modus Ponens rule we learned in Part III. In fact, in many proofs (including my example proof above) we abbreviate the rule “MP” instead of “E”. To remind you, this rule has the following form:
22.214.171.124. Disjunction Introduction and Elimination
Or-introduction is also fairly straightforward and takes the form:
Or-elimination, on the other hand, is considerably more complex:
All this says, though is that if you have a disjunction and can derive the same sentence () from either disjunct, then regardless of whichever between or is true, must also hold. It is also important to notice that we are using two different sub-proofs. The sub-proof where we assume is completely separated from the one where we assume . So, for example, we couldn’t use the reiteration rule to bring into the second sub-proof because this would involve reiterating it to a less deep scope level. It can be a bit of a tricky one to wrap your head around, but once you’ve played around with it a bit, it begins to make sense.
126.96.36.199. Negation Introduction and Elimination
Negation introduction and elimination have exactly the same form. The only difference between the two is the location of the symbol:
In fact, we technically don’t even need both of these rules. Instead we could forgo the elimination rule and simply use the double negation rule (see below). We would do this by assuming , deriving , concluding using I and then rewriting it to using the double negation rule. If you understand what I just said, you’re doing great so far!
188.8.131.52. Biconditional Introduction and Elimination
These two rules basically just formalize the notion that a biconditional is literally a conditional that goes both ways. Its introduction rule has the form:
184.108.40.206. Bottom Introduction and Elimination
Recall that the symbol is a logical constant which represents a sentence which is always false. Basically, its only use is to enable the negation introduction and elimination rules. You might wonder how we could possibly derive a sentence which is always false when we’re assuming that our sentences are always true. Well that’s kind of the point. Observe:
The only way to actually derive is to derive a contradiction. This will usually only happen in the context of some sub-proof which allows us to use the negation introduction and elimination rules that say basically “if your assumption leads to a contradiction, then your assumption is incorrect”. This is also a form of argument which is known as “reductio ad absurdum” where you grant an opponent’s premise to show that it is logically inconsistent with some other point.
The bottom elimination rule does exist, but is seldom used. It has the form:
All this does is return us to the notion that anything is derivable from a contradiction. Regardless of what actually is, if we already have a contradiction, we can infer it. It’s not like we can make it more inconsistent!
Once we start involving quantifiers in our sentences, our rules start getting a bit muckier, because we have to deal not only with sentences containing predicates, but also terms. Because of this, we need to have something called a substitution, which I’ll explain briefly.
As the name suggests, a substitution takes one term out of a sentence and replaces it with a completely different term. This is written as where is some arbitrary sentence, is the term we are replacing, and is what we are replacing it with. Here are some examples:
- evaluates to
- evaluates to
- evaluates to
- evaluates to
- evaluates to
We may sometimes wish to talk about what terms appear in a sentence. I will use the notation to mean that the term appears in the sentence and the notation to mean that it does not. With that in mind, let’s move on.
220.127.116.11. Existential Introduction and Elimination
Existential introduction is a way to weaken a particular statement. It takes the form:
At any given time, we can take a concrete statement and make it abstract. To give a particular example, if we had the claim “the Earth is round” we could weaken it to say “something is round”. Which, while less helpful, is still true. We must, however, take care that the variable we are attaching to our quantifier does not already appear in our sentence. For example, if one had the sentence , we could derive but not , because already appears in our formula.
Existential elimination is a bit more straightforward, but also has a bit more bookkeeping involved with it:
If we know that there is some for which is true, then we can give it a concrete name. The only thing we have to be careful about is what name we give it. In general, we can’t give it a name that is already assigned to an item. So if we had and then we could derive . Once we’ve done so, however, we can no longer derive but we can derive (assuming has not already been used somewhere else within the same scope). A constant term that has not yet been used is called “fresh”.
18.104.22.168. Universal Introduction and Elimination
Universal introduction is weird, to the point where I don’t like to use it because it’s confusing. In the example proof I linked to above, I actually avoid using it and instead use a combination of negation elimination and a version of DeMorgan’s laws (see below). But for the sake of completeness, here it is:
Once again, we have to make sure that the variable we are binding to our quantifier doesn’t appear in the rest of our sentence. But there’s another catch to this rule: in order to use it, you must be outside of any scope lines on your proof. In my example proof above, I could have used this rule to make another line at the end of the proof, but at no other point in my proof would this rule have been viable.
Universal elimination is much more straightforward:
22.214.171.124. Identity Introduction and Elimination
Our last (yay!) rule of inference involves the identity relation. Remember that this is a relation between two terms that indicates that they refer to the same entity. The introduction rule is as follows:
… wait, that’s it? Yep. Without having anything in advance, you can always deduce that any arbitrary term is identical to itself. No baggage required. The elimination rule is a bit more complicated, but really not by much:
4.2.2. Fitch-Rewriting Rules
The following rules allow you to rewrite sentences (or parts of sentences) in Fitch-style derivations. Actually, these rules will pretty much apply to any system of first-order logic. Once again, if you replace the with a biconditional then the resulting sentence is a tautology and will always be valid. Next time, I’ll even teach you how to prove this.
As we saw in Part III conjunction, disjunction and biconditionals are all commutative. Thus we have the following rewriting rules:
Additionally, the variables in a repeated existential or universal quantification are also commutative. This gives us:
It is important to note, however, that the following are not rewriting rules:
There’s an enormous difference between saying “everyone loves someone” and “someone loves everyone”, which is why the order of terms can only be arranged if they are under the same quantifier.
Again, as we saw in Part III conjunction, disjunction and biconditionals are associative, meaning that it doesn’t matter what order you evaluate them in:
There’s no real difference here between propositional and predicate logics.
126.96.36.199. DeMorgan’s Laws
Also as seen in Part III, we have DeMorgan’s laws which identify the relationship between conjunction, disjunction and negation and allow us to define them in terms of each other.
However, we also have a variation DeMorgan’s Laws which operate on quantifiers:
Seriously think about these until they make sense to you. It may help to think of an existential statement as a giant disjunction (ie: “something is round” means “either electrons are round or the Earth is round or squares are round or birds are round…”) and a universal statement as a giant conjunction (ie: “electrons have mass and the Earth has mass and squares have mass and birds have mass”).
188.8.131.52. Double Negation
Double negation is another one that’s straight out of last time:
If isn’t not true, then it must be true. Simple.
184.108.40.206. Definition of
Once again, can be defined in terms of and as follows:
Make sure you understand why this is true as well.
220.127.116.11. Symmetry and Transitivity of Identity
Finally, we have a couple of rules regarding identity. The first is easy enough:
You might ask yourself why this doesn’t fall under commutativity and the answer is a little bit confusing. Commutativity is a rule which applies between sentences. It says that you can swap the order of the sentences on either side of certain connectives. Symmetry is slightly different. It says that you can swap the order of the terms on either side of certain relations. It’s a bit of a technical distinction, but it’s one that’s important to maintain.
Transitivity of identity isn’t really a re-writing rule, but I like to put it here with symmetry because they go well together. Basically it states that if you know that and also that then you can also conclude that . It can be made into a rewriting rule by:
4.3. Future and Questions
It turns out that there’s actually a lot that needs to be said about first-order logic. As such, I’m going to split it into two (well, two-and-a-half) different entries. Next time I’ll talk about proving validity using Fitch-style systems as well as some strategies for proving things in first-order logic and translations of more complicated sentences. I also want to go over my example derivation and explain step-by-step what’s going on (in fact, I’ll probably do that part first).
In the meantime, here is a handy reference for all of the rules we went over today. See if you can answer the following questions:
- Translate the following sentences into first-order logic:
- All circles are round.
- Some buildings are tall.
- Phil said hello to everyone at the market.
- Joe knows a guy who can get you cheap concert tickets.
- She is her mother’s daughter.
- She got married to someone who owned a farm.
- John is taller than Suzie, but shorter than Pete. Frank is taller than all of them.
- There is at least one apple in that basket.
- There are exactly two apples in that basket.
- There are no more than three apples in that basket.
- “is the father of”
- “is related to”
- “is an ancestor of”
- “is the brother of”
- “is the sibling of”
- “is descended from”