# 127: Philosophical logic Logic exercises and philosophy tasks127: Philosophical logic Logic...

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127: Philosophical logicLogic exercises and philosophy tasks

James Studd

Hilary term 2018

Contents

1 How to use these exercises 2

2 Logic exercises 3Week 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 3Week 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 6Week 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 8Week 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 10Week 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 11Week 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 13Week 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 15Week 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Philosophy tasks 17Task A . . . . . . . . . . . . . . . . . . . . . . . . . . 17Task B . . . . . . . . . . . . . . . . . . . . . . . . . . 18Task C . . . . . . . . . . . . . . . . . . . . . . . . . . 19Task D . . . . . . . . . . . . . . . . . . . . . . . . . . 20Task E . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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Preliminaries HT18

1 How to use these exercises

Logic exercises vs. philosophy tasks

This booklet comprises eight sets of logic exercises and five philosophy tasks, sup-plementing the exercises in the course text book, Ted Sider’s Logic for Philosophy(LfP).

The logic exercises generally ask you to construct formal or informal proofs. Thesewill help you prepare for the problem-based part of the exam.

The philosophy tasks are short writing exercises. These will help you prepare for thephilosophical assessment part of the exam.

Each set of logic exercises deals with the material covered in the corresponding lecture.The five philosophy tasks correspond to the material covered in the lectures in weeks4–8 (which have shorter sets of logic exercises):

Week 4 Task AWeek 5 Task BWeek 6 Task CWeek 7 Task DWeek 8 Task E

The philosophy tasks may however be taught separately from the logic exercises. Asusual, your tutor will advise you on what to cover when.

Note on : and ‹

Questions marked ‹ are more mathematically involved (often using induction).

I recommend that students who haven’t studied Elements of Deductive Logic omitthese on their first pass. Students who have studied Elements of Deductive Logicshould be able to attempt the ‹’d questions, and may omit the questions flagged with: instead.

The :’s and ‹’s will quickly diminish as term goes on and we close the EDL-gap.

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127 Exercises: Week 1 HT18

2 Logic exercises

Week 1

Recommended supplementary reading

Tim Williamson, Vagueness (Routledge, 1994). Chapters 4.1–4.6.

Michael Tye, Sorites Paradoxes and the Semantics of Vagueness, Philosophical Per-spectives 8 (1994), 189–206 xhttp://www.jstor.org/stable/2214170y

[These will help with the last question.]

Propositional Logic (LfP 2.1–2.4)

1.: Let VI be a PL-valuation for PL-interpretation I .

(a) Give informal semantic arguments in the style of Example 2.1 (LfP, 32) to provethe following:

i. VI p„φÑ ψq “ 1 iff VI pφq “ 1 or VI pψq “ 1

ii. VI p„pφÑ „ψqq “ 1 iff VI pφq “ 1 and VI pψq “ 1

iii. VI p„ppφÑ ψq Ñ „pψ Ñ φqqq “ 1 iff VI pφq “ VI pψq

[Feel free to omit the I -subscripts when no confusion will arise.]

(b) The truth conditions for the usual connectives are as follows:

i. VI p„φq “ 1 iff VI pφq “ 0

ii. VI pφÑ ψq = 1 iff VI pφq “ 0 or VI pψq “ 1

iii. VI pφ_ ψq “ 1 iff VI pφq “ 1 or VI pψq “ 1

iv. VI pφ^ ψq “ 1 iff VI pφq “ 1 and VI pψq “ 1

v. VI pφØ ψq “ 1 iff VI pφq “ VI pψq

Briefly explain, in each case, why the clause holds on Sider’s presentation of thesemantics of PL. How does this differ from Halbach’s presentation in The LogicManual?

2. Give informal semantic arguments in the style of Example 2.2 (LfP, 35) to demonstratethe following facts about semantic consequence in PL. (Don’t use truth tables.)

(a) (PL φÑ pψ Ñ φq

(b) (PL pφÑ pψ Ñ χqq Ñ ppφÑ ψq Ñ pφÑ χqq

(c) (PL p„ψ Ñ „φq Ñ pp„ψ Ñ φq Ñ ψq

(d) φ, φÑ ψ (PL ψ

3.‹ Show that if φ contains at most one occurrence of any sentence letter, then *PL φ(LfP, Ex 2.8, 55)

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Variations of PL (LfP 3.1)

4.: The ‘Peirce arrow’ Ó (“nor”) is the connective such that

VI pφ Ó ψq “ 1 iff VI pφq “ 0 and VI pψq “ 0

(a) Write down a truth-table for Ó. Why “nor”?

(b) Write down formulas whose only connective is Ó which symbolize the truth func-tions symbolised by „P1 and P1 Ñ P2.

(c) Deduce that tÓu is adequate. Explain your answer.

Deviations from PL (LfP 3.3–3.4)

5. (a) Are the following claims true for Kleene’s three-valued logic? Justify your an-swers. (You may use three-valued truth-tables.)1

i. P, pP Ñ Qq (K Q (modus ponens)

ii. (K P Ñ ppP Ñ Qq Ñ Qq (modus ponens)

iii. P,„P (K Q (ex falso quodlibet)

iv. (K pP ^„P q Ñ Q (ex falso quodlibet)

(b) Are they true for Lukasiewicz’s logic? (Replace “(K” with “( L”.)

(c) What about Priest’s Logic of Paradox, LP? (Replace “(K” with “(LP”.)

6.‹ (a) Stipulate that # “ 12 , so that 1 ą # ą 0. Show that:

LVI pφÑ ψq “

#

1 if LVI pψq ě LVI pφq

1´ p LVI pφq ´ LVI pψqq if LVI pψq ă LVI pφq

(b) Let φ and ψ be distinct atomic formulas (i.e. sentence letters). Show that:

KVI pφÑ ψq “ SVI pφÑ ψq

[Hint: show KVI pφÑ ψq “ 1 implies SVI pφÑ ψq “ 1. Repeat for 0 and #.]

Does this still hold when φ and ψ are non-distinct or non-atomic? Explain.

(c) Show that Γ (PL φ iff Γ (S φ. [Hint: show Γ *PL φ iff Γ *S φ]

1Following Sider, Kleene’s three-valued logic refers to Strong Kleene.

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127 Exercises: Week 1 HT18

7. Let Pn formalize ‘n-year olds are young’. Consider a ‘Sorites’ argument:

P0, P0 Ñ P1, P1 Ñ P2, . . . , P99 Ñ P100; so P100

Say that a trivalent interpretation I is faithful if the following conditions hold:

• I pP0q “ 1; I pP100q “ 0

• if I pPn`1q “ 1, then I pPnq “ 1, for 0 ď n ă 100

• if I pPnq “ 0, then I pPn`1q “ 0, for 0 ď n ă 100

(a) What, intuitively, is faithful about faithful interpretations?

(b) Order the three truth-values 1 ą # ą 0 (as in question 6).

Show that if I is a faithful interpretation then:

1 “ I pP0q ě I pP1q ě . . . ě I pP99q ě I pP100q “ 0

(c) Consider the following statements:

α The conclusion of the argument is not a semantic consequence of its pre-misses.

β Some premise or other is false under every faithful interpretation.

Determine which of α and β hold for the following logics.

i. Classical propositional logic, PL

ii. Kleene’s three-valued logic

iii. Priest’s Logic of Paradox, LP

(d) What, in your view, if anything, is wrong with upholding α?

(e) What, in your view, if anything, is wrong with upholding β?

(f) Repeat part (c.ii) for a second Sorites argument.

∆P0, ∆P0 Ñ ∆P1, ∆P1 Ñ ∆P2, . . . , ∆P99 Ñ ∆P100, so ∆P100

(g) Discuss the philosophical significance of these results.

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127 Exercises: Week 2 HT18

Week 2

Semantics for MPL (LfP 6.3)

1. (a) Give informal semantic arguments to demonstrate the following validities:2

i. (K 2pφÑ ψq Ñ p2φÑ 2ψq

ii. (D 2φÑ 3φ

iii. (T 2φÑ φ

iv. (B φÑ 23φ

v. (S4 2φÑ 22φ

vi. (S5 3φÑ 23φ

(b) Specify models that demonstrate the following invalidities. You may (but neednot) use the method outlined in LfP, 6.3.3

ii. *K 2φÑ 3φ

iii. *D 2φÑ φ

iv. *S4 φÑ 23φ

v. *B 2φÑ 22φ

vi. *S4 3φÑ 23φ and *B 3φÑ 23φ.

Tense logic (LfP 7.3)

2. (a) Determine whether each of the following is valid in Priorean tense logic (PTL).Provide an informal semantic argument or counterexample in each case:

i. φÑ HFφ

ii. PPφÑ Pφ

iii. pFφ^ Fψq Ñ pFpφ^ Fψq _ Fpψ ^ Fφqq

(b) For each invalid formula, propose a plausible further constraint on ď that rendersit valid.3 Demonstrate this with a semantic argument.

2We indulge in a slight abuse of notation: (a.i) should be understood to mean that every instance of theformula-schema 2pφ Ñ ψq Ñ p2φ Ñ 2ψq is valid in K; (b.ii) to mean that some instance of 2φ Ñ 3φis invalid in K. See LfP 2.4.1 for discussion.

3i.e. such that the formula is true at all times in all models in which ď meets the further constraint.

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127 Exercises: Week 2 HT18

3. Introduce the following abbreviation: Cφ is short for 3φ^3„φ

(a) Provide an idiomatic English gloss of C. (Be as concise as you can.)

(b) Give conditions for VI pCφ,wq to be 1 in terms of VI pφ, uq for appropriate u.

(c) Show that:

i. (S5 CφÑ „CCφ

ii. (S5 „CφÑ „C„Cφ

(d) Show that (i) fails when we replace S5 with S4

(e) Show that (ii) fails when we replace S5 with B.

(f) ‘This shows that S5 is the correct modal logic.’ Discuss.

4.‹ This question concerns (non-empty) finite strings of 2s and 3s (e.g. 223232). Wecall these simply ‘strings’, and say that two strings O1 and O2 express the samemodality in S if (S pO1φØ O2φq. This is symbolised as O1 ”S O2.

(a) Show that:

i. if O1 ”S O2, then OO1 ”S OO2.4

ii. if O1 ”S O2, then O1O ”S O2O.

[You may assume without proof that if ( φ1 Ø φ2, then ( χpφ1q Ø χpφ2q, whereχpφq is the result of substituting φ for P in the formula χpP q]

(b) Show that:

i. Infinitely many modalities are expressed by strings in B.

ii. Exactly two modalities are expressed by strings in S5.

iii. Exactly six modalities are expressed by strings in S4.

Induction on complexity (LfP 2.7)5

5.: Prove the following claims carefully using induction:

(a) The number of occurrences of parentheses in an MPL-sentence is twice the num-ber of occurrences of Ñ.

(b) Let I ` be the interpretation such that I `pαq “ 1 for each sentence letter α.Show that for any sentence φ with no occurrences of negation VI `pφq “ 1.

(c) Show that if φ contains at most one occurrence of each sentence letter then thereis an interpretation I such that VI pφq “ 1 and an interpretation J such thatVJ pφq “ 0. Hence, or otherwise, complete Week 1, ex. 3.

6. LfP exs. 3.7–3.9

4OO1 is the string that results from concatenating O to the left of O1. e.g. for O “ 22 and O1 “ 333,OO1 “ 22333. Similarly for the other cases.

5Don’t worry about Soundness (yet), just the material on induction, pp. 50–3

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127 Exercises: Week 3 HT18

Week 3

Axiomatic proofs in PL (LfP, 2.6)

1. Demonstrate the following by giving axiomatic proofs:

(a) $PL P Ñ pP Ñ P q

(b) $PL P Ñ P

(c) $PL p„P Ñ P q Ñ P

Don’t use Sider’s “toolkit” in this question. You should give full, unabbreviated proofs,save that you need not repeat subproofs for parts you’ve proved in the early parts inthe later ones.

DT (LfP, 2.8)

2. Use DT to establish the following (you may also use CUT, but don’t assume withoutproof any of the results from example 2.11 onwards).

(a) $PL φÑ ppφÑ ψq Ñ ψq

(b) φ $PL ψ Ñ φ

(c) $PL „φÑ pφÑ ψq

(d) $PL „„φÑ φ

Axiomatic proofs in MPL (LfP, 6.4)

In the following questions, you may follow Sider in suppressing PL-steps (see LfP, 160)but don’t use the metatheorems from 6.4.7.

3. Give axiomatic proofs to establish the following:

(a) $T 22P Ñ P

(b) $T P Ñ 3P [i.e. $T P Ñ „2„P ]

(c) $S4 33P Ñ 3P

(d) $S4 2P Ñ 232P

(e) $S5 2p2P Ñ 2Qq _2p2QÑ 2P q

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127 Exercises: Week 3 HT18

4. Consider a variant axiomatization of K that deletes the axiom (K) and adds two othersinstead

Axiomatic system for PL˚

• Rules: MP and NEC

• Axioms all instances of PL1–PL3, plus

2pφÑ ψq Ñ p3φÑ 3ψq (K3)

„3„φØ 2φ (3df)

Write $K3 φ to mean φ is provable in the variant axiomatization. Show that:

(a) i. $K3 3pφ^ ψq Ñ 3φ^3ψ

ii. $K3 2pφÑ ψq Ñ p2φÑ 2ψq

(b) Deduce that the same MPL-sentences are provable in both systems.

[You may assume without proof $K 2pφÑ ψq Ñ p3φÑ 3ψq and $K „3„φØ 2φ.]

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127 Exercises: Week 4 HT18

Week 4

Soundness (LfP, 6.5.1)

1. Carefully prove using induction that S5 is sound.

Meta-Rules (LfP 2.8, 6.4)

2. (a) Prove Cut1 (see lecture 3). Deduce Cut.

(b) Prove DT for PL.

3. This question concerns meta-rules of the form:φ1 . . . φn

Rψ

We say that a rule R is K-admissible iff it preserves K-derivability—i.e. $K ψ holdswhenever $K φ1 and . . . and $K φn.

Prove that the following meta-rules are K-admissible:6

(a)φÑ ψ

Becker, where O is a finite string of 2s and 3sOφÑ Oψ

(b)χpαq

Subst1, where χpφq uniformly substitutes φ for α in χpαqχpφq

(c)φ1 Ø φ2 Subst, where χpφiq uniformly substitutes φi for α in χpαq

χpφ1q Ø χpφ2q

(d)φ1 ¨ ¨ ¨φn PL, where pφ1 Ñ ¨ ¨ ¨ pφn Ñ ψqq ¨ ¨q is a MPL- tautology

ψ

For part (d) you may assume that the axiomatic system for PL is complete.

Maximally consistent sets (LfP 6.6.1)

4. Let Γ $K φ be defined as per LfP 176, and say Θ is maximally consistent (in K) iff:

• Θ &K K, for K “ „pP Ñ P q, and

• φ P Θ or „φ P Θ, for each MPL-wff φ

Let Θ be a maximally consistent set. Show that:

(a) φ P Θ iff Θ $K φ

(b) i. „φ P Θ iff φ R Θ

ii. φÑ ψ P Θ iff φ R Θ or ψ P Θ

iii. 2φ P Θ iff, for every maximally consistent Σ s.t. RΘΣ, φ P Σ

where R is defined as on LfP 176.

6Here α is a sentence letter, and χpαq is a MPL-wff containing zero or more occurrences of α.

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127 Exercises: Week 5 HT18

Week 5

Validity in PC (LfP 4.1–3)

1. Let β be a term that is free for term α in PC-wff φ.7

(a) Prove that:

i. if g and h agree on the free variables in φ, Vgpφq “ Vhpφq.

ii. if rαsg “ rβsg, then Vgpφq “ Vgpφpβ{αqq.8

(b) Hence, show that:

i. (PC @αφÑ φpβ{αq

ii. (PC @αpφÑ ψq Ñ pφÑ @αψq whenever α does not occur free in φ

iii. If (PC φ, then (PC @αφ

Extensions of PC (LfP 5.1–5.5)

2. This question refers to the following languages:

• L“ is the language of PC with just = added.

• Lι is the language of PC with ι added (but not =).

• Lι,λ is the language of PC with ι and λ added (but not =).

Consider the following sentence and dictionary:

(˚) The king of France is not bald

K : . . . is king of France B : . . . is bald

(a) Using this dictionary, apply Russell’s theory of descriptions to obtain two seman-tically non-equivalent symbolizations of (˚) in L“.9

(b) Using the same dictionary, symbolize (˚) in Lι. Show that this symbolization issemantically equivalent to one of the symbolizations from part (a).

(c) Using the same dictionary, show that the other symbolization may be captured(up to semantic equivalence) in Lι,λ. Demonstrate the semantic equivalence witha semantic argument.

(d) Compare and contrast the symbolizations in part (a) and part (c).

7i.e. no occurrence of α occurs free within the scope of an occurrence of @β.8The formula φpβ{αq is the result of replacing each free occurrence of α with β. See LfP 99-100.9Sentences are said to be semantically equivalent if they are true in the same models.

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127 Exercises: Week 5 HT18

3. (a) Determine which of the following binary generalized quantifiers can be symbolizedin L“, the language of PC enriched with =:

VM ,gppNoα :φqψq “ 1 iff |φM ,g,α X ψM ,g,α| “ 0

VM ,gppAt least twoα :φqψq “ 1 iff |φM ,g,α X ψM ,g,α| ě 2

VM ,gppFinitely manyα :φqψq “ 1 iff |φM ,g,α X ψM ,g,α| is finite

VM ,gppMostα : φqψq “ 1 iff |φM ,g,α X ψM ,g,α| ą |φM ,g,α ´ ψM ,g,α|

Justify your answers with suitable semantic arguments.

[You may take it for granted that L“ is compact; you may also assume that anyset of L“ sentences with an infinite model has a countably infinite model.10]

(b) Which of these generalized quantifers can be captured in the language of second-order logic? Explain.

10We say that a model is a model of a set Γ if each member of Γ is true in the model.

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127 Exercises: Week 6 HT18

Week 6

Validity and invalidity in SQML (LfP 9.1–9.4)

1. Determine which of the following schemas are SQML-valid:

(a) 2@αφÑ @α2φ

(b) @α2φÑ 2@αφ

(c) 2DαφÑ Dα2φ

(d) Dα2φÑ 2Dαφ

In each case, give a semantic argument or counterexample as appropriate. (There isno need to prove that your counterexample is a counterexample.)

Proofs in SQML (LfP 9.7)

2. Let φ be a QML-formula and let α and β be terms. For term γ, let:

γpβ{αq “

#

β if γ “ α

γ if γ ‰ α

Say that β is substitutable for α in φ if α does not occur free in any subformula of φbeginning with @β. If β is not substitutable for α, leave φpβ{αq undefined; otherwise,define φpβ{αq as follows:

`

Πnγ1, . . . , γn˘

pβ{αq “ Πnγ1pβ{αq, . . . , γnpβ{αq, for Πn an n-place predicate`

„φqpβ{αq “ „pφpβ{αqq`

φÑ ψqpβ{αq “ φpβ{αq Ñ ψpβ{αq`

2φqpβ{αq “ 2pφpβ{αqq`

@αφqpβ{αq “ @αφ`

@βφqpβ{αq “ @βφ`

@γφqpβ{αq “ @γpφpβ{αqq if γ is distinct from α and β

(a) Compute:

i. pPxqpx{xq

ii. pPx^2Qyqpx{yq

iii. ppQyqpx{yqqpz{xq

iv. p2@xPxqpy{xq

v. p2@xPyqpx{yq

vi. p@xPx^Rxyqpy{xq

(b) Briefly explain why this definition of φpβ{αq agrees with Sider’s account of “cor-rect substitution” (LfP, 100) in the case of (non-modal) PC-formulas.

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127 Exercises: Week 6 HT18

3. Construct abbreviated proofs to demonstrate the following:

(a) $SQML @xFxÑ @yFy

(b) $SQML @αpφÑ ψq Ñ p@αφÑ @αψq

(c) $SQML p2@αφ^3Dαψq Ñ 3Dαpφ^ ψq

(d) $SQML 2@αφÑ @α2φ

(e) $SQML @α2φÑ 2@αφ

(f) $SQML 2@α2Dβpβ “ αq

Soundness for SQML

4. Show that the axiomatic proof system for SQML is sound for its semantics, i.e.:

$SQML φ only if (SQML φ

[There is no need to reestablish from scratch the validities that have already been estab-lished in earlier sheets (for PL, MPL and PC). Instead state the validities you require,and briefly explain how your earlier arguments may be generalized to establish them.]

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127 Exercises: Week 7 HT18

Week 7

Validity in VDQML (LfP 9.6)

1. Let β be a term that is substitutable for term α in QML-wff φ.11

(a) Let M “ xW ,R,D ,Q,I y be a VDQML model. Show that the results provedin week 5 generalize to QML:

i. if g and h agree on the free variables in φ, then VM ,gpφ,wq “ VM ,hpφ,wq, foreach w P W

ii. if rαsM ,g “ rβsM ,g, then VM ,gpφ,wq “ VM ,gpφpβ{αq, wq, for each w P W .12

(b) Which of the following are VDQML-valid?

i. @αφÑ φpβ{αq

ii. @αφÑ pDγpγ “ βq Ñ φpβ{αqq with γ ‰ α, β

iii. @αpφÑ ψq Ñ pφÑ @αψq whenever α does not occur free in φ

Provide semantic arguments or counterexamples.

2. Let a frame be a quadruple consisting of the first four components of a VDQML-model. Say that a QML-sentence is valid on a frame xW0,R0,D0,Q0y iff it is valid inevery model xW0,R0,D0,Q0,I y whose first four components are those of the frame.

(a) Show that all instances of (CBF) 2@αφ Ñ @α2φ are valid on a frame F “xW ,R,D ,Qy iff F is increasing (i.e. Ruw implies Du Ď Dw).

(b) Show that all instances of (CBF) 2@αφ Ñ @α2φ and (B) 32φ Ñ φ are validon a frame F “ xW ,R,D ,Qy only if F is locally constant (i.e. Ruw impliesDu “ Dw).

Two-dimensional Modal Logic (LfP, 10)

3. Determine whether or not the following formula-schemas are (i) 2D-valid and (ii)generally 2D-valid:

(a) φØ 2@φ

(b) 2pφØ 2@φq

(c) 2XpφØ 2@φq

4. Symbolize the following sentences in the language of QML with @ and X. Capture asmany English readings as possible, and comment on difficulties or points of interest:

(a) Some non-actual thing could exist.

(b) It could be the case that all non-actual things exist.

(c) It is necessary that all the red things could have been pink, and vice versa.

(d) Unless no one invented the zip, the actual inventor of the zip did.

11i.e. no occurrence of α occurs free within the scope of an occurrence of @β.12Recall that the formula φpβ{αq is the result of replacing each free occurrence of α with β.

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127 Exercises: Week 8 HT18

Week 8

Counterfactuals (LfP, 8)

1. Let M “ xW ,ĺ,I y be an SC-model.

(a) Show that we can define a ‘selection function’ f from wffs and worlds to worlds,such that:

VM pφ� ψ,wq “ 1 iff VM pφ, uq “ 0 for all u P W or VM pψ, fpφ,wqq “ 1

(b) Hence, or otherwise, show that (SC pφ� pψ _ χqq Ñ pφ� ψ _ φ� χq.

(c) Show that *LC pφ� pψ _ χqq Ñ pφ� ψ _ φ� χq.

2. (a) Demonstrate the following semantic consequences for the material conditional:

i. ψ (SC φÑ ψ

ii. φÑ ψ (SC „ψ Ñ „φ

iii. φÑ ψ (SC pφ^ χq Ñ ψ

iv. pφ^ ψq Ñ χ (SC pφÑ pψ Ñ χqq

(b) Demonstrate the following semantic non-consequences for Stalnaker’s conditional:

i. ψ *SC φ� ψ

ii. φ� ψ *SC „ψ� „φ

iii. φ� ψ *SC pφ^ χq� ψ

iv. pφ^ ψq� χ *SC pφ� pψ� χqq

(c) Do the inferences corresponding to (i)–(iv) above preserve-truth for the Englishcounterfactual conditional? Give an explanation or counterexample in each case.

3. (a) Formalize the following argument in the language of SC so that its conclusion isan SC-semantic consequence of its premisses. Demonstrate this with an informalsemantic argument.

Had I flipped the coin, it would have either landed heads or tails. Butit’s not the case that it would have definitely landed heads if flipped.So, if I’d flipped it, the coin would have landed tails.

(b) Specify a countermodel to show that it is not also LC-valid.

(c) Is the English argument intuitively valid? Briefly explain your answer.

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127 Exercises: Task A HT18

3 Philosophy tasks

Task A

What is the correct logic for metaphysical necessity?

Reading

‹ Nathan Salmon, The Logic of What Might have Been, The Philosophical Review 98(1989), 3–34.

xhttp://www.jstor.org/stable/2185369y

Tim Williamson, Modal Logic as Metaphysics (OUP, 2013) sections 3.1–3.3.

xhttp://doi.org/10.1093/acprof:oso/9780199552078.001.0001y

(‹ = compulsory.)

Assignment

For each of the following statements, briefly expound and critically evaluate an argumentfor it. (Maximum 500 words each.)

(1) S5 is the correct logic for metaphysical necessity.

(2) S5 and S4 are ‘fallacious systems for reasoning about what might have been.’ (SALMON)

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127 Exercises: Task B HT18

Task B

Is second-order logic logic?

Reading

‹ Quine, Philosophy of Logic (2nd ed., Harvard UP, 1986), Ch. 5, 61–70.

‹ Boolos, On Second-Order Logic, Journal of Philosophy 72 (1975), 509–527. Reprintedin his Logic, Logic and Logic.

xhttp://www.jstor.org/stable/2025179y

Further reading

Shapiro, Foundations without Foundationalism, chs. 4, 5 and 7.

(‹ = compulsory.)

Assignment

1. List the main differences between first- and second-order logic.

2. Briefly expound and critically assess what you take to be the best argument for

(a) taking second-order logic to be logic (500 words)

(b) taking second-order logic not to be logic (500 words)

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127 Exercises: Task C HT18

Task C

Is everything necessarily something?

Reading

‹ Williamson, Bare Possibilia, Erkenntnis 48 (1998), 257–273.

xhttp://www.jstor.org/stable/20012844y

‹ Hayaki, Contingent Objects and the Barcan Formula, Erkenntnis 64 (2006), 75–83.

xhttp://www.jstor.org/stable/20013380y

Further reading

Tim Williamson, Modal Logic as Metaphysics (OUP, 2013), chs 1, 2.1-2, 3, 4.1.

Sider, Williamson’s Many Necessary Existents, Analysis 69 (2009), 250–258.

xhttp://www.jstor.org/stable/40607569y

(‹ = compulsory.)

Assignment

Briefly expound and critically assess what you take to be the best argument for:

(a) accepting the Barcan Formula and the Converse Barcan Formula (500 words)

(b) rejecting the Barcan Formula and the Converse Barcan Formula (500 words)

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127 Exercises: Task D HT18

Task D

The contingent a priori.

Reading

‹ LfP 10.4

‹ Davies and Humberstone, Two Notions of Necessity, Philosophical Studies 38 (1980),1–30.

xhttp://www.jstor.org/stable/4319391y

Evans, Reference and Contingency, The Monist 62 (1979), 161–89.

xhttp://www.jstor.org/stable/27902586y

(‹ = compulsory.)

Assignment

Consider the following sentence: (σ) @xp@FxÑ Fxq

(a) Specify an idiomatic English sentence σEng that translates σ using the following dictio-nary:

F : . . . is valuable.

(b) Show that:

(a) (2D σ

(b) *2D 2σ

(c) (2D F@σ

(c) Assess what support, if any, these results give to the following claims (1000 words, total)

(a) What σEng says is knowable a priori.

(b) What σEng says is only contingently the case.

(c) The truth of an utterance of σEng does not make any substantive demand of theworld.

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127 Exercises: Task E HT18

Task E

Conditionals

Reading

‹ LfP 8.7

‹ Lewis, Counterfactuals (Blackwell, 1973), 3.4, ‘Stalnaker’s theory’.

‹ Stalnaker, Inquiry (Cambridge University Press, 1984), ch. 7 , ‘Conditional proposi-tions’.

(‹ = compulsory.)

Assignment

Discuss whether we should accept the following: (500 words each)

(a) If Γ ( C then tA� B : B P Γu ( A� C.

(b) ( pA� Cq _ pA� „Cq.

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