PHILOSOPHY OF LOGIC

PHIL323 First Semester, 2014 City Campus

Contents

1. COURSE INFORMATION

2. GENERAL OVERVIEW

3. COURSEWORK AND ASSESSMENT

4. READINGS

5. LECTURE SCHEDULE WITH ASSOCIATED READINGS

6. TUTORIAL EXERCISES

7. ASSIGNMENTS

1. Assignment 1: First Problem Set

2. Assignment 2: Essay

3. Assignment 3: Second Problem Set

8. ADMINISTRIVIA CONCERNING ALL ASSIGNMENTS

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1. COURSE INFORMATION

Lecturer and course supervisor: Jonathan McKeown-Green

Office: Room 201, Level 2, Arts 2, 18 Symonds Street, City Campus

Office hour: Wednesday, 11am-1pm

Email: j.mckeown-green@auckland.ac.nz

Telephone: 373 7599 extension 84631

Classes

Lectures are on Wednesdays 3pm – 5pm,

and

Tutorials are on Thursdays 9-10am,

2. GENERAL OVERVIEW

This course is an introduction to two closely related subjects: the philosophy of logic and logic’s

relevance to ordinary reasoning. The philosophy of logic is a systematic inquiry into what it is

for and what it is like. Taster questions include:

Question 1

Must we believe, as classical logic preaches, that every statement whatsoever is entailed by a

contradiction?

Question 2

Are there any good arguments that bad arguments are bad?

As noted, our second subject is the connection between ordinary, everyday reasoning, on the

one hand, and logic, as it is practised by mathematicians, philosophers and computer scientists,

on the other. Taster questions include:

Question 3

Should logical principles be regarded as rules for good thinking?

Question 4

Are logical principles best understood as truths about a reality that is independent of us or are

they really only truths about how we do, or should, draw conclusions?

The close relationship between our two subject areas becomes evident once we note that much

philosophical speculation about what logic is and what it is like arises because we care about

the relationship between logic and ordinary reasoning. Questions like (iii) and (iv) above arise

from such speculation.

All of questions 1-4 above are controversial and this is part of why they are interesting to

pursue. Our pursuit of these issues presupposes some familiarity with classical propositional

and predicate logic - exactly the sort of familiarity you get from completing some first and

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second year logic courses. If you have done more logic than this, you may also find that

experience useful. Even so, you will only occasionally be asked to write proofs or draw truth

trees. Most of the issues we confront require careful thought about logic, not direct applications

of logic. Some of the assigned readings are difficult to follow, but there should be sufficient

guidance given in class to help you get the most out of them.

Important!

You will need a pretty good grasp of both spoken and written English to be able to follow many

of the issues discussed.

3. COURSEWORK AND ASSESSMENT

There is no final examination. There will be three assignments. One of them will be worth 40%

of the final mark and each of the other two will be worth 30% of the final mark. The

assignments will initially be marked out of 100. The assignment for which you get the highest

percentage (whichever that one is) will be the one that contributes 40% to your final mark.

All assignments will be submitted electronically to my email address:

j.mckeown-green@auckland.ac.nz

Format should be .doc or .docx or rtf.

Assignment 1

First Problem Set.

Due Monday, 14 April (first day of mid-semester break), at 5 pm.

(The questions and instructions are included in this file.)

Assignment 2

1500 word essay

Due Monday, 19 May (start of week 10), at 5 pm.

(A selection of suggested topics is included in this file. If you want to write on something else,

talk to me.)

Assignment 3

2nd

Problem Set.

Due Monday, 9 June (the first Monday after classes end), at 5 pm.

(Again, the questions and instructions are included in this file.)

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READING FOR THE COURSE

What to read, and when?

The lecture schedule, which can be found later in these notes, lists the readings for each topic.

There are some required readings for each topic, and sometimes further recommended readings

for the keen at heart. The required readings are found either in the course book or in the

recommended text:

Stephen Read (1994): Thinking about logic: an introduction to the philosophy of logic, Oxford

University Press, ISBN: 019289238X.

The Stephen Read text is on short loan in the Information Commons. Each chapter contains a

final section with a guide to even more reading.

Readings in the online course booklet

1. Girle, Roderick (1976): Notes on the History of Logic, Notes for PD220: Applied and Modal

Logic, Philosophy Dept, University of Queensland.

2. Johnson-Laird, Philip et al. (2000): “Illusions in reasoning about consistency”, Science, 288,

pp. 531-2.

3. Harman, Gilbert (1984): “Logic and Reasoning”, Synthese, 60, pp. 107-127.

4. Knorpp, William Max (1997): “The Relevance of Logic to Reasoning and Belief Revision:

Harman On ‘Change in View’”, Pacific Philosophical Quarterly, 78, pp. 78-92.

5. Hunter, Geoffrey (1971): “Introduction: General Notions”, Metalogic: An Introduction to

the Metatheory of Standard First Order Logic, Macmillan, London, pp. 3-21.

6. Carroll, Lewis (1972): “What Achilles Said to the Tortoise”, Readings on Logic, Copi and

Gould (ed.), Macmillan, New York, pp. 117-119.

7. Massey, Gerald (1975): “Are There Any Good Arguments that Bad Arguments are Bad?”

Philosophy in Context, 4, pp. 61-77.

8. Kripke, Saul (1980): Naming and Necessity, Blackwell, Oxford, pp. 15-20.

9. Lewis, David (1985): “Counterparts or Double Lives?” (Chapter 4), On the Plurality of

Worlds, Blackwell, Oxford, pp. 192-198.

10. Stalnaker, Robert (1976): “Propositions”, in The Philosophy of Language, Martinich, A. P.

(ed), Oxford, 1990, pp.79-91.

11. Cartwright, Richard (1962): “Propositions”, in Philosophical Essays, Cartwright (ed), MIT,

1987, pp. 33-53.

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12. Haack, Susan (1978): “Sentence Connectives” (Chapter 3), Philosophy of Logics,

Cambridge University Press, Cambridge, pp. 28-38.

13. Quine, W. V. (1982): “Truth Functions” (Chapter 2), Methods of Logic, fourth edition,

Harvard University Press, Cambridge, Mass, pp. 7-13.

14. Grice, Paul (1975): “The Logic of Conversation” in The Logic of Grammar, Donald

Davidson and Gilbert Harman (eds), Dickenson, Encino, pp. 64-75.

15. Harman, Gilbert (1986): “The meanings of logical constants”, in Truth and Interpretation:

Perspectives on the Philosophy of Donald Davidson, Ernest Le Pore (ed), Blackwell, Oxford,

1986, pp. 125-134. http://www.nyu.edu/gsas/dept/philo/courses/concepts/meaning.html

16. Grice, Paul (1989): “Indicative Conditionals” (Chapter 4), Studies in the Way of Words,

Harvard University Press, Cambridge, Mass, pp. 58-85.

Additional resources

Online encyclopaedias provide invaluable back-up. Entries under formal systems,

counterfactuals, Frege, conditionals, etc, are worth consulting.

The Stanford Encyclopaedia of Philosophy

http://plato.stanford.edu

This contains thorough, lengthy, survey articles. If you are writing your essay on a particular

suite of topics, look them up in the Stanford.

The Internet Encyclopaedia of Philosophy

http://www.iep.utm.edu/

This contains shorter articles than the Stanford does, but they are still thorough and are often at

the right level of generality for an exploration of concepts with which you should be familiar,

but on which you do not need to be an expert.

The Routledge Encyclopaedia of Philosophy can be accessed through the library.

http://www.library.auckland.ac.nz/databases/learn_database/public.asp?record=routledge

The articles here are much shorter. The Routledge is a good repository of quick references and

reminders.

The following books can also be found on short loan or in the General Library (or online) and

contain useful background or further reading on general issues dealt with in this course:

Beall, J. C. and Restall, Greg (2006): Logical Pluralism, Clarendon, Oxford, chapter 2. (online)

Grayling, A.C. (1997): An introduction to philosophical logic, Blackwell, Oxford.

Haack, Susan (1978): Philosophy of Logics, Cambridge University Press, Cambridge.

van Eemeren, et al. (1996): Fundamentals of Argumentation Theory, L. Erlbaum, Mahwah, N.J.

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van Eemeren et al.(2001): Crucial Concepts in Argumentation Theory, Amsterdam University

Press, Amsterdam.

In addition to the required and recommended readings we have given you may wish to consult

other material if you are writing an essay on a particular topic. You should discuss this with me.

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4. LECTURE SCHEDULE (with associated readings)

Week 1 (3 March)

Logic and ordinary reasoning: What on earth is Philosophy of Logic? Can anybody remember

what we did in Stage I logic? What is the connection, if any, between logical theory and

ordinary, everyday reasoning? Aristotle’s approach; some history of logic; logic as a describer,

evaluator, or justifier of ordinary reasoning; logical (and other) mistakes in everyday reasoning;

Psychologism versus Anti-Psychologism; input from cognitive psychology.

Required Readings

(a) Girle, Roderick (1976): Notes on the History of Logic, Notes for PD220: Applied and Modal

Logic, Philosophy Dept, University of Queensland (in coursebook).

(b) Johnson-Laird, Philip et al. (2000): “Illusions in reasoning about consistency”, Science, 288,

pp. 531-2 (in coursebook).

The following two readings are difficult, but they are useful if you wish to pursue this topic.

Harman argues that logic has no special relevance to ordinary reasoning and Knorpp disagrees.

(c) Harman, Gilbert (1984): “Logic and Reasoning”, Synthese, 60, pp. 107-127 (in coursebook).

(d) Knorpp, William Max (1997): “The Relevance of Logic to Reasoning and Belief Revision:

Harman On ‘Change In View’”, Pacific Philosophical Quarterly 78, pp. 78-92 (in coursebook).

I have also provided a written expansion of my material for this class, which is a separate

document on CECIL:

(e) Lecture Notes on the relevance of Logic to ordinary reasoning

Week 2 (9 March)

Formal languages; semantics for formal languages.

Required Reading

(a) Ch. 1 of Hunter, Geoffrey (1971): Metalogic: An Introduction to the Metatheory of

Standard First Order Logic, Macmillan, London (in coursebook).

Tutorials begin in week 2.

Week 3 (16 March)

Formal systems and formal logics; syntax and semantics; what sorts of formal logic are there:

traditional, propositional, quantificational, modal, non-classical, … What else?

Required Reading

(a) Ch. 1 of Hunter’s book again (in coursebook).

Recommended Reading

(b) Ch. 12 of Haack, Susan (1978): Philosophy of Logics, Cambridge University Press,

Cambridge (on short loan)

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Week 4 (24 March)

Formal systems and formal logics; metalanguages and object languages; why do logicians

typically resort to (artificial) formal machinery and eschew ordinary language? What are the

advantages and disadvantages of formalism? Precision, computational tractability and

abstraction.

Required Readings

(a) Carroll, Lewis (1972): “What Achilles Said to the Tortoise”, Readings on Logic, Copi and

Gould (ed.), Macmillan, New York, pp. 117-119 (in coursebook). This is not strictly on the

topics discussed in week 4, but it is good preparation for week 5.

(b) Ch. 1 of Haack, Susan, Philosophy of Logics (on short loan).

(c) Ch. 9 of Haack, Susan, Philosophy of Logics (on short loan).

Week 5 (31 March)

What is logical consequence? Consequence, validity, consistency and logical truth; Aristotle

revisited; the classical account of logical consequence; refinements and criticisms of the

classical account of logic: introducing non-classical systems. Can current logical theory tell us

whether an argument is good or bad? Massey’s, and other, arguments.

Required Readings

(a) Ch. 2 of Read, Stephen, Thinking About Logic (on short loan).

(b) Ch. 2 of Beall & Restall(2006): Logical Pluralism (electronic copy of book available

through library catalogue)

(c) Massey, Gerald (1975): “Are There Any Good Arguments that Bad Arguments are Bad?”

Philosophy in Context, 4, pp. 61-77 (in coursebook).

Recommended Readings

(c) Ch. 2 of Haack, Susan: Philosophy of Logics (on short loan).

(d) Ch. 6 of van Eemeren et al (2001): Crucial Concepts in Argumentation Theory, Amsterdam

University Press, Amsterdam (in coursebook).

Week 6 (7 April)

Possibility, necessity and possible worlds: What are possibility and necessity; possible worlds in

modal model theory; possible worlds in metaphysics and the semantics of natural languages;

rival theories about the nature of possible worlds and the ways that they fulfil their theoretical

role.

Required readings

(a) Preface (excerpt only) from Kripke, Saul (1980): Naming and Necessity, Blackwell, Oxford,

pp. 15-20 (in coursebook).

(b) Ch 4.1 of Lewis, David K (1985): On the Plurality of Worlds, Blackwell, Oxford (in

coursebook).

(c) Ch 4 of Read, Stephen, Thinking About Logic (on short loan).

Further Recommended Reading

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(d) Ch. 10 of Haack, Susan, Philosophy of Logics (on short loan).

Mid-Semester Break (2 weeks) starting 14th

April.

Due date and time for Assignment 1 (First Problem Set) 5pm, Monday 14th

April.

Week 7 (28 April) Sentences, statements and propositions; the theoretical role played by propositions; rival

accounts of propositions.

Required readings

(a) Stalnaker, Robert (1976): “Propositions”, in The Philosophy of Language, Martinich, A. P.

(ed), Oxford, 1990 (in coursebook).

(b) Cartwright, Richard (1962): “Propositions”, Philosophical Essays, Cartwright, MIT, 1987,

pp. 33-53 (in coursebook).

Recommended reading

(c) Ch. 6 of Haack, Susan, Philosophy of Logics (on short loan).

Week 8 (5 May) Only connect: what are truth functions; truth functional & non-truth functional connectives;

Polish notation; truth-functional connectives in multi-valued logics.

Required Readings

(a) Ch. 3 of Haack, Susan: “Sentence Connectives”, Philosophy of Logics (in coursebook).

(b) Ch. 2 of Quine, W. V. (1982): Methods of Logic, fourth edition, Harvard University Press,

Cambridge, Mass, pp. 16-21(in coursebook).

Recommended reading

(c) Ch. 11 of Hack, Susan: Philosophy of Logics (on short loan).

Week 9 (12 May)

Truth functional connectives and ordinary language: objections to, and defences of, truth

functional explications of ordinary language connectives.

Required Readings

No readings this week. Chance to revise readings for Weeks 7 and 8.

Due date and time for Assignment 2 (Essay): 5 pm, Monday, 19 May.

Week 10 (19 May)

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‘If’-iculties: introducing conditionals. A few bits of unfinished business regarding truth

functional connectives. Are conditional statements truth functional?

Required reading

(a) Grice, Paul (1975): “The Logic of Conversation” in The Logic of Grammar, Donald

Davidson and Gilbert Harman (eds), Dickenson, Encino (in coursebook).

Week 11 (26 May)

Conditionals and ‘if’-iculties with them: truth functional accounts of conditionals; different uses

of the conditional.

Required reading

(a) Ch. 3 of Read, Stephen: Thinking About Logic (on short loan).

Week 12 (2 June)

Indicative and counterfactual conditionals; non-classical accounts of conditionals; the classicist

fights back.

Required reading

(a) Ch. 4 of Grice, Paul (1989): Studies in the Way of Words, Harvard University Press,

Cambridge, Mass (in coursebook).

Due date and time for Assignment 3 (2nd

Problem Set): 5 pm, Monday, 9 June.

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6. TUTORIAL EXERCISES

Tutorials begin in the second week of the semester. What follows is a list of exercises to try in

preparation for each tutorial. It is important that you try at least some of the exercises before

attending the tutorial, as there will be an opportunity in the tutorial to discuss the answers and

work on problems that have arisen. Many of the exercises help directly with assignment

questions and all of them are designed to facilitate understanding of matters discussed in

class. Note however that the tutorials will not be confined to treatments of these exercises.

Other issues will be addressed in the tutorials, including any questions or comments that you

bring along yourself.

Week 2 (10 March)

1. Translate the following into first-order predicate logic:

Somebody said something which upset everyone in the room except for Brian and

Celia who were not upset and laughed at it.

2. Suppose that some typical human subjects are given the problem below. What answer will

they give, according to Philip Johnson-Laird, and what is his story about why they give this

answer? Is the predicted answer straightforwardly correct? If not, does Johnson-Laird’s story

offer an account of why the typical subjects go wrong?

Esther is seated with Arthur sitting to her left and Tina sitting to her right. Is Tina sitting to the

left or to the right of Arthur?

3. Set out the following argument in standard form and explain why it is invalid.

GIGO (‘garbage in, garbage out’) is a wise saying. It reminds us that what we get out

of something depends on what we put into it. In particular, if a machine or a

bureaucrat or some other process gives you a result that you don’t like, you have no

right to blame the process; the original input is your culprit. We all receive dumb

computer-generated correspondence that is clumsy in style, inappropriately friendly,

or just inappropriate, but GIGO entails that machine error is always traceable to

human error. It’s no good blaming the machine!

4. Invent a language that has the following alphabet, but that is not the language of

propositional logic.

∑PL = {p, q, r, ', ~, &, V, , , (, )}

5. Using the alphabet in 2 above, check that you know how to form the language of

propositional logic and complete a definition of the truth conditions for all wffs of that

language.

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6. Come up with a different (non-classical) interpretation of the language you have been

exploring in questions 2 and 3.

7. See if you can come up with a way of generating a formal language for predicate logic.

Week 3 (17 March)

1. Take each of the following English sentences and, if possible, represent its meaning

accurately in the language of first-order quantificational logic. Provide a key to show how you

are using any sentential, individual or predicate constants. If you believe that the sentence

cannot be accurately represented in the language of first-order logic, don’t try to represent it.

Instead, explain why it can’t be done.

(a) There are only two types of music: good music and bad music; and I love both

kinds!

(b) There is nothing you can name that is anything like a dame.

(c) Every day is a good day to start reading the Bible every day.

(d) Any week is a good week in which to start learning a new word each week.

(e) If Brian ate a piano, then Brian ate that piano deliberately.

2. The English quantifier ‘most’ cannot be represented adequately in first-order

quantificational logic (QT) with its standard semantics. More precisely, it is impossible to

extend QT by adding some new operator and interpreting that operator in a ‘most’-like way,

so that every English sentence (S) in which the word ‘most’ is used can be translated into a

sentence in the extended system which has the same truth conditions as (S).

(We will not supply a rigorous proof for this here, but such a proof is available.)

Consider one possible extension of QT.

Let ‘M’ be added to QT in order to represent the English quantifier ‘most’. Give ‘M’ the

following interpretation:

for any open sentence α(x) in the language of QT (where ‘x’ is free in ‘α(x)’),

‘Mx α(x)’ is true if and only if ‘α(x)’ is true of more individuals in the domain than

it is false of.

Consider the sentence:

(S) Most vegetables are nutritious.

Here are two unsuccessful attempts to translate the sentence (S) into our extended system.

i. Mx(VEGETABLEx & NUTRITIOUSx)

ii.Mx(VEGETABLEx NUTRITIOUSx)

(a) Explain why each of i. and ii. above is an unsuccessful attempt.

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(Hint: show either that there is a situation in which (S) would be true, but in which the

suggested translation would be false, or that there is a situation in which (S) would be false,

but in which the suggested translation would be true.)

(b) Suppose we have a satisfactory answer to (a) above. Have we yet shown that it is

impossible to translate at least one English sentence in which ‘most’ is used into a sentence of

our extended system (QT plus the operator ‘M’ with its specified interpretation).

(c) Suppose we have shown that it is impossible to translate at least one English sentence in

which ‘most’ is used into a sentence of our extended system (QT plus the operator ‘M’ with

its specified interpretation). Have we yet shown that it is impossible to represent ‘most’ by

adding a single extra operator to QT? Explain.

3. Consider each of the following claims. State whether each claim is true. If the claim you are

considering is true, invent an argument in ordinary English which illustrates why it is true. If the

claim is false, explain why.

(a) There are some intuitively valid arguments whose validity can be demonstrated in

traditional (syllogistic) logic but not in propositional logic.

(b) There are some intuitively valid arguments whose validity can be demonstrated in

propositional logic but not in traditional logic.

(c) There are some intuitively valid arguments whose validity can be demonstrated in first-order

quantificational logic but not in propositional logic.

(d) There are some intuitively valid arguments whose validity can be demonstrated in

propositional logic but not in first-order quantificational logic.

(e) There are some intuitively valid arguments whose validity can be demonstrated in first-order

quantificational logic with identity but not in first-order quantificational logic without identity.

(f) There are some intuitively valid arguments whose validity can be demonstrated in first-order

quantificational logic with identity, but not by any mechanical method of computation.

(g) There are some intuitively valid arguments whose validity cannot be demonstrated in first-

order quantificational logic with identity.

(h) There are some arguments which are intuitively invalid and whose invalidity can be

established using the resources of propositional logic.

(i) There are some intuitively valid arguments whose validity can be demonstrated in traditional

logic but not in first-order quantificational logic with identity.

(j) There are arguments which are QT-invalid (that is, which are invalid according to first-order

quantificational logic), but whose QT-invalidity cannot be demonstrated using the resources of

first-order quantificational logic.

4. Invent and carefully define two formal systems S and S’ which, though distinct, have

exactly the same expressions as theorems. The set T of theorems must be an infinitely large

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set. (You only need to specify a syntax for the two systems. No semantic interpretation is

required. You can use any symbols and rules at all and specify them in any way that permits

the generation of a formal system; they do not need to be the sorts of symbols and rules that

we meet in standard logics. However, the two systems must not be mere alphabetic variants of

each other. In other words, it should not be possible to map S onto S’ simply by replacing

symbols in the alphabet of S with different symbols.)

Week 4 (24 March)

1. Explain informally how the following paradox can be derived.

There both is and is not such a thing as:

the least natural number that cannot be referred to by an expression consisting of

fewer than 100 words.

(You can assume that English has only finitely many words.)

Does the distinction between object languages and metalanguages help us to avoid this

paradox? If so, how?

NOTE: The first three exercises here have the same format as Problem Two in Assignment

One.

2. If any of the following is either obviously false or nonsensical as it stands, remedy matters by

adding quotation marks. Look for best solutions: solutions that involve adding as few pairs of

quotation marks as possible. (If you make a decision that you think might need defending, feel

free to add a note explaining your reasoning and why you think the case is controversial.)

(a) Palmerston is south of Palmerston North, but it is silly to claim that Palmerston

North is north of Palmerston.

(b) The last single-word expression in the proposed solution for (a) above is

Palmerston.

(c) Palmerston’s wife called Palmerston Palmerston, but it is not the case that

Palmerston’s wife called Palmerston Palmerston’s surname.

(d) We are using (d) as a name for this very sentence, but we are not using (d) as a

name for this very sentence.

(e) The last word of (f) is obscene.

(f) The last word of (f) is obscene.

(g) The last word of (g) is English.

(h) Logic is part of philosophy, part of mathematics and part of computer science.

(i) Freedom is a big problem for philosophers.

(j) In first-order logic with identity, the symbol = is treated as a logical word: it is

always assigned the identity relation (which every item in the domain of

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quantification bears to itself). In first-order logic without identity, = is treated as

a schematic letter or as an ordinary predicate symbol.

(Note: By a single-word expression, we mean either a single word or a single word enclosed

within one or more pairs of quotation marks.)

3. If any of the statements in the following passage are either obviously false or nonsensical as

they stand, remedy matters by adding quotation marks wherever they are needed. Rewrite the

whole passage adding the requisite quotation marks. Look for best solutions: solutions that

involve adding as few pairs of quotation marks as possible. (If you make a decision that you

think might need defending, feel free to add a note explaining your reasoning and why you

think the case is controversial.)

There’s a riddle which goes like this: I am the beginning of eternity that ends both time and

space, the start of every finish and both ends of every place. What am I?

The solution to this riddle is, E. So if we call our riddle R, we can make the following true

statements.

(i) E is the solution to R.

(ii) There is at least one e in R.

(iii) There is no e in R.

(iv) There is no i in team, but there is an i in: there’s no i in team.

4. Whenever a sentence in the following passage is either obviously false or nonsensical as it

stands, try to turn it into a true sentence by adding quotation marks wherever they are needed.

Rewrite the whole passage adding the requisite quotation marks. Look for best solutions:

solutions that involve adding as few pairs of quotation marks as possible. (If you make a

decision that you think might need defending, feel free to add a note explaining your reasoning

and why you think the case is controversial.)

I don’t know where I belong, but I know where I belongs: in In. Of course, there is a

place for me in the next sentence. It is at the end of time. That sentence, the one with

me but not myself in it, is called S. It does not contain a single word. Of course, S is

true. To put it more precisely, It is at the end of time is true if it refers to the place for

me. Note also that It is in it. Furthermore, It is at the end of time. is true is true, so S is

true. is true. S is not true, because S is not a sentence, even though S is. S is called

something, but it is not called anything.

5. For each of the following ambiguous English sentences:

(a) Paraphrase it in two different ways, each of which expresses, in careful English, a different

disambiguation.

(b) Explain the sorts of resources that a formal language would need to have in order clearly to

distinguish the two meanings of the sentence that you have distinguished in part (a).

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(c) Select two of the sentences below. For each of those sentences translate the English

paraphrases you created in part (a) into predicate logic. (Warning: be careful which sentences

you choose. Not all of the English sentences below can easily be captured in predicate logic.)

(i) The chief executive can’t bear children.

(ii) Preston doesn’t know a lot of songs.

(iii) John was seen by some people near the scene of the crime.

(iv) Either philosophy or English is compulsory if you do an Arts Degree.

6. Can you explain the syntactic ambiguity in the following argument in terms of differences

between translations of the disambiguations into the language of predicate logic?

Cold porridge is better than nothing; nothing is better than a hot bath; therefore, cold porridge is

better than a hot bath.

Week 6 (7 April)

1. Let S be any set of sentences in the language of quantificational logic. (Remember: this

includes all the sentences of propositional logic (with connectives) and also sentences with

quantifiers and the identity symbol.)

Let s1 and s2 each be sentences of the language (s1 and s2 can be different sentences or the

same sentences). This enables us to define some relationships as follows:

To say that s1 is a consequence of s2 is to say that every interpretation of the language that

makes s1 true also makes s2 true. (Other ways to say the same thing: s1 entails s2, the argument

with s1 as its only premise and s2 as its conclusion is valid.)

Example: p is a consequence of p & q.

(Note: if there is no interpretation that makes s1 true, then s2 is automatically a consequence of

s1 (because s2 is true in every interpretation -- of which there are none -- that makes s1 true).

For example: every sentence whatsoever is a consequence of p & ~p.

s1 is a consequence of the set S of sentences if and only if any interpretation that makes S true

also makes s1 true. Example: p & q is a consequence of the set {p, q, p v q, r}

s1 is inconsistent if and only if there is no interpretation which makes s1 true. For example: p &

~p is inconsistent because there is no way of assigning truth values which makes it true. (We

see this from its truth table.)

S is inconsistent if and only if there is no interpretation that makes all of the sentences in S true.

For example: the set {p, q, ~(p&q)} is inconsistent.

s1 is inconsistent with s2 if and only if there is no interpretation that makes s1 and s2 true. For

example: p is inconsistent with ~p.

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S is inconsistent with s1 if and only if there is no interpretation that makes all the sentences in S

true and also makes s1 true. For example: {p, q, r} is inconsistent with ~p.

Say for each of the following claims whether it is true or false and explain your answer.

(a) If s1 is a consequence of S, then s1 must be a consequence of s for any s in S.

(b) If s1 is a consequence of s for some sentence in S, then s1 must be a consequence of S.

(c) If s1 is inconsistent with S, then s1 must be inconsistent with some s in S.

(d) If s1 is inconsistent with some s in S, then s1 must be inconsistent with S.

(e) If s1 is a sentence in S, then s1 must be consistent with S.

2. Below are three bad arguments. For each argument, consider the following issues.

i. What is the conclusion of the argument?

ii. What are the premises? (Note that some of the premises may not be stated explicitly.)

iii. Why is the argument bad? (It will either be (a) because the premises, even if true, fail to give

adequate support to the conclusion or (b) because one or more of the premises is not an obvious

fact.

iv. Can the badness of the argument be traced to, or described as, a formal mistake according to

classical logic?

v. If so, use the resources of classical logic to explain what the formal mistake is - that is, to

explain how the mistake in reasoning arises. If not, speculate on whether there might be, now

or one day, some other kind of formal logic with the resources to analyse or explain the

mistake.

(a) Many kings of England were crueller than Saddam Hussein, but I notice that the

Americans have not declared war on Britain.

(b) Making a will makes you live longer. That’s the conclusion reached by legacy specialists

Live and Let Live, who compared the mortality figures of those who had made a will with

those who hadn’t. The average age of death for people who hadn’t made a will was 72 years, 6

months. However, bequeath your personal possessions on paper and life expectancy shoots up

to 80 years, 5 months. Want to live even longer? Leave some cash to charity: generous donors

lasted until the ripe old age of 83 years.

(c) Picasso was the greatest artist of the twentieth century. We know this because art critics

tell us so. We should believe these art critics, because they have better taste than the average

person. We can tell that they have better taste than the average person from the fact that they,

unlike the general public, recognise that Picasso is the greatest artist of the twentieth century.

3. Consider the following single-premise argument form.

x ~x = x

\

x y ~x = y

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(a) Are arguments of this form valid when evaluated with respect to the standard semantics for

first-order logic with identity?

(b) Are arguments of this form valid when evaluated with respect to the standard semantics for

first-order logic without identity?

Suppose we modify classical first-order semantics in two ways.

First modification: the domain of quantification is always the class of all items in the actual

world. (We hereby abolish Tarski’s innovation of allowing new instances of argument forms to

be generated by moving to a new domain of quantification. The only way to instantiate an

argument form is to replace the schematic letters in it with names for items in the real world and

let the variables range over all items in the real world.)

Second modification: every predicate symbol must be assigned a non-empty extension; that is,

if R is a two-place predicate symbol, then there are some a and b (not necessarily distinct) in the

domain of quantification such that a is related to be by the relation assigned to R.

(c) Are arguments of the form displayed above valid when evaluated with respect to the doubly-

modified semantics for first-order logic with identity?

(d) Are arguments of the form displayed above valid when evaluated with respect to the doubly-

modified semantics for first-order logic without identity?

Accompany your rulings on each of (a) to (d) with a short explanation -- as formal or informal

as you like -- of why this is the right ruling.

Week 7 (28 April)

1. Consider the conditional:

If Edmund were brave, he would be a mountaineer.

David Lewis would evaluate this conditional by appealing to a possible worlds semantics. He is

a counterpart theorist and a modal realist (or Modal Platonist, as Stephen Read would put it).

Read raises an objection to the Lewis analysis:

We wondered what would be possible if Edmund had been brave; we find that it

would not be Edmund, but some doppelgänger, who would take up mountaineering.

(p100)

Explain how Lewis could respond to this objection. What do you think of his response?

Should it satisfy Read?

2. Consider the following arguments.

(a) All day, I have been looking forward to serving the last customer of the day. You are the

last customer of the day. Therefore, all day, I have been looking forward to serving you.

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(b) You are my last customer for today. So, if you had come this morning, I could have closed

early and had at least half a day off.

Say for each of these arguments whether you think it is intuitively valid or whether this

depends on issues to do with how you interpret it. What sorts of distinctions would a logic

need to be sensitive to if it is going to deliver the right ruling about the validity of these

arguments?

Week 8 (5 May)

1. The following sentences are often used to strongly suggest propositions other than the

propositions that they literally express. In each case, state one or more strongly suggested

propositions. Also try to paraphrase the sentence so that it literally expresses the same

proposition as the original without carrying the strong suggestion with it. Would it be possible

to cancel the strong suggestion by accompanying the original sentence with another

disclaiming sentence?

(a) Boys will be boys.

(b) I’m not made of money!

(c) I have two daughters.

(d) Lewis Carroll is Charles Dodgson.

(e) Water is H2O.

(f) Either my father is at home or he is at work.

Week 9 (12 May)

No exercises this week. This is a good chance to discuss assignment essay issues. It is also a

chance to catch up on old tutorial exercises not previously discussed.

Week 10 (19 May)

1. Find the disjunctive normal form of the following formula:

(p q) (r & s)

2. Consider the following formulas.

(a) (p & q)

(b) (p & q) & r

(c) (p q) & (p r)

i. Translate each of the above three formulas into an equivalent formula whose only truth

functional connectives are members of the set {~, v}

ii. Translate each of the above three formulas into Polish notation.

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Polish notation:

Let ‘a’ and ‘b’ stand for arbitrary formulas of propositional logic.

For ~ write: Na

For & write: Kab

For v write: Aab

For write: Cab

For write : Eab

iii. For each of the three formulas find an English sentence which has the same form.

NOTE: The next two exercises are similar to a number of problems that you will find in the

third assignment.

3. Each of the following is a formula of the language of propositional logic.

(i) p (p & q)

(ii) ~ p v q

(iii) ~ p & ~ q

(iv) ~(~ p & ~ q)

(v) p & (q v r)

(a) Translate each of (i)-(v) into an equivalent formula which contains only one truth

functional operator. (That operator may occur as often as necessary.) Document the procedure

you are using to preserve the truth conditions of the formula; that is, set out your answer as a

proof.)

(b) Let “a” and “b” stand for arbitrary formulas in the language of propositional logic.

In the three-valued semantics of Łukasiewicz, compound formulas take the following values.

VAL (~ ) = 1-VAL()

VAL ( & ) = MIN(VAL(),VAL())

VAL ( v ) = MAX(VAL(),VAL())

VAL ( ) = MIN(1, (1-VAL())+VAL())

VAL ( ) = MIN(1, (1-VAL())+VAL(), (1+VAL())-VAL())

Suppose we are evaluating formulas of the language of propositional logic in accordance with

the above three-valued semantics of Łukasiewicz. Let val(p) = 1, val(q) = 0 and val(r) = 1/2.

Evaluate each of (i)-(v). (Show your working by, for instance, drawing up a truth table.)

(c) Translate each of (i)-(v) from problem two into Polish notation.

4. Consider the following expressions. For each one, translate it from Polish notation into a

formula of the language of propositional logic if that is possible. If it’s not possible, explain

why.

(vi) NApq

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(vii) KpNr

(viii) ENKpqArsNt

(ix) ACNpqNrs

(x) CCpqNr

5. For each of the following formulas of propositional logic, find an equivalent formula in

disjunctive normal form.

(i) p & ~q

(ii) p (p v q)

(iii) (p q) v p

(iv) q

(v) (p q) (r & s)

6. Translate each of (i)-(v) from Problem Two above into an equivalent formula that contains

only one truth functional operator. (That operator may occur as often as necessary.) Document

the procedure you are using to preserve the truth conditions of the formula; that is, set out your

answer as a step-by-step proof.)

Note: the single connective you are left with after transforming a particular formula could be

any of the standard propositional connectives: not, and, or, hook, or tribar, but it could also be

the Sheffer stroke (nand ‘|’) or the Peirce arrow (nor ‘↓’).

7. Let ‘’ and’ ‘’ stand for arbitrary formulas in the language of propositional logic.

In the three-valued semantics of Łukasiewicz, compound formulas take the following values.

VAL (~ ) = 1-VAL()

VAL ( & ) = MIN(VAL(),VAL())

VAL ( v ) = MAX(VAL(),VAL())

VAL ( ) = MIN(1, (1-VAL())+VAL())

VAL ( ) = MIN(1, (1-VAL())+VAL()), (1+VAL())-VAL())

Suppose we are evaluating formulas of the language of propositional logic in accordance with

the above three-valued semantics of Łukasiewicz. Let VAL(p) = 1, VAL(q) = 0 and VAL(r) =

0.5. Evaluate each of (i)-(v) from Problem Two above. (Show your working by, for instance,

drawing up a truth table.)

Week 11 (26 May)

The following exercise is similar to Problem Three in the third assignment.

Here are some sentences of ordinary English. Imagine each one being used in an ordinary,

natural, conversational context. Say, for each sentence, whether you think it expresses a truth

21

functional proposition or not. If you think it does, describe the relevant truth function, either

in words or with a truth table. If you think the sentence doesn’t express a truth functional

proposition, explain why not. (You might like to supply a counter-example to the claim that it

is a truth functional proposition.) If you like, you can explain how the sentence might be

regarded as either truth functional or not truth functional depending on which account of the

meanings of ordinary language connectives is assumed.

(a) There will be not only pavlova, but also a fine selection of vegan desserts.

(b) Found my coat and grabbed my hat. Made the bus in seconds flat.

(c) I’m tired because I’ve been yelling over the sounds of rivet guns all day.

(d) If you throw a stone into a bucket of water, the ripples spread outwards.

(e) Since I’ve stopped smoking, I’ve felt like a new man.

(f) If I knew then what I know now, I’d be a rocket scientist.

(g) Whether or not there’s a decent wind, we’ll sail home on Wednesday.

(h) It’s a long way. Nevertheless, I think we’ll get there.

(i) It’s hard work, so don’t be surprised if you decide to give up halfway through.

(j) I hear that you want to train to be a DJ.

(k) If you cook a chook for long enough, it bursts into flames.

(l) I would have rung you if I’d heard anything.

(m) Brian is far from happy.

(n) Brian is not happy.

(o) Pack your bags and leave town!

(p) Nobody moved.

(q) Brian and his wife went home early.

(r) Most of the story is pretty boring. There’s a twist at the end, though.

(s) Only a few tickets left, so book now!

(t) I believe I can fly

Weeks 12 (2 June)

Revision. No new exercises.

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7. ASSIGNMENTS

(All of the assignments are included here.)

Assignment 1: First Problem Set

Due Monday, 14 April at 5 pm.

Answer all of the following. Each of the five problems is worth 20% of the final mark for the

assignment. For some questions, a well argued, informed, thoughtful opinion is enough to earn

full marks.

Problem One

Recall: a formal language must be specifiable without reference to any particular

interpretation of it. However, when devising a formal language, one usually has some purpose

or application in mind, so it is perfectly in order to design the language so that it will fulfil

that purpose or serve that application.

Bearing this in mind, consider the following suggestion for an alphabet for the language of

propositional modal logic.

{p, ~, , , &, v, , ≡, (, )}

(a) Discuss the following claim. Is it correct or even coherent? Explain your answer.

Claim: The above is not an adequate alphabet from which to develop a language for

propositional modal logic. We can devise an adequate formal language in which to do

propositional logic without including symbols corresponding to all of the 5 standard truth

functions and without including both modal operators. A smaller set of truth functional

connectives will suffice, as will a single modal operator. (5 marks)

(b) Discuss the following claim. Is it correct? Explain your answer.

Claim: Suppose that, when we give the language an interpretation from the class of intended

interpretations, each of the symbols {p, , ~, &, v, ≡} gets its standard interpretation. Then

this language cannot possibly express all the theorems of propositional modal logic, because

there is only one propositional letter in the language and there are infinitely-many atomic

propositions which that single letter has to represent. (5 marks)

(c) Invent a new formal language with the alphabet above but with different formation rules

from the language of propositional modal logic. (This language must not be a mere alphabetic

variant of the language of propositional modal logic.) (5 marks)

(d) Invent a formal system whose language is the language you invented under (c) above and

ensure that the system has infinitely-many theorems. (5 marks)

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Problem Two

Whenever a sentence in the following passage is either obviously false or nonsensical as it

stands, try to turn it into a true sentence by adding quotation marks wherever they are needed.

Rewrite the whole passage adding the requisite quotation marks, even though this will soil the

riddle mentioned in the passage. Look for best solutions: solutions that involve adding as few

pairs of quotation marks as possible. (If you make a decision that you think might need

defending, feel free to add a note explaining your reasoning and why you think the case is

controversial.) Aim to add exactly 20 sets of quotation marks.

There’s a riddle that goes: What occurs once in a minute, twice in a moment, never in a

thousand years and only once in a million?

The answer to this riddle is m.

If we call the riddle M, then all of the following sentences are true: m is the answer to M., m

occurs four times in M and m never occurs in M., never occurs once in M and never in M.,

occurs once occurs only once in M and only once occurs only once in M, but never in M.

Problem Three

Below are four ambiguous English sentences. For each of these sentences, perform the

following four tasks.

(a) paraphrase the sentence in two different ways, each of which expresses, in careful

English, a different disambiguation. (1 mark with each successful paraphrase)

(b) Explain the sorts of resources that a formal language would need in order to

distinguish clearly the two meanings of the sentence that you have captured in part

(a). (1 mark per English sentence)

(c) If you think it can be done, translate the English paraphrases you created in part (a)

into the formal language of predicate logic. The differences between your two

translations should make clear that there are two quite different meanings intended. If

you think the translation cannot be done (or cannot easily be done), don’t attempt it.

Instead, explain why you don’t think that obvious translations are available. (2 marks

per sentence)

i Mildred wrote an article on a wall-hanging.

ii An illegal act is performed in New Plymouth every hour.

iii Some economists work in almost every area of government.

iv Please pass on your suggested changes to the committee

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Problem Four

Let Γ be any set of sentences and let A be any sentence.

The classical account of consequence says that A is a consequence of Γ if and only if any

interpretation that makes all of the sentences in Γ true also makes A true. (Example: p & q is a

consequence of the set {p, q, p v q, r})

(Note: if there is no interpretation that makes all the sentences in Γ true, then A is automatically

a consequence of Γ (because A is true in every interpretation – of which there are none – that

makes all the sentences in Γ true.)

Classically, A is a logical truth if and only if A is true in every interpretation.

(a) Show that each of the following two principles holds according to the classical account of

consequence. (10 marks)

i. Ex quodlibet verum: If A is a logical truth, then every sentence whatsoever entails A.

ii. (Thinning) For any sentence A, if A is entailed by a set Γ of premises, then A is entailed by

any set Γ+ of premises which has Γ as a subset.

(b) In your opinion, do either (or both) of these principles (ex quodlibet verum and thinning)

undermine the claim that the classical account of consequence can be used to justify certain

types of good reasoning (that is, to explain why that reasoning is good)? Explain. (10 marks)

Problem Five

For each of the following arguments, answer these questions.

(a) Is the argument intuitively valid? (1 mark per argument)

(b) Is the argument a valid syllogism according to Aristotle? (1 mark per argument)

(c) Can the argument be shown to be valid in propositional logic? Explain. (Offer a (valid or

invalid) formalisation in propositional logic to support your answer.) (1.5 marks per argument)

(d) Can the argument be shown to be valid in first-order quantificational logic (with its normal

Tarskian semantics)? Explain. (If the argument can be shown to be valid in this way, provide an

appropriate translation into predicate logic.) (1.5 marks per argument)

i. Everybody loves somebody nice; therefore, everybody loves somebody.

ii. All public servants should wear suits; most of the former students at our school are

public servants; therefore, most of the former students at our school should wear

suits.

iii. Anything that somebody wants, Lola gets; therefore, there is at least one thing.

iv. I destroyed the bed by lying down on it while it was on fire; therefore, the bed was

destroyed.

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Assignment 2 - 1500 word Essay

Due on Monday, 19 May at 5 pm.

Choose one of the following topics. Alternatively, if you have a pet topic of your own, make

sure you clear it with lecturer or tutor.

1. What is Gilbert Harman’s opinion about the relationship between logic and ordinary

reasoning? What are his arguments for his opinion? Should we agree with him? Does a

consideration of Harman’s views shed any interesting light on the true relationship between

logic and ordinary reasoning?

2. The classical account of logical consequence says that logical consequence is formal, truth

preserving and compact. Choose and explain one of these three criteria, consider some

objections to that criterion and decide whether you think, in the end, the objections you consider

undermine the claim that logical consequence should be construed classically. (See especially

Read, Ch. 2)

3. Does formal logic yield any good arguments that bad arguments are bad? Does it yield any

good arguments that good arguments are good? In the light of your (well-argued) views on

these matters, what can you say about the relationship between logic and the evaluation of

argumentation in ordinary life?

4. “When philosophers, logicians and others talk about possible worlds, they seem to be

engaging in fantasy. Even if they are not, they are not talking about anything that matters for our

understanding of what reality is like or about how we do or should reason. Possible worlds are a

tool that serves no useful purpose.” Discuss. (See especially the required readings for Week 6

and the Stalnaker reading required for Week 7.)

5. Some argue that the Classical (truth functional) definitions of the logical connectives are

inappropriate because translations of ordinary language arguments into symbols often distort the

original arguments. What sorts of phenomena do these objectors have in mind? What responses

are available to those who think we should retain the classical definitions? Assess the relative

merits of the competing arguments.

6. Here is a sequence of propositions paraphrased from a book by Neil Smith called “Chomsky:

Ideas and Ideals”, (p. 11). “In physics, claims are put forward which are either right or wrong

and we can test whether they are right or wrong. In logic, on the other hand, this is not so. If a

logician says “nothing is both an F and a non-F” this is not formulated as a hypothesis to be

evaluated and tested by her colleagues. The observations of a logician may be useful or

insightful, but they are not either correct or incorrect and they are not empirical.” Discuss.

(See especially Haack, Ch. 10 and Ch. 12.)

7. Consider the following English sentence (L):

The sentence (L), that is, the sentence you are considering is false.

Carefully explain why this English sentence is paradoxical, that is, how it is apparently both

true and false. Explain carefully how recourse to the distinction between meta-language and

object language can prevent the liar problem from arising for a sentence in a formal language.

27

How relevant is this formalistic solution to the problem posed by this sort of sentence in

ordinary English? (See especially Read, Ch. 6 and Haack, Ch. 8.)

8. What reasons have been given (and could reasonably be given) for claiming that there are

more than two truth-values? What do you think about these proposals? In what situations, if

any, do logics with truth-values other than the classic TRUE and FALSE represent inferences

more adequately than logics that employ only those two truth-values. (Haack, Ch. 11 is a good

starting point.)

Some General Notes about the Essay Assignment

1. Essay Drafts

You are encouraged to present a version of your essay to the one of us well before the due date.

We will take a look and give you feedback.

2. Reading for essays

The notes and source readings in this course booklet should provide a sufficient background for

any of the essay topics above. Refer to the lists of readings associated with the various lecture

topics as a guide, but bear in mind that the essay topics are designed to overlap the lecture

topics somewhat. Do not hesitate to contact us for advice about specific reading in the course

booklet or about additional readings which might be relevant to your take on a topic. However,

there is no point in over-reading. We are looking for thoughtful, well-argued responses to the

issues, not for wide-ranging research.

3. The 1500-word guideline applies to the main text of your essay only; it excludes footnotes

and bibliography.

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Assignment 3 - Second Problem Set

Due Monday, 9 June at 5 pm.

Answer all of the following. Each of the five problems is worth 20% of the final mark. For

some questions, a well argued, informed, thoughtful opinion is enough to earn full marks.

Problem One

Consider the English sentence (S) below.

(S) Either I left my keys in the door, or I forgot to bring them.

(a) Write down an English sentence other than (S) which, necessarily, is true if and only if (S) is

true. (2 marks)

(b) Write down a sentence that (expresses a proposition that) entails, but is not entailed by, (the

proposition expressed by) (S). (2 marks)

(c) Write down a sentence that (expresses a proposition that) is entailed by, but does not entail,

(the proposition expressed by) (S). (2 marks)

(d) Write down two sentences each of which would normally be (conversationally) implicated

by an utterance of (S). (2 marks)

(e) Write down a sentence that is a contrary of (S). (2 marks)

(f) Write down a sentence that is a contradictory of (S). (2 marks)

(g) Write down a sentence that expresses a propositional attitude towards the proposition

expressed by (S). (2 marks)

(h) Is (S) a truth function of any of its components? (Try to explain your answer in a sentence or

two.) (2 marks)

Problem Two

(a) Translate each of the following formulae of propositional logic into Polish notation. (10

marks)

(For Polish notation, see below.)

(i) p & ~ q

(ii) p (p v q)

(iii) (p q) v p

(iv) q

(v) (p q) (r & s)

29

(b) Translate each of the following strings of Polish notation into a formula of the language of

propositional logic, if that is possible. If it’s not possible, explain why. (10 marks)

(i) EpNq

(ii) ANrNp

(iii) CNpEKpqr

(iv) NCEApqrsNt

(v) AAApNqrp

Polish notation

Let ‘α’ and’ ‘β’ stand for arbitrary formulas of propositional logic.

For ~α write: Nα

For α & β write: Kαβ

For α v β write: Aαβ

For α β write: Cαβ

For α β write: Eαβ

Problem Three

Here are some sentences of ordinary English. Imagine each one being used in an ordinary,

natural, conversational context. Say, for each sentence, whether or not its meaning in the

ordinary context can be fully described as a truth function of the meanings of one or more of

its components. If you think it can, describe the relevant truth function, either in words or with

a truth table. If you think the sentence doesn’t express a truth functional proposition, explain

why not. (You might like to supply a counter-example to the claim that it is a truth functional

proposition.) Your verdict might be that the sentence could be regarded as either truth

functional or not truth functional, depending on which account of the meanings of ordinary

language connectives is assumed. This verdict is fine, so long as you explain why you hold

that view.

(a) We will go carol singing unless our chooks win the raffle.

(b) If you would like to hear the programme again, you can download the audio and listen to it

as a podcast.

(c) Phillippa and I got married.

(d) Phillippa and I are Aucklanders.

(e) Phillippa and I are probably not westies.

(f) Phillippa and I are not Westies, but Vanya is.

(g) I know that some Westies are nice to their chooks.

(h) I’d forget my own head if it wasn’t screwed on.

30

(i) I’m leaving you because you buy me too many raffle tickets.

(j) If you think you know the answer already, why are you asking me?

Problem Four

The connective ‘|’ is called ‘the Sheffer Stroke’ and is usefully pronounced ‘nand’ or ‘not both

... and ...’. Let P+ be the language of propositional logic augmented to include the Sheffer

Stroke ‘|’ (nand) and the Peirce Arrow ‘↓’(nor) as additional logical symbols.

Here is the truth table for |.

P q p | q

T T F

T F T

F T T

F F T

Here is the truth table for ↓.

P Q p q

T T F

T F F

F T F

F F T

(a) Explain carefully how, if we assume the standard classical semantics for the standard

truth-functional connectives, we can show that every truth function whatsoever can be

expressed by a formula of P+ which has the Sheffer stroke ‘|’ as its only connective. (The

Scheffer stroke may occur as many times as necessary in the formula expressing a truth

function.)

You do not need to provide a complete proof and you may assume any results formally proved

in class, (in particular, you may assume that any dyadic truth function can be expressed by a

formula of P+ which has ‘~’ (not) and ‘&’ (and) as its only connectives and you may assume

that every truth function can be expressed by a formula that has the Peirce Arrow ‘↓’ (nor), as

its only connective). However, you may need to prove some small steps. (10 marks)

(b) Let ‘α’ and’ ‘β’ stand for arbitrary formulas in the language of propositional logic.

Determine whether, if we assume the three-valued semantics of Łukasiewicz, “(α ⊃ β)”

expresses the same truth function as (~α v β). (5 marks)

In the three-valued semantics of Łukasiewicz, compound formulas take the following values.

VAL (~ ) = 1-VAL()

VAL ( & ) = MIN(VAL(),VAL())

VAL ( v ) = MAX(VAL(),VAL())

31

VAL ( ) = MIN(1, (1-VAL())+VAL())

VAL ( ) = MIN(1, (1-VAL())+VAL(), (1+VAL())-VAL())

(c) Does your result in part (b) above have any ramifications for the issue of whether the

inference rules governing the conditional, as it is ordinarily used, can be explained in terms of

those for negation and disjunction? (If you offer an informed, careful, opinion on this matter,

you will get full marks.) (5 marks)

Problem Five

(a) Consider the following pair of arguments.

(i) P1 The oldest lion-tamer in Brisbane is perfect.

P2 I’m the oldest lion-tamer in Brisbane.

Therefore,

C I’m perfect.

(ii) P1 Nobody is perfect.

P2 I’m nobody.

Therefore,

C I’m perfect.

Use your understanding of predicate logic to explain why we should regard the first argument as

valid and the second as invalid, even though they may appear to be structurally similar in

English. (It may help to write yourself a dictionary and formalise the two arguments in predicate

logic.) (5 marks)

(b) Now consider the following pair of arguments.

(iii) A bike is better than nothing, but a car is better than a bike. So, of course, a car is better

than nothing.

(iv) Nothing is better than a hot bath. A cold shower is better than nothing. So, a cold shower is

better than a hot bath.

Again, use your understanding of predicate logic to explain why we should regard the first

argument as valid and the second as invalid, even though they may appear to be structurally

similar in English. (As before, it may help to write a dictionary and formalise the two arguments

in predicate logic. If you do this, bear in mind that you may need to add to both arguments a true

suppressed premise which tells something about the logic of the word “better”.) (5 marks)

(c) Consider the following pair of arguments.

(v) P1 If I get to rule the world, certain people will have to jack up their ideas.

P2 If I ever become the greatest trombonist of all time, then I will get to rule the world.

Therefore,

32

C If I ever become the greatest trombonist of all time, then certain people will have to

jack up their ideas.

(vi) P1 If I were licking milk from a bowl and crunching my way through fine, high-quality,

Kittibix, then I’d be thrown into an institution for the wayward.

P2 If I were a cat, I would be licking milk from a bowl and crunching my way through

fine, high-quality, Kittibix.

Therefore,

C If I were a cat, I’d be thrown into an institution for the wayward.

Use your knowledge of the various logical systems that are appropriate for evaluating

arguments involving different sorts of conditionals to explain why we can reasonably regard (v)

as valid but (vi) as invalid. (It might be rather difficult to formalise these arguments, so a

general description of the logic involved is fine.) (5 marks)

(d) Think about all of the invalid arguments in parts (a) to (c). Explain why, according to Gerald

Massey, we can never use formal logic to demonstrate that they are invalid. (5 marks)

33

8. ADMINISTRIVIA CONCERNING ALL THE ASSIGNMENTS

1. Please type your assignments.

2. Hand in all assignments by emailing to j.mckeown-green@auckland.ac.nz, preferably

as a doc or docx word document or as an rtf.

4. In your course work, clearly indicate all passages that are not your own work by

placing quotation marks around the passage, or by indenting the passage. Then refer to

the place where the passage can be found. This is best done by a footnoted reference to

the page(s) of a work listed in a bibliography placed at the end of your assignment. In

the bibliography include where possible the author, book title, or name of journal, and

publisher and place and date of publication. (See books listed in the lecture schedule

above as an example of citation style.) If your work contains passages from the

writings of others not indicated in this way then you might be penalised. Penalties can

be imposed for unoriginal work submitted as your own work, either intentionally or

not. What an unpleasant thought to end on! Sorry about that.