forall x: Calgary. Solutions to Selected Exercises

1 ForallxCALGARYS olutions toSelected ExercisesP. D. MagnusTim Buttonwith additions byJ. Robert LoftisRobert Truemanremixed and revised byAaron Thomas-BolducRichard ZachFall 2021 iiThis booklet is based on the Solutions bookletforallx:Cambridge, byTim ButtonUniversity of Cambridgeused under a CC BY license, which is based in turn onforallx, MagnusUniversity at Albany, State University of New Yorkused under a CC BY license,which was remixed & expanded byAaron Thomas-Bolduc & Richard ZachUniversity of CalgaryIt includes additional material fromforallxby Magnus, usedunder a CC BY license, and fromforallx:Lorain County Remix,by Cathal Woods and J. Robert Loftis, and fromA Modal Logic Primerby Robert Trueman, used with work is licensed under a Creative Commons Attribution li-cense.

2 You are free to copy and redistribute the material in any mediumor format, and remix, transform, and build upon the material for anypurpose, even commercially, under the following terms: You must give appropriate credit, provide a link to the license,and indicate if changes were made. You may do so in any rea-sonable manner, but not in any way that suggests the licensorendorses you or your use. You may not apply legal terms or technological measures thatlegally restrict others from doing anything the license LATEX source for this book is available on GitHub. This version isrevision aa59cc9 (2021-06-18).The preparation of this textbook was made possible by a grant fromthe Taylor Institute for Teaching and arguments23 Other logical notions45 Connectives86 Sentences of TFL1511 Complete truth tables1712 Semantic concepts2513 Truth table shortcuts3214 Partial truth tables3516 Basic rules for TFL4318 Additional rules for TFL5219 Proof-theoretic concepts5620 Derived rules6521 Soundness and completeness6723 Sentences with one quantifier6924 Multiple generality7325 Identity7826 Sentences of FOL7927 Definite descriptions8030 Truth in FOL8432 Using Interpretations8734 Basic rules for FOL8936 Conversion of quantifiers10837 Rules for identity11438 Derived rules12541 Natural deduction for

3 ML12842 Semantics for ML14143 Normal forms144iiiCHAPTER 1 ArgumentsHighlight the phrase which expresses the conclusion of each of thesearguments:1. It is sunny. So I should take my It must have been sunny. I did wear my sunglasses, after No one but you has had their hands in the cookie-jar. And thescene of the crime is littered with cookie-crumbs. You re the cul-prit!4. Miss Scarlett and Professor Plum were in the study at the timeof the murder. And Reverend Green had the candlestick in theballroom, and we know that there is no blood on his hands. HenceColonel Mustard did it in the kitchen with the lead-piping. Recall,after all, that the gun had not been of the following arguments is valid?

4 Which is invalid?1. Socrates is a All men are carrots. Socrates is a Abe Lincoln was either born in Illinois or he was once Abe Lincoln was never president. Abe Lincoln was born in If I pull the trigger, Abe Lincoln will I do not pull the trigger. Abe Lincoln will not Lincoln might die for some other reason: someone else mightpull the trigger; he might die of old Abe Lincoln was either from France or from Abe Lincoln was not from Luxemborg. Abe Lincoln was from If the world were to end today, then I would not need to get uptomorrow I will need to get up tomorrow morning. The world will not end 2. VALID ARGUMENTS31. Joe is now 19 years Joe is now 87 years old.

5 Bob is now 20 years argument is valid if and only if it is impossible for all thepremises to be true and the conclusion false. It is impossible forall the premises to be true; so it is certainly impossible that thepremises are all true and the conclusion is there be:1. A valid argument that has one false premise and one truepremise? : the first argument, A valid argument that has only false premises? : Socrates is a frog, all frogs are excellent pianists, there-fore Socrates is an excellent A valid argument with only false premises and a false conclusion? same example will An invalid argument that can be made valid by the addition of anew premise? of examples, but let me offer a more general canalwaysmake an invalid argument valid, by adding a con-tradiction into the premises.

6 For an argument is valid if and onlyif it is impossible for all the premises to be true and the conclu-sion false. If the premises are contradictory, then it is impossiblefor them all to be true (and the conclusion false).5. A valid argument that can be made invalid by the addition of anew premise? argument is valid if and only if it is impossible for all thepremises to be true and the conclusion false. Adding anotherpremise will only make it harder for the premises all to be each case: if so, give an example; if not, explain why 3 Other each of the following: Is it necessarily true, necessarily false, orcontingent?1. Caesar crossed the Someone once crossed the No one has ever crossed the If Caesar crossed the Rubicon, then someone Even though Caesar crossed the Rubicon, no one has ever crossedthe false6.

7 If anyone has ever crossed the Rubicon, it was Caesar. each of the following: Is it a necessary truth, a necessary false-hood, or contingent?1. Elephants dissolve in Wood is a light, durable substance useful for building If wood were a good building material, it would be useful forbuilding I live in a three story building that is two stories If gerbils were mammals they would nurse their of the following pairs of sentences are necessarily equivalent?1. Elephants dissolve in you put an elephant in water, it will 3. OTHER LOGICAL NOTIONS52. All mammals dissolve in you put an elephant in water, it will George Bush was the 43rd Obama is the 44th Barack Obama is the 44th Obama was president immediately after the 43rd Elephants dissolve in mammals dissolve in of the following pairs of sentences are necessarily equivalent?

8 1. Thelonious Monk played Coltrane played tenor Thelonious Monk played gigs with John Coltrane played gigs with Thelonious All professional piano players have big player Bud Powell had big Bud Powell suffered from severe mental piano players suffer from severe mental John Coltrane was deeply Coltrane viewed music as an expression of the following sentences:G1 There are at least four giraffes at the wild animal There are exactly seven gorillas at the wild animal There are not more than two Martians at the wild animal Every giraffe at the wild animal park is a consider each of the following collections of sentences. Whichare jointly possible? Which are jointly impossible?

9 1. Sentences G2, G3, and G4 Jointly possible2. Sentences G1, G3, and G4 Jointly impossible3. Sentences G1, G2, and G4 Jointly possible4. Sentences G1, G2, and G3 Jointly the following All people are 3. OTHER LOGICAL NOTIONS6M2 Socrates is a Socrates will never Socrates is combinations of sentences are jointly possible? Mark each pos-sible or impossible. 1. Sentences M1, M2, and M32. Sentences M2, M3, and M43. Sentences M2 and M34. Sentences M1 and M45. Sentences M1, M2, M3, and of the following is possible? If it is possible, give an it is not possible, explain A valid argument that has one false premise and one true premiseYes: All whales are mammals (true).

10 All mammals are plants(false). So all whales are plants. 2. A valid argument that has a false conclusionYes. (See example from previous exercise .)3. A valid argument, the conclusion of which is a necessary false-hoodYes: 1+1=3. So1+2=4. 4. An invalid argument, the conclusion of which is a necessary truthNo. If the conclusion is necessarily true, then there is no way tomake it false, and hence no way to make it false whilst making allthe premises A necessary truth that is contingentNo. If a sentence is a necessary truth, it cannot possibly be false,but a contingent sentence can be Two necessarily equivalent sentences, both of which are necessarytruthsYes: 4 is even , 4 is divisible by 2.