Ockham’s Razor

By Ricardus Sapiens

ENTIA NON SUNT MULTIPLICANDA PRAETER NECESSITATEM. – Entities are not to be multiplied beyond necessity, the methodological principle underlying scientific thought known as OCKHAM'S RAZOR which is also the title of this collection of four stories in the form of direct speech, consistent with the requirements of logical possibility. It is presented as a programme of ongoing assignments given to a group of philosophy students (of varying ages) by their Professor, to be conducted as a collective endeavour. 
The stories, or dramatic tableaux, woven together by a common thread, interlace the realms of art and philosophy inextricably, forging an intricate bond.
Ockham’s Razor, poised at the intersection of methodological rigour and narrative splendour, unfurls as a circular odyssey. It is a journey of profound reflection upon weighty themes, including the enigma of the human condition and the nature of truth itself. In its essence, Ockham's Razor metamorphoses into a carousel, a cyclical excursion that finds resonance in the immortal words of T.S. Eliot, we shall not cease from exploration and the end of all our exploring – the end of the book in this case - will be to arrive where we began and know the place for the first time.
Ockham's Razor is a book for everyone and especially suitable for book clubs, for a collective experience just like that of the characters.

Marcus Aurelius Antoninus has remarked in one of the notes that he addressed to himself (6.44): “As Antoninus my city and country is Rome, as a man [a human being] it is the world.” It is a statement that the author strongly identifies with.

Ricardus Sapiens (a 'nom de plume') was born in Margaret River, a country town in the southwest of Western Australia, and has lived in Australia all his life, apart from several short trips to Russia. Having been trained, originally, as a general music teacher he has acquired much experience in the areas of music composition, music teaching - private piano teaching especially - and music history. Two pieces of his for orchestra have been performed publicly - one by the WASO (Western Australian Symphony Orchestra) and the other by the Fremantle Orchestra. He understands Homeric and Classical Greek and Latin a little and is familiar with several other languages but speaks none of them fluently. He currently resides, with his partner, Louise Pain, in Melbourne and is working on a second book, MORE LIMPID THAN THE DAWN. 
(Ricardus Sapiens is a Latinized version of Richard Wise.)

• Language: English
• Publisher: Europa Edizioni
• Release date: Nov 22, 2023
• ISBN: 9791220148238

Book preview

PART I

HINCHENER LANE

A Tale of Two Towns 

in two acts with a prologue for the theatre of the mind – a play for reading

[includes Allegri’s Miserere (excerpts), Debussy’s Chansons de

Bilitis, Brennan’s poem 1897]

Playscript

[219 pages]

‘Never trust the artist. Trust the tale.’

[D. H. Lawrence: Studies in Classic American Literature, Chapter One]

CHARACTERS

JONATHAN Ainsworth, Lecturer of Philosophy at Vadum

      University

LEONARD Carnegie-Smith, Lecturer of Mathematics at Pons

    University

EDWARD Carter, philosophy student at Pons University

CONRAD Goggarty, philosophy student at Pons University

PROFESSOR/JULIAN Grenfell, Research Fellow of Hartley

College and Lecturer of Philosophy at Pons University

THEODORE Grenfell, Lecturer of Philosophy at Vadum

    University

LEWIS Irving, Provost of Vadum University

GILBERT Mainwaring, philosophy student at Pons University

MAGNUS Moellendorf, Chancellor of Vadum University

PHILLIPS, private secretary to MAGNUS Moellendorf

RADIO NEWS READER (voice only)

TAXI-DRIVER

DAVID Vaughan, Senior Lecturer of Philosophy at Vadum

    University

1st and 2nd YOUNG MAN; 3rd YOUNG MAN (a younger

JULIAN Grenfell)

1st OLDER MAN; 2nd OLDER MAN (a younger MAGNUS

    Moellendorf)

ANNA BOGNAR, arts student at Vadum University and daughter      of Maric

JUDITH Carnegie-Smith, wife of LEONARD Carnegie-Smith

ERASMUS Dalrymple, teacher at Haverstock (Catholic)

Grammar School

ADRIENNE Dewhurst, Heiress Apparent to the throne of Patria

    Felix, incognito

ELLIDA Forsythe, philosophy student at Pons University

ELIZABETH Grenfell, wife of THEODORE and mother of

    JULIAN Grenfell

HOLLY Stokes, philosophy student at Pons University

CLEANER/PENELOPE Thostlewaite, cleaner at Pons

    University

1st and 2nd YOUNG WOMAN

1st OLDER WOMAN; 2nd OLDER WOMAN (a younger

PENELOPE Thostlewaite)

TWO STAGEHANDS

THE STUDENTS in Act One, Scene 1 (who also play the other      parts above excepting JULIAN and the CLEANER)

PLACE

The imaginary state of Patria Felix – the scene of the action alternates between the university towns of Pons and Vadum.

TIME

Unspecified.

NOTE

at the commencement of that scene; (v) vv.17-19 during the transition to 2.5; (vi) vv.20-21 after 2.7.

    The complete setting (vv.1-21) is to be played while the      audience is leaving the theatre after the play.

HINCHENER LANE

A Tale of Two Towns

PROLOGUE

[The report of five pistol shots rings out: four in quick succession,  followed by a fifth after a pause.]

RADIO NEWS READER [voice only]: Good evening. This is the      seven o’clock news. Heading tonight’s bulletin: A state of      emergency has been called in the troubled constitutional      monarchy of Nepal today following the outbreak of renewed      conflict with Communist Party rebels. The renewed conflict      follows earlier outbreaks whose origins go as far back as the      unrest in November 2001, after the alleged murder, by the      Crown Prince Dipendra, of ten members of his immediate      family – including his father King Birendra and his mother      Queen Aishwarya – after which the Crown Prince is reported to      have killed himself. With regard to the current situation, an      attempt to establish a ceasefire and a fresh start to peace-talks      between the Government of Nepal and the Communist Party      rebels are already under way. [Fade.] Other news to come in this      bulletin includes...

[Night.

Vadum.

MAGNUS Moellendorf’s bedroom at Vadum University.  MAGNUS asleep in bed; a dressing-gown hangs over a chair and  an alarm clock sits on a bedside table.

PHILLIPS enters, in a dressing-gown, lighted by a candle he is  holding. He moves to MAGNUS; gently shakes him...]

PHILLIPS: Chancellor. Chancellor.

MAGNUS [waking up]: Mm… mm… what is it?

PHILLIPS: Chancellor.

MAGNUS [opens his eyes, turns to PHILLIPS]: Oh, it’s you, 

    Phillips. What is it?

PHILLIPS: I’m sorry to disturb you, Chancellor, but I have just      received a message from the Ministry of Home Affairs. Some      terrible news.

MAGNUS: The Ministry of Home Affairs? [Looks at the clock on  the bedside table.] Phillips, it’s two o’clock in the morning!      What on earth are you doing? Go back to bed. [Turns away from

    PHILLIPS.]

PHILLIPS: The Crown Prince has shot himself.

MAGNUS [turns to PHILLIPS]: What?

PHILLIPS: The Crown Prince has shot himself, and the rest of the      Royal Family.

MAGNUS: What! Our Crown Prince? [Sits up, rubs his eyes, fully  awake.]

PHILLIPS: The same.

MAGNUS: I don’t believe it. Where?

PHILLIPS: At the Palace, about six hours ago, I’m told, at the       dinner-table. He is reported to have pulled out a revolver from      his pocket after the soup had been served and shot dead his      mother the Queen Regent, her brother, and his own younger      brother and sister, before shooting himself.

MAGNUS [sits up further]: Good Lord! Turn the light on.

PHILLIPS: I can’t. There’s been a power failure.

MAGNUS: Oh, God, has there? This isn’t your idea of a joke, is it?

PHILLIPS: As you said yourself, Chancellor, it is two o’clock in      the morning; and it is cold. As a matter of fact, I took the      opportunity of ringing a friend of mine who has a position at the      Palace: it is true.

[MAGNUS stands, puts on the dressing-gown hanging over  the chair, very concerned.]

MAGNUS: But this is terrible. Terrible! You’re certain?

MAGNUS: The Crown Prince has shot himself and the rest of the      Royal Family; and the Queen Regent’s brother, you say?

PHILLIPS: That is what I have been told.

MAGNUS: What are we going to do? What sort of reaction is there      going to be here in the University? There hasn’t been a press       release yet?

PHILLIPS: Not to my knowledge.

MAGNUS: No doubt there will be soon enough. The whole world      will know by this time tomorrow morning; and then heaven help      us! What’s going to happen at Vadum?... But why did he do it?

PHILLIPS: For love.

MAGNUS: Love!

PHILLIPS: My informant at the Palace tells me that the Crown      Prince and the Queen Regent had a terrible argument two days      ago concerning a young woman the Crown Prince had been      seeing: he wanted to marry the young lady apparently, but his      mother refused to give her consent.

MAGNUS: And so he shot her!

PHILLIPS: It would seem so.

MAGNUS: And his own brother and sister!

PHILLIPS: It is hard to fathom what love will do to a person.

MAGNUS: To a trigger-happy spoilt brat! God damn him! This      gets worse… more ridiculous… by the minute! ([Recalling.] It’s       like that incident years ago; in Nepal, wasn’t it. Do you      remember? There was a Crown Prince involved there, too. He      shot all the members of his family as well; and it might even      have been for love, come to think of it.

PHILLIPS: Yes. I do remember something about that: how unusual      and out of place it was somehow, in our day and age. What      happened; afterwards, I mean?

MAGNUS: I can’t remember. It was way back in 2001, wasn’t it?        [Purposefully.] Find out.

[PHILLIPS nods assent.]

MAGNUS [very irritated, alluding to his own Crown Prince]: God      damn him! What will the Vis Mundi think of it, I wonder?      [Authoritatively.] Get me the Vice-Chancellor.

PHILLIPS: Yes, sir.

[PHILLIPS turns to exit.]

MAGNUS: Wait! I’ll come with you.

[PHILLIPS and MAGNUS exit together.]

[Optional: Using some sort of ‘special effect’ the tableau/stage set melts,  dissolves, wavers momentarily; or, alternatively, there is a change in the  lighting and/or a glissando on a musical instrument such as the harp, violin,  viola, cello, double bass, celeste, glockenspiel, flute, oboe, clarinet or kettle  drum. This is followed by the BLACKOUT below and verse 1 of Allegri’s   setting of Psalm 51, the MISERERE.]

BLACKOUT

ACT ONE Scene One

[Afternoon.

A lecturing room at a university.

The only items of furniture are a desk, rubbish bin, stack-chairs  and two large whiteboards – the whiteboards will remain on the  stage throughout the play. The PROFESSOR is discovered leaning  easily against the front of the desk and speaking clearly to his  students (THE STUDENTS), who are of all ages, seated in front  of him taking notes. One student sits conspicuously apart from the  other students and wears a greatcoat, the collar up, a cap and  glasses… ]

PROFESSOR: "ENTIA NON SUNT MULTIPLICANDA

    PRAETER NECESSITATEM. – Entities are not to be      multiplied beyond necessity." Ockham’s Razor… One of you      asked, not long ago, what I considered to be the pillars on which      modern logic has been founded. By that I assume you wanted      the key concepts and underlying principles that support it; the      building blocks as it were. In the time that remains today I      would like to go into it – it’s important; and follow it up – as a      matter of fact – with an assignment that I want you all to do      together, as a joint exercise: I think, I hope, you will find it      interesting. [Stands, with animation] Ockham’s Razor… I am      going to briefly summarize them for you – the principles of      logic, I mean. If you require more detail we can go into it      another time – on Thursday, perhaps, in ‘discussion group’.

Meaning.

[The PROFESSOR stands and, with a marking pen, writes the  word ‘Meaning’ on the first of the two whiteboards as a  summary heading – further whiteboarded work follows on  from this down the first and onto the second whiteboard.]

    What is meaning? It is the extension and intension that is      signified by a given term – the general or class term that denotes      the objects to which it may be correctly applied – and the  collection or class of these objects constitutes the ‘extension’ or ‘denotation’ of the term.

[Whiteboards ‘extension/denotation’.]

    To understand a term is to know how to apply it correctly, its      usage; but for this it is not necessary to know all of the objects      to which it may be applied: it is only required that we have a      criterion for deciding of any given term whether it falls within      the extension of that term or not. Now, all objects in the      extension of a given term have some common properties or      characteristics which lead us to use the same term to denote      them. The properties possessed by all of the objects in a term’s       extension are called the ‘intension’ or ‘connotation’ of the term.

[Whiteboards ‘intension/connotation’.]

    General or class terms have both and intensional or connotative      meaning and an extensional or denotative meaning. Thus, the      intension or denotation of the term ‘bridge’ consists of the      properties common and peculiar to all structures that span a      body of water and are used for the purposes of crossing it; while      the extension or denotation of the term is the class containing      London Bridge, the Sydney Harbour Bridge, and any of the      other myriad bridges that exist around the world. It is the      conventional intension or connotation of a term that is its most      important aspect for purposes of definition and communication,      since it is both public and can be known by people who are not      omniscient.

Don’t ever say of a sentence, Maybe it has some meaning, but      a meaning we can’t understand; perhaps it is too profound for      us. If we say this, we forget the fundamental point that      meanings are given to words, not inherent in words. The word      ‘bridge’ has meaning – it has been given one – in the context of      describing an object, and apart from that context it has no      meaning, in its literal sense. A sentence containing a word in      which that context is not present, either explicitly or implicitly,      is meaningless. Why? Because we are mixing categories.

Genus and Species.

[The PROFESSOR writes the heading on the whiteboard.]

    Classes having members may have their membership divided      into subclasses according to whatever criterion of specific      difference it is decided to implement. For example, the class of      all bridges might be further divided into three non-empty      subclasses: those primarily made of wood, those primarily made      of steel and those primarily made of concrete respectively. The      terms ‘genus’ and ‘species’ are often used in this connection: the      class whose membership is divided into subclasses is the      ‘genus’, the various subclasses are the ‘species’.

[Summarizes the meanings of ‘genus’ and ‘species’ on the  whiteboard.]

    As used here the words ‘genus’ and ‘species’ are relative terms:      for example, as well as being a genus the class of all bridges is      also a species, or subclass, of the higher genus, or class, of all      man-made structures.

Definition.

[The PROFESSOR writes the heading on the whiteboard.]

    Definitions are always of symbols, for only symbols have      meanings for definitions to explain.

[Summarizes on the whiteboard.]

    We can define the word ‘chair’, since it has a meaning, but we      cannot define a chair itself, for a chair is an article of furniture,      not a symbol which has a meaning for us to explain. A definition      can be expressed in either of two ways: by talking about the      symbol to be defined, or by talking about its referent. Thus, we      can equally well say either [writes on the whiteboard]

"The word ‘bridge’ means a man-made structure spanning a        body of water for the purposes of crossing it."

or,

"A bridge is – by definition – a man-made structure spanning a        body of water for the purposes of crossing it."

    One point should be made here concerning the question of      ‘existence’. A definition has nothing to do with the question of      whether it defines a ‘real’ or ‘existent’ thing. The following      definition [Writes on the whiteboard.]

The word ‘unicorn’ means an animal like a horse but having a        single straight horn projecting from its forehead.

    Is a ‘real' definition, and a true one, because ‘unicorn’ is a word      with long-established usage and means exactly what is meant by      its definition. Yet the definition does not name or denote any      existent – [Writes on the whiteboard next to the definition of ‘unicorn’.]

No class membership.

    There is no class membership – since there are no unicorns.      Such a term as ‘unicorn’ shows that in some cases meaning      pertains more to intension than to extension: that is, to      connotation than to denoting.

    There are two limitations of the technique for defining terms.      [Notes this on the whiteboard and continues to summarize as he  goes.] Firstly, words connoting universal properties – if they      may be called that – such as the words ‘being’, ‘entity’, ‘existent’, ‘object’ and the like, cannot be defined by the      method of genus and difference – that is, genus and species –  because the class of all ‘entities’, for example, is not a species of      some broader genus; ‘entities’ themselves constitute the very      highest genus.

    The same applies to words for ultimate metaphysical categories,      such as ‘substance’ or ‘property’. Secondly, the method is      applicable only to words which connote complex properties. If      there are any simple, unanalysable properties – and whether      there really are such properties or not is open to question – then      the words connoting them are not susceptible of definition by      genus and difference. Examples of such properties, that I would      suggest, are the sensed qualities of specific shades of colour.      The colour ‘red’, for instance – [Writes ‘red’ on the  whiteboard.] –...‘Red’ is one of those ultimate words in terms of      which others can be defined, but which cannot itself be defined      by means of other words. Here language comes into direct      contact with the world, and using more words won’t help. For      terms like ‘red’ are the rock bottom of language, as are many      words of sense experience… fear, pain and the like, also… (Optional: [The tableau ‘melts’ momentarily.]...) – I have a toothache;      you have a toothache. To say that is to mean two different 

    things. But what does it mean? (Optional: [The tableau ‘stabilizes’.]...)

Propositions.

[The PROFESSOR writes the heading on the whiteboard.]

    It is the proposition that is true or false, but the sentence that has      meaning or fails to have it. A sentence is only a vehicle of      meaning, and only when we know what that meaning is can we      know whether the proposition it expresses is true or false. A      proposition has, indeed, often been defined as Anything that is       true or false. – [Writes the definition on the whiteboard.]... A      statement can mean either the proposition expressed or the      sentence expressing it.

    The propositions Black cats are black and Three feet is three feet are ‘analytical propositions’ – [Writes on the whiteboard.]      – that is, the ‘logical predicate’ of the sentence merely repeats      what is already contained in the subject of the sentence: "AB is

A and A is A" – [Writes on the whiteboard.]

Analytical propositions are of interest in that they are

‘axiomatic’: that is, they are known to be true without any      further investigation – specifically, without any observation of      the world. Those propositions whose truth or falsity cannot be      determined without at least some observation of the world –      most of the propositions we believe in – are called ‘synthetic      propositions’. [Summarizes on the whiteboard.]

    Examples of propositions that are not explicitly analytic are All brothers are male – [Points at the whiteboard.] – AB is A, and A yard is three feet – [points at the whiteboard] – A is      A. That is, key terms need to be defined in order to make the      sentence analytic: ‘brother’ equals ‘male siblings’, ‘yard’ equals      ‘three feet’. Blackbirds are black birds and Business is      business are not analytic – [Writes on the whiteboard.]

    Two definitions of ‘analytic proposition’ are

          [Summarizes on the whiteboard.]

    Mathematical propositions are analytic – [Writes on the  whiteboard.] 2+2 = 4, or (1+1) + (1+1) = 1+1+1+1.

    That is, their denial is self-contradictory.

    So, in summary: if to say My brother is male is analytic, to      say My brother is tall is synthetic.

    Analytical statements are definitional, synthetic statements are      not. [Summarizes on the whiteboard.]

Truth.

[The PROFESSOR writes the heading on the whiteboard.]

What is truth? When a sentence is used to report a ‘state-of    affairs’, and the ‘state-of-affairs’ the sentence is used to report is      actual – that is, exists, or is truth-functionally true – then the      proposition that the sentence expresses is true, and, we may add,      any other sentence that is used to express the same ‘state-of-      affairs’ will also express a true proposition. [Summarizes as a  definition on the whiteboard.] To say A proposition is true until      it is proved false and A proposition is false until it is proved      true reveal equally obvious mistakes. If it has been shown to be      false, then one should disbelieve it; if it has been shown to be      true, one should believe it; and if it has not been shown to be      either, one should neither believe or disbelieve it – that is, one      should suspend belief. There is a well-known paradox of truth:      What I am now saying is false. This appears to be both true      and false, which is absurd. What makes it so paradoxical is its        ‘context sensitivity’. For the purposes of logic, then, the several      different accounts of the nature of truth – truth as      correspondence, truth as coherence, truth as what ‘works’, and       the rest – do not apply, or are at most second-order predicates:      ‘existence’ is all.

    To continue. [Summarizes on the whiteboard as he goes,  including examples.] A proposition that is ‘necessarily true’ is      true in all possible worlds and its negation would be necessarily      false – examples: One cannot be in two places at the same      time, What has shape has size, 2 plus 2 equals 4, Black      cats are black, and all other analytical propositions. A      proposition that is ‘contingently true’ is true contingent on what      the universe happens to be like and its negation would be      contingently false – examples: Some dogs are white, There       are men and women in the room, My brother is tall, and all      other synthetic propositions. ‘Necessary truth’ and ‘truth       knowable A PRIORI’ are interchangeable expressions – they are      knowable A PRIORI because they necessarily hold for all cases.      An A PRIORI statement is one whose truth is knowable A  PRIORI – it needs no verification by further experience – and is      analytic. A contingent statement is one that needs to be tested      further to see whether it holds for future cases and is knowable      only A POSTERIORI – that is, from experience – and is      synthetic. Any true statement whose truth cannot be known A  PRIORI is knowable, if at all, only A POSTERIORI.

Validity.

[The PROFESSOR writes the heading on the whiteboard.]

    Someone define ‘syllogism’ for me.

STUDENT 1/GILBERT: A syllogism is a form of reasoning in      which from two given or assumed propositions called the      ‘premises’, and having a common or middle term, a third is      deduced called the ‘conclusion’ from which the middle term is      absent.

[The PROFESSOR summarizes STUDENT 1’s definition of

‘syllogism’ on the whiteboard.]

PROFESSOR: Very good. Thank you. Logic is the study of valid      reasoning and it attempts to show why some types of argument      are valid and other types are not. But it is important to      distinguish ‘validity’ from ‘truth’. In a valid argument the      premises – the two propositions from which the third is to be      inferred – need not be true: it is only required that the inference      or conclusion follows logically – that is, that the conclusion is      not inconsistent with the premises. – [Writes a summary  definition of ‘validity’ on the whiteboard and follows it with the  examples below.] –

    Example a) All dogs are mammals.

      All mammals are animals.

      Therefore, All dogs are animals.

    Example b) All cows are green.

    I am a cow.

      Therefore, I am green.

    Both of these syllogisms are ‘valid’ but only one is ‘true’.

    Example c) All dogs are mammals.

      All cats are mammals.

      Therefore, All dogs are cats.

    This is ‘invalid’. Why?

STUDENT 2/CONRAD: It commits The Fallacy of the

Undistributed Middle Term.

PROFESSOR: Correct. Deductive logic is the study of validity, not      of truth. [Writes on the whiteboard.]

STUDENT 3: What about inductive logic?

PROFESSOR: Not relevant.

STUDENT 3: Why?

JULIAN: Because it is not demonstrable – that is, an inductive      argument does not prove its conclusions the way a deductive      argument does. [Continuing with what he was saying before  being interrupted.] – Remember: a proposition that we do not      know on the basis of sense-experience, and that we do not know      through reason either, is called a ‘truth of reason’ – [Points at  the whiteboard.] – a ‘necessary truth’. Examples: A book is a book, a thing cannot be in two places at the same time, What has shape has size, A valid conclusion is one that is      drawn from two premises sharing a correctly distributed middle      term.

Material Implication.

[The PROFESSOR writes the heading on the whiteboard.]

‘q’."

‘r’, then ‘p’ implies ‘r’ –

i.e.  (p → q)

(q → r)

(p → q)...verbalized as ‘p’ therefore ‘r’.

[Writing on the whiteboard.] Given All members of the crew were drowned and Smith was a member of the crew we are entitled to assert that Smith was drowned.

[The PROFESSOR finishes writing on the whiteboard before  continuing.]

    If Smith... [Points to the appropriate part of the proof above.]      ...‘p’, then a member of the crew... [Points.] ...‘q’. If a member       of the crew... [Points.] … ‘q’, then drowned... [Points.] ...‘r’.       Therefore, if Smith... [Points.] ...‘p’, then drowned... [Points.]      ...‘r’. If ‘p’ then ‘q’, if ‘q’ then ‘r’; therefore, if ‘p’ then ‘r’.

    This is an example of deductive reasoning – of thinking – and      these... [Indicates 1) and 2) on the whiteboard.] ...are the two      simple rules of logical inference employed, in this case, to make      the assertion that Smith was drowned. All logical inference is founded on ‘material implication’.

    For the purposes of logical analysis ‘material implication’ can      also be expressed algebraically by the conjunction – [Writes        (p→q) = ~ (p . ~p) on the whiteboard, points to ~(p . ~p).] –      verbalized as not both ‘p’ and not-‘q’ where ‘p’ and ‘q’       represent the propositions that make up any ‘if-then’ statement.      In other words, for any conditional If ‘p’ then ‘q’ to be true,      the negation... [Points to the ‘~’ in front of the bracket.]...of the      conjunction of its antecedent...[Points.]...‘p’ with the negation of      its consequent... [Points.]...‘ ~q’ must be true – that is, every      conditional statement means to deny that its antecedent is true      and its consequent is false. Using De Morgan’s Theorem ‘material implication’ can also be expressed algebraically as the      disjunction – [Adds = (~p ˅ q) to the equation already on the  whiteboard so that it reads: (p→q) = ~(p . ~q) = (~p ˅ q).] – as      well, verbalized as Either not-‘p’ or ‘q’. However, in order to      test for the validity of ‘material implication’ – its truth      functional truth – the variables must be reduced ultimately to a      conjunctive form in which the propositions that make it up can      be clearly seen to be true or false. This brings us to the Laws of  Thought.

    There are three fundamental Laws of Thought and they were      originally developed by Aristotle.

[Writes on the whiteboard.]

The Law of Identity: A is A.

The Law of Non-contradiction: Nothing can be both A and      not-A.

The Law of Excluded Middle: Everything is either A or not-A.

    Formulated as truths about propositions they can be stated as –

      [Writes on the whiteboard next to the appropriate Law of  Thought.] 

If ‘p’ then ‘q’; (p→q).

Not both ‘p’ and not-‘p’; ~(p . ~p).

Either ‘p’ or not-‘p’; p ˅ ~p.

    in which case they are ‘tautologies’, from the Greek – τὸ αὐτό,      or ταὐτό – meaning ‘the same’; that is, a propositional form in      which all the statements we get by substituting sentences for the      symbols are true. 

    Also, apart from this, each formulation is a different way of      saying the same thing. – [Writes on the whiteboard next to the  appropriate Law of Thought.] –

If it is ice then it is ice.

Not both ice and not ice: equivalent to saying If it is ice then it is ice.

Either it is ice or it is not ice: also equivalent to saying If it is          ice then it is ice.

    I repeat, all logical inference is founded on ‘material      implication’.

Knowing.

[The PROFESSOR writes the heading on the whiteboard.]

What is ‘knowing’? The ‘objective requirement’ in saying that       you know ‘p’ is knowing that ‘p’ is true. The ‘subjective      requirement’ in saying that you know ‘p’ is that not only must 

‘p’ be true, you must believe that ‘p’ is true: to do this you must  have evidence for ‘p’ – that is, a reason to believe ‘p’.

[Summarizes on the whiteboard.]

There is a strong and a weak sense of ‘know’. In the ‘weak’     sense I know a proposition when I believe it, have good reason      for believing it, and it is true – this is the sense of ‘know’ in daily life. In the ‘strong’ sense, in order to know a proposition, it      must be true, I must believe it, and I must have absolutely      conclusive evidence in favour of it – this is more the      philosophical sense of ‘know’. [Summarises on the whiteboard.]      So much for knowing.

    To conclude,

Possibility and Impossibility.

[The PROFESSOR writes the heading on the whiteboard.]

    A state-of-affairs is said to be ‘logically possible’ whenever the       proposition that this state-of-affairs exists is not self-      contradictory, and ‘logically impossible’ when the state-of-      affairs is self-contradictory. Other forms of ‘possibility’ and ‘impossibility’ are ‘empirical’ and ‘technical’.

[The PROFESSOR writes all that he has had to say on  ‘possibility’ and ‘impossibility’ on the whiteboard, puts down  the marking pen he has been using, resumes his position  leaned against the front of the desk as at the beginning.]

    If a state-of-affairs is logically impossible, then it is impossible      in the other senses, too – that is, empirically and technically –      but not necessarily vice versa. Many contemporary      philosophers, if not all, illustrate their meaning by drawing their      examples from logical possibility: when we say that      propositions are logically possible, we do not mean that we      expect them to happen, or that we think there is the remotest      empirical possibility that they will happen; we only mean that if      we asserted that they did happen, or would happen, our assertion      would not be self-contradictory, even though it would be false.

Let me present you with an example that may cause  considerable confusion in unthinking minds.

Is it logically possible to go back in time – say to 490 B.C.E. – and help the Athenians defeat the Persians at the Battle of Marathon? That is, is ‘time-travel’ possible?

    We can speak very easily and literally about going backwards      and forwards in space, and it is tempting to use the same      language about time as we do about space, and to assume that      time-language is meaningful in all of the same contexts as      space-language. But this is a dangerous assumption. Let us be      aware, then, that when we talk about going back to 490 B.C.E.       we mean it literally – we are going to do it. Taken in a figurative      sense there is no problem, for we can certainly imagine      ourselves as being at far distant places in space and at various      eras in time. We can imagine ourselves being there at the Battle      of Marathon. But, if we imagine it then surely it is logically      possible.

    Of course, this too – the logical possibility – has been imagined      on numerous occasions in feature films, in T.V. series, in short      stories and in novels – H.G. Wells’ THE TIME MACHINE      springs immediately to mind – and every one of us is able to      imagine it as well by means of them. But let us also be aware of      just what it is that we are imagining. We can imagine ourselves      being born in a different era and as being with the Athenians at      Marathon, but can we imagine ourselves now, in the 21st century      C.E. – and not merely in our imagination – as being in 490      B.C.E.? How can we be in the 21st century C.E. and the 5th      century B.C.E. at the same time? Here already is one      contradiction. We cannot be in the 21st century C.E. and not be      in the 21st century C.E. – for example, in another century, like      the 5th century B.C.E. – at the same time. It is not logically      possible to be in one century of time and in another century of      time at the same time.

But, one of you might object, this is not the situation we are      imagining. What we are imagining is being one day in the 21st      century C.E. and then moving backward in time so that the next      day we are living in the year 490 B.C.E. – and on that day we      are no longer in the 21st century C.E. But let us be careful:  suppose the day you are talking about is January 1st, 2025, and  that on January 2nd, 2025, you use a time machine and go back 

to some day in 490 B.C.E. Isn’t there a contradiction here again? For the next day after January 1st, 2025, is January 2nd, 2025.

The day after Monday is Tuesday – this is analytic: ‘Tuesday’ is      defined as the day that follows Monday – and the day after      January 1st, 2025 is January 2nd, 2025 – this is also analytic. So it 

    is logically impossible to go from January 1st, 2025 to any other      day except January 2nd, 2025 – that is, to the following day of      the same year. To be living on January 2nd, 2025, and at the      same time, January 2nd, 490 B.C.E., is a contradiction in terms,      and hence logically impossible.

That’s true, you might say, "but you miss the point. The point      is that we go backward in time – not to the next day, but to a day      over 2,500 years earlier. So we don’t go to the next day – if we      did it would be January 2nd, 2025 – we go to a previous day."

    But isn’t the real point that it is the nature of time to go forward;      that time goes forward is analytic? What else can time do but      go forward? People can walk backwards in space, but what      would going backwards in time literally mean? And if you      continue to live, what can you do but get one day older every      day? Isn’t ‘getting younger every day’ a contradiction in terms? – unless, of course, it is meant figuratively, as in "My       dear,you’re getting younger every day," where it is still taken for      granted that the person, while looking younger every day, is still      getting older every day?

I’m still not convinced, I can hear someone say. "The       situation I am proposing is that of going from one particular day      in 2025 – let us say January 1st – not to the following day in      2025 – that is, January 2nd – but to a day in 490 B.C.E. And I      don’t see how that is logically impossible, though it may be      empirically impossible."

    To begin again. Many centuries ago the Battle of Marathon was      fought and when it happened you were not there – you weren’t       even born. It happened long before you were born, and it      happened without your assistance or even your observation. This      is an unchangeable fact: you can’t change the past. That is the  crucial point: the past is what has happened, and you can’t make what has happened not have happened, for that is a logical impossibility. When you say that it is logically possible for you

– literally – to go back to 490 B.C.E. and help the Athenians      defeat the Persians at Marathon you are faced with the question:      did you help the Athenians or did you not? The first time it      happened, you did not. All you can say then would be that the      second time it happened you were there – and that there was at      least one difference between the first time and the second time:      the first time you weren’t there and the second time you were.      But now we are speaking of two different times, the first time      being in 490 B.C.E. and the second time being 2025 C.E.

    Now it is logically possible that history might start repeating      itself, but time would still be going forward – if you want to use      that expression – and the day after January 1st, 2025, the day of      the sudden transformation, would not be a day in 490 B.C.E. –      that day is long past and gone, and irrecoverable, like everything      else in the past; no, the day after January 1st, 2025 would be      January 2nd, 2025.

    Once you are convinced of the logical impossibility of changing      the past – or making what has happened un-happen – you will      doubtless see the logical impossibility of literally going back in      time to 490 B.C.E. We are inclined to be misled into thinking      that it is logically possible because we read books and see      theatrical works and feature films and T.V. programmes in      which it is presented as if logically possible: we believe the story; and we like to be deceived.

STUDENT 4/HOLLY: Does that mean that it is not possible to go      forward into the future?

PROFESSOR: I did not say that. It is obviously perfectly possible      to go forward into the future – it is what we are always doing –      so long as the continuity of self-identity is not broken: so long as      there is consistency. But if you mean is it possible to go twenty       years or two thousand years, or whatever, into the future, no.

STUDENT 5/EDWARD: Why?

PROFESSOR: Because it is treating the future as though it is a past      already written and, that being the case, the same principles      apply as going back into the past. But the future is not a past      already written: to say as much is self-contradictory.

[The PROFESSOR is thoughtful for a moment. The  CLEANER enters wheeling a cleaning trolley; stops,  surprised at seeing the others.]

CLEANER: Oh dear. Oops! Sorry, Professor, I thought you had      already finished.

PROFESSOR: And we almost have.

CLEANER: I’ll wait outside. Excuse me.

PROFESSOR: We won’t be long.

[The CLEANER exits.]

PROFESSOR [stands, animated]: Now then, I have a task for you,      lest this brief summary of the elements of logic be in vain. – Pay      attention: Using the information that I have presented to you      today – [Indicates the whiteboards.] – I want you, as a joint      exercise, mind, to devise a story – a play – using the elements,      or some of them, that have made up what I have spoken about. –      But... there is a stipulation: you must make it – I expect a joint,      co-operative effort, remember – you must make it strictly within      the realms of logical possibility, that is the key, the essential      thing – the whole thing must be logically possible! Invent, be as      imaginative and improbable as you like – the more diverse and      wild the better: stretch credibility to the limits, if you can – but      all within the realms of logical possibility. Go! Be young      creative philosophers, of the possible! Do it! Together !

STUDENT 4/HOLLY: Can we count on your co-operation if we      need it? We might need some help.

PROFESSOR: Certainly. If by that you mean you might need me to      participate – to play a part – I expect nothing less. Any other      extras that you might need we should also be able to accommodate... within reason. If need be, we will enlist the help of volunteers!

[The PROFESSOR sits on a chair in front of the class and  crosses his legs, is thoughtful… Using the same ‘special  effect’ as at the end of the Prologue the tableau melts,  dissolves, wavers momentarily; or, alternatively, there is a  change in the lighting and/or a glissando on a musical  instrument such as the harp, violin, viola, cello, double bass,  celeste, glockenspiel, flute, oboe, clarinet or kettle drum.  When stability is re-established, or the lighting has returned  to normal, the situation is the same – a professor giving a  lecture to his students – but it is now set within the story/play  that THE STUDENTS have been asked to devise. All the  members of the original class, except those playing the  characters of ADRIENNE, ELLIDA, HOLLY, CONRAD,  EDWARD and GILBERT have left the stage. One student –  ADRIENNE – sits conspicuously apart from the other  students and wears a greatcoat, the collar up, a cap and  glasses – but not the same student as at the beginning of the  scene...]

[Monday.

Afternoon.

Pons.

A lecturing room at Pons University.]

JULIAN [quietly, calmly, without animation, self-absorbed,]: Do      any of you have any further questions?

CONRAD: ‘Is’, the verb ‘to be’... it’s important in logic, isn’t it?

JULIAN: Yes, but surely you realize that. Material implication,      equivalence, the existential and universal quantifiers: they’re all founded on it. Nevertheless, bear in mind that the ‘is’ of identity       is not the same as the ‘is’ of predication, and that the ‘is’ of       existential quantification is different again from both of the      other two.

CONRAD: But ‘is’ always means ‘being’, doesn’t it?

JULIAN: Yes. Perhaps a better way of putting it is to say that it      always describes a mode of existence. What are you getting at?

CONRAD: I’m not sure; probably nothing. I need to think about it      some more.

JULIAN: Anyone else?

[None of the students respond.]

JULIAN: Very well. [Stands.] Let me finish today by posing an old      riddle for you to exercise your minds on – [Writes the riddle on  the whiteboard.] – What happens when an irresistible force      meets an immovable object? I will even provide you with an      argument to illustrate it – [Writes on the whiteboard.] –

"If the helicopter had engine trouble it would have landed at      Vadum University.

    If the helicopter did not have engine trouble it would have      landed at Pons University.

    The helicopter did not land at either Vadum or Pons University.

    Therefore the helicopter must have landed at Quorsum."

    If you can understand the reason why that argument is invalid      then you have the solution to the riddle.

    [Puts marking pen down.] 

    Thank you, everyone, that will do for now. Let me see you all      again on Wednesday. Consider the illustration carefully.

[JULIAN reseats himself, thoughtful.]

ELLIDA [to HOLLY, still writing]: Isn’t he great!

HOLLY: Oh yes, he’s wonderful! [Closes her file and packs up her  books.]

[CONRAD, HOLLY, EDWARD and ADRIENNE exit.

GILBERT waits for ELLIDA who, on finishing her notes,  goes up to JULIAN, wanting to speak to him; GILBERT  joins her. On seeing how abstracted JULIAN is ELLIDA  decides not to disturb him; ELLIDA and GILBERT leave  together. PENELOPE re-enters with her cleaning trolley.] PENELOPE: I’m sorry about that, Professor.

JULIAN [shaking off abstraction]: Mm? Sorry about what?

PENELOPE: Barging in before you had finished.

JULIAN: Oh, that! It’s no matter.

PENELOPE: I do it too often. I never think; not like you university      people.

JULIAN: Don’t let it bother you, Margaret, –

PENELOPE [correcting him]: Penelope.

JULIAN [not registering PENELOPE’s interjection]: ...it doesn’t       bother me. I went overtime anyway. [Stands, gathers his papers  together on the desk.]

PENELOPE: You haven’t forgotten that your friend from 

    Haverstock is arriving this afternoon, have you?

JULIAN [obviously has]: Ah, no.

PENELOPE: You asked me to remind you.

JULIAN: Yes, I remember.

PENELOPE: It must be nice having friends drop in from all over      the place, like you do. Me and my Tom, we hardly know      anybody that lives outside of Pons; and them that we do know      never visit.

JULIAN: Yes, it can be very pleasant. Oh, by the way, Margaret –

PENELOPE: Penelope.

JULIAN: I would like to give a dinner party in my rooms here at      the University on Friday night –

PENELOPE: For your friend?

JULIAN: Yes. Would you be able to take care of it for me again?

PENELOPE: This Friday. Yes, that should be alright.

JULIAN: Very good.

PENELOPE: How many people?

JULIAN: Just four, at this stage.

PENELOPE: What time?

JULIAN: Could you have it ready by about seven?

PENELOPE: Fine. Leave it to me, Professor.

JULIAN: Thank you.

LEONARD [offstage]: Julian? Julian? [Enters.] Aha! There you      are!

JULIAN: Oh, hello, Leonard.

LEONARD: There’s a message been left for you in your faculty      staffroom; see that you find out what it is before you leave.

[JULIAN nods.]

LEONARD: Well, I’m off, old chap.

[LEONARD extends his hand; JULIAN shakes it.]

JULIAN: Where are you going?

LEONARD: Don’t tell me you’ve forgotten! I’m going up to      Pulchrum Nobis to speak at the seminar on The Crisis  Concerning Contemporary Mathematics.

JULIAN: Oh, I see.

LEONARD [to PENELOPE]: He has a very good memory really.