# Disjunction

*First published Sat 6 Jan, 2001*

Disjunction is a binary truth-function, the output of which is a
sentence true if at least one of the input sentences (disjuncts) is
true, and false otherwise. Disjunction, together with negation, provide
sufficient means to define all other truth-functions. Its supposed
connection with the *or* words of natural language has intrigued
and mystified philosophers for many centuries, and the subject has
inspired much creative myth-making, particularly since the advent of
truth-tables early in the twentieth century. In this article some of
those myths are set out and dispelled.

- Introduction
- Syntax
- Proof Theory
- Semantics
- Inclusive and Exclusive Disjunctions
- Natural Language
- The Myth of
*Vel*and*Aut* - The
*Or*of Natural Language - Bibliography
- Other Internet Resources
- Related Entries

## Introduction

A disjunction is a kind of compound sentence historically associated
by English-speaking logicians and their students with indicative
sentences compounded with *either* ... *or*, such as

Either I am very rich or someone is playing a cruel joke.But nowadays the term

*disjunction*is more often used in reference to sentences (or well-formed formulae) of associated form occurring in formal languages. Logicians distinguish between

(a) the abstractedformof such sentences and the roles that sentences of that form play in arguments and proofs,(b) the

meaningsthat must be assigned to such sentences to account for those roles.

The former represents their *syntactic* and
*proof-theoretic* interests, the latter their
*semantic* or *truth-theoretic* interest in
disjunction. Introductory logic texts are sometimes a little unclear
as to which should provide the defining characteristics of
disjunction. Nor are they clear as to whether disjunctions are
primarily features of natural or of formal languages. Here we
consider formal languages first.

## Syntax

The definition of a formal system, either axiomatic or natural deductive, requires the definition of a language, and here the formal vocabulary of disjunction makes its first appearance. If the disjunctive constant (historically suggestive of Latin*vel*(

*or*)) is a primitive constant of the language, there will be a clause, here labeled [] in the inductive definition of the set of well-formed formulae (wffs). Using and as metalogical variables, ranging over wffs, such a clause would read:

[] If is a wff and is a wff, then is a wff

perhaps accompanied by an instruction that
is to be referred to as the *disjunction* of
the wffs
and
, and read as "[name of first wff]
vel (or ‘vee’, or ‘or’) [name of second
wff]". Thus, on this instruction, the wff
*p*
*q* is the *disjunction*
of *p* and
*q*, and is pronounced as ‘pea vel
queue’ or ‘pea vee queue’ or ‘pea or queue’.
In this case, *p* and
*q* are the disjuncts of the
disjunction.

If is a non-primitive constant of the language, then typically it will be introduced by an abbreviative definition. In presentations of classical systems in which the conditional constant or and the negational constant are taken as primitive, the disjunctive constant might be introduced in the abbreviation of a wff (or ) as . Alternatively, if the conjunctive & has already been introduced either as a primitive or as a defined constant, might be introduced in the abbreviation of a wff ( & ) as .

## Proof Theory

Much as we would understand the conversational significance of vocabulary more generally if we had a complete set of instructions for initiating its use in a conversation, and for suitable responses to its introduction by an interlocutor, we give the proof-theoretic significance of a connective by providing rules for its introduction into a proof and for its elimination. In the case of , these might be the following:[-introduction] For any wffs and , a proof having a subproof of from an ensemble of wffs, can be extended to a proof of from .Intuitively, the former would correspond to a rule of conversation that permitted us to assert[-elimination] For any wffs , , , a proof that includes

can be extended to a proof of from .

- a subproof of from an ensemble of wffs ,
- a subproof of from an ensemble {}, and
- a subproof of from an ensemble {},

*A*or

*B*(for any

*B*) given the assertion that

*A*. Thus if we are told that Nicholas is in Paris, we can infer that Nicholas is either in Paris or in Toulouse.

Intuitively, the latter rule would correspond to a rule that, given
the assertion that *A* or *B*, would permit the
assertion of anything that is permitted both by the assertion of
*A* and by the assertion of *B*. For example, given the
assertion on certain grounds that Nicholas is in Paris or Toulouse,
we are warranted in asserting on the same grounds plus some
geographical information, that Nicholas is in France, since that
assertion is warranted (a) by the assertion that Nicholas is in Paris
together with some of the geographical information and (b) by the
assertion that Nicholas is in Toulouse together with the rest of the
geographical information. More generally we may sum the matter up by
saying that the rule corresponds to the conversational rule that lets
us extract information from an *or*-sentence without the
information of either of its clauses. In the example, we are given
the information that Nicholas is in Paris or Toulouse, but we are
given neither the information that Nicholas is in Paris nor the
information that he is in Toulouse.

## Semantics

In its simplest, classical, semantic analysis, a disjunction is understood by reference to the conditions under which it is true, and under which it is false. Central to the definition is a*valuation*, a function that assigns to every atomic, or unanalysable sentence of the language a value in the set {1, 0}. In general the inductive truth-definition for a language corresponds, clause by clause to the definition of its well-formed formulae. Thus for a propositional language it will take as its basis, a clause according to which an atom is true or false accordingly as the valuation maps it to 1 or to 0. In systems in which is a primitive constant, the clause corresponding to disjunction takes to be true if at least one of , is true, and takes it to be false otherwise. Where is introduced by either of the definitions earlier mentioned, that truth-condition can be computed for from those of the conditional ( or ) or conjunction (&) and negation ().

Now the truth-definition can be regarded as an extension of the valuation from the atoms of the language to the entire set of wffs with 1 understood as the truth-value, true, and 0 understood as the truth-value, false. Thus, classically, disjunction is semantically interpreted as a binary truth-function from the set of pairs of truth-values to the set {0, 1}. The tabulated graph of this function, as dictated by the truth-definition, is called the truth-table for disjunction. That table is the following:

1 1

1 0

0 1

0 01

1

1

0

## Inclusive and Exclusive disjunctions

Authors of introductory logic texts generally take this opportunity to distinguish the disjunction we have been discussing from another binary truth-function whose graph is tabulated by the table:

1 1

1 0

0 1

0 00

1

1

0

where
is
read
*xor*
. This truth-function is referred to
variously as exclusive disjunction, as 0110 disjunction (after the
succession of values in its main column), and as logical
difference. The wff
is true when exactly one of
,
is true; false otherwise. To make matters explicit, the earlier
discussed truth-function
is called inclusive, or non-exclusive
or 1110 disjunction.

## Natural Language

It is an assumption, at any rate a claim, of many textbook authors that there are both uses of*or*in English that correspond to 1110 disjunction and uses that correspond to 0110 disjunction, and this supposition generally motivates the introduction and discussion of the xor connective. Since we are following the usual order of textbook exposition, this is perhaps the moment to make a few observations on this score. The first are purely syntactic. The

*or*of English that such authors cite is a coordinator (or coordinating conjunction). It can coordinate syntactic elements of virtually any grammatical type, not merely whole sentences. Moreover, if we consider only its uses joining whole sentences, we must notice that it can join sentences of virtually any mood: interrogative sentences and imperatives as well as indicative sentences can be joined by

*or*in English. And again, if we restrict our attention to its uses joining indicative sentences, we must note that

*or*is by no means restricted to the binary cases in this role. Indeed, there is no theoretical finite limit to the number of clauses that it can join. This is perhaps the most fundamental relevant syntactic difference between

*or*on the one hand and and on the other. The sentence

Nathalie has been and gone or Nathalie will arrive today or Nathalie will not arrive at allis a perfectly correct sentence and not ambiguous as between

(Nathalie has been and gone or Nathalie will arrive today) or Nathalie will not arrive at alland

Nathalie has been and gone or (Nathalie will arrive today or Nathalie will not arrive at all).By contrast, the wff

*p*

*q*

*r*, far from being ambiguous as between (

*p*

*q*)

*r*and

*p*(

*q*

*r*), is, on the inductive definition of well-formedness, not a wff. If the parenthesis-free notation is tolerated in general logical exposition, this is because is

*associative*, that is, the wffs (

*p*

*q*)

*r*and

*p*(

*q*

*r*) are syntactically interderivable, and semantically have identical truth-conditions. The formal account of disjunction could readily be liberalized to accommodate that fact, and even conveniently in languages in which was primitive. In that case our inductive definition of the language could permit any such string as (

_{1}, ... ,

_{i}, ... ,

_{n}) to be well-formed if

_{1}, ... ,

_{i}, ... and

_{n}are. The relevant clause of the truth-definition would accordingly be modified in such a way as to give (

_{1}, ... ,

_{i}, ... ,

_{n}) the maximum of the truth-values of

_{1}, ... ,

_{i}, ... and

_{n}. Moreover, this accords well with such cases as the one cited in which

*or*joins more than two simple clauses: such a sentence is true if at least one of its clauses is true; false otherwise.

The fact that English *or* is not binary does not accord so
well with the claim made by many textbook authors that there are uses
of *or* that require representation by 0110 disjunction. To be
sure,
is associative, so that a notational
liberalization would be possible, parallel to the one described for
. But, as Hans Reichenbach
seems first to have pointed out
(in Reichenbach [1947]), the
truth-definition for
(_{1},
... ,
_{i}, ... ,
_{n}) would have
to be such as to give it the value 1 if any odd number of
_{1}, ... ,
_{i}, ... ,
_{n} have the
value 1; the value 0 otherwise. The result is evident from the
truth-table where *n* > 2. For *n* = 3, suppose that
has the value 1. The
truth-definition as given by the table requires that exactly one of
,
has the value 1. Let
have the value 1; then
has
the value 0. Then
and
have the same value. That is, either
both
and
have the value 0, or both
and
have the value 1. In the former case
exactly one of
,
,
has the value 1; in the latter,
all three have the value 1. That is, the disjunction will take the
value 1 if and only if an odd number of disjuncts have the value 1. A
simple induction will prove that this result holds for an exclusive
disjunction of any finite length. It is sufficient for present
purposes to note that, in the case where *n* = 3,
(_{1},
_{2},
_{3}) will be true if all
of its disjuncts are true. Now there is no naturally occurring
coordinator in any natural language matching the truth-conditional
profile of such a connective. There is certainly no use of
*or* in English in accordance with which five sentences
*A*, *B*, *C*, *D*, and *E* can be
joined to form a sentence *A* or *B* or *C* or
*D* or *E*, which is true if
and only if either exactly one of the component sentences is true, or
exactly three of them are true or exactly five of them are true.

Most of the texts make no claims about exclusive disjunctive uses of
either English or Latin *or*-words beyond the two-disjunct
case. But it is a fair presumption that the belief in exclusive
disjunctive uses of *or* in English includes just such
three-disjunct uses of *or*. Such a use of *or*, would
be one in accordance with which three sentences *A*,
*B*, and *C* can be joined to form a sentence
*A* or *B* or *C*, which is true if and only if
exactly one of the component sentences is true. Though not a
0110-disjunctive use of *or*, this would be a general use
representable as 0110 disjunction in the two-disjunct case.

The question as to whether there is such a use of *or* in
English, or any other natural language goes to the very heart of the
conception of truth conditional semantics. For it seems certain that
there are conversational uses of *or* that invite the
inference of exclusivity, but which do not seem to require
exclusivity for their truth. Thus, for example, if one says (as in
Tarski [1941], 21) ‘We are going on a
hike or we are going to a theater’, even with charged emphasis
upon the *or*, one will have spoken falsely if in the event we
do both, unless, as in Tarski's example, one has also denied the
conjunction.

Some authors have sought examples of 0110 disjunction in
*or*-sentences whose clauses are mutually exclusive. For
example, Kegley and Kegley discuss the case
(Kegley and Kegley [1978], 232):

John is at the play, or he is studying in the libraryof which the authors remark, "There is no mistaking the sense of

*or*here: John cannot be in both places at once". If their example were an example of exclusive disjunction, we could safely infer from it that the play is not being performed in the library, that the theatre is not in the library, that John is not swotting in the stalls between acts while his companion fights her way to the bar to fetch the drinks. In fact, even, perhaps particularly, when the disjuncts are genuinely mutually exclusive, there are no grounds for the supposition that the

*or*represents 0110 disjunction. Were there such grounds the of formal logic would require distinct semantic accounts for the wffs

*p*

*q*and

*p*

*p*. As Barrett and Stenner point out (Barrett and Stenner [1971]), the case requires quite the reverse. Since the truth-tables of and differ exactly in the output value of the first row, what alone would clinch the case for the existence of an exclusive

*or*would be a sentence in which both disjuncts were true, and the disjunction therefore false. No author has yet produced such an example.

## The Myth of *Vel* and *Aut*

If the logic texts dictate the structure and content of our
discussion, it is perhaps as well to dispel another current myth --
namely that the notational choice of
,
(read as *vel*) as the connective of inclusive disjunction, and the claim that the English

*or*has 0110-disjunctive uses are supported by the facts of the Latin language. I. Copi is as explicit as any (Copi [1971], 241):

The Latin word "vel" expresses weak or inclusive disjunction, and the Latin word "aut" corresponds to the word "or" in its strong or exclusive sense.

The idea is, first, that whereas English has only one
*or*-word, Latin has two: *vel* and *aut*, and
secondly, that the uses of *vel* in Latin would be
representable as 1110 disjunction and the uses of *aut* as
0110 disjunction. As to the first, the very shape of the claim is
likely to mislead. The case is not that Latin had two words for
*or*, but rather that Latin had more than one word that gets
translated into English as *or*. In fact, Latin had
*many* words that are translated into English as *or*,
including, besides the two listed, at least *seu, sive* and
the enclitic *ve*. So does English have many words that can be
translated into English as *or*, including *unless, if
... not, but* (It does not rain but it pours) and so on. All
vocabulary has a history, and languages accumulate vocabulary that
becomes adapted to nuanced uses.

Now the supposition that Latin had a 0110 coordinator must suffer
from the same implausibilities as the corresponding supposition about
English. What of the two-disjunct case? If any general tendency can be
detected in actual Latin usage, say in the classical period, that would
distinguish the uses of *vel* from those of *aut*, it is
that *aut* tended to be brought into use in the formation of
lists of disjoint or contrasted or opposed items, categories or classes
or states, as for example

The difficulty with these examples is that the exclusiveness of the states independendently of the choice of connective must mask any disjointness that the connective could itself impose. That it does not imposeOmne enuntiatum aut verum aut falsum est[Every statement is either true or false] (Cicero,De Fato, 222).

*any*disjointness itself is best seen in its list-forming uses. Consider the list (Cicero,

*De Officiis*):

to be sure the categories are disjoint, and this fact might be supposed to contribute to the selection oftribunos aut plebes[the magistrates or the mob, (accusative plural)]

*aut*. But the mutual exclusion in such cases need not survive the addition of a verb.

does not exclude the case in which one feared both. However, what clinches the refutation of this mythical supposition is that if that whole clause is brought within the scope of a negator, the resulting sentence will expect a reading along the lines of 1110 disjunction.Timebat tribunos aut plebes[one feared the magistrates or the mob]

just means no one feared either. It does not mean everyone either feared neither or feared both. Since the negation of a 0110 disjunction is a 1110 disjunction (either both disjuncts are true or both disjuncts are false), this use ofNemo timebat tribunos aut plebes[No one feared the magistrates or the mob]

*aut*cannot be a 0110 disjunctive use.

In fact, in classical Latin, *aut* was favoured over
*vel* in constructions involving negations, and in that use,
*aut* behaves analogously to
. But pretty well anywhere an
*aut* could be used, a *vel* could be substituted, and
vice versa. The resulting sentence would have a different flavour,
and in some instances would be mildly eccentric, but would not have a
different truth condition. The uses of *vel* reflected its
origins as an imperative form of *volo*. The flavour of

would be closer to that ofNemo timebat vel tribunos vel plebes

Name which group (of the two) you will: no one feared them.

*Aut*was adversative: no one feared either social extremity. (For more examples and a more detailed discussion, see Jennings [1994], 239-251.)

## The *Or* of Natural Language

There are undoubtedly disjunctive uses of *or*in English, and of corresponding vocabulary in other natural languages. But the uses of

*or*after the pattern of the logic texts:

Either Argentina will boycott the conference or the value of lead will diminishand so on constitute only a very small proportion, certainly fewer than 5% of the occurrences of

*or*in English, and, it can be supposed, of corresponding words in all other natural languages as well. It is therefore not surprising that it should be some of these non-disjunctive uses that have been misidentified as instances of exclusive disjunction. The example cited (in Richards [1978], 84) is a good representative example of one such common misidentification:

So how can we find a clear-cut case of the exclusive ‘or’? Imagine a boy who asks for ice creamOnce again there is a difficulty in trying to account for the exclusivity by reference to truth-conditions, though, if we are permitted to consult the intentions of the speaker (as Richards himself does) we may be in no doubt as to the prohibition of strawberries and icecream, however curious such a prohibition might seem. But this example, in company with the many others like it (which this author has sometimes referred to collectively asandstrawberries for tea. He is told as a sort of refusal:’You can have ice creamHere there is no doubt: not both may be had.orstrawberries for tea’.

*the argument from confection*) suffers from the even more serious flaw that it is not a disjunction at all. The problem is not that the

*or*does not join whole clauses. Even if we expand the example to

’You can have ice cream for teathe sentence cannot be construed as a disjunction. The reason is that the child would be correct in inferring that he can have ice cream for tea, and would be correct in inferring that he can have strawberries for tea. Such sentences are elliptical for conjunctions, not for disjunctions, even on a truth-conditional construal. It just happens that for such conjunctions, questions of exclusivity, or rather non-combinativity also arise.oryou can have strawberries for tea’,

Not every *or* of English (nor every counterpart of
*or* in other languages) is disjunctive, even among those that
join pairs of indicative sentences.

## Bibliography

- Barrett, Robert B., and Stenner, Alfred J.,
"The Myth of the Exclusive ‘Or’",
*Mind*,**80 (317)**, 1971, 116-121. - Cicero, Marcus Tullius,
*De Fato*, Translated by H. Rackham. Cambridge, Mass 1942. - Cicero, Marcus Tullius,
*De Officiis*, Translated by Walter Miller. Cambridge, Mass 1975. - Copi, I.M.,
*Introduction to Logic*, New York, 1971. - Jennings, R.E.,
*The Genealogy of Disjunction*, New York: Oxford University Press, 1994. - Kegley, Charles W., and Kegley, Jacquelyn Ann,
*Introduction to Logic*, Lanham, Md., 1978. - Reichenbach, Hans,
*Elements of Symbolic Logic*, New York: MacMillan, 1947. - Richards, T.A.,
*The Language of Reason,*Rushcutters Bay, NSW, 1978. - Alfred Tarski,
*Introduction to Logic and to the Methodology of the Deductive Sciences,*New York 1941, (Revised 1946 edition).