1.1 Validity

Logic is concerned with good arguments and, most especially, with proofs (arguments that are important in mathematics and abstract theory-building). It is good for arguments to be clear and economical, but logic is not concerned with those stylistic features. Logic is concerned with a standard of correctness for arguments, known as validity.

Here is a typical example of a valid argument.

This argument starts with premises (1.1) and (1.2) and ends with the concluding sentence “All dogs are animals.” The conclusion is marked with the special symbol “∴.” We can read this symbol as expressing the phrase “Therefore….” To check whether this argument is valid we need to evaluate the connection between its premises and conclusion.

When we say that an argument is valid (correct), it means that the right kind of connection is really there; if the connection is missing, the argument is invalid (incorrect). Standard textbook logic says that Example 1 is valid; the premises of the argument really do support its conclusion; the conclusion follows from the premises. These are three different ways of expressing that the argument has the right logical properties.

The roots of this field of study in Western philosophy trace back to Aristotle: “A deduction is speech (logos) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so” (Prior Analytics I.2, 24b18–20, trans. Smith Reference Smith1989).

The translation of this passage uses the word “deduction” to refer to roughly the same thing as a valid argument. Aristotle suggests that when we make certain assumptions, lo and behold, something else results of necessity. A deduction or valid argument expresses this kind of airtight connection between its premises and conclusion.

Good arguments and reasonable conclusions are just as important in empirical science as they are in logic. However, there is a difference in emphasis. Science is much more concerned with actual, observable facts. For example, a physicist might ask: Is it actually true that dark matter is composed of axion particles? This is a question about how things work in the physical world. Reasoning and argument are indirectly involved only because they can help scientists find good answers to their questions.

Logic directly concerns itself with reasoning and argument themselves. We can see logical questions as hypotheticals. A typical question of logic is something like this: assuming that everything in the universe is composed of axion particles, does it follow that dark matter is composed of axion particles? We do not need to know real physics to answer this question. It is not a question about matter or particles. The logical question is about what follows from what. Hypothetical thinking along these lines has an important role in all walks of life, but logic makes this into its primary topic of study.

If the conclusion of an argument really does follow from the premises by an airtight, necessary connection, then the argument is valid.

It would be nice to know which arguments are valid. In modern logic, we build systematic accounts of validity. There are different accounts, hence different logics.

The main job of logic is to classify valid arguments. Logical pluralism says that there is more than one correct way to do this. In order to discuss this philosophical view, it might help to know something about the methods of modern logic.

The rest of this section introduces the methods of modern, formal logic. The goal is to showcase how minor variations and choices in these methods produce a variety of different logical systems. This will make certain aspects of logical pluralism much clearer. If you already know the difference between first-order logic and second-order logic, or that between classical logic and non-classical logic, you can skip ahead to the summary of key results in Section 1.5. For everyone else: read on.

1.2 Forms, Rules, Models

Perhaps the most important idea in modern logic is this: Validity is often determined by the internal structure or logical form of arguments. Some examples should help to make the idea of logical form clearer. Here is an example of an argument.

Here is another argument with the same form.

When we say that Examples 2 and 3 have the same form, what are we talking about? It probably seems quite obvious, but let’s spell it out. Each argument has two premises. The conclusion is a more complex sentence that joins the premise sentences together with the word “and” between them. That pattern is exactly the same between both arguments. Formal logic uses such patterns to analyze which arguments are valid.

We use formal languages to make this sort of analysis clear. A formal language represents sentence parts with symbols. In this Element, we will use the lowercase letters $p,q,r$ to stand for whole, declarative sentences. We also have special logical symbols:

• The negation connective $\mathrm{\neg }$ is read as the English word “not.”

• The conjunction connective $\wedge$ is read as the English word “and.”

• The disjunction connective $\vee$ is read as the English word “or.”

• The conditional connective $\to$ is read as the English phrase “if…, then….”

The rules for putting these symbols together create formulas. Each formula represents an underlying structure that occurs in many sentences, based on the parts of the sentence and how those parts are put together. An argument form is simply the collection of formulas that represent the logical forms of the sentences in the argument. For example, here is the argument form shared by Examples 2 and 3.

This should be easy enough to understand. The formula $p\wedge q$ represents a sentence in which the simpler sentence $p$ is joined with $q$ by putting the word “and” between them, just like we saw in Examples 2 and 3.

The type of logic that draws on this formal language is called propositional logic. It only focuses on the aspects of argument form that involve the placement of whole, declarative sentences and the logical connectives. We could imagine various ways to expand this formal language to represent other operators on sentences. For example, we could include symbols for modalities. In the formal language of modal logic, $\square p$ says “$p$ is necessary” and $◊p$ says “$p$ is possible.” This allows us to represent a wider range of argument forms than is possible with just the basic language of propositional logic.

Here is another pair of examples to think about.

The following argument has the same form as the preceding one.

When we say that Examples 5 and 6 have the same form, what are we talking about? Each argument has two premises that ascribe a property to some specific, named object. The conclusion sentence has the word “and” in the middle. Each half of the conclusion sentence talks about one of the same properties from the premises, but instead of ascribing that property to a specific, named object, it says more generically that something has the salient property. Again, this pattern is exactly the same between both arguments.

To represent this kind of pattern we need a formal language that can represent the internal parts of sentences. We use the lowercase letters $a,b,c$ to stand for names of specific people, places, or things. We use the uppercase letters $F,G,H$ to stand for predicates, which express properties of things. We also need more logical symbols:

• The existential quantifier $\mathrm{\exists }$ is read as the English phrase “there is….”

• The universal quantifier $\mathrm{\forall }$ is read as the English phrase “for all….”

• The variables $x,y,z$ make clear how the quantifiers are used.

Here is the argument form shared by Examples 5 and 6.

We read this as follows. $Fa$ represents the form of a sentence that applies predicate $F$ to name $a$, so it ascribes a property to some specific, named object. Likewise for $Gc$. This is just like the premises in Examples 5 and 6. $\mathrm{\exists }xFx$ attaches the quantifier phrase to a variable, which you can think about as a sort of “test” to see whether any chosen object has property $F$. In other words, $\mathrm{\exists }xFx$ represents the form of a sentence that says there is at least one thing that satisfies the $Fx$ “test,” or in more natural English, it just says something has property $F$. Likewise, $\mathrm{\exists }yGy$ says that something has property $G$.

The type of logic that draws on this formal language is called first-order logic. It focuses not only on sentences and logical connectives, but other aspects of argument form that involve the placement of names, predicates, and objectual quantifiers. We could imagine various ways to expand this formal language to represent other types of quantifiers. For example, we could add second-order quantifiers to generalize over predicate position. In the language of second-order logic, $\mathrm{\forall }XXa$ says “All properties apply to $a$” and $\mathrm{\exists }XXc$ says “Some property applies to $c$.” This allows us to represent a wider range of argument forms than is possible with just the basic language of first-order logic.

Formal languages draw our attention to specific aspects of sentence structure, which is extremely helpful for logical analysis, but it also involves a choice. Should our logical tools pay attention to modal operators or second-order quantifiers? These are not part of standard, introductory logic books. Are modal operators or second-order quantifiers inappropriate for logical study? We will revisit this question about the logical vocabulary in Section 2.2 and consider whether it may be connected with logical pluralism.

Creating a formal language is only the first step into formal logic. The next natural question is: What do we do with this structural information? How can we use argument forms to classify valid arguments? There are two prominent techniques for doing this. One of them uses proof rules, the other uses semantic models.

Outside of formal logic, a proof is just a piece of conclusive reasoning. Most of us encounter proofs in mathematics. Just think about examples you have seen, like the proof that $\sqrt{2}$ is irrational or the proof of the Pythagorean theorem which states that $a^2+b^2=c^2$ for the edges of a right triangle. You may recall that these proofs involve a lot of steps. This is a key idea in proof theory, an important method for classifying valid arguments.

A formal analysis of proofs, also known as a derivation, is built up from individual steps. Each step involves a transition from input formulas to output formulas. The essence of proof theory is given by proof rules that tell us what steps are allowed. In some sense, our goal is to choose rules that are so secure that we can blindly follow them without ever making a mistake. The derivable arguments are all valid.

There is a second important technique for working with argument forms. Recall how Aristotle said that the conclusion of a valid argument results of necessity. This may suggest the idea that validity comes from an external relationship between sentences and the circumstances that make sentences true. A semantic model is a kind of precise representation or a mathematical abstraction of such a circumstance. This is the key idea in model theory, which is yet another method for classifying valid arguments.

In broad outline, models assign truth-values to formulas. A formula might be true in some models and not in others. We think of each model as a logically possible circumstance. Now, we can look for relationships between arguments and models. If every model that makes the premises true also makes the conclusion true, we say that such an argument is truth-preserving. The truth-preserving arguments are all valid.

In the next few sections, we will look at some details of proof theory and model theory to understand how these techniques are used to define different logics. Before moving on, there are a few more useful preliminary remarks to make.

In principle, we can take any formulas $p_1,p_2,\dots$ and any single formula $q$ and think of them, respectively, as the premises and conclusion of an argument. This determines a class of argument forms. A specific formal technique divides that class into two parts: The valid arguments and the invalid arguments. Simply put, the result of using proof theory or model theory is to produce a demarcation of the set of valid argument forms. This is what we call an extensional definition of validity. It sometimes happens that we can use two different techniques to get exactly the same extensional definition. One of the most famous mathematical results in logic is called the Completeness Theorem, which basically shows that a proof theory and a model theory are extensionally equivalent.

In the next section, we will compare some famous formal logics known as classical, intuitionistic, and relevant logics. These “named logics” are not identified with just one specific proof theory or one specific model theory. When we talk about a logical system by name, what matters is the extensional definition of validity that goes along with that type of logic. There are different ways of doing proof theory or model theory that produce exactly the same logic in the extensional sense. When we find two different tools that define the same logic, like in the Completeness Theorem, then we have at least two ways of understanding the nature of that single notion of validity.

Logics are used to analyze arguments in the first place, but they also have repercussions for how we think about theories. A theory is a bunch of sentences, so we can think about what would happen if we treated those sentences as the premises of an argument. If conclusion $C$ follows from a theory, we say that $C$ is one of the commitments of that theory. These commitments are part of the implicit worldview of the theory. A theory can only be true if all of its commitments are true.

Adding information to a theory often causes its commitments to expand. If we add too much information, we might end up with a theory that overgenerates commitments. The most extreme outcome is a trivial theory that is committed to everything whatsoever.

Triviality is extremely bad. Since a trivial theory has indiscriminate implications, it cannot be true (on pain of everything being true). The source of this defect is the internal structure of the theory itself. For this reason, trivial theories are widely considered to be the paradigm examples of nonsensical or uninterpretable theories.

The boundary between trivial and non-trivial theories indicates a difference between minimally acceptable theories and useless, nonsensical theories. Since this depends on logic, we can compare how different logics categorize trivial theories. This is one more way to compare and contrast different logical systems.

1.3 Proof Theory

In this section, we will briefly explore proof theory. We will only discuss propositional logics with a standard set of connectives as indicated in Section 1.2. We will highlight some details of proof theory for classical, intuitionistic, and relevant logics.

Proof theory gives a precise definition of derivations. A derivation might look like a list or a graph with formulas written at each step and transitions determined by the proof rules of the system. If we can construct a derivation that leads from the premise formulas $p_1,p_2,\dots$ to the conclusion $q$, we write something like this:

$p_1,p_2,\dots ⊢q.$

The single turnstile $⊢$ is a symbol for the proof-theoretic definition of validity. The preceding displayed string says “from $p_1,p_2,\dots$ it is valid to infer $q$.” However, different logics give different theories of validity. So, we usually add a subscript $\mathsf{L}$ to the turnstile ${⊢}_{\mathsf{L}}$ and read this as saying “… it is valid according to logic $\mathsf{L}$….”

There are a few styles of proof theory. For now, we will look at an approach that was popularized by the mathematician David Hilbert. This name will come up again in one important connection to eclectic pluralism (Section 2.6). In a Hilbert system, each derivation is a numbered list of formulas. It looks something like this:

Each step contains a specific formula. On the right side, we list the justification of each step. The justification comes from the proof rules of the system. We are only allowed to make a step if it is somehow permitted by those rules.

Hilbert systems have several rules. One type of rule is a “proper” rule in the sense that it takes input from earlier steps of a derivation. We will look at one such proper rule known as modus ponens. This rule is a part of the proof theory for many logics.

• Rule of Modus Ponens Suppose you are building a derivation. At step $n$ you have the formula $p\to q$. At step $m$ you have the formula $p$. Then you are allowed to write $q$ at the next step with justification [MP, $n$, $m$]. This says that you reached the current step by applying modus ponens (MP) to the input steps $n$ and $m$.

It is important to understand that $p$ and $q$ can be replaced by any two formulas. The pattern is what matters. For example, the formulas $(p\vee s)\to r$ and $p\vee s$ could be input formulas. Modus ponens would then give us $r$ as output. The formulas $(p\to q)\to (s\to t)$ and $p\to q$ could be input formulas. Modus ponens would then give us $s\to t$ as output.

We need earlier steps in a derivation in order to generate later steps using MP, so where do the earlier steps come from? Some rules directly permit us to write down certain formulas. One of these rules just tells us that if we are reasoning with a particular premise, we can write that premise down at any time in our derivation.

• Rule of Assumptions If you are trying to derive something from premise $p$, you are allowed to write down $p$ at any step and mark it as an [Assumption].

Another important rule deals with formulas called axioms. Every Hilbert system comes with its own distinct set of axioms. They act as “free” pieces of information for derivations. The axioms of system $\mathsf{L}$ are the main engine of derivations in $\mathsf{L}$.

• Rule of Axioms If there is an axiom called Axiom-A, then you are allowed to write down this kind of formula at any step with the justification [Axiom-A].

Now, this is all very abstract and we would perhaps like to know more about why these rules are chosen. Most especially, what are axioms? Why are some formulas chosen as axioms while others are not?

A helpful perspective, known as inferentialism, says that the meaning of an expression is determined by the inferences we may draw from sentences using that expression (Brandom, Reference Brandom1994; Dummett, Reference Dummett1991). For example, MP seems to capture an important aspect of the meaning of the conditional insofar as it permits the following step of reasoning: When you assert “If $p$, then $q$” and you assert $p$, you may infer $q$. In this spirit, axioms are often taken to be the forms of analytic sentences. Since the axioms of a logical system are what enable certain steps of reasoning, they also reflect a theory of meaning about the logical connectives in those axioms. For the time being, we leave this philosophical issue aside, but it will come up again in Sections 2.5 and 3.3.

Turning now to specific formal systems, we will first look at a set of axioms involving conjunction and disjunction. When we look at the Hilbert systems for classical, intuitionistic, and relevant logic, all three of them accept Ax1–Ax4.

[Ax1]

$(p\wedge q)\to p$

[Ax2]

$(p\wedge q)\to q$

[Ax3]

$p\to (p\vee q)$

[Ax4]

$q\to (p\vee q)$

To be a bit more careful, these are technically axiom schemata. If we take a schema and uniformly replace the occurrences of each sentence letter ($p$, $q$, etc.) with any formula, then we have an axiom. Any formula that follows the pattern of one of the schemata counts as an axiom. For example, here are three different formulas that follow the pattern of Ax1: $(s\wedge t)\to s$, $((p\vee q)\wedge r)\to (p\vee q)$, $(p\wedge (r\to q))\to p$. If we are working in a Hilbert system that accepts Ax1, then we are allowed to write down any of those formulas at any time.

Classical logic is the most widely taught logic. It is what you will find in most logic textbooks. The influence of classical logic on contemporary philosophy can be traced back to two philosophical mathematicians, Gottlob Frege (Reference Frege1879) and Bertrand Russell (Reference Russell1903). We write ${⊢}_{\mathsf{C}\mathsf{L}}$ for derivability in classical logic. This is defined as follows.

Definition. (Classical Derivability)

$p_1,p_2\dots {⊢}_{\mathsf{C}\mathsf{L}}q$ means that, in the Hilbert system for classical logic, it is possible to construct a derivation where each premise $p_i$ is marked with [Assumption], the other steps of the derivation come from the classical axioms or from MP, and the final step is the conclusion formula $q$.

Since classical logic accepts Ax1–Ax4, we can show that $p\wedge q{⊢}_{\mathsf{C}\mathsf{L}}q\vee r$.

Example 7.02

1.	$p\wedge q$	[Assumption]
2.	$(p\wedge q)\to q$	[Ax2]
3.	$q$	[MP, 1, 2]
4.	$q\to (q\vee r)$	[Ax3]
5.	$q\vee r$	[MP, 3, 4]

There is one premise $p\wedge q$, correctly marked with [Assumption]. The other steps in the derivation come from classical axioms or from MP. This meets all of the requirements for a classical derivation. It shows that the argument is classically derivable.

To give a complete Hilbert system for classical logic we need to fill in more axioms. We are not going to worry about completeness, but we will look at a few interesting classical axioms involving negation and (embedded) conditionals.

[Ax5]

$(p\to (p\to q))\to (p\to q)$

[Ax6]

$(p\to q)\to ((q\to r)\to (p\to r))$

[Ax7]

$p\to (q\to p)$

[Ax8]

$p\to (\mathrm{\neg }p\to q)$

[Ax9]

$(p\to \mathrm{\neg }q)\to (q\to \mathrm{\neg }p)$

[Ax10]

$(p\to \mathrm{\neg }p)\to \mathrm{\neg }p$

[Ax11]

$\mathrm{\neg }\mathrm{\neg }p\to p$

With Ax7 we can use any formula of the form “If $p$, then if $q$, then $p$,” and with Ax11, we can use any formula of the form “If not-not-$p$, then $p$.” The second principle is known as double negation elimination (DNE). What can we do with these axioms?

Here is a derivation to show that $p{⊢}_{\mathsf{C}\mathsf{L}}\mathrm{\neg }\mathrm{\neg }p$.

Example 7.03

1.	$p$	[Assumption]
2.	$(\mathrm{\neg }p\to (\mathrm{\neg }p\to \mathrm{\neg }p))\to (\mathrm{\neg }p\to \mathrm{\neg }p)$	[Ax5]
3.	$\mathrm{\neg }p\to (\mathrm{\neg }p\to \mathrm{\neg }p)$	[Ax7]
4.	$\mathrm{\neg }p\to \mathrm{\neg }p$	[MP, 1, 2]
5.	$(\mathrm{\neg }p\to \mathrm{\neg }p)\to (p\to \mathrm{\neg }\mathrm{\neg }p)$	[Ax9]
6.	$p\to \mathrm{\neg }\mathrm{\neg }p$	[MP, 4, 5]
7.	$\mathrm{\neg }\mathrm{\neg }p$	[MP, 1, 6]

This is the principle of double negation introduction (DNI). Since classical logic accepts both DNI and DNE, it treats “not-not-$p$” as logically equivalent to $p$. In other words, it considers these two sentences to be able to prove all the same things.

We can show that $p\wedge \mathrm{\neg }p{⊢}_{\mathsf{C}\mathsf{L}}q$ as follows.

Example 7.04

1.	$p\wedge \mathrm{\neg }p$	[Assumption]
2.	$(p\wedge \mathrm{\neg }p)\to p$	[Ax1]
3.	$p$	[MP, 1, 2]
4.	$p\to (\mathrm{\neg }p\to q)$	[Ax8]
5.	$\mathrm{\neg }p\to q$	[MP, 3, 4]
6.	$(p\wedge \mathrm{\neg }p)\to \mathrm{\neg }p$	[Ax2]
7.	$\mathrm{\neg }p$	[MP, 1, 6]
8.	$q$	[MP, 5, 7]

This is the principle of explosion, often expressed in Latin as ex contradictione quodlibet, which describes the inference pattern: from a contradiction, anything follows. Classical and intuitionistic logic both have explosive consequence relations, though not all logics possess this feature. Those who accept contradictions, but don’t want their theories to be trivial, or who do not think that everything follows from a contradiction, will instead opt for a logic with a nonexplosive consequence relation.

There is a another classical principle $p{⊢}_{\mathsf{C}\mathsf{L}}q\vee \mathrm{\neg }q$ that looks like explosion flipped upside down. This is the principle of excluded middle. It captures the idea that either $q$ or $\mathrm{\neg }q$ must hold come what may. The derivation of excluded middle is complicated. We will just sketch two components that make it work. For the first component, we can use Ax5–Ax9 to derive $\mathrm{\neg }(q\vee \mathrm{\neg }q)\to \mathrm{\neg }q$ and then $\mathrm{\neg }q\to \mathrm{\neg }\mathrm{\neg }(q\vee \mathrm{\neg }q)$, then use Ax6 and Ax10 to derive $\mathrm{\neg }\mathrm{\neg }(q\vee \mathrm{\neg }q)$. From here, Ax11 aka DNE gives us the conclusion $q\vee \mathrm{\neg }q$. For the second component, we observe that this reasoning only appeals to axioms so we can correctly repeat the same steps under any assumption. These two components explain why $q\vee \mathrm{\neg }q$ is derivable from any premise $p$ according to classical logic.

This particular example is a perfect derivation to highlight how intuitionistic logic and relevant logic diverge from classical logic. Both of these logics reject the derivation of excluded middle that we just explained, but for different reasons.

Intuitionistic logic is sometimes motivated by a background view on the nature of mathematics. L.E.J. Brouwer (Reference Brouwer1908) considered mathematics to depend on a mental activity that he called construction. For example, if you think about the concepts of a natural number and the less-than relation, pick any number, and keep reducing by one less, you will eventually reach $0$. This construction results in a specific object that demonstrates the theorem “there is a smallest natural number.” For a negative mathematical statement to hold $\mathrm{\neg }p$, we must be able to perform a special construction: We must be able to refute $p$ by showing that every method of trying to construct $p$ fails.

From this perspective, double negation elimination and excluded middle are questionable. Consider the following argument about a mathematical quantity used in the study of prime numbers known as Legendre’s constant (L). Remember that “irrational number” is just an abbreviation for the expression “not a rational number.”

You don’t need a lot of mathematics to understand this example. All you need to know is that (8.1) really does mathematically imply (8.2). Since the assumption “$\mathrm{\neg }$(L is rational)” leads to absurd results, the conclusion “$\mathrm{\neg }\mathrm{\neg }$(L is rational)” follows. Both classical logic and intuitionistic logic can agree on those steps.

Double negation elimination would allow us to go further and infer “L is rational,” but this is unwarranted from an intuitionistic perspective. Example 8 just shows that we can refute ($\mathrm{\neg }$) every method of trying to refute ($\mathrm{\neg }$) “L is rational.” That is different from giving a construction of “L is rational.” To reach that conclusion, we need to know the value of L, which our argument has not yet shown. These ideas about logical principles were implemented in a modern, formal system by Arendt Heyting (Reference Heyting1930).

We can build a Hilbert system for intuitionistic logic with the following axioms for negation and (embedded) conditionals. Notice, these are mostly the same as the set of classical axioms. The only difference is that we removed Ax11 aka DNE.

[Ax5]

$(p\to (p\to q))\to (p\to q)$

[Ax6]

$(p\to q)\to ((q\to r)\to (p\to r))$

[Ax7]

$p\to (q\to p)$

[Ax8]

$p\to (\mathrm{\neg }p\to q)$

[Ax9]

$(p\to \mathrm{\neg }q)\to (q\to \mathrm{\neg }p)$

[Ax10]

$(p\to \mathrm{\neg }p)\to \mathrm{\neg }p$

The symbol for derivability in intuitionistic logic ${⊢}_{\mathsf{I}\mathsf{L}}$ is defined as follows.

Definition. (Intuitionistic Derivability)

$p_1,p_2\dots {⊢}_{\mathsf{I}\mathsf{L}}q$ means that, in the Hilbert system for intuitionistic logic, it is possible to construct a derivation where each premise $p_i$ is marked with [Assumption], the other steps of the derivation come from the intuitionistic axioms or from MP, and the final step is the conclusion formula $q$.

Since intuitionistic logic accepts Ax1–Ax10 and otherwise has exactly the same rules as classical logic, many of the derivations we gave earlier are still acceptable. For example, we still have $p\wedge q{⊢}_{\mathsf{I}\mathsf{L}}q\vee r$. We still have double negation introduction $p{⊢}_{\mathsf{I}\mathsf{L}}\mathrm{\neg }\mathrm{\neg }p$ in intuitionistic logic. And we have the principle of explosion $p\wedge \mathrm{\neg }p{⊢}_{\mathsf{I}\mathsf{L}}q$. There is substantial overlap between intuitionistic logic and classical logic.

Unsurprisingly, the principle of excluded middle fails here $p{⊬}_{\mathsf{I}\mathsf{L}}q\vee \mathrm{\neg }q$. Remember that the classical derivation of this result has two components. The first component is a series of steps using only axioms. We can still use Ax5–Ax10 to derive $\mathrm{\neg }\mathrm{\neg }(q\vee \mathrm{\neg }q)$. However, we cannot derive $q\vee \mathrm{\neg }q$ because we can’t use DNE. That is why the derivation of excluded middle fails in intuitionistic logic. Any logical system that considers excluded middle to be invalid is called a paracomplete logic. Intuitionistic logic is the most famous example.

There is a systematic relationship between intuitionistic and classical logic. In a complete Hilbert system for intuitionistic logic, the rules are a proper subset of the rules for classical logic. The only difference is in the axioms. This means that any intuitionistic derivation has to be acceptable from the classical point of view:

$p_1,p_2,\dots {⊢}_{\mathsf{I}\mathsf{L}}q⟹p_1,p_2,\dots {⊢}_{\mathsf{C}\mathsf{L}}q.$

It does not work the other way around:

$p_1,p_2,\dots {⊢}_{\mathsf{C}\mathsf{L}}q̸⟹p_1,p_2,\dots {⊢}_{\mathsf{I}\mathsf{L}}q.$

That means that, in classical logic, we can often derive more conclusions from a certain set of premises assumptions than we can using intuitionistic logic. For this reason we say that intuitionistic logic is weaker than classical logic.

Finally, we will look at a Hilbert system for relevant logic. This type of logic is motivated by the view that explosion and excluded middle are degenerate forms of reasoning. These arguments look as if they pull unrelated sentences out of thin air. For example, here is an argument in plain English that has the form of explosion.

Imagine that Jan is on the fence about having lunch. He can’t make up his mind and he says, “I’m hungry, but then again I’m not.” Your friend overhears this and jumps in with the remark, “Ah ha! That implies that the moon is made of green cheese!” Classical logic tells us that that this argument is valid, but relevant logics disagree.

This example is kind of silly, but it highlights an important point. Although the premises of this argument contradict each other, it is even more noticeable that the conclusion talks about a topic that has nothing to do with those premises. How could a claim about the moon follow from claims that only talk about Jan? The inventors of relevant logic, Anderson and Belnap (Reference Anderson and Belnap1962, Reference Anderson and Belnap1975), claim that this kind of argument is invalid because the premises are irrelevant to the conclusion.

There are two aspects of a Hilbert system that formalize the notion of relevance. First, we go back to the classical axioms and remove Ax7 and Ax8 that generate classical derivations like $p,q{⊢}_{\mathsf{C}\mathsf{L}}p$. We replace these axioms with a much simpler Ax12.

[Ax5]

$(p\to (p\to q))\to (p\to q)$

[Ax6]

$(p\to q)\to ((q\to r)\to (p\to r))$

[Ax9]

$(p\to \mathrm{\neg }q)\to (q\to \mathrm{\neg }p)$

[Ax10]

$(p\to \mathrm{\neg }p)\to \mathrm{\neg }p$

[Ax11]

$\mathrm{\neg }\mathrm{\neg }p\to p$

[Ax12]

$p\to p$

Next, we will define an extra rule that basically says: Every assumption in a derivation must be used at some point, otherwise the derivation fails. All of this imposes a tight connection between sentences. The only acceptable derivations are those where the premises and conclusion have shared parts or shared content.

To formulate the extra rule for this Hilbert system, we can introduce a practice of flagging assumptions. A flag is a symbol. It could be anything. We attach a flag to each marked [Assumption]. In the following examples we use $\mathrm{♯}$ to stand for a flag. When we apply MP we pass any flags from inputs to outputs. The conclusion of a derivation must inherit a flag from each assumption, otherwise we have unused assumptions.

• Rule of Relevance: Each time you mark an [Assumption] you must flag it. Each time you use MP you must pass any flags from input formulas to the output formula. The final step of a derivation must have a flag from each [Assumption].

The symbol for derivability in relevant logic ${⊢}_{\mathsf{R}\mathsf{L}}$ is defined as follows.

Definition. (Relevant Derivability)

$p_1,p_2\dots {⊢}_{\mathsf{R}\mathsf{L}}q$ means that, in the Hilbert system for relevant logic, it is possible to construct a derivation where each premise $p_i$ is marked with [Assumption], the other steps of the derivation come from the relevant axioms or from MP, the final step is the conclusion formula $q$, and the whole derivation satisfies the rule of relevance.

Here is a relevant derivation of $p\wedge q{⊢}_{\mathsf{R}\mathsf{L}}q\vee r$.

Example 9.01

1.	$p\wedge q$	[Assumption]	$\mathrm{♯}$
2.	$(p\wedge q)\to q$	[Ax2]
3.	$q$	[MP, 1, 2]	$\mathrm{♯}$
4.	$q\to (q\vee r)$	[Ax3]
5.	$q\vee r$	[MP, 3, 4]	$\mathrm{♯}$

This looks almost exactly like the classical derivation. All of the axioms used in this derivation are accepted by both relevant and classical logic. The only extra detail is the column that keeps track of flags, which shows us that the derivation satisfies the rule of relevance, so it is completely acceptable in relevant logic.

Here is a relevant derivation of $p{⊢}_{\mathsf{R}\mathsf{L}}\mathrm{\neg }\mathrm{\neg }p$.

Example 9.02

1.	$p$	[Ax12]	$\mathrm{♯}$
2.	$\mathrm{\neg }p\to \mathrm{\neg }p$	[Ax12]
3.	$(\mathrm{\neg }p\to \mathrm{\neg }p)\to (p\to \mathrm{\neg }\mathrm{\neg }p)$	[Ax8]
4.	$p\to \mathrm{\neg }\mathrm{\neg }p$	[MP, 1, 2]	$\mathrm{♯}$
5.	$\mathrm{\neg }\mathrm{\neg }p$	[MP, 1, 2]	$\mathrm{♯}$

Although the steps are slightly different from the classical derivation, the end result is the same. The flags show us that this derivation satisfies the rule of relevance. So, relevant logic accepts both DNI and DNE. It treats “not-not-$p$” as equivalent to $p$.

The classical derivation of explosion used Ax8, but there is no way to reach a similar result in relevant logic $p\wedge \mathrm{\neg }p{⊬}_{\mathsf{R}\mathsf{L}}q$. Any logical system that considers explosion to be invalid is called a paraconsistent logic. Relevant logic is the most famous example.

More interesting is why excluded middle fails $p{⊬}_{\mathsf{R}\mathsf{L}}q\vee \mathrm{\neg }q$. We used two components to explain the classical derivation. First, we explained how to derive $q\vee \mathrm{\neg }q$ using axioms. Second, we notice that it is innocent to repeat those derivation steps under any assumption. In relevant logic, the second component is wrong. Suppose we start with a flagged assumption $p$. Then we introduce a bunch of axioms like $q\to (q\vee \mathrm{\neg }q)$ or $\mathrm{\neg }(q\vee \mathrm{\neg }q)\to \mathrm{\neg }(q\vee \mathrm{\neg }q)$. This eventually leads to $q\vee \mathrm{\neg }q$, but the final step only depended on the $q$-based axioms and not on the initial assumption $p$. The flag from $p$ will never be passed down, so this “attempted” derivation violates the rule of relevance.

In a complete Hilbert system for relevant logic, the axioms are a proper subset of the axioms for classical logic (Ax12 can be derived from Ax5 and Ax7). The main difference is the extra rule of relevance, which filters out a bunch of classical derivations. This means that any relevant derivation has to be acceptable from the classical point of view:

$p_1,p_2,\dots {⊢}_{\mathsf{R}\mathsf{L}}q⟹p_1,p_2,\dots {⊢}_{\mathsf{C}\mathsf{L}}q.$

It does not work the other way around:

$p_1,p_2,\dots {⊢}_{\mathsf{C}\mathsf{L}}q̸⟹p_1,p_2,\dots {⊢}_{\mathsf{R}\mathsf{L}}q.$

That means that, in classical logic, we can often derive more conclusions from a certain set of premises assumptions than we can using relevant logic. For this reason we say that relevant logic is weaker than classical logic.

That wraps up our discussion of proof theory for now. Hopefully, this section clearly illustrated the kind of methods that are unique to proof theory. You can find a summary of key result in Section 1.5. We will now turn to a different formal technique.

1.4 Model Theory

In this section, we will briefly present the model theory for classical, intuitionistic, and relevant propositional logics. On the model-theoretic approach, logic has an important connection with external phenomena like truth or verification.

Model theory gives a precise definition of abstract structures that we call semantic models. Each model represents a circumstance where someone might be reasoning. Formulas receive semantic values: $1$ stands for true or verified and $0$ stands for false or unverified. If it always happens (in every model) that when the premises $p_1,p_2,\dots$ receive value $1$, the conclusion $q$ also receives value $1$, we write something like the following:

$p_1,p_2,\dots \models q.$

The double turnstile $\models$ is a symbol for the model-theoretic definition of validity. The displayed string says “from $p_1,p_2,\dots$ it is valid to infer $q$.” However, different logics give different theories of validity. So, we add a subscript $\mathsf{L}$ to the turnstile ${\models }_{\mathsf{L}}$ and read this as saying “… it is valid according to logic $\mathsf{L}$….”

Model theory treats validity as the preservation of truth or verification. On this approach, valid arguments are airtight because the conclusion “automatically” comes out true whenever the premises are true. This is why valid reasoning cannot go wrong.

In proof theory, the choice of axioms and rules made all the difference. Model theory has a similar feature, which we call evaluation policies (and supporting “apparatus”). This is sometimes connected to philosophy of mind and language, where sentence meaning has been identified as a set of truth-conditions (Heim and Kratzer, Reference Heim and Kratzer1998). The idea behind truth-conditional semantics is that certain details determine the semantic values of complex sentences in any circumstance. However, to pose a challenging question: What are the circumstances in which a conditional sentence “If $p$, then $q$” is true? The value of $p$ might be important and the value of $q$ might be important, but is that enough information about a circumstance to tell us whether the conditional sentence is true? Different model theories offer very different answers to this question.

Classical logic has a truth-functional model theory. A model $M$ is literally built from a mathematical function $[[.]{]}^M$ that maps certain inputs to outputs. The input side is always a formula and the output side is one of the semantic values, $1$ or $0$. The outputs represent the facts – what is true and what is false – in the circumstance represented by $M$:

$[[p]{]}^M=1$	says that $p$ is true in $M$.
$[[p]{]}^M=0$	says that $p$ is false in $M$.

The evaluation policies for complex formulas are also based on this simple representation of facts in a model. Classical models, thus, suggest the idea that “truth here and now” is the only thing about a circumstance that matters to logic.

When we state evaluation policies we write “iff” as an abbreviation for the ordinary English phrase “if and only if,” which is an expression of definitional equivalence. The following policies tell us how conjunction and disjunction formulas are evaluated in classical logic. Much like we saw with proof theory, the classical treatment of conjunction and disjunction will carry over to other logics in this section.

Definition. (Classical Evaluation of $\wedge$ and $\vee$)

For any formula built from $\wedge$ or $\vee$, a classical model $M$ must assign semantic values based on the following policies:

$[[p\wedge q]{]}^M=1$	iff	both $[[p]{]}^M=1$ and $[[q]{]}^M=1$.
$[[p\vee q]{]}^M=1$	iff	either $[[p]{]}^M=1$ or $[[q]{]}^M=1$.

We can read these policies as follows: A true conjunction requires both of its parts to be true, whereas a true disjunction requires only one of its parts to be true.

Once we have defined the range of classical models, then the symbol for semantically valid arguments in classical logic ${\models }_{\mathsf{C}\mathsf{L}}$ can be defined as follows:

Definition. (Classical Semantic Validity)

$p_1,p_2\dots {\models }_{\mathsf{C}\mathsf{L}}q$ means that, in all classical models $M$, if $[[p_1]{]}^M=1$ and $[[p_2]{]}^M=1$ and…, then $[[q]{]}^M=1$.

For example, based on just the preceding details, we can easily see that $p\wedge q{\models }_{\mathsf{C}\mathsf{L}}q\vee r$. Just think about how the evaluation policies work in classical models. Consider any classical model $M$ where $[[p\wedge q]{]}^M=1$. Then we must have $[[q]{]}^M=1$ by the conjunction policy. So, we must have $[[q\vee r]{]}^M=1$ by the disjunction policy.

The most interesting question is how to handle negation and the conditional. For negation, we swap values. In any classical model, “not-$p$” has the opposite value from “p.” For the conditional, we adopt something called the material conditional. According to this concept, the conditional sentence “if $p$, then $q$” has the same value as “not-$p$ or $q$.”

The material conditional treats an “if” statement as a statement purely about the facts here and now. Are there good reasons to do this? Here is an argument: The only type of circumstance that obviously falsifies “if $p$, then $q$” is a circumstance where $p$ is true and $q$ is false, so any other circumstance must make the sentence “if $p$, then $q$” true. If we accept that argument, then we will accept the following classical evaluation policies.

Definition. (Classical Evaluation of $\mathrm{\neg }$ and $\to$)

For any formula built from $\mathrm{\neg }$ or $\to$, a classical model $M$ must assign semantic values based on the following policies:

$[[\mathrm{\neg }p]{]}^M=1$	iff	$[[p]{]}^M=0$.
$[[p\to q]{]}^M=1$	iff	either $[[p]{]}^M=0$ or $[[q]{]}^M=1$.

With these definitions in place, we can check several argument forms we have not seen before. The following are called DeMorgan principles: $\mathrm{\neg }(p\wedge q){\models }_{\mathsf{C}\mathsf{L}}\mathrm{\neg }p\vee \mathrm{\neg }q$  and  $\mathrm{\neg }p\wedge \mathrm{\neg }q{\models }_{\mathsf{C}\mathsf{L}}\mathrm{\neg }(p\vee q)$  and  $\mathrm{\neg }(p\vee q){\models }_{\mathsf{C}\mathsf{L}}\mathrm{\neg }p\wedge \mathrm{\neg }q$  and  $\mathrm{\neg }p\vee \mathrm{\neg }q{\models }_{\mathsf{C}\mathsf{L}}\mathrm{\neg }(p\wedge q)$.

Here is an explanation for the first DeMorgan principle. Take any classical model $M$. Suppose that $[[\mathrm{\neg }(p\wedge q)]{]}^M=1$. Then we have $[[p\wedge q]{]}^M=0$. And that means either $[[p]{]}^M=0$ or $[[p]{]}^M=0$. So, we have $[[\mathrm{\neg }p]{]}^M=1$ or $[[\mathrm{\neg }q]{]}^M=1$. Thus, $[[\mathrm{\neg }p\vee \mathrm{\neg }q]{]}^M=1$. Similar explanations work for the other DeMorgan principles.

Each classical axiom in Section 1.3 is true in every classical model. For example, you can check for yourself that we always have $[[(p\to (p\to q))\to$ $(p\to q)]{]}^M=1$ and $[[p\to (q\to p)]{]}^M=1$ and $[[(p\to \mathrm{\neg }p)\to \mathrm{\neg }p]{]}^M=1$ in every model $M$.

If we look at the classically derivable arguments in Section 1.3, we can see that they are all semantically valid as well. For example, double negation elimination $\mathrm{\neg }\mathrm{\neg }p{\models }_{\mathsf{C}\mathsf{L}}p$, double negation introduction $p{\models }_{\mathsf{C}\mathsf{L}}\mathrm{\neg }\mathrm{\neg }p$, the principles of explosion $p\wedge \mathrm{\neg }p{\models }_{\mathsf{C}\mathsf{L}}q$, and excluded middle $p{\models }_{\mathsf{C}\mathsf{L}}q\vee \mathrm{\neg }q$. An invalid argument requires at least one model where the premises are true and the conclusion is untrue (a countermodel). The classical definition of negation guarantees that $[[p\wedge \mathrm{\neg }p]{]}^M=0$ in every model $M$ so there is no countermodel for explosion. The definition of negation also guarantees that $[[q\vee \mathrm{\neg }q]{]}^M=1$ in every model $M$ so there is no countermodel for excluded middle. With no countermodel, these arguments are considered valid in classical logic.

This is a good example to highlight how intuitionistic logic and relevant logic diverge from classical logic in their model theory just as much as their proof theory. Both of these logics reject excluded middle, but for different reasons.

As mentioned in Section 1.3, a historic motivation for intuitionistic logic was the idea that mathematics depends on creative thinking (mental construction). This idea has been modified and extended into a more general account of truth that can hold in many domains (Dummett, Reference Dummett1959; Wright, Reference Wright1992). On this view, truth is said to be epistemically constrained: In some sense, what we cannot know cannot be true. Logical principles like excluded middle fail because negation is tied to such epistemic limitations.

We can build models that capture these intuitionistic ideas, but we need a more complex apparatus than before. Each model comes with a set of items $@,s_1,s_2,s_3,\dots$ that represent potential stages in a creative thinking process. $@$ is the current stage of knowledge. There is a function $[[.]{]}_s^M$ for each stage $s$ that maps formulas to $1$ or $0$. The output tells us the facts – what is verified and what is unverified – at each individual stage within the overall picture represented by model $M$:

$[[p]{]}_s^M=1$	says that $p$ is verified-at-$s$ in $M$.
$[[p]{]}_s^M=0$	says that $p$ is unverified-at-$s$ in $M$.

The evaluation policies for conjunction and disjunction look familiar.

Definition. (Intuitionistic Evaluation of $\wedge$ and $\vee$)

For any formula built from $\wedge$ or $\vee$, an intuitionistic model $M$ must assign semantic values at each stage $s$ based on the following policies:

$[[p\wedge q]{]}_s^M=1$	iff	both $[[p]{]}_s^M=1$ and $[[q]{]}_s^M=1$.
$[[p\wedge q]{]}_s^M=1$	iff	either $[[p]{]}_s^M=1$ or $[[q]{]}_s^M=1$.

Importantly, stages are related to one another. $s_1\prec s_2$ means that $s_1$ is included in $s_2$. This is meant to portray how knowledge can develop across potential stages, but always builds upon itself. In more abstract terms, we say that inclusion is:

• Pointed: Every stage includes the actual stage. (We always have $@\prec s$.)

• Reflexive: Every stage includes itself. (We always have $s\prec s$.)

• Transitive: Inclusions persist over inclusions. (If $s_1\prec s_2$ and $s_2\prec s_3$, then $s_1\prec s_3$.)

• Cumulative: Everything that is verified at an earlier stage continues to be verified at later stages in which it is included. (If $[[p]{]}_{s_1}^M=1$ and $s_1\prec s_2$, then $[[p]{]}_{s_2}^M=1$.)

This relation forms a tree-like structure branching out from $@$, where every stage includes the facts that are known at $@$, but it can also add more facts. This suggests the notion of “truth unfolding through a creative thinking process” as a means of understanding logical properties. Notice, a single model or single picture of a circumstance now includes a whole structure of stages. This is important to negation and conditional formulas.

Definition. (Intuitionistic Evaluation of $\mathrm{\neg }$ and $\to$)

For any formula built from $\mathrm{\neg }$ or $\to$, an intuitionistic model $M$ must assign semantic values at each stage $s$ based on the following policies:

$[[\mathrm{\neg }p]{]}_s^M=1$	iff	for all $s^{\prime }$ such that $s\prec s^{\prime }$, we have $[[p]{]}_{s^{\prime }}^M=0$.
$[[p\to q]{]}_s^M=1$	iff	for all $s^{\prime }$ such that $s\prec s^{\prime }$, if $[[p]{]}_{s^{\prime }}^M=1$, then $[[q]{]}_{s^{\prime }}^M=1$.

Informally, $\mathrm{\neg }p$ is verified when there is no further, potential stage that verifies $p$. So, negation expresses refutability. The conditional $p\to q$ is verified when, at all further, potential stages of knowledge, if $p$ is verified, then $q$ is verified.

The symbol for intuitionistically valid arguments ${\models }_{\mathsf{I}\mathsf{L}}$ is defined as follows.

Definition. (Intuitionistic Semantic Validity)

$p_1,p_2\dots {\models }_{\mathsf{I}\mathsf{L}}q$ means that, in all intuitionistic models $M$ and for all stages of the model $s$, if $[[p_1]{]}_s^M=1$ and $[[p_2]{]}_s^M=1$ and…, then $[[q]{]}_s^M=1$.

We can check a few important results of this definition.

Each of the intuitionistic axioms in Section 1.3 is verified at all stages of intuitionistic models. This relies on features of the inclusion relation. For example, $[[p\to (q\to p)]{]}_s^M=1$ always holds because of the cumulative feature of inclusion.

If we look at the intuitionisticly derivable arguments in Section 1.3, they are all semantically valid here. For example, the principle of explosion $p\wedge \mathrm{\neg }p{\models }_{\mathsf{I}\mathsf{L}}q$ holds because the premises can never be verified. The principle of double negation introduction $p{\models }_{\mathsf{I}\mathsf{L}}\mathrm{\neg }\mathrm{\neg }p$ may not be obvious. Consider this explanation. Take any model with $s$ where $[[p]{]}_s^M=1$, then $\mathrm{\neg }p$ is forever unverified, which means that $[[\mathrm{\neg }\mathrm{\neg }p]{]}_s^M=1$. Why is $\mathrm{\neg }p$ forever unverified? When $[[p]{]}_s^M=1$ and $s\prec s^{\prime }$ we get $[[p]{]}_{s^{\prime }}^M=1$ at all future stages, by reflexivity they all see that $p$ is not refutable, thus $[[\mathrm{\neg }p]{]}_{s^{\prime }}^M=0$ at all future stages.

The following diagram of an intuitionistic model helps us to see how this semantics invalidates DNE and excluded middle. Each dot is a stage. $@$ is on the far left. The formulas written above a stage are verified at that stage. The inclusion relation follows the arrows. In this model, $p$ is not yet verified by the initial stage $[[p]{]}_{@}^M=0$, but there is some later evolution of thought where $p$ eventually becomes verified.

Example 9.03

Figure 1.01

Call the ultimate stage $s$ where $[[p]{]}_s^M=1$. Because $@\prec s$, we have $[[\mathrm{\neg }p]{]}_{@}^M=0$. Since $[[p]{]}_{@}^M=0$, we have $[[p\vee \mathrm{\neg }p]{]}_{@}^M=0$. This is a countermodel to excluded middle $q{⊭}_{\mathsf{I}\mathsf{L}}p\vee \mathrm{\neg }p$. This model represents a circumstance where a potential, linear expansion of verified facts undermines the claim that $p$ must be verified or refuted (it is neither).

The preceding model exhibits a form of coherent incompleteness. It portrays the kind of circumstance where the current stage of knowledge “refutes the refutation of $p$” without verifying $p$. If we think that this portrays a genuine possible circumstance of reasoning, then we might agree with the intuitionistic logical principles.

A somewhat surprising feature of intuitionistic logic is that one of the DeMorgan principles fails in this logic: $\mathrm{\neg }(p\wedge q){⊭}_{\mathsf{I}\mathsf{L}}\mathrm{\neg }p\vee \mathrm{\neg }q$. For a countermodel, just imagine a circumstance where there is a potential expansion of verified facts in two different branches. $p$ is not yet verified but eventually becomes verified at the end of one branch, and $q$ is not yet verified but eventually becomes verified at the end of a different branch. This results in $[[\mathrm{\neg }(p\vee q)]{]}_{@}^M=1$ and $[[\mathrm{\neg }p]{]}_{@}^M=0$ and $[[\mathrm{\neg }q]{]}_{@}^M=0$.

Finally, we will discuss a model theory for relevant logic. We will present a semantic framework where the relevance requirement is a byproduct of information flow between states of affairs (Restall, Reference Restall, Seligman and Westerståhl1996; Mares, Reference Mares2004).

One of the tools for this semantics is that we treat $[[.]]$ as a loose relation instead of a strict function. Think of $1$ and $0$ as independent properties. The fact that there is no elephant in my apartment might be an example of a primitive, negative “non-elephant-ish” fact. It is independent from a list of all the positive facts about my apartment: the floors are made of wood, the walls are green, the windows are large, etc. If we use semantic values in this way, then a specific formula $p$ might get the value $1$, it might get $0$, it might not get any value, it might even get both values at once. These options might represent something like incomplete or inconsistent entries in a database (Belnap, Reference Belnap, Anderson, Belnap and Dunn1992).

Using this tool, we will build up relevant models. Each model comes with a set of items $@,s_1,s_2,s_3,\dots$ that represent potential states of affairs. $@$ is the actual state. There is a loose relation $[[.]{]}_s^M$ between each $s$ and the semantic values $1$ and $0$:

$[[p]{]}_s^M\cong 1$	says	that $p$ is true-at-$s$ in $M$.
$[[p]{]}_s^M\cong 0$	says	that $p$ is false-at-$s$ in $M$.

The symbol $\cong$ means “at least has this value.” The evaluation policies for conjunction and disjunction are familiar, but we have to write separate policies for each possible value.

Definition. (Relevant Evaluation of $\wedge$ and $\vee$)

For any formula built from $\wedge$ or $\vee$, a relevant model $M$ must assign semantic values at each state $s$ based on the following policies:

$[[p\wedge q]{]}_s^M\cong 1$	iff	both $[[p]{]}_s^M\cong 1$ and $[[q]{]}_s^M\cong 1$.
$[[p\wedge q]{]}_s^M\cong 0$	iff	either $[[p]{]}_s^M\cong 0$ or $[[q]{]}_s^M\cong 0$.
$[[p\vee q]{]}_s^M\cong 1$	iff	either $[[p]{]}_s^M\cong 1$ or $[[q]{]}_s^M\cong 1$.
$[[p\vee q]{]}_s^M\cong 0$	iff	both $[[p]{]}_s^M\cong 0$ and $[[q]{]}_s^M\cong 0$.

States are related in relevant models. $s_1{s}_2⊐s_3$ means that the information in $s_1$ and $s_2$ is realized in $s_3$. This portrays how states can be joined together to generate information that was not available in them separately.

Consider a realistic example: There was a state where Antoni Gaudí came up with a design such that, if it were funded, the Sagrada Família would be the most beautiful church in Barcelona, and there was a state in which the project was eventually funded, but the combination of these states was only realized later when the Sagrada Família was built and it really did became the most beautiful church in Barcelona. This productive relation creates $n$-chains of information with the following pattern:

$s_1{s}_2⊐x_1\text{ and }{x}_1{s}_3⊐x_2\text{ and ldots and }{x}_{(n-1)}{s}_n⊐s_{(n+1)}.$

When we have this kind of pattern in a model, we call the $s$ parts sites and we call the $x$ parts channels that together form an information chain from $s_1$ to $s_{(n+1)}$. Now, in abstract terms, we say that the realization relation is:

• Pointed: $@$ does not alter the information of any state ($@s⊐s$).

• Commutative: the realization of two joined states does not depend on the order in which those states are joined together (whenever $s_1{s}_2⊐s_3$, then $s_2{s}_1⊐s_3$).

• Extensible: realization chains can always be extended through an intermediary channel (whenever $s_1{s}_2⊐s_3$, there is some $x$ such that $s_1{s}_2⊐x$ and $x{s}_2⊐s_3$).

This suggests the notion of “truth across information chains” as a means of understanding logical properties. Notice, a single model or single picture of a circumstance now includes a whole structure of states. This is important to negation and conditional formulas.

Definition. (Relevant Evaluation of $\mathrm{\neg }$ and $\to$)

For any formula built from $\mathrm{\neg }$ or $\to$, a relevant model $M$ must assign semantic values at each state, $s$, based on the following policies:

	$[[\mathrm{\neg }p]{]}_s^M\cong 1$	iff	$[[p]{]}_s^M\cong 0$.
	$[[\mathrm{\neg }p]{]}_s^M\cong 0$	iff	$[[p]{]}_s^M\cong 1$.
	$[[p\to q]{]}_s^M\cong 1$	iff	for all $s^{\prime },s^{\mathrm{\prime }\mathrm{\prime }}$ such that $s{s}^{\prime }⊐s^{\mathrm{\prime }\mathrm{\prime }}$, if $[[p]{]}_{s^{\prime }}^M\cong 1$, then $[[q]{]}_{s^{\mathrm{\prime }\mathrm{\prime }}}^M\cong 1$.
	$[[p\to q]{]}_s^M\cong 0$	iff	for some $s^{\prime },s^{\mathrm{\prime }\mathrm{\prime }}$ we have $s^{\prime }{s}^{\mathrm{\prime }\mathrm{\prime }}⊐s$ and $[[p]{]}_{s^{\prime }}^M\cong 1$ and $[[q]{]}_{s^{\mathrm{\prime }\mathrm{\prime }}}^M\cong 0$.

Informally, $\mathrm{\neg }p$ is true when $p$ is false, and vice versa. This is extremely similar to the classical definition. The relevant conditional $p\to q$ is evaluated by the realization relation. This makes perfect sense based on what we said earlier: $p\to q$ is true in the kind of states that can always be combined with $p$-states to realize $q$-states.

The symbol for relevantly valid arguments ${\models }_{\mathsf{R}\mathsf{L}}$ is defined as follows.