Classical Logic

1. Introduction

Today, logic is a branch of mathematics and a branch of philosophy. In most large universities, both departments offer courses in logic, and there is usually a lot of overlap between them. Formal languages, deductive systems, and model-theoretic semantics are mathematical objects and, as such, the logician is interested in their mathematical properties and relations. Soundness, completeness, and most of the other results reported below are typical examples. Philosophically, logic is at least closely related to the study of correct reasoning. Reasoning is an epistemic, mental activity. So logic is at least closely allied with epistemology. Logic is also a central branch of computer science, due, in part, to interesting computational relations in logical systems, and, in part, to the close connection between formal deductive argumentation and reasoning (see the entries on recursive functions, computability and complexity, and philosophy of computer science).

This raises questions concerning the philosophical relevance of the various mathematical aspects of logic. How do deducibility and validity, as properties of formal languages – sets of strings on a fixed alphabet – relate to correct reasoning? What do the mathematical results reported below have to do with the original philosophical issues concerning valid reasoning? This is an instance of the philosophical problem of explaining how mathematics applies to non-mathematical reality.

Typically, ordinary deductive reasoning takes place in a natural language, or perhaps a natural language augmented with some mathematical symbols. So our question begins with the relationship between a natural language and a formal language. Without attempting to be comprehensive, it may help to sketch several options on this matter.

One view is that the formal languages accurately exhibit actual features of certain fragments of a natural language. Some philosophers claim that declarative sentences of natural language have underlying logical forms and that these forms are displayed by formulas of a formal language. Other writers hold that (successful) declarative sentences express propositions; and formulas of formal languages somehow display the forms of these propositions. On views like this, the components of a logic provide the underlying deep structure of correct reasoning. A chunk of reasoning in natural language is correct if the forms underlying the sentences constitute a valid or deducible argument. See for example, Montague [1974], Davidson [1984], Lycan [1984] (and the entry on logical form).

Another view, held at least in part by Gottlob Frege and Wilhelm Leibniz, is that because natural languages are fraught with vagueness and ambiguity, they should be replaced by formal languages. A similar view, held by W. V. O. Quine (e.g., [1960], [1986]), is that a natural language should be regimented, cleaned up for serious scientific and metaphysical work. One desideratum of the enterprise is that the logical structures in the regimented language should be transparent. It should be easy to “read off” the logical properties of each sentence. A regimented language is similar to a formal language regarding, for example, the explicitly presented rigor of its syntax and its truth conditions.

On a view like this, deducibility and validity represent idealizations of correct reasoning in natural language. A chunk of reasoning is correct to the extent that it corresponds to, or can be regimented by, a valid or deducible argument in a formal language.

When mathematicians and many philosophers engage in deductive reasoning, they occasionally invoke formulas in a formal language to help disambiguate, or otherwise clarify what they mean. In other words, sometimes formulas in a formal language are used in ordinary reasoning. This suggests that one might think of a formal language as an addendum to a natural language. Then our present question concerns the relationship between this addendum and the original language. What do deducibility and validity, as sharply defined on the addendum, tell us about correct deductive reasoning in general?

Another view is that a formal language is a mathematical model of a natural language in roughly the same sense as, say, a collection of point masses is a model of a system of physical objects, and the Bohr construction is a model of an atom. In other words, a formal language displays certain features of natural languages, or idealizations thereof, while ignoring or simplifying other features. The purpose of mathematical models is to shed light on what they are models of, without claiming that the model is accurate in all respects or that the model should replace what it is a model of. On a view like this, deducibility and validity represent mathematical models of (perhaps different aspects of) correct reasoning in natural languages. Correct chunks of deductive reasoning correspond, more or less, to valid or deducible arguments; incorrect chunks of reasoning roughly correspond to invalid or non-deducible arguments. See, for example, Corcoran [1973], Shapiro [1998], and Cook [2002].

There is no need to adjudicate this matter here. Perhaps the truth lies in a combination of the above options, or maybe some other option is the correct, or most illuminating one. We raise the matter only to lend some philosophical perspective to the formal treatment that follows.

2. Language

Here we develop the basics of a formal language, or to be precise, a class of formal languages. Again, a formal language is a recursively defined set of strings on a fixed alphabet. Some aspects of the formal languages correspond to, or have counterparts in, natural languages like English. Technically, this “counterpart relation” is not part of the formal development, but we will mention it from time to time, to motivate some of the features and results.

2.1 Building blocks

We begin with analogues of singular terms, linguistic items whose function is to denote a person or object. We call these terms. We assume a stock of individual constants. These are lower-case letters, near the beginning of the Roman alphabet, with or without numerical subscripts:

We envisage a potential infinity of individual constants. In the present system each constant is a single character, and so individual constants do not have an internal syntax. Thus we have an infinite alphabet. This could be avoided by taking a constant like \(d_{22}\), for example, to consist of three characters, a lowercase “\(d\)” followed by a pair of subscript “2”s.

We also assume a stock of individual variables. These are lower-case letters, near the end of the alphabet, with or without numerical subscripts:

In ordinary mathematical reasoning, there are two functions terms need to fulfill. We need to be able to denote specific, but unspecified (or arbitrary) objects, and sometimes we need to express generality. In our system, we use some constants in the role of unspecified reference and variables to express generality. Both uses are recapitulated in the formal treatment below. Some logicians employ different symbols for unspecified objects (sometimes called “individual parameters”) and variables used to express generality.

Constants and variables are the only terms in our formal language, so all of our terms are simple, corresponding to proper names and some uses of pronouns. We call a term closed if it is not a variable. In general, we use \(v\) to represent variables, and \(t\) to represent a closed term, an individual constant. Some authors also introduce function letters, which allow complex terms corresponding to: “\(7+4\)” and “the wife of Bill Clinton”, or complex terms containing variables, like “the father of \(x\)” and “\(x/y\)”. Logic books aimed at mathematicians are likely to contain function letters, probably due to the centrality of functions in mathematical discourse. Books aimed at a more general audience (or at philosophy students), may leave out function letters, since it simplifies the syntax and theory. We follow the latter route here. This is an instance of a general tradeoff between presenting a system with greater expressive resources, at the cost of making its formal treatment more complex.

For each natural number \(n\), we introduce a stock of \(n\)-place predicate letters. These are upper-case letters at the beginning or middle of the alphabet. A superscript indicates the number of places, and there may or may not be a subscript. For example,

are three-place predicate letters. We often omit the superscript, when no confusion will result. We also add a special two-place predicate symbol “\(=\)” for identity.

Zero-place predicate letters are sometimes called “sentence letters”. They correspond to free-standing sentences whose internal structure does not matter. One-place predicate letters, called “monadic predicate letters”, correspond to linguistic items denoting properties, like “being a man”, “being red”, or “being a prime number”. Two-place predicate letters, called “binary predicate letters”, correspond to linguistic items denoting binary relations, like “is a parent of” or “is greater than”. Three-place predicate letters correspond to three-place relations, like “lies on a straight line between”. And so on.

The non-logical terminology of the language consists of its individual constants and predicate letters. The symbol “\(=\)”, for identity, is not a non-logical symbol. In taking identity to be logical, we provide explicit treatment for it in the deductive system and in the model-theoretic semantics. Most authors do the same, but there is some controversy over the issue (Quine [1986, Chapter 5]). If \(K\) is a set of constants and predicate letters, then we give the fundamentals of a language \(\LKe\) built on this set of non-logical terminology. It may be called the first-order language with identity on \(K\). A similar language that lacks the symbol for identity (or which takes identity to be non-logical) may be called \(\mathcal{L}1K\), the first-order language without identity on \(K\).

2.2 Atomic formulas

If \(V\) is an \(n\)-place predicate letter in \(K\), and \(t_1, \ldots,t_n\) are terms of \(K\), then \(Vt_1 \ldots t_n\) is an atomic formula of \(\LKe\). Notice that the terms \(t_1, \ldots,t_n\) need not be distinct. Examples of atomic formulas include:

The last one is an analogue of a statement that a certain relation \((A)\) holds between three objects \((a, b, c)\). If \(t_1\) and \(t_2\) are terms, then \(t_1 =t_2\) is also an atomic formula of \(\LKe\). It corresponds to an assertion that \(t_1\) is identical to \(t_2\).

If an atomic formula has no variables, then it is called an atomic sentence. If it does have variables, it is called open. In the above list of examples, the first and second are open; the rest are sentences.

2.3 Compound formulas

We now introduce the final items of the lexicon:

We give a recursive definition of a formula of \(\LKe\):

A formula corresponding to \(\neg \theta\) thus says that it is not the case that \(\theta\). The symbol “\(\neg\)” is called “negation”, and is a unary connective.

The ampersand “\(\amp\)” corresponds to the English “and” (when “and” is used to connect sentences). So \((\theta \amp \psi)\) can be read “\(\theta\) and \(\psi\)”. The formula \((\theta \amp \psi)\) is called the “conjunction” of \(\theta\) and \(\psi\).

The symbol “\(\vee\)” corresponds to “either … or … or both”, so \((\theta \vee \psi)\) can be read “\(\theta\) or \(\psi\)”. The formula \((\theta \vee \psi)\) is called the “disjunction” of \(\theta\) and \(\psi\).

The arrow “\(\rightarrow\)” roughly corresponds to “if … then … ”, so \((\theta \rightarrow \psi)\) can be read “if \(\theta\) then \(\psi\)” or “\(\theta\) only if \(\psi\)”.

The symbols “\(\amp\)”, “\(\vee\)”, and “\(\rightarrow\)” are called “binary connectives”, since they serve to “connect” two formulas into one. Some authors introduce \((\theta \leftrightarrow \psi)\) as an abbreviation of \(((\theta \rightarrow \psi) \amp(\psi \rightarrow \theta))\). The symbol “\(\leftrightarrow\)” is an analogue of the locution “if and only if”.

The symbol “\(\forall\)” is called a universal quantifier, and is an analogue of “for all”; so \(\forall v\theta\) can be read “for all \(v, \theta\)”.

The symbol “\(\exists\)” is called an existential quantifier, and is an analogue of “there exists” or “there is”; so \(\exists v \theta\) can be read “there is a \(v\) such that \(\theta\)”.

Clause (8) allows us to do inductions on the complexity of formulas. If a certain property holds of the atomic formulas and is closed under the operations presented in clauses (2)–(7), then the property holds of all formulas. Here is a simple example:

We next define the notion of an occurrence of a variable being free or bound in a formula. A variable that immediately follows a quantifier (as in “\(\forall x\)” and “\(\exists y\)”) is neither free nor bound. We do not even think of those as occurrences of the variable. All variables that occur in an atomic formula are free. If a variable occurs free (or bound) in \(\theta\) or in \(\psi\), then that same occurrence is free (or bound) in \(\neg \theta, (\theta \amp \psi), (\theta \vee \psi)\), and \((\theta \rightarrow \psi)\). That is, the (unary and binary) connectives do not change the status of variables that occur in them. All occurrences of the variable \(v\) in \(\theta\) are bound in \(\forall v \theta\) and \(\exists v \theta\). Any free occurrences of \(v\) in \(\theta\) are bound by the initial quantifier. All other variables that occur in \(\theta\) are free or bound in \(\forall v \theta\) and \(\exists v \theta\), as they are in \(\theta\).

For example, in the formula \((\forall\)x(Axy \(\vee Bx) \amp Bx)\), the occurrences of “\(x\)” in Axy and in the first \(Bx\) are bound by the quantifier. The occurrence of “\(y\)” and last occurrence of “\(x\)” are free. In \(\forall x(Ax \rightarrow \exists\)xBx), the “\(x\)” in \(Ax\) is bound by the initial universal quantifier, while the other occurrence of \(x\) is bound by the existential quantifier. The above syntax allows this “double-binding”. Although it does not create any ambiguities (see below), we will avoid such formulas, as a matter of taste and clarity.

The syntax also allows so-called vacuous binding, as in \(\forall\)x\(Bc\). These, too, will be avoided in what follows. Some treatments of logic rule out vacuous binding and double binding as a matter of syntax. That simplifies some of the treatments below, and complicates others.

Free variables correspond to place-holders, while bound variables are used to express generality. If a formula has no free variables, then it is called a sentence. If a formula has free variables, it is called open.

2.4 Features of the syntax

Before turning to the deductive system and semantics, we mention a few features of the language, as developed so far. This helps draw the contrast between formal languages and natural languages like English.

We assume at the outset that all of the categories are disjoint. For example, no connective is also a quantifier or a variable, and the non-logical terms are not also parentheses or connectives. Also, the items within each category are distinct. For example, the sign for disjunction does not do double-duty as the negation symbol, and perhaps more significantly, no two-place predicate is also a one-place predicate.

One difference between natural languages like English and formal languages like \(\LKe\) is that the latter are not supposed to have any ambiguities. The policy that the different categories of symbols do not overlap, and that no symbol does double-duty, avoids the kind of ambiguity, sometimes called “equivocation”, that occurs when a single word has two meanings: “I’ll meet you at the bank.” But there are other kinds of ambiguity. Consider the English sentence:

John is married, and Mary is single, or Joe is crazy.

It can mean that John is married and either Mary is single or Joe is crazy, or else it can mean that either both John is married and Mary is single, or else Joe is crazy. An ambiguity like this, due to different ways to parse the same sentence, is sometimes called an “amphiboly”. If our formal language did not have the parentheses in it, it would have amphibolies. For example, there would be a “formula” \(A \amp B \vee\) C. Is this supposed to be \(((A \amp B) \vee C)\), or is it \((A \amp(B \vee C))\)? The parentheses resolve what would be an amphiboly.

Can we be sure that there are no other amphibolies in our language? That is, can we be sure that each formula of \(\LKe\) can be put together in only one way? Our next task is to answer this question.

Let us temporarily use the term “unary marker” for the negation symbol \((\neg)\) or a quantifier followed by a variable (e.g., \(\forall x, \exists z)\).

The proof proceeds by induction on the number of instances of (2)–(7) used to construct the formula, and we leave it as an exercise.

We are finally in position to show that there is no amphiboly in our language.

This result is sometimes called “unique readability”. It shows that each formula is produced from the atomic formulas via the various clauses in exactly one way. If \(\theta\) was produced by clause (2), then its main connective is the initial “\(\neg\)”. If \(\theta\) was produced by clauses (3), (4), or (5), then its main connective is the introduced “\(\amp\)”, “\(\vee\)”, or “\(\rightarrow\)”, respectively. If \(\theta\) was produced by clauses (6) or (7), then its main connective is the initial quantifier. We apologize for the tedious details. We included them to indicate the level of precision and rigor for the syntax.

3. Deduction

We now introduce a deductive system, \(D\), for our languages. As above, we define an argument to be a non-empty collection of sentences in the formal language, one of which is designated to be the conclusion. If there are any other sentences in the argument, they are its premises.[1] By convention, we use “\(\Gamma\)”, “\(\Gamma'\)”, “\(\Gamma_1\)”, etc, to range over sets of sentences, and we use the letters “\(\phi\)”, “\(\psi\)”, “\(\theta\)”, uppercase or lowercase, with or without subscripts, to range over single sentences. We write “\(\Gamma, \Gamma'\)” for the union of \(\Gamma\) and \(\Gamma'\), and “\(\Gamma, \phi\)” for the union of \(\Gamma\) with \(\{\phi\}\).

We write an argument in the form \(\langle \Gamma, \phi \rangle\), where \(\Gamma\) is a set of sentences, the premises, and \(\phi\) is a single sentence, the conclusion. Remember that \(\Gamma\) may be empty. We write \(\Gamma \vdash \phi\) to indicate that \(\phi\) is deducible from \(\Gamma\), or, in other words, that the argument \(\langle \Gamma, \phi \rangle\) is deducible in \(D\). We may write \(\Gamma \vdash_D \phi\) to emphasize the deductive system \(D\). We write \(\vdash \phi\) or \(\vdash_D \phi\) to indicate that \(\phi\) can be deduced (in \(D)\) from the empty set of premises.

The rules in \(D\) are chosen to match logical relations concerning the English analogues of the logical terminology in the language. Again, we define the deducibility relation by recursion. We start with a rule of assumptions:

We thus have that \(\{\phi \}\vdash \phi\); each premise follows from itself. We next present two clauses for each connective and quantifier. The clauses indicate how to “introduce” and “eliminate” sentences in which each symbol is the main connective.

First, recall that “\(\amp\)” is an analogue of the English connective “and”. Intuitively, one can deduce a sentence in the form \((\theta \amp \psi)\) if one has deduced \(\theta\) and one has deduced \(\psi\). Conversely, one can deduce \(\theta\) from \((\theta \amp \psi)\) and one can deduce \(\psi\) from \((\theta \amp \psi)\):

The name “&I” stands for “&-introduction”; “&E” stands for “&-elimination”.

Since, the symbol “\(\vee\)” corresponds to the English “or”, \((\theta \vee \psi)\) should be deducible from \(\theta\), and \((\theta \vee \psi)\) should also be deducible from \(\psi\):

The elimination rule is a bit more complicated. Suppose that “\(\theta\) or \(\psi\)” is true. Suppose also that \(\phi\) follows from \(\theta\) and that \(\phi\) follows from \(\psi\). One can reason that if \(\theta\) is true, then \(\phi\) is true. If instead \(\psi\) is true, we still have that \(\phi\) is true. So either way, \(\phi\) must be true.

For the next clauses, recall that the symbol, “\(\rightarrow\)”, is an analogue of the English “if … then … ” construction. If one knows, or assumes \((\theta \rightarrow \psi)\) and also knows, or assumes \(\theta\), then one can conclude \(\psi\). Conversely, if one deduces \(\psi\) from an assumption \(\theta\), then one can conclude that \((\theta \rightarrow \psi)\).

This elimination rule is sometimes called “modus ponens”. In some logic texts, the introduction rule is proved as a “deduction theorem”.

Our next clauses are for the negation sign, “\(\neg\)”. The underlying idea is that a sentence \(\psi\) is inconsistent with its negation \(\neg \psi\). They cannot both be true. We call a pair of sentences \(\psi, \neg \psi\) contradictory opposites. If one can deduce such a pair from an assumption \(\theta\), then one can conclude that \(\theta\) is false, or, in other words, one can conclude \(\neg \theta\).

By (As), we have that \(\{A,\neg A\}\vdash A\) and \(\{\)A,\(\neg\)A\(\}\vdash \neg A\). So by \(\neg\)I we have that \(\{A\}\vdash \neg \neg A\). However, we do not have the converse yet. Intuitively, \(\neg \neg \theta\) corresponds to “it is not the case that it is not the case that” . One might think that this last is equivalent to \(\theta\), and we have a rule to that effect:

The name DNE stands for “double-negation elimination”. There is some controversy over this inference. It is rejected by philosophers and mathematicians who do not hold that each meaningful sentence is either true or not true. Intuitionistic logic does not sanction the inference in question (see, for example Dummett [2000], or the entry on intuitionistic logic, or history of intuitionistic logic), but, again, classical logic does.

To illustrate the parts of the deductive system \(D\) presented thus far, we show that \(\vdash(A \vee \neg A)\):

The principle \((\theta \vee \neg \theta)\) is sometimes called the law of excluded middle. It is not valid in intuitionistic logic.

Let \(\theta, \neg \theta\) be a pair of contradictory opposites, and let \(\psi\) be any sentence at all. By (As) we have \(\{\theta, \neg \theta, \neg \psi \}\vdash \theta\) and \(\{\theta, \neg \theta, \neg \psi \}\vdash \neg \theta\). So by \((\neg\)I), \(\{\theta, \neg \theta \}\vdash \neg \neg \psi\). So, by (DNE) we have \(\{\theta , \neg \theta \}\vdash \psi\) . That is, anything at all follows from a pair of contradictory opposites. Some logicians introduce a rule to codify a similar inference:

If \(\Gamma_1 \vdash \theta\) and \(\Gamma_2 \vdash \neg \theta\), then for any sentence \(\psi, \Gamma_1, \Gamma_2 \vdash \psi\)

The inference is sometimes called ex falso quodlibet or, more colorfully, explosion. Some call it “\(\neg\)-elimination”, but perhaps this stretches the notion of “elimination” a bit. We do not officially include ex falso quodlibet as a separate rule in \(D\), but as will be shown below (Theorem 10), each instance of it is derivable in our system \(D\).

Some logicians object to ex falso quodlibet, on the ground that the sentence \(\psi\) may be irrelevant to any of the premises in \(\Gamma\). Suppose, for example, that one starts with some premises \(\Gamma\) about human nature and facts about certain people, and then deduces both the sentence “Clinton had extra-marital sexual relations” and “Clinton did not have extra-marital sexual relations”. One can perhaps conclude that there is something wrong with the premises \(\Gamma\). But should we be allowed to then deduce anything at all from \(\Gamma\)? Should we be allowed to deduce “The economy is sound”?

A small minority of logicians, called dialetheists, hold that some contradictions are actually true. For them, ex falso quodlibet is not truth-preserving.

Deductive systems that demur from ex falso quodlibet are called paraconsistent. Most relevant logics are paraconsistent. See the entries on relevance logic, paraconsistent logic, and dialetheism. Or see Anderson and Belnap [1975], Anderson, Belnap, and Dunn [1992], and Tennant [1997] for fuller overviews of relevant logic; and Priest [2006a,b], for dialetheism. Deep philosophical issues concerning the nature of logical consequence are involved. Far be it for an article in a philosophy encyclopedia to avoid philosophical issues, but space considerations preclude a fuller treatment of this issue here. Suffice it to note that the inference ex falso quodlibet is sanctioned in systems of classical logic, the subject of this article. It is essential to establishing the balance between the deductive system and the semantics (see §5 below).

The next pieces of \(D\) are the clauses for the quantifiers. Let \(\theta\) be a formula, \(v\) a variable, and \(t\) a term (i.e., a variable or a constant). Then define \(\theta(v|t)\) to be the result of substituting \(t\) for each free occurrence of \(v\) in \(\theta\). So, if \(\theta\) is \((Qx \amp \exists\)xPxy), then \(\theta(x|c)\) is \((Qc \amp \exists\)xPxy). The last occurrence of \(x\) is not free.

A sentence in the form \(\forall v \theta\) is an analogue of the English “for every \(v, \theta\) holds”. So one should be able to infer \(\theta(v|t)\) from \(\forall v \theta\) for any closed term \(t\). Recall that the only closed terms in our system are constants.

The idea here is that if \(\forall v \theta\) is true, then \(\theta\) should hold of \(t\), no matter what \(t\) is.

The introduction clause for the universal quantifier is a bit more complicated. Suppose that a sentence \(\theta\) contains a closed term \(t\), and that \(\theta\) has been deduced from a set of premises \(\Gamma\). If the closed term \(t\) does not occur in any member of \(\Gamma\), then \(\theta\) will hold no matter which object \(t\) may denote. That is, \(\forall v \theta\) follows.

This rule \((\forall \mathbf{I})\) corresponds to a common inference in mathematics. Suppose that a mathematician says “let \(n\) be a natural number” and goes on to show that \(n\) has a certain property \(P\), without assuming anything about \(n\) (except that it is a natural number). She then reminds the reader that \(n\) is “arbitrary”, and concludes that \(P\) holds for all natural numbers. The condition that the term \(t\) not occur in any premise is what guarantees that it is indeed “arbitrary”. It could be any object, and so anything we conclude about it holds for all objects.

The existential quantifier is an analogue of the English expression “there exists”, or perhaps just “there is”. If we have established (or assumed) that a given object \(t\) has a given property, then it follows that there is something that has that property.

The elimination rule for \(\exists\) is not quite as simple:

This elimination rule also corresponds to a common inference. Suppose that a mathematician assumes or somehow concludes that there is a natural number with a given property \(P\). She then says “let \(n\) be such a natural number, so that \(Pn\)”, and goes on to establish a sentence \(\phi\), which does not mention the number \(n\). If the derivation of \(\phi\) does not invoke anything about \(n\) (other than the assumption that it has the given property \(P)\), then \(n\) could have been any number that has the property \(P\). That is, \(n\) is an arbitrary number with property \(P\). It does not matter which number \(n\) is. Since \(\phi\) does not mention \(n\), it follows from the assertion that something has property \(P\). The provisions added to \((\exists\)E) are to guarantee that \(t\) is “arbitrary”.

The final items are the rules for the identity sign “=”. The introduction rule is about a simple as can be:

This “inference” corresponds to the truism that everything is identical to itself. The elimination rule corresponds to a principle that if \(a\) is identical to \(b\), then anything true of \(a\) is also true of \(b\).

The rule \(({=}\mathrm{E})\) indicates a certain restriction in the expressive resources of our language. Suppose, for example, that Harry is identical to Donald (since his mischievous parents gave him two names). According to most people’s intuitions, it would not follow from this and “Dick knows that Harry is wicked” that “Dick knows that Donald is wicked”, for the reason that Dick might not know that Harry is identical to Donald. Contexts like this, in which identicals cannot safely be substituted for each other, are called “opaque”. We assume that our language \(\LKe\) has no opaque contexts.

One final clause completes the description of the deductive system \(D\):

Again, this clause allows proofs by induction on the rules used to establish an argument. If a property of arguments holds of all instances of (As) and \(({=}\mathrm{I})\), and if the other rules preserve the property, then every argument that is deducible in \(D\) enjoys the property in question.

Before moving on to the model theory for \(\LKe\), we pause to note a few features of the deductive system. To illustrate the level of rigor, we begin with a lemma that if a sentence does not contain a particular closed term, we can make small changes to the set of sentences we prove it from without problems. We allow ourselves the liberty here of extending some previous notation: for any terms \(t\) and \(t'\), and any formula \(\theta\), we say that \(\theta(t|t')\) is the result of replacing all free occurrences of \(t\) in \(\theta\) with \(t'\).

Theorem 8 allows us to add on premises at will. It follows that \(\Gamma \vdash \phi\) if and only if there is a subset \(\Gamma'\subseteq \Gamma\) such that \(\Gamma'\vdash \phi\). Some systems of relevant logic do not have weakening, nor does substructural logic (See the entries on relevance logic, substructural logics, and linear logic).

By clause (*), all derivations are established in a finite number of steps. So we have

Theorem 11 allows us to chain together inferences. This fits the practice of establishing theorems and lemmas and then using those theorems and lemmas later, at will. The cut principle is, some think, essential to reasoning. In some logical systems, the cut principle is a deep theorem; in others it is invalid. The system here was designed, in part, to make the proof of Theorem 11 straightforward.

If \(\Gamma \vdash_D \theta\), then we say that the sentence \(\theta\) is a deductive consequence of the set of sentences \(\Gamma\), and that the argument \(\langle \Gamma,\theta \rangle\) is deductively valid. A sentence \(\theta\) is a logical theorem, or a deductive logical truth, if \(\vdash_D \theta\). That is, \(\theta\) is a logical theorem if it is a deductive consequence of the empty set. A set \(\Gamma\) of sentences is consistent if there is no sentence \(\theta\) such that \(\Gamma \vdash_D \theta\) and \(\Gamma \vdash_D \neg \theta\). That is, a set is consistent if it does not entail a pair of contradictory opposite sentencess.

Define a set \(\Gamma\) of sentences of the language \(\LKe\) to be maximally consistent if \(\Gamma\) is consistent and for every sentence \(\theta\) of \(\LKe\), if \(\theta\) is not in \(\Gamma\), then \(\Gamma,\theta\) is inconsistent. In other words, \(\Gamma\) is maximally consistent if \(\Gamma\) is consistent, and adding any sentence in the language not already in \(\Gamma\) renders it inconsistent. Notice that if \(\Gamma\) is maximally consistent then \(\Gamma \vdash \theta\) if and only if \(\theta\) is in \(\Gamma\).

Notice that this proof uses a principle corresponding to the law of excluded middle. In the construction of \(\Gamma'\), we assumed that, at each stage, either \(\Gamma_n\) is consistent or it is not. Intuitionists, who demur from excluded middle, do not accept the Lindenbaum lemma.

4. Semantics

Let \(K\) be a set of non-logical terminology. An interpretation for the language \(\LKe\) is a structure \(M = \langle d,I\rangle\), where \(d\) is a non-empty set, called the domain-of-discourse, or simply the domain, of the interpretation, and \(I\) is an interpretation function. Informally, the domain is what we interpret the language \(\LKe\) to be about. It is what the variables range over. The interpretation function assigns appropriate extensions to the non-logical terms. In particular,

If \(c\) is a constant in \(K\), then \(I(c)\) is a member of the domain \(d\).

Thus we assume that every constant denotes something. Systems where this is not assumed are called free logics (see the entry on free logic). Continuing,

Define \(s\) to be a variable-assignment, or simply an assignment, on an interpretation \(M\), if \(s\) is a function from the variables to the domain \(d\) of \(M\). The role of variable-assignments is to assign denotations to the free variables of open formulas. (In a sense, the quantifiers determine the “meaning” of the bound variables.)

Let \(t\) be a term of \(\LKe\). We define the denotation of \(t\) in \(M\) under \(s\), in terms of the interpretation function and variable-assignment:

If \(t\) is a constant, then \(D_{M,s}(t)\) is \(I(t)\), and if \(t\) is a variable, then \(D_{M,s}(t)\) is \(s(t)\).

That is, the interpretation \(M\) assigns denotations to the constants, while the variable-assignment assigns denotations to the (free) variables. If the language contained function symbols, the denotation function would be defined by recursion.

We now define a relation of satisfaction between interpretations, variable-assignments, and formulas of \(\LKe\). If \(\phi\) is a formula of \(\LKe, M\) is an interpretation for \(\LKe\), and \(s\) is a variable-assignment on \(M\), then we write \(M,s\vDash \phi\) for \(M\) satisfies \(\phi\) under the assignment \(s\). The idea is that \(M,s\vDash \phi\) is an analogue of “\(\phi\) comes out true when interpreted as in \(M\) via \(s\)”.

We proceed by recursion on the complexity of the formulas of \(\LKe\).

If \(t_1\) and \(t_2\) are terms, then \(M,s\vDash t_1 =t_2\) if and only if \(D_{M,s}(t_1)\) is the same as \(D_{M,s}(t_2)\).

This is about as straightforward as it gets. An identity \(t_1 =t_2\) comes out true if and only if the terms \(t_1\) and \(t_2\) denote the same thing.

If \(P^0\) is a zero-place predicate letter in \(K\), then \(M,s\vDash P\) if and only if \(I(P)\) is truth.

If S\(^n\) is an \(n\)-place predicate letter in \(K\) and \(t_1, \ldots,t_n\) are terms, then \(M,s\vDash St_1 \ldots t_n\) if and only if the \(n\)-tuple \(\langle D_{M,s}(t_1), \ldots,D_{M,s}(t_n)\rangle\) is in \(I(S)\).

This takes care of the atomic formulas. We now proceed to the compound formulas of the language, more or less following the meanings of the English counterparts of the logical terminology.

The idea here is that \(\forall v\theta\) comes out true if and only if \(\theta\) comes out true no matter what is assigned to the variable \(v\). The final clause is similar.

\(M,s\vDash \exists v\theta\) if and only if \(M,s'\vDash \theta\), for some assignment \(s'\) that agrees with \(s\) except possibly at the variable \(v\).

So \(\exists v\theta\) comes out true if there is an assignment to \(v\) that makes \(\theta\) true.

Theorem 6, unique readability, assures us that this definition is coherent. At each stage in breaking down a formula, there is exactly one clause to be applied, and so we never get contradictory verdicts concerning satisfaction.

As indicated, the role of variable-assignments is to give denotations to the free variables. We now show that variable-assignments play no other role.

By Theorem 14, if \(\theta\) is a sentence, and \(s_1, s_2\), are any two variable-assignments, then \(M,s_1 \vDash \theta\) if and only if \(M,s_2 \vDash \theta\). So we can just write \(M\vDash \theta\) if \(M,s\vDash \theta\) for some, or all, variable-assignments \(s\). So we define

\(M\vDash \theta\) where \(\theta\) is a sentence just in case \(M,s\vDash\theta\) for all variable assignments \(s\).

In this case, we call \(M\) a model of \(\theta\).

Suppose that \(K'\subseteq K\) are two sets of non-logical terms. If \(M = \langle d,I\rangle\) is an interpretation of \(\LKe\), then we define the restriction of \(M\) to \(\mathcal{L}1K'{=}\) to be the interpretation \(M'=\langle d,I'\rangle\) such that \(I'\) is the restriction of \(I\) to \(K'\). That is, \(M\) and \(M'\) have the same domain and agree on the non-logical terminology in \(K'\). A straightforward induction establishes the following:

In short, the satisfaction of a sentence \(\theta\) only depends on the domain of discourse and the interpretation of the non-logical terminology in \(\theta\).

We say that an argument \(\langle \Gamma,\theta \rangle\) is semantically valid, or just valid, written \(\Gamma \vDash \theta\), if for every interpretation \(M\) of the language, if \(M\vDash\psi\), for every member \(\psi\) of \(\Gamma\), then \(M\vDash\theta\). If \(\Gamma \vDash \theta\), we also say that \(\theta\) is a logical consequence, or semantic consequence, or model-theoretic consequence of \(\Gamma\). The definition corresponds to the informal idea that an argument is valid if it is not possible for its premises to all be true and its conclusion false. Our definition of logical consequence also sanctions the common thesis that a valid argument is truth-preserving – to the extent that satisfaction represents truth. Officially, an argument in \(\LKe\) is valid if its conclusion comes out true under every interpretation of the language in which the premises are true. Validity is the model-theoretic counterpart to deducibility.

A sentence \(\theta\) is logically true, or valid, if \(M\vDash \theta\), for every interpretation \(M\). A sentence is logically true if and only if it is a consequence of the empty set. If \(\theta\) is logically true, then for any set \(\Gamma\) of sentences, \(\Gamma \vDash \theta\). Logical truth is the model-theoretic counterpart of theoremhood.

A sentence \(\theta\) is satisfiable if there is an interpretation \(M\) such that \(M\vDash \theta\). That is, \(\theta\) is satisfiable if there is an interpretation that satisfies it. A set \(\Gamma\) of sentences is satisfiable if there is an interpretation \(M\) such that \(M\vDash\theta\), for every sentence \(\theta\) in \(\Gamma\). If \(\Gamma\) is a set of sentences and if \(M\vDash \theta\) for each sentence \(\theta\) in \(\Gamma\), then we say that \(M\) is a model of \(\Gamma\). So a set of sentences is satisfiable if it has a model. Satisfiability is the model-theoretic counterpart to consistency.

Notice that \(\Gamma \vDash \theta\) if and only if the set \(\Gamma,\neg \theta\) is not satisfiable. It follows that if a set \(\Gamma\) is not satisfiable, then if \(\theta\) is any sentence, \(\Gamma \vDash \theta\). This is a model-theoretic counterpart to ex falso quodlibet (see Theorem 10). We have the following, as an analogue to Theorem 12:

5. Meta-theory

We now present some results that relate the deductive notions to their model-theoretic counterparts. The first one is probably the most straightforward. We motivated both the various rules of the deductive system \(D\) and the various clauses in the definition of satisfaction in terms of the meaning of the English counterparts to the logical terminology (more or less, with the same simplifications in both cases). So one would expect that an argument is deducible, or deductively valid, only if it is semantically valid.

Even though the deductive system \(D\) and the model-theoretic semantics were developed with the meanings of the logical terminology in mind, one should not automatically expect the converse to soundness (or Corollary 19) to hold. For all we know so far, we may not have included enough rules of inference to deduce every valid argument. The converses to soundness and Corollary 19 are among the most important and influential results in mathematical logic. We begin with the latter.

A converse to Soundness (Theorem 18) is a straightforward corollary:

Our next item is a corollary of Theorem 9, Soundness (Theorem 18), and Completeness:

Soundness and completeness together entail that an argument is deducible if and only if it is valid, and a set of sentences is consistent if and only if it is satisfiable. So we can go back and forth between model-theoretic and proof-theoretic notions, transferring properties of one to the other. Compactness holds in the model theory because all derivations use only a finite number of premises.

Recall that in the proof of Completeness (Theorem 20), we made the simplifying assumption that the set \(K\) of non-logical constants is either finite or denumerably infinite. The interpretation we produced was itself either finite or denumerably infinite. Thus, we have the following:

Corollary 23. Löwenheim-Skolem Theorem. Let \(\Gamma\) be a satisfiable set of sentences of the language \(\LKe\). If \(\Gamma\) is either finite or denumerably infinite, then \(\Gamma\) has a model whose domain is either finite or denumerably infinite.

In general, let \(\Gamma\) be a satisfiable set of sentences of \(\LKe\), and let \(\kappa\) be the larger of the size of \(\Gamma\) and denumerably infinite. Then \(\Gamma\) has a model whose domain is at most size \(\kappa\).

There is a stronger version of Corollary 23. Let \(M_1 =\langle d_1,I_1\rangle\) and \(M_2 =\langle d_2,I_2\rangle\) be interpretations of the language \(\LKe\). Define \(M_1\) to be a submodel of \(M_2\) if \(d_1 \subseteq d_2, I_1 (c) = I_2 (c)\) for each constant \(c\), and \(I_1\) is the restriction of \(I_2\) to \(d_1\). For example, if \(R\) is a binary relation letter in \(K\), then for all \(a,b\) in \(d_1\), the pair \(\langle a,b\rangle\) is in \(I_1 (R)\) if and only if \(\langle a,b\rangle\) is in \(I_2 (R)\). If we had included function letters among the non-logical terminology, we would also require that \(d_1\) be closed under their interpretations in \(M_2\). Notice that if \(M_1\) is a submodel of \(M_2\), then any variable-assignment on \(M_1\) is also a variable-assignment on \(M_2\).

Say that two interpretations \(M_1 =\langle d_1,I_1\rangle, M_2 =\langle d_2,I_2\rangle\) are equivalent if one of them is a submodel of the other, and for any formula of the language and any variable-assignment \(s\) on the submodel, \(M_1,s\vDash \theta\) if and only if \(M_2,s\vDash \theta\). Notice that if two interpretations are equivalent, then they satisfy the same sentences.

Another corollary to Compactness (Corollary 22) is the opposite of the Löwenheim-Skolem theorem:

Combined, the proofs of the downward and upward Löwenheim-Skolem theorems show that for any satisfiable set \(\Gamma\) of sentences, if there is no finite bound on the models of \(\Gamma\), then for any infinite cardinal \(\kappa\), there is a model of \(\Gamma\) whose domain has size exactly \(\kappa\). Moreover, if \(M\) is any interpretation whose domain is infinite, then for any infinite cardinal \(\kappa\), there is an interpretation \(M'\) whose domain has size exactly \(\kappa\) such that \(M\) and \(M'\) are equivalent.

These results indicate a weakness in the expressive resources of first-order languages like \(\LKe\). No satisfiable set of sentences can guarantee that its models are all denumerably infinite, nor can any satisfiable set of sentences guarantee that its models are uncountable. So in a sense, first-order languages cannot express the notion of “denumerably infinite”, at least not in the model theory. (See the entry on second-order and higher-order logic.)

Let \(A\) be any set of sentences in a first-order language \(\LKe\), where \(K\) includes terminology for arithmetic, and assume that every member of \(A\) is true of the natural numbers. We can even let \(A\) be the set of all sentences in \(\LKe\) that are true of the natural numbers. Then \(A\) has uncountable models, indeed models of any infinite cardinality. Such interpretations are among those that are sometimes called unintended, or non-standard models of arithmetic. Let \(B\) be any set of first-order sentences that are true of the real numbers, and let \(C\) be any first-order axiomatization of set theory. Then if \(B\) and \(C\) are satisfiable (in infinite interpretations), then each of them has denumerably infinite models. That is, any first-order, satisfiable set theory or theory of the real numbers, has (unintended) models the size of the natural numbers. This is despite the fact that a sentence (seemingly) stating that the universe is uncountable is provable in most set-theories. This situation, known as the Skolem paradox, has generated much discussion, but we must refer the reader elsewhere for a sample of it (see the entry on Skolem’s paradox and Shapiro 1996).

6. The One Right Logic?

Surely, logic has something to do with correct reasoning, or at least correct deductive reasoning. The details of the connection are subtle, and controversial – see Harman [1984] for an influential study. It is common to say that someone has reasoned poorly if they have not reasoned logically, or that a given (deductive) argument is bad, and must be retracted, if it is shown to be invalid.

Some philosophers and logicians have maintained that there is a single logical system that is uniquely correct, in its role of characterizing validity. Among those, some, perhaps most, favor classical, first-order logic as uniquely correct, as the One True Logic. See, for example, Quine [1986], Resnik [1996], Rumfitt [2015], Williamson [2017], and a host of others.

That classical, first-order logic should be given this role is perhaps not surprising. It has rules which are more or less intuitive, and is simple for how strong it is. As we have seen in section 5, classical, first-order logic has interesting and important meta-theoretic properties, such as soundness and completeness, that have lead to many important mathematical and logical studies.

However, as noted, the main meta-theoretic properties of classical, first-order logic lead to expressive limitations of the formal languages and model-theoretic semantics. Key notions, like finitude, countability, minimal closure, natural number, and the like cannot be expressed.

Barwise [1985, 5] once remarked:

As logicians, we do our subject a disservice by convincing others that logic is first-order and then convincing them that almost none of the concepts of modern mathematics can really be captured in first-order logic.

And Wang [1974, 154]:

When we are interested in set theory or classical analysis, the Löwenheim-Skolem theorem is usually taken as a sort of defect... of the first-order logic... [W]hat is established [by these theorems] is not that first-order logic is the only possible logic but rather that it is the only possible logic when we in a sense deny reality to the concept of [the] uncountable...

Other criticisms of classical, first-order logic have also been lodged. There are issues with its ability to deal with certain paradoxes (see, for example, the entry on Russel’s paradox ), its apparent overgeneration of beliefs (see the entry on (the normative status of logic), and some argue that it has some arguments that do not match with the way we normally think we think (see for example, the entry on relevance logic).

There are two main options available to those who are critical of classical, first-order logic, as the One True Logic. One is to propose some other logic as the One True Logic. Priest [2006a] describes the methodology one might use to settle in the One True Logic.

The other main option is to simply deny that there is a single logic that qualifies as the One True Logic. One instance of this is a kind of logical nihilism, a thesis that there is no correct logic. Another is a logical pluralism, the thesis that a variety of different logical all qualify as correct, or best, or even the true logic, at least in various contexts.

Of course, this is not the place to pursue this matter in detail. See Beall and Restall [2006] and Shapiro [2014] for examples of pluralism, and the entry on logical pluralism for an overview of the terrain for both logical pluralism and logical nihilism.

We close with brief sketches of some of the main alternatives to classical, first-order logic, providing references to other work and entries to this Encyclopedia. See also the second half of Shapiro and Kouri Kissel [2022].

6.1 Rivals to classical, first-order logic

In recent years, some work has been done to "approximate" classical logic. The idea is to get as close to classical logic as possible, in order to preserve some of the benefits, while at the same time removing some limitations of classical logic, like being closer to intuitive inference or applying to things like vagueness and paradoxes.

For example, Barrio, Pailos and Szmuc [2020] show that we can approximate classical logic in something called the ST-hierarchy (ST for strict-tolerant, from Cobreros, Egre, Ripley and van Rooij [2012a,b]). This allows them to avoid certain classical problems at each level of the hierarchy, like some of the paradoxes, while at the same time maintaining many of the benefits of the strength of classical logic when considering the full hierarchy.

Dave Ripley [2013] provides a multi-sequent calculus version of “classical logic” that he argues solves some of the paradoxes. Notably, he claims it solves at least the Sorites and Liar Paradoxes (see the entries on the sorites paradox and liar Paradox). The system conservatively extends classical logic. Ripley claims that this is what makes it classical. However, the system is not transitive, and does not have a Cut rule.

There are, of course, some questions about whether these new logics are really classical, but it is informative work nonetheless.

One way to extend classical, first-order logic is to add additional operators to the underlying formal language. Modal logic adds operators which designate necessity and possibility. So, we can say that a proposition is possibly true, or necessarily true, rather than just true.

W. V. O Quine [1953] once argued that it is not coherent for quantifiers to bind variables inside modal operators, but opinion on this matter has since changed considerably (see, for example, Barcan [1990]). There is now a thriving industry of developing modal logics to capture various kinds of modality and temporal operators. See the entry on modal logic.

All of the formal languages sketched above have only one sort of variable. These are sometimes called first-order variables. Each interpretation of the language has a domain, which is the range of these first-order variables. It is what the language is about, according to the given interpretation. Second-order variables range over properties, sets, classes, relations, or functions of the items in that domain. Third-order variables range over properties, classes, relations of whatever is in the range of the second-order variables. And it goes on from there.

A formal language is called second-order if it has second-order variables and first-order variables, and no others; Third-order if it has third-order, second-order, and first-order variables and no others, etc. A formal language is higher-order if it is at least second-order.

As noted, it is not an exaggeration to say that classical, first-order logic is the paradigm of contemporary logical theory. Most textbooks do not mention higher-order languages at all, and most of the rest give it scant treatment.

A number of different deductive systems and model-theoretic semantics have been proposed for second- and higher-order languages. For the semantics, the main additional feature of the model-theory is to specify a range of the higher-order variables.

In Henkin semantics, each interpretation specifies a specific range of the higher-order variables. For monadic second-order variables, each interpretation specifies a non-empty subset of the powerset of the domain, for two-place second-order variables, a non-empty set of ordered pairs of members of the domain, etc. The system has all of the above limitative meta-theoretic results. There is a deductive system that is sound and complete for Henkin semantics; the logic is compact; and the downward and upward Löwenheim-Skolem theorems all hold.

In so-called standard semantics, sometimes called full semantics, monadic second-order variables range over the entire powerset of the domain; two-place second-order variables range over the entire class of ordered pairs of members of the domain, etc. It can be shown that second-order languages, with standard semantics, can characterize many mathematical notions and structures, up to isomorphism. Examples include the notions of finitude, countability, well-foundedness, minimal closure, and structures like the natural numbers, the real numbers, and the complex numbers. As a result, none of the limitative theorems of classical, first-order logic hold: there is no effective deductive system is both sound and complete, the logic is not compact, and both Löwenheim-Skolem theorems fail. Some, such as Quine [1986], argue that second-order logic, with standard semantics is not really logic, but is a form of mathematics, set theory in particular. For more on this, see Shapiro [1991] and the entry on higher-order logic, along with the many references cited there.

One might also consider generalized quantifiers as an expansion of classical first-order logic (see the entry on generalized quantifiers). These quantifiers allow from an expansion between the classical “all” and “some” , and can accommodate quantifiers like “most” , “less than half” , “usually” , etc. They are useful from both a logical and linguistic perspective. For example, Kennedy and Väänänen [2021] use generalized quantifiers to argue that “ uncountable” is a logical notion.