oxford logic

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THE LOGIC MAN UAL

for Introduction to Logic/

Volker Halbach

Oxford

th August

is text is to be used by candidatesin their first year in /. eset text for Literae Humaniores stu-

dents sitting Moderations in isHodgessLogic.


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. Notational Conventions

. Formalisation

e Semantics of Predicate Logic . Structures . Truth . Validity, Logical Truths, and Contradictions . Counterexamples

Natural Deduction . Propositional Logic

. Predicate Logic

Formalisation in Predicate Logic . Adequacy . Ambiguity . Extensionality . Predicate Logic and Arguments in English

Identity and Definite Descriptions . Qualitative and Numerical Identity . e Syntax ofL

=

. Semantics . Proof Rules for Identity . Uses of identity . Identity as a Logical Constant

Natural Deduction Rules


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eLogic Manualis a relatively brief introduction to logic. I have tried tofocus on the core topics and have neglected some issues that are coveredin more comprehensive books such as Forbes (), Guttenplan (),Hodges (), Smith (), and Tennant (). In particular, I havetried not to include material that is inessential to Preliminary Examina-tions and Moderations in Oxford. For various topics, I could not resistadding footnotes offering extra information to the curious reader.

Logic is usually taught in one term. Consequently, I have divided thetext into eight chapters:

. Sets, Relations, and Arguments. Syntax and Semantics of Propositional Logic. Formalisation in Propositional Logic. e Syntax of Predicate Logic. e Semantics of Predicate Logic. Natural Deduction. Formalisation in Predicate Logic. Identity and Definite Descriptions

If the reader wishes to read selectively, chapters constitute a self-con-tained part, to which Section . (Natural Deduction for propositional

logic) can be added; and chapters yield an introduction to predicatelogic without identity.I have set the core definitions, explanations, and results in italics like

this. is might be useful for revision and for finding important passagesmore quickly.

In some cases, the end of an example or a proof is marked by a square. e word iff is short for if and only if.

I have written an Exercises Booklet that can be used in conjunctionwith this Logic Manual. It is available from WebLearn. ere also someadditional teaching materials may be found such as further examples of

proofs in the system of Natural Deduction.

I am indebted to colleagues for discussions and comments on previous


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versions of the text. In particular, I would like to thank Stephen Blamey,

Paolo Crivelli, Geoffrey Ferrari, Lindsay Judson, Ofra Magidor, DavidMcCarty, Peter Millican, Alexander Paseau, Annamaria Schiaparelli, Se-bastian Sequoiah-Grayson, Markakkar, Gabriel Uzquiano, and DavidWiggins. I am especially grateful to Jane Friedman and Christopher vonBlow for their help in preparing the final version of the Manual.


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Sets, Relations, and Arguments

.

Set theory is employed in many disciplines. As such, some acquaintancewith the most basic notions of set theory will be useful not only in logic,

but also in other areas that rely on formal methods. Set theory is a vastarea of mathematical research and of significant philosophical interest.For the purposes of this book, the reader only needs to know a fragmentof the fundamentals of set theory.

A set is a collection of objects.ese objects may be concrete objectssuch as persons, cars and planets or mathematical objects such as numbersor other sets.

Sets are identical if and only if they have the same members. erefore,the set of all animals with kidneys and the set of all animals with a heartare identical, because exactly those animals that have kidneys also havea heart and vice versa. In contrast, the property of having a heart isusually distinguished from the property of having kidneys, although both

properties apply to the same objects.at a is an element of the set M can be expressed symbolically by

ere are various mathematical introductions to set theory such as Devlin (),Moschovakis () or the more elementary Halmos (). In contrast to rigorousexpositions of set theory, I will not proceed axiomatically here.

I have added this footnote because there are regularly protests with respect to thisexample. For this example, only complete and healthy animals are being considered. Ihave been told that planarians (a type of flatworms) are an exception to the heartkidney

rule, so, for the sake of the example, I should exclude them as well.

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writing a M. If a is an element of M, one also says that a is in M or

that M contains a.ere is exactly one set that contains no elements, namely, the empty

set.Obviously, there is only one empty set, because all sets containingno elements contain the same elements, namely none.

ere are various ways to denote sets.One can write down names of the elements, or other designations of

the elements, and enclose this list in curly brackets.e set{London, Munich}, for instance, has exactly two cities as its

elements.e set{Munich, London}has the same elements. erefore,the sets are identical, that is:

{London, Munich}={Munich, London}.

us, if a set is specified by including names for the elements in curlybrackets, the order of the names between the brackets does not matter.

e set{the capital of England, Munich}is again the same set be-cause the capital of England is just another way of designating London.{London, Munich, the capital of England}is still the same set: addinganother name for London, namely, the capital of England, does not adda further element to{London, Munich}.

is method of designating sets has its limitations: sometimes onelacks names for the elements. e method will also fail for sets withinfinitely many or even just impractically many elements.

Above I have designated a set by the phrase the set of all animalswith a heart. One can also use the following semi-formal expression todesignate this set:

{ x xis an animal with a heart }

is is read as the set of all animals with a heart. Similarly, { x xis anatural number bigger than } is the set of natural numbers bigger than ,

and{ x xis blue all over orxis red all over }is the set of all objects that

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are blue all over and all objects that are red all over.

.

e expression is a tiger applies to some objects, but not to others.ereis a set of all objects to which it applies, namely the set{ x xis a tiger }containing all tigers and no other objects. e expression is a bigger citythan, in contrast, does not apply to single objects; rather it relates twoobjects. It applies to London and Munich (in this order), for instance,because London is a bigger city than Munich. One can also say that theexpression is a bigger city than applies to pairs of objects. e set of all

pairs to which the expression is a bigger city than applies is called thebinary relation ofbeing a bigger city than or simply the relation ofbeinga bigger city than. is relation contains all pairs with objects dand esuch thatdis a bigger city than e .

However, these pairs cannot be understood simply as the sets{d, e},such that dis a bigger city thane, because elements of a set are not orderedby the set: as pointed out above, the set{London, Munich}is the sameset as{Munich, London}. So a set with two elements does not have afirst or second element. Since London is bigger than Munich, but not

vice versa, only the pair with London as first component and Munich as

e assumption that any description of this kind actually describes a set is problematic.e so-called Russell paradox imposes some limitations on what sets one can postulate.See Exercise ..

By the qualification binary one distinguishes relations applying to pairs from relationsapplying to triples and strings of more objects. I will return to non-binary relations inSection ..

Oen philosophers do not identify relations with sets of pairs. On their terminologyrelations need to be distinguished from sets of ordered pairs in the same way propertiesneed to be distinguished from sets (see footnote ). In set theory, however, it is common

to refer to sets of ordered pairs as binary relations and I shall follow this usage here.

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second component should be in the relation ofbeing a bigger city than,

but not the pair with Munich as first component and London as secondcomponent.

erefore, so-called ordered-pairs are used in set theory. ey aredifferent from sets with two elements. Ordered pairs, in contrast to setswith two elements, have a first and a second component (and no furthercomponents). e ordered pairLondon, Munichhas London as its firstcomponent and Munich as its second.Munich, Londonis a differentordered pair, because the two ordered pairs differ in both their first andsecond components. More formally, an ordered paird, eis identicalwithf,gif and only ifd= f ande =g. e ordered pairthe largest

city in Bavaria, the largest city in the UKis the same ordered pair asMunich, London, because they coincide in their first and in their secondcomponent. An ordered pair can have the same object as first and secondcomponent:London, London, for instance, has London as its first andsecond component.Munich, LondonandLondon, Londonare twodifferent ordered pairs, because they differ in their first components. SinceI will not be dealing with other pairs, I will oen drop the qualificationordered from ordered pair.

.. A set is a binary relation if and only if it contains onlyordered pairs.

According to the definition, a set is a binary relation if it does notcontain anything that is not an ordered pair. Since the empty set doesnot contain anything, it does not contain anything that is not an orderedpair. erefore, the empty set is a binary relation.

e binary relation of being a bigger city than, that is, the relationthat is satisfied by objectsdande if and only ifdis a bigger city than e isthe following set:

Using a nice trick, one can dispense with ordered pairs by defining the ordered paird,eas{{d}, {d,e}}. e trick will not be used here.

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London, Munich, London, Oxford, Munich, Oxford,Paris, Munich, . . .

In the following definition I will classify binary relations. Later, I shallillustrate the definitions by examples. Here, and in the following, I shall

use iff as an abbreviation for if and only if .

.. A binary relation R is

(i) reflexive on a set S iff for all d in S the paird, dis an element of R;(ii) symmetric iff for all d, e: ifd, e R thene , d R;

(iii) asymmetric iff for no d, e:d, e R ande , d R;(iv) antisymmetric iff for no two distinct d, e:d, e R ande, d R;

(v) transitive iff for all d, e ,f : if d, e R ande ,f R, then alsod,f R;

(vi) an equivalence relation on S iff R is reflexive on S, symmetric andtransitive.

In the following I shall occasionally drop the qualification binary.As long as they are not too complicated, relations and their properties

such as reflexivity and symmetry can be visualised by diagrams. Forevery component of an ordered pair in the relation, one writes exactlyone name (or other designation) in the diagram. e ordered pairs in therelation are then represented by arrows. For instance, the relation

{France, Italy, Italy, Austria, France, France,Italy, Italy, Austria, Austria}

has the following diagram:

France

Austria

Italy

e arrow from France to Italy corresponds to the pair France,Italy, and the arrow from Italy to Austria corresponds to the pairItaly,

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Austria. e three loops in the diagram correspond to the three pairsFrance, France,Italy, Italy,Austria, Austria.

Since France, Italy and Austria all have such a loop attached tothem, the relation is reflexive on the set{France, Italy, Austria}. erelation is not reflexive on the set{France, Italy, Austria, Spain}, becausethe pairSpain, Spainis not in the relation.

e relation is not transitive. For transitivity it is required that if thereis an arrow from a pointdto a pointeand one frometo fin the diagram,then there must be a shortcut, that is, a (direct) arrow fromdto f. In thediagram above there is an arrow from France to Italy and an arrow fromItaly to Austria, but there is no arrow from France to Austria. Hence

the relation is not transitive. If the additional pairFrance, Austriawereadded to the relation, then a transitive relation would be obtained.

If a relation is symmetric, then there are no one-way arrows. atis, if there is an arrow fromdto e , then there must be an arrow back todfrome . e relation above is not symmetric. For instance, the pairFrance, Italyis in the relation, but not the pairItaly, France. at is,in the diagram there is an arrow from France to Italy but no arrow backfrom Italy to France.

e relation is also not asymmetric. If a relation is asymmetric andd, eis in the relation, thene, dcannot be in the relation. e pair

France, Franceis in the relation, but the pair with its elements reversed,that is,France, France(which happens to be the same ordered pairagain), is in the relation as well, thereby violating the condition for asym-metry.

Generally, in the diagram of an asymmetric relation there are onlyone-way arrows: there is never an arrow from an objectdto an objecteand then an arrow back frome tod. is implies that in the diagram ofan asymmetric relation there cannot be any loops, because if there is anarrow fromdtod, there is also, trivially, an arrow back fromdtod: the

very same arrow.e relation in the diagram on page is antisymmetric: in an anti-

symmetric relation there must not be two different objects with arrows

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in both directions between them. us, antisymmetry is the same as

asymmetry except that in an antisymmetric relation elements may haveloops attached to them. In the above diagram there are objects with loops,but no two different objects with arrows in both directions between them.

erefore, the relation is antisymmetric.I turn to another example, a relation with the following diagram:

Mars

Pluto

Venus

Mercury

is relation is reflexive on the set{Mars, Pluto, Venus, Mercury}; it isalso symmetric. It fails to be transitive since direct arrows are missing,for instance, from Mars to Venus. e relation is not asymmetric orantisymmetric since there are pairs of objects such as Mars and Mercury

that have arrows going back and forth between them .Both relations discussed so far are not equivalence relations, as they

are not transitive.e relation has some peculiar properties: its diagram is empty. It is

reflexive on the empty set

, but on no other set. It is symmetric, as thereis no arrow for which there is not any arrow in the opposite direction.But it is also asymmetric and antisymmetric because there is no arrowfor which there is an arrow in the opposite direction. is also transitive.Consequently, is an equivalence relation on .

e relation with the diagram below is not reflexive on the set withthe two elements Ponte Vecchio and Eiffel Tower, because there is no loopattached to Eiffel Tower.

Eiffel Tower

Ponte Vecchio

e relation is symmetric, but not asymmetric or antisymmetric. It isalso not transitive: there is an arrow from Eiffel Tower to Ponte Vecchio

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and an arrow back from Ponte Vecchio to Eiffel Tower, but there is no

shortcut from Eiffel Tower directly to Eiffel Tower, that is, there is noloop attached to Eiffel Tower.

Now I turn to a relation that cannot easily be described by a diagramor by listing the pairs in the relation, namely to the relation that obtainsbetween personsdand e if and only ifdis at least as tall ase , that is, therelation that contains exactly those pairsd, esuch thatdis at least astall ase . is relation is reflexive on the set of all persons because everyperson is at least as tall as themselves. e relation is not symmetric: I amtaller than my brother, so I am at least as tall as he is, but he is not at leastas tall as I am. us the pairVolker Halbach, Volker Halbachs brother

is an element of the relation, whileVolker Halbachs brother, VolkerHalbachis not an element of the relation. e relation is transitive: ifdis at least as tall ase and e is at least as tall as f, then surelydis at leastas tall as f. Since the relation is not symmetric it is not an equivalencerelation.

e relation oflovingcontains exactly those ordered pairsd, esuchthatdlovese . is relation is presumably not reflexive on the set of allpersons: some people do not love themselves. Much grief is caused by thefact that this relation is not symmetric, and the fortunate cases of mutuallove show that the relation is also not asymmetric or antisymmetric. It

clearly fails to be transitive: there are many cases in whichdlovese andeloves f, but in many casesddoes not love his or her rival f.e relation ofnot having the same hair colouris the set containing

exactly those pairsd, esuch thatddoes not have the same hair colouras e . is relation is surely not reflexive on the set of all persons, butit is symmetric: ifds hair colour is different from e s hair colour, thensurelye s hair colour is different fromds hair colour. e relation fails tobe transitive: my hair colour is different from my brothers hair colourand his hair colour is different from mine. If the relation were transitive,then I would have a hair colour that differs from my own hair colour.More formally, the pairsVolker Halbach, Volker Halbachs brotherandVolker Halbachs brother, Volker Halbachare in the relation, while

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Volker Halbach, Volker Halbachis not. is example illustrates againthat in the definition of transitivity it is not presupposed thatdmust bedifferent from f.

e relation of being born on the same day is reflexive on the set ofall persons; it is also symmetric and transitive. us it is an equivalencerelation on the set of all persons.

I will now turn to another very important kind of relation. It is soimportant that it deserves a section of its own.

.

.. A binary relation R is a function iff for all d, e,f : if d, e R andd,f R then e = f .

us a relation is a function if for everydthere is at most one e suchthatd, eis in the relation.

In the diagram of a function there is at most one arrow leaving fromany point in the diagram. In order to illustrate this, I will consider thefunction with the following ordered pairs as its elements:France, Paris,Italy, Rome,England, London, andthe United Kingdom, London.e function has the following diagram:

France

Paris

Italy Rome

England London

the United Kingdom

In this diagram, there are arrows from France, Italy, England, andthe United Kingdom. e set containing France, Italy, England and the

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United Kingdom is called the domain of the function. e names of the

three cities receive arrows; the set of these three cities is called the rangeof the function.

..

(i) e domain of a function R is the set{ d: there is an e such thatd, e R }.

(ii) e range of a function R is the set{e: there is a d such thatd, eR }.

(iii) R is a function into the set M if and only if all elements of the rangeof the function are in M.

e elements of the domain serve as inputs or arguments of thefunction; the elements of the range are outputs or values.

In the above example the set containing France, Italy, England and the

United Kingdom is the domain of the function, while the set with Paris,Rome and London as its elements is the range. According to (iii) of theabove definition, the function is a function into the set of all Europeancities, for instance.

.. If d is in the domain of a function R one writes R(d)forthe unique object e such thatd, eis in R.

e relation containing all pairs d, e such thatdhaseas a biological

mother is a function: ifdhas e as biological mother and dhas f asbiological mother, thene and fmust be identical. Its domain is the setof all people and animals, its range the set of all female animals withoffspring.

In contrast, the relation containing all pairsd, esuch thatdis thebiological mother ofe is not a function: my brother and I have the samebiological mother, yet we are not identical.

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. -

e relations I have considered so far are binary; they contain only orderedpairs. Expressions such as dlovese express binary relations; the expres-sion dlovese expresses the relation that contains exactly those orderedpairsd, esuch thatdlovese . In contrast, the expression dpreferseover f expresses a ternary (-place) relation rather than a binary one. Inorder to deal with ternary relations, ordered triples (or triples for short)are used. Triples are very much like ordered pairs.

A tripled, e ,fis identical with a tripleg, h,iif and only if theyagree in the first, second and third component, respectively, that is, if and

only if d=g , e = h and f =i .

Ternary relations are sets containing only triples.Besides ordered pairs and triples there are also quadruples and so on.

is canbe generalised to even higher arities n: an n-tuple d, d, . . . , dnhas n components. An n-tuple d, d, . . . , dn and an n-tuple e, e, . . . , enare identical if and only ifd = eand d = eand so on up to dn = en.Nown-tuples allow one to deal withn-place relations:

An n-place relation is a set containing only n-tuples. An n-place rela-tion is called a relation of arity n.

For instance, there is the relation that contains exactly those-tuples

d, e,f,g, hsuch thatdkillede with f ingwith the help ofh.

is isa-ary relation, which, for instance, contains among others the-tupleBrutus, Caesar, Brutus knife, Rome, Cassius.

I also allow-tuples as a special case. I stipulate that dis simplyditself. us a -place or unary relation is just some set.

As has been remarked in footnote above, one can define ordered pairs as certain sets.Similarly one can define the tripled,e,fusing ordered pairs asd,e,f. So in theend only sets are needed.

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. ,

In logic usually sentences are taken as the objects that can be true or false.Of course not every sentence of English can be true or false: a commandor a question is neither true nor false.

Sentences that are true or false are called declarative sentences. In whatfollows I will focus exclusively on declarative sentences.I will oen dropthe restriction declarative, because I will be concerned exclusively withdeclarative sentences.

Whether a sentence is true or not may depend on who is uttering thesentence, who is addressed, where it is said and various other factors. e

sentence I am Volker Halbach is true when I say it, but the same sentenceis false when uttered by you, the reader. It is raining might be true inOxford but false in Los Angeles at the same time. So the truth of thesentence depends partly on the context, that is, on the speaker, the place,the addressee and so on. Dealing with contexts is tricky and logicianshave developed theories about how the context relates to the truth of asentence. I will try to use examples where the context of utterance doesnot really matter, but for some examples the context will matter. Even inthose cases, what I am going to say will be correct as long as the contextdoes not shi during the discussion of an example. is will guarantee

that a true sentence cannot become false from one line to the other.We oen draw conclusions from certain sentences, and a sentenceis oen said to follow from or to be a consequence of certain sentences.Words like therefore, so, or hence, or phrases such as it follows thatoen mark a sentence that is supposed to follow from one or more sen-tences. e sentences from which one concludes a sentence are calledpremisses, the sentence, which is claimed to be supported by the pre-misses is called conclusion. Together premisses and conclusion form anargument.

.. An argument consists of a set of declarative sentences (the

premisses) and a declarative sentence (the conclusion) marked as the con-

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cluded sentence.

ere is no restriction on how the conclusion is marked as such.Expressions like therefore or so may be used for marking the conclusion.

Oen the conclusion is found at the end of an argument. e conclusion,however, may also be stated at the beginning of an argument and thepremisses, preceded by phrases such as this follows from or for, followthe conclusion.

In an argument there is always exactly one conclusion, but there maybe arbitrarily many premisses; there may be even only one premiss or nopremiss at all.

e following is an argument with the single premiss Zeno is a tor-

toise and the conclusion Zeno is toothless.

Zeno is a tortoise. erefore Zeno is toothless.

A biologist will probably accept that the conclusion follows from the pre-miss Zeno is a tortoise, as he will know that tortoises do not have teeth.at the conclusion follows from the premiss depends on a certain biolog-ical fact. is assumption can be made explicit by adding the premiss thattortoises are toothless. is will make the argument convincing not onlyto biologists but also to people with no biological knowledge at all. ebiologist, if prompted for a more explicit version of the argument, wouldprobably restate the argument with the additional premiss on which hemay have implicity relied all along:

Zeno is a tortoise. All tortoises are toothless. erefore Zenois toothless.

Now no special knowledge of the subject matter is required to see thatthe conclusion follows from the premisses. e conclusion follows fromthe two premisses purely formally or logically: the conclusion is a conse-quence of the premisses independently of any subject-specific assump-tions. It does not matter who Zeno is, what tortoises are, what beingtoothless is, or which objects the argument is thought to be about.

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In this argument the conclusion follows from the premisses indepen-

dently of what the premisses and conclusion are about. Whatever theyare taken to be about, in whatever way the subject-specific terms are(re-)interpreted, the conclusion will be true if the premisses are. Argu-ments of this kind are called logically valid or formally valid. usin a logically valid argument the conclusion follows from the premissesindependently of the subject matter.

. ( ). An argument is logicallyvalid if and only if there is no interpretation under which the premisses areall true and the conclusion is false.

In particular, if all terms are interpreted in the standard way, then,according to Characterisation ., the conclusion is true if the premissesare true. us the conclusion of a logically valid argument is true if thepremisses are true.

e notion of an interpretation employed in Characterisation .needs some clarification: An interpretation will assign meanings to thesubject-specific terms such as Zeno, tortoise, and iridium. It will alsodetermine which objects the argument is taken to be about. e logicalterms, that is, the subject-independent terms, such as all are not subjectto any (re-)interpretation.ese logical terms belong to the form of theargument and they are not affected by interpretations.

In later chapters I shall provide an exact definition of interpretationsor structures, as I shall call them in the case of formal languages. eseformal accounts of logical validity can also be seen as attempts to elucidatethe notion of logical validity in natural languages such as English at leastfor those parts of English that can be translated into the formal languages.

According to the above characterisation of logical validity, the mean-ings of the subject-specific terms do not matter for the logical validityof the argument. us, one can replace these terms by other terms and

A precise and informative definition of the logical validity of an argument is not so easy

to give. Sainsbury (, chapter ) provides an critical introductory discussion.

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thereby obtain a logically valid argument again. e following argument

has been obtained from the argument about Zeno by such a substitutionof nonlogical, that is, subject-specific terms:

Iridium is a metal. All metals are chemical elements. ere-fore iridium is a chemical element.

Both the argument about Zeno and the argument about iridium have the

same pattern; they share the same form. e conclusion follows from thepremisses solely in virtue of the form of the argument. is is the reasonfor calling such arguments formally valid.

e notion of logical or formal validity is occasionally contrasted withother, less strict notions of validity, under which more arguments comeout as valid. Some arguments in which the truth of the premisses doesguarantee the truth of the conclusion are not formally valid. Here is anexample:

Hagen is a bachelor. erefore Hagen is not married.

In this argument the conclusion is bound to be true if the premiss is true,but it is not logically or formally valid, that is, valid in virtue of its form.Hagen is not married follows from Hagen is a bachelor in virtue of themeaning of the word bachelor, which is subject-specific.

Also, arguments in which the premisses do not guarantee the truth of

the conclusion are oen called valid. Here is an example:

All emeralds observed so far have been green. erefore allemeralds are green.

e premiss may support the conclusion in some sense, but it does notguarantee the truth of the conclusion. Such arguments as the argumentabove are said to be inductively valid. In logically valid arguments, incontrast, the truth of the premisses guarantees the truth of the conclusion.

Logically valid arguments, are also called deductively valid.

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In this book I will focus on logical validity and not consider other, less

stringent kinds of validity. erefore, I shall oen drop the specificationlogical or formal: validity will always be understood as logical validity.

ere are good reasons to focus on logically valid arguments. Philoso-phers oen suppress premisses in arguments because they think thatthese premisses are too obvious to state. However, one philosophersobvious premiss can be another philosophers very contentious premiss.Trying to make an argument logically valid forces one to make all hiddenassumptions explicit. is may unearth premisses that are not obviousand uncontroversial at all. Also, there is usually not a unique way toadd premisses to render an argument logically valid, and it may remain

controversial which premisses were implicitly assumed by the originalauthor, or whether he relied on any implicit premisses at all. At any rate,if an argument is formally valid, then the validity does not rely on anypotentially controversial subject-specific assumptions: all the assump-tions needed to establish the conclusion will be explicitly laid out forinspection.

is is not to say that logical validity is always obvious: all requiredpremisses may have been made explicit, but it might not be obvious thatthe conclusion follows from the premisses, that is, one might not be ableto see easily that the argument is logically valid. Characterisation . of

logical validity does not demand an obvious connection between thepremisses and the conclusion that is easy to grasp. Almost all of theexamples of logically valid arguments considered in this book are toyexamples where it will be fairly obvious that they are logically valid, butshowing that an argument is logically valid can be extremely difficult.Mathematicians, for instance, are mainly concerned with establishingthat certain sentences (theorems) follow from certain premisses (axioms),

that is, with showing that certain arguments are logically valid. Of courseone can try to break up valid arguments into chains of short and obvioussteps. In Chapter this task is taken up and a formal notion of proof isdeveloped.

A valid argument need not have a true conclusion. In the following

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example the non-logical terms of the logically valid argument about Zeno

(or iridium) have been replaced in such a way as to make the conclusionfalse:

Water is a metal. All metals are chemical elements.ereforewater is a chemical element.

Although the conclusion Water is a chemical element is false, the argu-ment is logically valid: the conclusion still follows from the premisses.In a logically valid argument the conclusion may be false as long as atleast one premiss is false. In this case Water is a metal is false. erefore,one cannot refute the validity of an argument by merely pointing out afalse conclusion. If the conclusion of an argument is false, then either atleast one of the premisses is false or the argument is not logically valid(or both).

So far I have used only one argument form (argument pattern) in myexamples. Here is an argument of a different pattern:

EitherCO-emissions are cut or there will be more floods. Itis not the case thatCO-emissions are cut. erefore therewill be more floods.

e argument is logically valid according to Characterisation . of logi-cally valid arguments since the validity of the argument does not dependon the subject-specific terms such as CO-emissions and floods. e

validity of the argument depends on the logical terms either . . . or . . . and it is not the case that ....

In the argument about Zeno I could replace various terms, but notcomplete sentences. In the present example one can replace entire sen-tences. In this case the argument will still be valid aer replacing thesentences CO-emissions are cut and ere will be more floods withsome other sentences. e pattern of the valid argument is a pattern ofwhole sentences. Valid arguments of this kind are said to be proposi-

tionally valid. us an argument is propositionally valid if and only if

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there is no (re-)interpretation of the sentences in the argument such that

all premisses are true and yet the conclusion is false. ese patterns ofpropositionally valid arguments are studied in sentential or propositionallogic. Propositional validity will be treated in Chapters and .

e argument about Zeno can be adequately analysed in predicatelogic only, and not in propositional logic. Predicate logic is based onpropositional logic; from the technical point of view it is a refinement ofpropositional logic. us I shall start with propositional logic and thenmove on to predicate logic.

e notion of consistency is closely related to the notion of validity.

. (). A set of sentences is consistent if

and only if there is a least one interpretation under which all sentences ofthe set are true.

e negation of a sentence is obtained by writing It is not the casethat in front of the sentence (in English there are various stylisticallymore elegant ways to express negation). A sentence is false if and only ifits negation is true.

For a valid argument there is no interpretation under which the pre-misses are all true and the conclusion is false. us, for a valid argumentthere is no interpretation under which the premisses are all true and thenegation of the conclusion is also true. us, if an argument is valid, the

set obtained by adding the negation of the conclusion to the premisses isinconsistent; and if the set obtained by adding the negation of the conclu-

sion to the premisses is inconsistent, then there is no interpretation underwhich all sentences of that set are true, and, consequently, there is nointerpretation under which all the premisses are true and the conclusionis false. Hence one can define validity in terms of consistency:

An argument is valid if and only if the set obtained by addingthe negation of the conclusion to the premisses is inconsis-tent.

I have not imposed any restrictions on the number of premisses in

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an argument. In particular, there may be no premisses at all. Arguments

with no premisses may still be logically valid. e following argumentdoes not have any premisses but only a conclusion:

All metaphysicians are metaphysicians.

e sentence is true, and it is true for any interpretation of metaphysician,which is the only non-logical, subject-specific term in the sentence.ere-fore, there is no interpretation under which all premisses are true (thereis none) and the conclusion is false. erefore, the argument is logically

valid.e conclusion of a logically valid argument with no premisses isalso called logically true or logically valid.

. ( ). A sentence is logically true ifand only if it is true under any interpretation.

ere are also sentences that cannot be made true by any interpreta-tion. ese sentences are called logically false. ey are called contra-dictions.

.(). A sentence is a contradictionif and only if it is false under any interpretation.

If a sentence Afollows logically from a sentence Band B followslogically fromA, that is, if the argument withA as its only premiss and

Bas conclusion, and the argument withBas premiss andA as conclusion,are logically valid, then the sentencesAand B are logically equivalent.According to Characterisation ., the argument withAas premiss andBas conclusion and the argument with Bas premiss andA as conclusionare both logically valid if and only ifAand B are true under the sameinterpretations:

. ( ). Sentences are logicallyequivalent if and only if they are true under exactly the same interpreta-tions.

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. ,

In the following chapters I will examine formal languages. ese lan-guages are in many respects much less complicated than natural languagessuch as English or German. ey are intended to mirror certain prop-erties of natural languages. Some philosophers conceive of these formallanguages as models for natural languages.

Usually, in analysing either natural or formal languages one distin-guishes three aspects of a language: syntax, semantics and pragmatics.

In order to use a language competently, one must master all three aspectsof it.

Syntax is concerned with the expressions of a language bare of theirmeanings. In the syntax of a language it is specified what the words orsentences of the language are. In general, the grammar of a languagebelongs to the syntax of that language, and oen syntax is identified withgrammar. In order to use the language competently, one must know thegrammar of the language. In particular, one must know how to formsentences in the language.

Semanticssemantics may be described as the study of the meaningsof the expressions of a language. Clearly, to use a language one must notonly know what the words and the sentences of the language are; one

must also know what they mean.e expression Im Mondschein hockt auf den Grbern eine wildgespenstische Gestalt is a well-formed German declarative sentence. Ina syntactic analysis of that sentence one may remark that hockt is a verbin present tense, and so on. All this is merely syntactic information; itdoes not tell one anything about the meaning of that sentence. In orderto understand the sentence, you need information about meaning. Forinstance, it is a semantic fact of German that im Mondschein means inthe moonlight.

e trichotomy was introduced by Morris ().

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Syntax and Semantics of Propositional Logic

In this section I shall introduce the language of propositional logic. Allother formal languages that will be discussed in this manual are based onthe language of propositional logic.

Before introducing the syntax of this language I will briefly outline a

method for talking efficiently about the expressions of a language and fordescribing the syntax of a language. e method is by no means specificto the language of propositional logic.

.

By enclosing an expression in quotation marks one can talk about thatexpression. Using quotation marks one can say, for instance, that A isthe first letter of the alphabet and that Gli enigmi sono tre is an Italiansentence. e quotation of an expression is that very expression enclosed

in quotation marks.

Quotation marks allow one to designate single expressions. Describ-ing the syntax of a language usually makes it necessary to talk about alarge or infinite number of expressions. For instance, one would like tobe able to state that one can construct new sentences in English by com-bining sentences using and (ignoring capitalisation and punctuation).

Cappelen and LePore (Spring ) provide an overview of the intricacies of quotationand of proposed theories. A classical text on quotation is Quine ().

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Logicians would express that general claim by saying the following:

() If andare English sentences then and is anEnglish sentence.

It is raining is an English sentence. If one takes It is raining as bothandin the above rule, then the rule says that It is raining and it israining is also an English sentence (again we ignore the absence of thefull stop and the missing capitalisation). One can then use It is rainingand it is raining asagain and It is raining asto conclude from therule that also It is raining and it is raining and it is raining is an English

sentence. In this way one can construct longer and longer sentences andthere is no limit to the iterations.I think that () is fairly straightforward and should be easy to

understand.ere is, however, something puzzling about it as well: thepart of () claiming that and is an English sentence is decidedlynot about the expression in quotation marks. e letters and areGreek letters, and the expression with as first symbol, followed by aspace, followed by and and another space, followed by , is definitelynot an English sentence. Only once and are replaced by Englishsentences does and become an English sentence.

e Greek letters used in this way are metavariables or metalinguisticvariables.

us, the above rule may also be expressed in the following way:

An English sentence followed by and (in spaces) and anotheror the same English sentence is also an English sentence.

is way of rephrasing () does not rely on quotation marks but ontalking about expressions following one another. is method is perhapssafer than using () with its quotation marks and metavariables, but itis also more cumbersome when applied to intricate grammatical rules.us, I will present definitions in the style of () rather than talkingabout expressions following one another.

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Logicians hardly ever use the expressions of formal languages in

the way they use the expressions of their mother tongue, but they oentalk and write about the expressions of these formal languages. Sincethe expressions of the formal languages they are concerned with differfrom expressions of English, logicians usually drop the quotation marks.Instead of saying

(P (Q R)) is a sentence of the language of propositionallogic

they say,

(P (Q R))is a sentence of the language of propositionallogic.

I will follow this convention and usually drop quotation marks aroundthe expressions of formal languages in this manual. is also applies toexpressions containing metavariables.

.

Now I can describe the syntax of the languageLof propositional logic.

. ( ). P, Q, R, P, Q , R , P, Q, R, P,Q, R, and so on are sentence letters.

Using metavariables I will define the notion of a sentence of the lan-guageLof propositional logic.

. ( L).

(i) All sentence letters are sentences ofL.(ii) Ifandare sentences ofL, then ,( ),( ),( )

and( )are sentences ofL.(iii) Nothing else is a sentence ofL.

Given what I have said about metavariables, (ii) implies that ( )becomes a sentence of the language of propositional logic when the Greek

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letters and have been replaced by sentences of the language of

propositional logic. e Greek letters and themselves are notexpressions of the language L.

As I explained on page , I could have formulated part (ii) of Defini-tion . without using the metavariables and by expressing (ii) inthe following way:

e negation symbol followed by a sentence ofLis again asentence ofL. e symbol ( followed by a sentence ofL,followed by the symbol (or , , ), followed by asentence (not necessarily distinct from the first one), followed

by the symbol )

, is a sentence ofL.

I hope that (ii) is not only shorter but also easier to grasp.e last clause in Definition . says that only expressions one can

obtain by using clauses (i) and (ii) are sentences. Very oen this last clauseis omitted and the clauses (i) and (ii) are implicitly taken to be the onlymeans of arriving at sentences. At various points in this manual I willprovide definitions that are similar to Definition .. In those cases I willdrop the analogues of clause (iii) for the sake of simplicity.

Logicians also say the negation of rather than .In this termi-nology, is the negation of, and similarly()is the conjunction of

and, andis the disjunction ofand.

e sentence

(P Q),for instance, is the negation of(P Q). .. e following expressions are sentences of the language L:

((P P) Q),(R (P (P Q))),((P Q) P).

In the next example I show how to prove that the last sentence aboveis indeed a sentence ofL.

.. By Definition .(i),Pis a sentence ofL. us, by (ii),Pis also a sentence ofL. By (i) again,Qis a sentence ofL. By (ii) and

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by what has been said so far,(P Q) is a sentence, and by (ii) again andby what has been said so far,((P Q) P)is also a sentence ofL.

e symbols , , , , are called connectives. ey roughly cor-respond to the English expressions not, and, or, if ..., then ... and ifand only if, respectively.

name in English symbol alternativeused here symbols

conjunction and .,&disjunction or +,vnegation it is not the ,

case thatarrow if . . . then , (materialimplication,conditional)

double arrow, if and only if , (biconditionalmaterialequivalence)

e names in brackets and the symbols in the rightmost column are used

by some other authors; they will not be used here.e expressions in the in English column indicate how the connec-

tives are commonly read, rather than their precise meanings.

.

A sentence with many brackets can be confusing. For convenience I shallemploy certain rules for dropping brackets.ese rules are not revisionsof the definition of a sentence and they do not form part of the officialsyntax of the language L of propositional logic. ese rules are mereconventions that allow one to write down abbreviations of sentencesinstead of the sentences themselves in their official form.

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.

In Section . I gave a characterisation of the logical validity of argumentsin English. In this section I will define validity for arguments in thelanguage Lof propositional logic. e informal Characterisation .of validity for English arguments will be adapted to the language Lofpropositional logic and thereby be transformed into a precise definition.

In order to define logical validity for the language L, the notion ofan interpretation for the language Lneeds to be made precise. First, Ineed to say which expressions can be interpreted in different ways andwhich are always interpreted in the same way. e connectives are logical

symbols, the brackets merely auxiliary symbols; logical and auxiliarysymbols cannot be re-interpreted (insofar as one can speak of auxiliarysymbols being interpreted at all). e sentence letters are the only non-logical symbols ofL; they can be interpreted in different ways.

e interpretations of the sentence letters will be provided by so-calledL-structures.eseL-structures need only provide enough informationto determine whether a sentence is true or false. Now, under whichconditions is the sentence P Q true? If the connective functionslike and in English, then bothPand Q must be true for P Q to betrue; otherwiseP Qwill be false. Similarly, since works like not, the

sentence

Ris true if and only ifRis false. As

corresponds to or, thesentenceP Qis true if and only ifPorQ is true (or both are true).e arrow corresponds roughly to the English if ... then, but the

latter has a rather complicated semantics.e L-sentenceP Qis falseif and only ifPis true andQ is false; otherwise it is true. e phrase if . . .then, which corresponds to the arrow, doesnotalways behave like this.How the arrow is related to if . . . then will be discussed in Section ..

Generally, in L, the truth or falsity of a sentence ofLdepends onlyon the truth or falsity of the sentence letters occurring in the sentence;any further information is not relevant. erefore,L-structures needonly determine the truth and falsity of all sentence letters.

Instead of saying that a sentence is true, logicians say that the sentence

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has the truth-value True. is sounds like a philosophically profound

move since new objects are required: truth-values. However, truth-valuesare hardly mysterious objects. Some more mathematically minded logi-cians use the numberfor the truth-value True andfor the truth-valueFalse. In the end it matters only that True and False are distinct. It ispossible, although not very customary and technically less convenient,to develop the semantics ofLwithout truth-values by saying that a sen-tence is true (or false) instead of saying that it has the truth-value True (orFalse). I shall use the letter T as a name for the truth-value True and Ffor the truth-value False.

us an L-structure provides interpretations of all sentence letters

by assigning to every sentence letter exactly one truth-value, T or F.. (L-). AnL-structure is an assignment of ex-actly one truth-value(TorF)to every sentence letter of the language L.

One may think of an L-structure as an infinite list that provides avalue T or F for every sentence letter.e beginning of such a list couldlook like this:

P Q R P Q R P Q RT F F F T F T T F

Starting from the truth-values assigned to the sentence letters by anL-structure A, one can work out the truth-values for sentences con-taining connectives in the following way.e shortest sentences are thesentence letters; their respective truth-values are fixed directly by theL-structureA. For instance,Pcould be assigned the truth-value T andRcould be assigned the same truth-value. In this caseP Rwould receivethe truth-valueT, too. IfPis given the truth-valueFbyA, the sentenceP (P R)gets the truth-valueT, becauseP Ris true; ands beingtrue is sufficient to make a sentence true.

In more mathematical terms, an L-structure is a function into the set{T, F}with the

set of all sentence letters ofLas its domain.

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us, the truth-values of the shortest sentences, that is, of the sentence

letters, are fixed by theL-structureA, and then the truth-values for longersentences are determined successively by the truth-values of the sentencesthey are made up from.

I will writeAfor the truth-value ofthat is obtained on the basisofA; it is determined byAin the following way:

.( L-). LetAbe someL-structure.en . . . A assigns to every sentence ofL eitherT orF in the followingway:

(i) If is a sentence letter,Ais the truth-value assigned to by theL

-structure A.

(ii) A = Tif and only ifA = F.(iii) A = Tif and only ifA = TandA = T.(iv) A = Tif and only ifA = TorA = T.(v) A = Tif and only ifA = ForA = T.

(vi) A =Tif and only ifA =A.Instead of writingA =T, I will sometimes write thatis true in A

or that T is the truth-value ofin A.e definition of . . . Adoes not say explicitly when a sentence has the

truth-valueFin A. Nonetheless, extra clauses for falsity are not required,since a sentence has the truth-value F (inA) if and only if it does not havethe truth-valueT. In particular, a sentence letter has the truth-valueFifand only if it is not true in A. Similarly, for negation the following falsityclause follows from Definition .:

A = F if and only ifA = T.

Definition . also implies the following claim for conjunction (and simi-larly for the other connectives):

More formally, . . . Ais a function with the set of all L-sentences as its domain into

the set{T, F}. It properly extends theL

-structureA

, that is, it contains all the orderedpairs that the function Acontains and more besides them.

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( )T T TT F FF T TF F T

( )T T TT F FF T FF F T

Truth tables are also useful for calculating the truth-values of sen-tences with more than one connective. I will use the same example asabove to show how this can be done. e first step is to write below eachsentence letter the truth-value assigned to it by the L-structureA:

P Q (P Q) (P Q )T F T F T F

e next step is to calculate the truth-values of sentences directly built upfrom sentence letters according to the truth tables (in this case the tablesfor and are needed):

P Q (P Q) (P Q )T F T F F T F F

en one can go on to determine the truth-values for more and morecomplex sentences:

P Q

(P Q) (P

Q)T F T T F F T F F

Finally, one will obtain the truth-value for the entire sentence (here high-lighted in using boldface):

P Q (P Q) (P Q)T F T T F F F T F F

One can also use a (multi-line) truth table to work out the truth-valuesof a given sentence for all L-structures.

In a truth table one can also work out the truth-value of the sentence(P Q) (PQ)in any given L-structure. I employ again the

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sentence (P Q) (P Q)as an example; it contains two sentenceletters. In a given structure,Pcan be true or false, andQ can be true orfalse. us there are four possibilities: in any given L-structure, eitherbothPandQare true, orPis true andQ is false, orPis false andQis true,or both sentence letters are false. ese four possibilities are captured inthe two lemost columns of the truth table below. Now one can calculate

the truth-values of the sentence for all four possibilities, and, thereby, forallL-structures:

P Q (P Q) (P Q)T T F T T T T T T T

T F T T F F F T F FF T F F T T T F F TF F F F T F T F F F

Again, the column with the truth-value of the entire sentence is in boldfaceletters.I will call this column the main column.

Clearly, if there are onlyTs in the main column of a sentence thesentence is true in all L-structures; if there are onlyFs in the maincolumn the sentence is false in all L-structures. us one can use truthtables to determine whether a sentence is always true, or whether it isalways false, or whether it is true in some structures and false in others.

e notion of an L-structure corresponds to that of an interpretationof an English sentence in Section .. In that section I also used thenotion of an English sentence being true under an interpretation; thiscorresponds to the notion of anL-sentence being true in a structure. edefinitions of logical validity in English (Characterisation .), of logicaltruth in English (Characterisation .), and so on, can be adapted to thelanguage Lof propositional logic. e definitions are the same for Las for English, except that the informal notion of an interpretation fromSection . is replaced by the technical notion of an L-structure.

..

(i) A sentence ofLis logically true if and only if is true in allL-

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structures.

(ii) A sentence ofL is a contradiction if and only if is not true inanyL-structure.

(iii) A sentence and a sentenceare logically equivalent if andaretrue in exactly the sameL-structures.

Logically true sentences are also called tautologies.Logical truths, contradictions and logically equivalent sentences ofL

can also be characterised in terms of truth tables. In what follows, truth

tables are always understood as full truth tables with lines for all possiblecombinations of truth-values of all the sentence letters in the sentence.

..(i) A sentence ofLis logically true(or a tautology)if and only if there

are onlyTs in the main column of its truth table.(ii) A sentence is a contradiction if and only if there are onlyFs in the

main column of its truth table.(iii) A sentence and a sentenceare logically equivalent if they agree

on the truth-values in their main columns.

One of the main purposes of developing semantics for L was todefine the notion of a valid argument in Lthat would be analogous toCharacterisation . of validity for arguments in English.

.. Let be a set of sentences ofLand a sentence ofL.e argument with all sentences in as premisses and as conclusion isvalid if and only if there is no L-structure in which all sentences in aretrue andis false.

e phrase e argument with all sentences in as premisses andas conclusion is valid will be abbreviated by ; this is also oenread as follows from or as (logically) implies. us if andonly if the following holds for all L-structuresA:

IfA = T for all , thenA = T.

us an L-argument is not valid iff there is a structure that makes all

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premisses true and the conclusion false:

.. AnL-structure is a counterexample to the argumentwith as the set of premisses and as conclusion if and only if A =T

for all andA = F.erefore, an argument inLis valid if and only if it does not have a coun-terexample.

Following the pattern of the Definition . of consistency for sets ofsentences in English I will define consistency for sets ofL-sentences:

. ( ). A set ofL-sentences issemantically consistent if and only if there is anL-structure Asuch that

A = Tfor all sentences of. Semantic inconsistency is just the oppositeof semantic consistency: a setofL-sentences is semantically inconsistentif and only if is not consistent.

Aer Definition . I argued that an argument is valid if and only ifthe set obtained by adding the negation of the conclusion to the premissesis inconsistent. e argument carries over to L:

.. If and all elements of are L-sentences, then the fol-lowing obtains:

if and only if the set containing all sentences in and

is semantically inconsistent.

us, for an argument with, say, two premisses andand a con-clusion, this means that , if and only if the set{,, }issemantically inconsistent. e proof of the theorem is le to the reader.

Logicians usually allow infinite sets of premisses, but such infinitesets of premisses will not play an important role here. One can actuallyprove that if a sentence ofLfollows from a set of sentences, then

ere is an alternative way of defining the consistency of sets ofL-sentences, which isknown as syntactic consistency. Although the definition looks very different, both no-

tions of consistency coincide. Syntactic consistency will be introduced in Definition ..

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already follows from a finite set of sentences in . is result is known

as the Compactnesseorem of propositional logic.e setof premisses may also be empty. A sentence follows from the

empty set of premisses if and only if it is a tautology (that is, iff it is logically

true). is is fairly obvious; it is also a special case ofeorem . below.Ifhas only finitely many elements, one can use truth tables to check

whether . I will show how to answer the question whether bymeans of an example.

.. {P Q , Q} P.Claims like the one above may be abbreviated by dropping the curly

brackets around the premisses.Generally, , . . . , n, where, . . . ,nandare L-sentences, is short for {, . . . ,n} .So the claim of

Example . may be writtenP Q , Q P.First I draw a truth table for the premisses and the conclusion in the

following way:P Q P Q Q P

T T T F F T T F TT F T T T F F F TF T F T F T T T FF F F T T F F T F

(.)

Now I will check whether there is any line in which the entries in themain columns of the premisses all have Ts, while the conclusion has an F.In the first line of truth-values the first premiss receives an F, the seconda T, and the conclusion an F. e second and fourth lines also have Fsfor one premiss.e third line has Ts for both premisses, but also a T forthe conclusion. us, there is no line where all premisses receive Ts andthe conclusion an F. erefore, the argument is valid, that is, Pfollowsfrom{P Q , Q}or, formally,P Q , Q P.

For finite setsof premisses, one can reduce the problem of checkingwhether to the problem of checking whether a single sentence islogically true. To do this one combines all of the premisses, that is, allsentences in, using , and then one puts the resulting conjunction in

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front of an arrow followed by the conclusion. e resulting sentence is

logically true if and only if the argument is valid. is can be expressedmore succinctly as follows:

.. , . . . ,n ifand only if . . .n is a tautology(that is, iff . . . n is logically true).

I do not give a proof of this theorem here.I will applyeorem . to the example above: First, the two pre-

misses are combined into(P Q) Q. It is necessary to reintroducethe brackets aroundP Qbecause otherwise the result would be anabbreviation forP (Q Q)as binds more strongly than . Next,

the arrow is put between this conjunction and the conclusion.

is yields((P Q) Q) P. e brackets around the conjunction of the twopremisses are not really necessary since binds more strongly than ,but they might make the sentence easier to read. e truth table for thislong sentence looks like this:

P Q ((P Q) Q ) PT T T F F T F T T F TT F T T T F F F T F TF T F T F T T T T T FF F F T T F F F T T F

(.)

us, the sentence((P Q) Q) Pis valid, that is, it is a tautology.Byeorem ., it follows thatP Q , Q P. Of course, we knowthis already from truth table (.).

Drawing truth tables for arguments or sentences with many sentenceletters is cumbersome: every new sentence letter doubles the number oflines of the truth table, because for any already given line two possibilitiesmust be considered: the new sentence letter can have truth-value T or F.us, a sentence or argument withsentence letter requires onlylines,one withdifferent sentence letters requires, one withrequireslines,and so on. Generally, the truth table for a sentence or argument withndif-ferent sentence letters will haven lines of truth-values. In Exercise .

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there will be an argument in L with sentence letters. us, writing

down the corresponding truth table would require = lines.To show that an L-sentence is a tautology, one does not need to draw

a complete truth table. One only needs to show that there cannot be aline in the truth table that yields an F in the main column. In order torefute the existence of such a line for Example ., the best strategy is tostart with the assumption that the value in the main column is F:

P Q ((P Q) Q ) PF

A sentence of the form

has the truth-value F only ifhas truth-value T andhas truth-value F. us, I must have:

P Q ((P Q ) Q ) PT F F

I can continue to calculate truth-values backwards:

P Q ((P Q ) Q) PT T T F F T

I write the calculated truth-values also under the first two occurrences ofPandQ :

P Q ((P Q ) Q) PT T ? T T T F F T

But now there is no way to continue. e slot marked with a questionmark cannot be filled with a truth-value: there should be an F under thenegation symbol , asQ has truth-value T, but there should also be a T,because(P Q)andPhave Ts. It follows that there cannot be a linewith an F in the main column. erefore, in the full truth table with allthe lines, all truth-values in the main column are Ts. is proves againthat((P Q) Q) Pis a tautology.

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Since it is not easy (for an examination marker, for instance) to recon-

struct how the truth-values have been calculated, it is useful to record theorder in which the values were obtained:

P Q ((P Q) Q) PT T ? T T T F F T

Of course I could have written down the truth-values in a different order.For instance, aer arriving atF, I could have calculated the value in thelast column and only then have turned to the part preceding.

Now I will use the same method to show that(P Q) Pis nota tautology. As before, an F is written in the main column:

P Q (P Q ) PF

e following table results from the first backwards-calculation:

P Q (P Q ) PT F F

us,Pmust receive the truth-value T:

P Q (P Q ) PT F F T

us one can also write a T under the first occurrence ofP:

P Q (P Q ) PT T F F T

SinceP Qhas the truth-value T andPalso has the truth-valueT, thesentence Qreceives a T, and Q , accordingly, an F. Hence, the line canbe completed as follows (I will also insert the obligatory indices):

P Q (

P Q) P

T F T T T F F F T

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At any rate, when one has arrived at a possible line one should cal-

culate the truth-values from bottom to top (starting from the truth-valuesthat have been obtained for the sentence letters) to ensure that one has notmissed a column that cannot given a unique truth-value.Only once thisfinal check has been carried out, one knows that the line obtained is apossible line in a truth table.

e above backwards-calculations shows that the sentence(P Q) Phas truth-valueF ifPhas the truth-valueT and Q has thetruth-value F.

Technically speaking, ifA(P) = Tand A(Q) = Ffor a structure A,then(P Q) Pis false in A. So, by Definition .(i),(P Q)Pis not logically valid, that is, it is not a tautology.

Sometimes this method of calculating truth-values backwards re-quires more than one line. is is the case in the following example:

P Q (P Q) (Q P)F

If a sentence is false, there are two possibilities: could havetruth-value T andtruth-value F, or, could have F andcould havetruth-value T. As such, one has to take these two possibilities into account:

P Q (P Q ) (Q P)

T F F F F T

I have underlined the truth-values that cannot be uniquely determined,and so more than one possibility (line) needs to be checked.

e rest is routine. e indices in the table below indicate the orderin which I arrived at the truth-values. e order in which the valuesare calculated does not really matter, but the indexing makes it easier tofollow the reasoning.

P Q (P Q) (Q P)

F T ? T F

T F F F F F T F T T ?

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Neither of the two lines can be completed.is shows that(P Q)(Q P)is a tautology.

Of course it can happen that more lines are required and that differentcases under consideration have to be split up into further subcases.

e method of backwards-calculation can also be applied in order tocheck whether an argument is valid or not. To show that an argumentis not valid, one has to find a line (that is, a structure) where all of thepremisses are true and the conclusion is false. If there is no such line,the argument is valid. Here is an example of how to use the method todetermine whether an argument is valid. I have picked an example thatforces me to consider several different cases. So, I want to determine

whether

P Q , (P Q) (P P) (P Q) P.

I will start by writing the two premisses and the conclusion in one table.I have to check whether there can be a line in the table where the twopremisses come out as true while the conclusion is false. As such, I shouldstart by writingTs in the main columns of the premisses and an F in themain column of the conclusion:

P Q P PQ (P Q) (P P) (PQ) PT T F

Now I have to make a case distinction: the first sentence could be truebecausePis false or becauseQ is true. Similarly, in the case of the othersentences, there is no unique way to continue. Given that I can make acase distinction with respect to any of the three sentences, it is not clearhow to proceed. But some ways of proceeding can make the calculationsquicker and less complicated. It is useful to avoid as much as possiblepicking a sentence that will require a new case distinction in the nextstep.Ultimately though, so long as all possible cases are systematicallychecked, the order in which one proceeds will not affect the end result.

At this stage in the calculation, a case distinction cannot be avoided,and so I will pick the last sentence: (P Q) P can be false either

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becauseP Qis false or becausePis false. I will try to complete the

line for the latter case and leave the former for later:

P Q P PQ (P Q) (P P) (PQ) P T T F F T F FT T ? F F F

e second line of the table cannot be completed: the second premissmust be true, but it follows from my assumption that both (P Q)andP Pmust be false. is means that the rules tell me to write an Fin the slot marked by ?, because Pis false, but they also tell me to writea T for ?, because the entire sentence(P Q) (P P)is true and

(P Q)is already false.Line is more complicated, because P Q can be false for tworeasons: first,Pcan be true whileQis false or, second,Pcan be false whileQis true. us, I need to distinguish the subcases marked . and .:

P Q P PQ (P Q) (P P) (PQ) P. T T ? TT TF F F T F FT T ? F F F. T T FF T F

SincePis true andP Qis true,Q has to be true as well. But accordingto the assumption,Q is false. erefore, line.cannot be completed.

Only case . remains:

P Q P PQ (P Q) (P P) (PQ) P. T T ? TT TFF F T F FT T ? F F F. T F TT T F ? FFT F

Since the second premiss is true and (P Q)is false,P Pought tobe true. But this cannot be the case, becausePis false. So line. cannotbe completed either. Since this exhausts the possible ways of refuting theargument, the following claim has been established:

P Q , (P Q) (P P) (P Q) P.

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terms of and in the following way:

T T F F FT F F F TF T T F FF F T T T

Alternatively, one can re-express or defineas (). us, addingto the language Lwould not increase the expressive power ofL. econnective would only allow one to abbreviate some sentences. ere

are many more truth tables for whichL

does not have connectives. Sofar I have looked only at binary connectives (connectives conjoining twosentences) such as , ,, and , but there are also unary connectives(connectives taking one sentence) other than; and there are ternaryconnectives (connectives taking three sentences), and so on. Can all theseconnectives be expressed with the connectives ofL, that is, with , , ,, and ? e answer is yes: all truth tables can be produced with theold connectives ofL. is fact is called the truth-functional completenessofL. In fact, and together without any further connectives aresufficient for expressing all other connectives. And even on its ownwould do the trick. At any rate, adding more connectives toLthan thoseused here is not really necessary and would not increase the expressivepower ofL. I will not prove these results here.

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Formalisation in Propositional Logic

In the previous chapter I focused on the formal language L. e con-nectives , , , and ofLcan be used to combine sentences ofLtoform new compound sentences; the connective can be added in frontof a sentence ofLto build a new sentence.

English sentences can also be combined and modified with manydifferent connectives to form new sentences. For instance, one can writethe word and between two English sentences to obtain a new sentence.Or, because, although, but, while, if and many others can be usedin the same way. An expression that connects sentences can also bemore complex: the expression due to the fact that between two Englishsentences yields a new English sentence. Other expressions, such as if ...,then, connect sentences even though they are not written between twosentences.

Other expressions do not combine sentences, but rather modify a

sentence, as is the case with not, as is well known, John strongly be-lieves that, and regrettably. Not is special insofar as it oen cannot besimply inserted into a sentence, but rather requires the introduction ofthe auxiliary verb to do: the introduction of not into Alan went toLondon yields Alan did not go to London. In this respect not is morecomplicated than an adverb such as regrettably or the connective ofL.

In the previous chapter I defined the notion of a connective; now Iwill apply it to English as well:Expressions that can be used to combine ormodify English sentences are connectives.is definition is far from beingprecise, but an exact definition of the notion of a connective of English

is not easy to give because sometimes the connectives are not simply

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plugged into or written between sentences. Occasionally the sentences

themselves have to be modified, for instance, by introducing auxiliaryverbs, as the above examples of not shows.

. -

e connectives ofL, that is, , , , and, correspond to connectivesin English. e semantics of the connectives ofLis very simple; it isencompassed in their truth tables. In contrast, many connectives ofEnglish function in a much more intricate way.

As an example I will consider the connective because. Imagine that I

drop my laptop computer on the street. Its had it: the screen is broken.So my laptop computer does not work. e sentence

My computer does not work because I dropped my computer

is also true: the laptop would still be functional if I had not dropped it.Moreover, it is true that the computer does not work and it is true that Idropped it.us, because connects the two true sentences My computer

does not work and I dropped my computer together forming a new truesentence. In this respect it seems similar to and.

In other cases, however, one can use because to connect two true

English sentencesA andBand end up with a false sentence. Aer pickingup my broken laptop, I consider the following sentence:

My laptop computer does not work because it is not plugged

in.

In the situation I just described, it is true that my computer does not work,and it is true that it is not plugged in as I am standing in the street withmy broken laptop. Nevertheless the sentence that my laptop computerdoes not work because it is not plugged in is false: it would work if I hadnot dropped it. Even if it were now plugged in, it would not work. It does

not work because I dropped it, not because it is not plugged in. So this

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is a case in which using because to connect two true sentences yields a

false sentence.Nevertheless, the truth of A becauseB is not completely independent

of the truth and falsity of the English sentencesAandB. IfA orB (orboth) are false, then A becauseB is also false. ese dependencies canbe summarised in the following truth table for the English connectivebecause, whereA andBare declarative sentences of English:

A B A becauseB

T T ?T F F

F T FF F F

e question mark indicates that in this case the truth-value of Abe-causeB depends not only on the truth-values of the direct subsentences,that is, on the truth-values of the sentencesA and B that because connects.is means that when because is used to connect two true sentences,sometimes the resulting sentence is true and sometimes the resultingsentence is false; so the truth-value of the compound sentence is not de-termined by the truth-values of the sentences connected by because. Inthis respect because differs from and. If the truth-value of the compound

sentence is determined by the truth-value of the connected sentences,as is the case with and, then the connective is called truth-functional.Connectives like because are not truth-functional.

e following is a general, less than precise characterisation of truth-functionality :

. (-). A connective is truth-functional if and only if the truth-value of the compound sentence cannotbe changed by replacing a direct subsentence with another sentence havingthe same truth-value.

For instance, because is not truth-functional: replacing the truesentence I dropped my computer with the equally true sentence the

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computer is not plugged in does change the truth-value of the compound

sentence

My computer does not work because I dropped my computer

from True to False.us, the definition of truth-functionality of an English connective

canbe paraphrased in terms of truth tables: a connective is truth-functional

if and only if its truth table does not contain any question marks.If ... then is usually translated as the arrow. Some of its occur-

rences, however, are definitely not truth-functional. A sentence is true if is false oris true. In the following sentence, if ... thenfunctions differently:

If Giovanni hadnt gone to England, he would not have caught

a cold in Cambridge.

Assume that Giovanni really did go to England, but did not catch a coldin Cambridge. In this case one may hesitate to assign a truth-value tothe sentence: some people would say that the sentence is neither truenor false; others would say that it is false. At any rate, in that case thesentence is not true. But if the whole sentence is not true, then this is acase in which the first subsentence following if is false, but the wholesentence is also false. But according to the truth table for a sentencewith a false antecedent is true. is means that the arrow cannot beused to formalise the sentence correctly.

If-sentences describing what would have happened under circum-stances that are not actual are called subjunctives or counterfactuals.In these sentences if does not function like the arrow and cannotbe translated as the arrow. e proper treatment of counterfactuals isbeyond the scope of this book.

Indicative conditionals such as

Lewis () is a classic text on counterfactuals.

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If Jones gets to the airport an hour late, his plane will wait

for him

are oen formalised using the arrow, but it is somewhat questionablewhether really is appropriate.

Assume, for instance, that Jones arrives at the airport early and heeasily catches the plane. Suppose also that Jones is not a VIP and so theairline would not have waited for him. Using the arrow for if one cantry to translate this sentence asP Qwith the following dictionary:

P: Jones gets to the airport an hour late,Q: Joness plane will wait for Jones.

IfPA = F, that is, ifPis false in the structure A, thenP Q is truein A, that is,P QA = Tby Definition . or by the truth table of onpage . According to the assumptions, Jones gets to the airport an hourlate is actually false. us, if the formalisation is correct, the displayedEnglish if-sentence should be true. But it is highly questionable whether

it is true: one may hold that If Jones gets to the airport an hour late, hisplane will wait for him is simply false, even if Jones gets there on time.

ere is an extensive literature on the treatment of if-sentences. etreatment of if-sentences, including counterfactuals, has interesting

philosophical implications. I shall not go further into the details of thisdiscussion here. e above example should be sufficient to show thatformalising if by the arrow is problematic even in the case of indicativeconditionals. For most purposes, however, the arrow is considered tobe close enough to the if . . . then .. . of English, with the exception ofcounterfactuals.

e definition of truth-functionality also applies to unary connectives:a unary connective is truth-functional if and only if the truth-value of thesentence with the connective cannot be changed by replacing the direct sub-sentence with a sentence with the same truth-value.

It is necessarily the case that A or It is necessary that is a unaryconnective that is not truth-functional. IfAis a false English sentence,

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then It is necessary that A is false, but ifA is true, It is necessary thatA

may be either true or false:

It is necessary that all trees are trees.

is sentence is true: all trees are trees is logically true and thus necessary.But, if the true sentence All trees are trees is replaced by the true sentence

Volker has ten coins in his pocket then the resulting sentence

It is necessary that Volker has ten coins in his pocket

is not true, because I could easily had only nine coins in my pocket.Generally ifA is only accidentally true, It is necessary thatA will be false.

us the corresponding truth table looks like this:

A it is necessary thatA

T ?F F

Some other connectives likeBill believes that . . . have nothing butquestion marks in their truth tables. In contrast, Bill knows that . . . hasthe same truth table as it is necessary that.

.

In this section and the next I will show how to translate English sentences

into Lsentences. ese translations are carried out in two steps: Firstthe sentence is brought into a standardised form, which is called the(propositional) logical form. In the second step the English expressions

A more comprehensive account of truth-functionality is given by Sainsbury (,Chapter ).

In this chapter I will usually drop the specification propositional from propositionallogical form since I will not deal with any other kind of logical form for now.ere is

also a more complex predicate logical form of an English sentence.

e predicate logicalform will be studied in Chapter .

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nectives of the formal language L, respectively. Finally, one reapplies the

procedure to the sentence(s) from which the main sentence is built up.I give now the five steps of the procedure and then show how it works

by means of some examples:

. Check if the sentence can be reformulated in a natural way as a sen-tence built up from one or more sentences with a truth-functionalconnective. If this is not possible, then the sentence should be put inbrackets and not analysed any further.

. If the sentence can be reformulated in a natural way as a sentencebuilt up from one or more sentences with a truth-functional connec-tive, do so.

. If that truth-functional connective is not one of the standard con-nectives in Table ., reformulate the sentence using the standardconnectives.

. Enclose the whole

oxford logic

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