Logic is the study of inferences and rules of inference. When we make philosophical arguments, we’re offering some form of inference from a set of premises to our conclusion (namely, the thesis which your paper defends). We can use the tools of formal logic to analyse the structure of those arguments. By stripping away the content of the argument, we can see just the bare bones of how the different claims relate to one another. We can use formally-defined concepts such as validity and consistency, and logical methods such as truth tables and proof systems to analyse the argument in this abstract form. This has a few advantages – not least that if two arguments have the same underlying logical form, they will have the same properties. So if an argument with the same logical form as the one you want to make is valid, for instance, then so is your argument. The skill of seeing the logical form of an argument is not one that every philosopher acquires, but can be extremely valuable.
This guide will take you through the process of breaking down a basic, propositional argument into its logical form. It will presuppose that you’re already familiar with the definitions of validity, soundness, tautology and contradiction laid out in the guide to validity and soundness, so do read that first. Later guides will allow you to develop these skills further, to analyse more complex arguments and to apply logical tools and methods to your formalisation.
Propositions and Logical Form
We know that arguments consist entirely of declarative sentences. Some sentences are simple, like “Charles is tall” or “Eddie Vedder is the lead singer of Pearl Jam”. Others are more complicated, bringing together different claims in a multitude of ways. For example: “Charles is tall and Dana is Irish”, “Either Eddie Vedder is the greatest lead singer of all time or Chris Cornell is”, and “If man-made climate change is occurring, then more extreme weather events will occur in the coming years”.
Each of those more complex sentences could be broken down into simpler propositions. “Charles is tall and Dana is Irish” could clearly be broken down into two propositions: “Charles is tall” and “Dana is Irish”. The claim that “If man-made climate change is occurring, then more extreme weather events will occur in the coming years” involves two simpler propositions: “Man-made climate change is occurring”, and “More extreme weather events will occur in the coming years”. The claim that “Either Eddie Vedder is the greatest lead singer of all time or Chris Cornell is” is marginally more difficult to break down. We have “Eddie Vedder is the greatest lead singer of all time”, but the proposition “Chris Cornell is” makes no sense on its own. This is just quirky English grammar at work, though: we know the original sentence was just shorthand for “Either Eddie Vedder is the greatest lead singer of all time or Chris Cornell is the greatest lead singer of all time.” So the second proposition involved is: “Chris Cornell is the greatest lead singer of all time”.
We’ll call these simple sentences atomic propositions because they can be asserted on their own, independently of any other proposition, and (at least for the moment) they can’t be broken down into any simpler propositions. There’s no proposition which could be asserted on its own which forms part of the proposition “Chris Cornell is the greatest lead singer of all time”. It’s in the simplest form it can be (again, for now).
Going forward, we’ll represent an atomic proposition with a single lower-case letter of the alphabet. We could represent a proposition as ‘p’ or ‘q’ or ‘z’ or ‘a’. We’ll tend to use ‘p’ (for proposition) first, and go on from there. If we have two atomic propositions involved in our argument, we’ll give them a different letter each. For example, suppose my argument is:
1. If it is sunny, then I’m going for a run.
2. It is sunny.
Therefore, I’m going for a run.
There are two atomic propositions involved in this argument: “It is sunny” and “I’m going for a run”. We’ll give “It is sunny” the letter ‘p’, and “I’m going for a run” the letter ‘q’. Notice that we have some words left over, which we’ll just leave in place for now. Now the argument looks like this:
1. If p then q
We can call this the logical form of this argument. To find other arguments with the same logical form, all we have to do is substitute in another pair of sentences for ‘p’ and ‘q’. So if I substitute in ‘Santa Claus lives in Lapland’ for p and ‘Anthony is an exceptional giraffe’ for q, then we get:
1. If Santa Claus lives in Lapland, then Anthony is an exceptional giraffe.
2. Santa Claus lives in Lapland.
Therefore, Anthony is an exceptional giraffe.
Even though the two arguments have completely different content – one is entirely mundane and the other entirely absurd – they have identical logical form.
Moreover, both arguments are valid. In both cases, it is impossible for the premises to both be true while the conclusion is false (recap the guide to validity and soundness if this is unfamiliar). This brings us to our first logical theorem:
This gives us a straightforward way to demonstrate that a logical argument is invalid: to give an example of an argument which has the same logical form, and which has premises which are all true and a conclusion which is false. This is the basis of the notion of argumentative counterexamples. If you can find such an argument, then any argument with that form must be invalid.
So consider, for instance, this argument:
1. If idealism is the hallmark of Platonism, then Berkeley is the inheritor of Plato.
2. Berkeley is the inheritor of Plato.
Therefore, idealism is the hallmark of Platonism.
This inference is bamboozling at first, filled with technical terminology and names we might have to look up. Is it possible for the premises to be all true and the conclusion false? It’s hard to say at a single glance. But we don’t need to worry. Logical formalisation will strip away all that idealism nonsense and leave us with:
1. If p then q.
This looks a lot like our argument above, but rather than the second premise being p and the conclusion q, they’re the other way around. This is an invalid argument. To see that clearly, all we need to do is find a pair of sentences we can substitute in for p and q which make ‘If p then q’ and ‘q’ obviously true, but ‘p’ obviously false. Try to come up with a pair for yourself.
I chose p = “Bonn is the capital of Germany” and q = “The capital of Germany begins with ‘B'”. So:
1. If Bonn is the capital of Germany, then the capital of Germany begins with ‘B’.
2. The capital of Germany begins with ‘B’.
Therefore, Bonn is the capital of Germany.
Premise (1) and (2) are true – (1) is obvious and (2) is true as the capital of Germany is Berlin. But the conclusion is false. So not only is this inference invalid, so is every inference with the same logical form, including our one about Plato and Berkeley.
But notice that we can’t use this same approach to prove that a valid inference (like our one about it being sunny and my going for a run, above) is valid. That’s because we can’t prove that there aren’t any arguments of the same logical form which have true premises and a false conclusion. We could check a bunch of them, as many as you like, but there are infinitely many possible pairs of sentences, so we can’t look at all of them. We’ll need some more sophisticated logical machinery to do that.
It may seem strange to bring ‘truth’ and ‘falsity’ into this, when we’re looking just at abstract letters like ‘p’ and ‘q’. How could ‘p’ be true or false? A strength of formal reasoning is that we can set aside complex debates about the nature of truth and how we know whether claims are true or false. We can treat the truth or falsity of a claim as just another property of a proposition, albeit an important one, and focus on analysing how that property is transmitted between propositions.
We’ll assume for now that there are two “truth values” that a proposition could take: true or false, and we’ll denote them as ‘T’ and ‘F’, always in the upper case. What does it mean to say that p is true? Nothing more than that the truth-value of p is T.
When formalising arguments using this kind of propositional logic, we assume that each atomic proposition has its own truth value, either T or F. We’ll exclude any atomic propositions which don’t have a truth-value (if there are such propositions – a few have been suggested, like ‘The present king of France is bald’ or ‘Colourless green ideas sleep furiously’ – but we can ignore these for now). How we work out the truth value of an atomic proposition, or whether we even know what that truth value is, falls beyond our remit. What is the truth value of ‘God exists’? We can shrug that off. The proposition has a truth value even if we don’t know what that truth value is right now, or may never know.
We saw earlier that we can make more complicated sentences by joining together atomic propositions with linking words like “and”, “if … then … “, and “or”. We’ll call these tools for forming more complicated propositions connectives, and we’ll call a proposition which consists of one or more atomic propositions linked together by one or more connectives a compound.
The important thing about compounds in propositional logic is that their truth-value depends on the truth-value(s) of the atomic propositions which they contain. Each connective gives us a unique way to make a claim about the truth-values of the atomic propositions.
We’ve already seen three connectives in action. These are:
The sentence “Charles is tall and Dana is Irish” puts together two atomic propositions – ‘Charles is tall’ and ‘Dana is Irish’. Immediately, we can see the way in which this compound sentence’s truth value depends on the truth values of the atomic propositions. For the conjunction to be true, both of the conjuncts (the propositions joined by the ‘and’) must also be true.
Conjunction is sometimes called “and”, but in natural language takes many forms: “but”, “and also”, “yet”, “however”, and so on. Any way of saying that two propositions are both true is a conjunction.
We can now replace the word ‘and’ (or ‘but’, etc.) in our logical formalisations with a symbol to denote this connective. In different logic texts, different symbols are used to represent the various connectives. We’ll use simple-to-remember and easily found symbols where possible. For conjunction, we’ll use ‘&’, known as the ampersand. (In other texts, you’ll see the symbol ∧ used for conjunction, and elsewhere see ∙ used).
So now, to write the logical form of the sentence “Charles is tall and Dana is Irish”, we write: p & q. The logical form of “Eddie Vedder is the lead singer of Pearl Jam and Ten is the greatest rock album of all time” is exactly the same: p & q.
Other compound propositions represent the claim that one of the atomic propositions is true. For example: “Jack went up the hill or Jill went up the hill.” We’ll call the propositions on either side of the ‘or’ disjuncts. Things get a little more complicated here than with conjunction. In ordinary language, we use two kinds of disjunction. Sometimes when we use ‘or’ we mean that at least one of the disjuncts is true, but it could be both. Suppose I said “Paula will sell a car today or Quentin will sell a car today”. What happens if both Paula and Quentin sold a car? Is my sentence true or false? It’s not entirely clear. We can’t have that. Logic must be unambiguous.
At some point, the decision was taken that when we use ‘or’ in logic, we’ll mean inclusive or. This means that in logic, the proposition ‘p or q’ is true if and only if p is true, q is true, or both. If we analyse “Paula will sell a car today or Quentin will sell a car today” as ‘p or q’, then if both Paula and Quentin sold a car, the sentence would be true.
Sometimes we don’t want this. If I said “Either we’ll go to the cinema or we’ll go to the pub”, it’s clear that we won’t be doing both. This sentence would be false if we went to the cinema and the pub. Often, this is denoted in English by “either … or …”, whereas an inclusive ‘or’ lacks the ‘either’. This is called exclusive or or xor for short. For simplicity’s sake, we won’t introduce a connective for xor at this point, but we’ll see in a later guide how easy it would be to do it. We’ll also see that it’s easy enough to use our existing connectives to express an xor proposition, later.
We’ll represent an inclusive or compound proposition using the symbol ‘∨’. You can remember the meaning of this connective by thinking of ‘v.’ or ‘vs.’ in a sporting match. It’s Ali vs. Fraser, Manchester United v. Leicester City. But remember that unlike in a boxing ring or on the football field, both can ‘win’. So the claim that “Jack went up the hill or Jill went up the hill” can be formalised as: ‘p ∨ q’.
Conditional (“if … then … “):
Conditional form (or ‘implication’) covers those sentences where we claim that if one thing happens, then another thing will. We saw some of these above: “If Bonn is the capital of Germany, then the capital of Germany begins with ‘B'”, and “If it is sunny, then I’m going for a run”, for example. We can formalise this neatly using an arrow: ‘ → ‘. The sentence “If it’s sunny then I’m going for a run” can be formalised as ‘p → q’.
Working out what this means in terms of the truth of the antecedent (here “It’s sunny”) and the consequent (“I’m going for a run”) is a little tricky. It’s clear that this is true when it is sunny and I do go for a run. So when both the antecedent and the consequent are true, p → q is true. Similarly, if it was sunny but I didn’t go for a run, the claim would be false. So when the antecedent is true but the consequent is false, p → q is false. This might start to look like our analysis of deductive validity. For validity, our argument was valid if and only if the premises could not be true while the conclusion was false. A conditional is similar: the conditional is true if and only if the antecedent is not true while the consequent is false. So this handles the two more confusing cases. What if it isn’t sunny? Is the conditional “If it’s sunny, then I’ll go for a run” true or false? We might naturally say that we don’t know. We’d only get to the truth of the matter if it became sunny. But that’s no good for clean, unambiguous logic. So we’ll say that the proposition is true unless it is shown to be false. In other words, if it isn’t sunny, the conditional “If it’s sunny, then I’ll go for a run” is true. If the antecedent is false, then the conditional is true (whether or not the consequent is true).
So, let’s look at “If Bonn is the capital of Germany, then the capital of Germany begins with ‘B'”. Is this conditional true or false? We known the capital of Germany begins with ‘B’, so the consequent is true. We also know that the capital isn’t Bonn, so the antecedent is false. Any conditional with a false antecedent must be true. So “If Bonn is the capital of Germany, then the capital of Germany begins with ‘B'” is true.
Bi-conditional (“if and only if”)
There are two more connectives that we need to meet before we go any further. One is “if and only if” or ‘iff’. We’ve seen this a few times, especially when giving definitions. We call this the bi-conditional, because it works just like a pair of conditionals, p → q and q → p. A bi-conditional “p if and only if q” asserts that the truth values of p and q are the same. It’s true just when both p and q are true or when both p and q are false. If they have different truth values, the bi-conditional is false. We write a bi-conditional using an arrow with arrowheads on both ends, like this: ‘ ↔ ‘. So the sentence “You are dead if and only if your heart has stopped beating” would be analysed as: ‘p ↔ q’.
Bi-conditionals are very useful for expressing definitions and in proving theorems.
The final connective we’ll need is negation. Negation works a little differently to the other connectives. Conjunction, disjunction, conditionals and bi-conditionals are all binary connectives – they connect together two propositions in different ways. Negation is a unary connective: it relates to just one proposition. Negation serves the same function as adding ‘not’ to a sentence. For example, we might want to express: “I am not going to the for a run”. Formally, we will say that negation adds the clause “It is not the case that … ” to the start of a proposition. So “I am not going for a run” could be stated as “It is not the case that I am going for a run”. This helps clear up the issue of where in the sentence the ‘not’ needs to appear. We’ll use ‘ ¬ ‘ to symbolise negation. (Elsewhere, you might see ‘~’ used to represent negation, or a line drawn over the top of the proposition symbol). So the proposition “I am not going for a run” is formalised as: ‘¬ p’ (where p = “I am going for a run”).
Before you go on to the more complex logical formalisations, it’s a good idea to try out formalising some basic propositions using these five connectives. The WritePhilosophy quiz Logic: Basic Propositions will give you a chance to try this out.
Brackets and Complex Compounds
We now have six ways in which we could express a proposition:
An atomic proposition, like “I am going for a run”, which we express as: p
A conjunction, like “I am going for a run and then I will have a nap”, which we express: p & q
An inclusive disjunction, e.g. “I am going for a run or I will have a nap”, which is: p ∨ q
A conditional, such as “If I go for a run, then I will have a nap”, which is: p → q
A bi-conditional, like “I will go for a run if and only if I have a nap”, which is: p ↔ q
And finally, negation, such as “I am not going for a run”, which is: ¬ p
But how would we express a proposition like: “If I go for a run, then I will have a nap or I will go to the pub”? There isn’t a connective for this. We could create one (a three-part connective, which is interesting), but we don’t need to because we can express it using our existing conditional and disjunction forms. We can see that “I will have a nap or I will go the pub” is a disjunction. Let’s say that’s ‘q ∨ r’. That disjunction is the consequent of a conditional. So we analyse this sentence as: p → (q ∨ r)
We use brackets to make it clear how these compounds work. There’s an important difference between ‘p → (q ∨ r)’ and ‘(p → q) ∨ r’. The first says “If I go for a run, then I will have a nap or I will go to the pub”, the second says “If I go for a run then I will have a nap, or I will go to the pub”. We’re not so used to that latter type of statement, but they do come up from time to time. It might be more obvious if we reversed the order of the disjuncts, for example: “I will go to the pub or if I go for a run, I’ll have a nap.” That’s clearly ‘r ∨ (p → q)’.
Let’s see a few more. Suppose I assert: “If Eddie Vedder is not the best singer in rock music, then Chris Cornell or Kurt Cobain is.” How would you analyse this sentence? Try it out, before taking a look at the answer below.
First, we might want to spell out the sentence fully, removing some of the simplifying quirks of the English language. When fully stated, the sentence is: “If it is not the case that Eddie Vedder is the best singer in rock music, then Chris Cornell is the best singer in rock music or Kurt Cobain is the best singer in rock music.” The overall structure of the sentence is a conditional. But the antecedent and consequent are both compound propositions in their own rights. The antecedent, “it is not the case that Eddie Vedder is the best singer in rock music”, is a negation, while the consequent is a disjunction. So we can analyse the proposition as:
(¬ p) → (q ∨ r)
Notice that we put brackets around the ‘¬ p’ as well here. If we left those off, leaving this just as ¬ p → (q ∨ r), it wouldn’t be obvious whether the negation referred just to p, or to the whole of ‘p → (q ∨ r)’. What would this sentence mean if we wrote it instead as: ¬ (p → (q ∨ r))?
Let’s try one more: “If man-made climate change isn’t addressed and we continue to burn fossil fuels at the same rate, then we will experience more natural disasters and we will be unable to stop sea levels rising”. Try it yourself.
There are a couple of negations in here which are hidden away, without using the word ‘not’. We might think that “Man-made climate change isn’t addressed” is an atomic proposition. But really we should read this as “It is not the case that man-made climate change is addressed”. The same goes for “We will be unable to stop sea levels rising”, which is really the negation of “We will be able to stop sea levels rising”. So the analysis of this sentence is:
((¬ p) & q) → (r & (¬ s))
Usually we’ll be pedantic about including brackets. Perhaps we should always do it, even when expressing a sentence like “Jack and Jill went up the hill” as ‘p & q’, we really should write ‘(p & q)’. Sometimes, though, for simplicity’s sake we might omit the brackets when there’s no risk of ambiguity. Two of our connectives, conjunction and disjunction, allow us to chain together a whole bunch of conjuncts or disjuncts. A sentence like “Eddie Vedder, Chris Cornell and Kurt Cobain are great singers” is really a conjunction of three atomic propositions (“Eddie Vedder is a great singer”, “Chris Cornell is a great singer”, etc.). Technically, we defined conjunction as a binary connective, so we’d have to formalise this as ‘p & (q & r)’ or ‘(p & q) & r’. But since it doesn’t matter, we’ll usually be content to just write ‘p & q & r’.
Using this machinery, we can express extremely complex propositions, with arbitrarily many atomic propositions and all manner of connectives amongst them. We can also use our existing set of connectives to express other relationships between propositions, which we didn’t define a specific connective for. Remember xor? How would you express a sentence like “Either we are going to the cinema or we are going to be pub”, an exclusive disjunction, using only the connectives we already have? Try it out.
There are a few ways we could do this. The simplest might be to reformulate it as “We are going to the cinema or we are going to the pub, but not both”. That would be formalised as: ‘(p ∨ q) & ¬(p & q)’.
We’re now ready to define a proposition:
Propositions range from the simple – a single atomic proposition, like “I am going for a run” (p) – to the very complex, like “If you don’t go for a run and you eat fast food, then unless you have an incredible metabolism, then you will put on a lot of weight and won’t be healthy” (((¬p) & q) → (¬r → (s & (¬t))).
While we use lower-case letters to stand in for atomic propositions, we use upper case letters to stand for any proposition whatsoever. So if I write ‘P & Q’, for instance, then this is a conjunction of two propositions. But those propositions could themselves have some complex structure to them (or could just be simple atomic propositions).
You now have enough logical machinery at your disposal to analyse most sentences and work out the logical form of the proposition. In our next guide, we’ll explain how we can use this formalisation to work out the precise circumstances in which a proposition is true or false, and to demonstrate that an inference is valid or invalid. For now, it’s important to try out these skills. Formalise sentences around you, in newspaper articles and philosophy papers, at the pub, in conversation. Try out the WritePhilosophy compound logical formalisation quiz to make sure you’re fully confident in analysing propositions. Then, it’s time to bring truth and falsity back into the picture.
Latest edit: 05/03/2021 by CJ Blunt