Identity. We often use the word identical to simply mean looks the same. For instance:

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1 Identity 1. Introduction: In this lesson, we are going to learn how to express statements about identity; or, namely, we re going to learn how to say that two things are identical. We often use the word identical to simply mean looks the same. For instance: Ashley and Mary Kate are identical twins. Your iphone is identical to mine. The look on your two faces is identical right now. That s not how WE are going to use the term identical. In philosophy, identical means is one and the same object. For instance: Mark Twain is identical to Samuel Clemens. The capitol city of Virginia is identical to Richmond. The inventor of the bifocals is identical to Benjamin Franklin. Here, when we say identical, we mean, e.g., that Mark Twain is one and the same individual as Samuel Clemens. To differentiate these two uses, philosophers often say that Ashley and Mary Kate (two twins) are QUALITATIVELY identical, while Mark Twain and Samuel Clemens are NUMERICALLY identical. We re going to use a new symbol to express numerical identity, the equals sign =. Note that = is NOT an operator; rather it is a predicate. For, while we will express x is identical to y as x=y, we might just as easily have chosen I to express the predicate is identical to and have written Ixy instead. But, the predicate is so common that logicians decided to give it its own symbol. Here are some translations using this new symbol: (1) x is identical to y x=y (2) x is not identical to y (x=y) alternatively: x y 2. Except Statements: Identity is especially useful in translating except statements. Here are some examples: (3) Everyone except Dave was at the show. Sa (ꓯx)(x d Sx) Literally: Dave was not at the show, and, for all x, if that x is a person who is not Dave, then they were at the show. Note: Of course we don t usually mean that EVERYONE except Dave was at the show. That means over 7 billion people attended! But, typically, in conversation it is assumed that our 1

2 statement refers to a restricted domain. So, in this class it will be permissible to assume that the domain of the ꓯ is restricted to the relevant objects whenever it is implied by the context of the statement. Notice that I wrote Sa at the front of the wff. If I had only written (ꓯx)(x d Sx) then this could be true EVEN IF DAVE WAS ALSO AT THE SHOW. But, when we make except statements, we mean to EXCLUDE the exception. For instance, if I dismissed class by saying, Everyone except Alex may go, it would be quite cheeky of Alex to get up and leave as well, claiming that Everyone except Alex may go does not entail that Alex may NOT go. Of course it does. So, in this class, we will translate except phrases as conjunctions, where one of the conjuncts EXCLUDES the exception. NOTE THAT THIS IS IN DISAGREEMENT WITH YOUR TEXTBOOK (though it is in agreement with most other textbooks). By the way, Everyone except Alex may go gets translated: Ga (ꓯx)(x a Gx) Note: This goes for other than and but phrases as well. For instance, Everyone other than Alex may go and Everyone but Alex may go both get translated in the same way as the except statement above. Let s do another: (4) If every athlete except Betty was disqualified, then Betty won. (ꓯx){[(Ax x b) Dx] Db} Wb Literally: If all of the x s who are athletes and aren t Betty were disqualified AND if Betty herself wasn t also disqualified then Betty won. (Note also that the entire antecedent of this conditional is an except statement.) 3. Only Statements: Only statements are like inside-out except statements. For instance, if I said, Only Edward slept, it means the same thing as Everyone except Edward FAILED to sleep. (5) Only Edward slept. Se (ꓯx)(Sx x=e) (6) Everyone except Edward failed to sleep. Se (ꓯx)(x e Sx) As it turns out, (5) and (6) are logically equivalent. This should look vaguely familiar. Remember contraposition (contra.)? P Q Q P (S65) 2

3 Let s do another: (7) The only character who could open the golden snitch was Harry. [C, O, h] (Ch Oh) (ꓯx)[(Cx Ox) x=h] Literally: Harry was a character and Harry could open the golden snitch, AND, for all x, if x was a character who could open the snitch, then that x was Harry. 4. Superlative Adjectives: These are adjectives with most or -est. Translated as: (8) Jupiter is the largest planet. Pj (ꓯx)[(Px x j) Ljx] Translate Lxy as x is larger than y. Literally, the symbolized statement reads: Jupiter is a planet, and, for all x, if x is a planet that is not Jupiter, then Jupiter is larger than it. 5. Numerical Statements: Many statements refer to numbers. Let s translate some: (9) There is at least one president. (ꓱx)Px (10) There is at most one president. (ꓯx)(ꓯy)[(Px Py) x=y] Literally: For all x, and for all y, if x is president and y is president, then x is y. (11) There is exactly one president. (ꓱx)[Px (ꓯy)(Py y=x) Literally: There exists an x that is president, and for all y, if y is president, then y is x. (12) There are at least two bedrooms. (ꓱx)(ꓱy)[(Bx By) x y] Literally: There exists at least one x that is a bedroom and at least one y that is a bedroom, and x is not the same thing as y. (13) There are at most two bedrooms. (ꓯx)(ꓯy)(ꓯz){[(Bx By) Bz] [(x=y x=z) y=z]} Literally: For all x, y, and z, if x, y, and z are all bedrooms, then either x and y are the same bedroom, or x and z are, or y and z are (i.e., two of the 3 are really just 1 thing). (14) There are exactly two bedrooms. (ꓱx)(ꓱy){[(Bx By) x y] (ꓯz)[Bz (z=x z=y)]} Literally: There exists at least one x and one y that are bedrooms, and x is not y, AND, for all z, if z is a bedroom them z is either the same bedroom as x, or the same one as y. 3

4 Derivation Rules for Identity As with the operators that we ve learned, the identity predicate also has both an elimination (=E) and an introduction (=I) rule. 1. Identity Elimination (=E): The following inference is intuitively true: Superman can fly. Superman is Clark Kent. Therefore, Clark Kent can fly. We can write this as the following sequent: S217: Fs, s=c Ⱶ Fc (note that your textbook uses different letters) Basically, =E allows us to replace one proper name with another whenever we know that two proper names refer to the same individual. So, here is the derivation: 1 (1) Fs A 2 (2) s=c A 1, 2 (3) Fc 1, 2, =E Here is an instance of using =E for multi-place predicates: The Joker attacked Superman. Superman is Clark Kent. Therefore, the Joker attacked Clark Kent. Here s the sequent: S218: Ajs, s=c Ⱶ Ajc (again, your textbook uses different letters) 1 (1) Ajs A 2 (2) s=c A 1, 2 (3) Ajc 1, 2, =E Since we know from (2) that s=c, =E allows us to take any wff where s appears and rewrite that wff with c replacing s. The new line rests on the original wff as well as the line which tells us that s=c. Note #1: Note that you are NOT required to replace every occurrence of the proper name with its new name. For example, the following inference is valid: Superman can fly, and the Joker attacked Superman. Superman is Clark Kent. Therefore, Clark Kent can fly, and the Joker attacked Superman. S219: Fs Ajs, s=c Ⱶ Fc Ajs (again, I used different letters than the textbook) 4

5 1 (1) Fs Ajs A 2 (2) s=c A 1, 2 (3) Fc Ajs 1, 2, =E Notice that only the first s was replaced with a c above. Replacing both is not required. Note #2: Note that sometimes you will use =E to replace a variable with another variable. There is a restriction on this: (a) The original variable must be free AND (b) It must STILL be free after it has been replaced. For instance, the following is NOT acceptable: 1 (1) (ꓱy)Bxy A 2 (2) x=y A 1, 2 (3) (ꓱy)Byy 1, 2, =E NO! Here, we attempted to replace x in line (1) with y. Criterion (a) IS met: x is free in the original formula in line (1). So far, so good. However, criterion (b) is NOT met: y is NOT free in (1), so the variable BECOMES bound in line (3) once we make the replacement. This is NOT permitted. [Clearly it does not follow from some x betrays someone in (1) that someone betrays him/herself in line (3).] 2. Identity Introduction (=I): The following are tautologies: Mark Twain is Mark Twain. Superman is Superman. Benjamin Franklin is Benjamin Franklin. Remember that a tautology is a statement that cannot possibly be false. As such it is a logical truth, which you are permitted to introduce at any point in a derivation. Like this: 1 (1) blah blah A 2 (2) blah blah A - (3) a=a =I Notice that line (3) rests on no other statements it s a tautology! and also that, to the right, it involves no other lines; only identity-introduction. We can use =I in the following ways: a=b Ⱶ b=a 1 (1) a=b A - (2) a=a =I 1 (3) b=a 1, 2, =E 5

6 Note that just plain swapping a and b is not permitted in one line. b=a IS entailed by a=b but it s not a primitive rule that we can just swap their places. Here s another easy sequent: a=a (b=b Sc) Ⱶ Sc 1 (1) a=a (b=b Sc) A - (2) a=a =I 1 (3) b=b Sc 1, 2, E - (4) b=b =I 1 (5) Sc 3, 4, E 3. Practice Derivations: So, when would we use =E or =I? Let s do some examples. All authors are literate. Mark Twain is an author. Samuel Clemens is Mark Twain. Therefore, Samuel Clemens is literate. (ꓯx)(Ax Lx), Am, s=m Ⱶ Ls 1 (1) (ꓯx)(Ax Lx) A 2 (2) Am A 3 (3) s=m A??? 1, 2, 3 (n) Ls? What to do? Well, we can perform ꓯE on line (1), and since we know that m is in the domain, we can replace the x variables with m proper names: 1 (1) (ꓯx)(Ax Lx) A 2 (2) Am A 3 (3) s=m A 1 (4) Am Lm 1, ꓯE 1, 2 (5) Lm 2, 4, E - (6) s=s =I 3 (7) m=s 3, 6 1, 2, 3 (8) Ls 5, 7, =E Note: Why didn t we just go straight from (5) to (8)? Well, technically, the rules only permit you to replace m with s if there is a statement m=s where the original name is the FIRST name, and the new name is the SECOND name in the identity statement. But, since 6

7 we now know that a=b entails b=a, let s forget this technicality from now on. We can treat it as a sequent introduction variant of =E; just be aware that, strictly speaking, to jump straight from (5) to (8) IS a sequent introduction and not straightforward use of =E. Another: Donald is president. Becky is not president. Therefore, Donald is not Becky. Pd, Pb Ⱶ d b 1 (1) Pd A 2 (2) Pb A??? 1, 2 (n) d b? We should approach this by assuming the opposite of the conclusion for reductio. Note that d b is really just (d=b), so the opposite of the conclusion is just d=b. 1 (1) Pd A 2 (2) Pb A 3 (3) d=b Ass. (Red.) 1, 3 (4) Pb 1, 3, =E 1, 2, 3 (5) Pb Pb 2, 4, I 1, 2 (6) d b 3, 5, I One More: The Roman general who defeated Pompey conquered Gaul. Julius Caesar is a Roman general, and he defeated Pompey. Therefore, Julius Caesar conquered Gaul. 1 (1) (ꓱx)({[(Rx Dxp) (ꓯy)[(Ry Dyp) y=x]} Cxg) A 2 (2) Rj Djp A??? 1, 2 (n) Cjg? Because the first claim is that THE Roman general who, the implication is that there is only ONE such person. That is why we added the (ꓯy)[(Ry Dyp) y=x] claim. Now, clearly we re going to have to get rid of the ꓱ in (1), so an ꓱE is in order. Let s start that, and a series of E s to dig out the universally quantified wff: 7

8 1 (1) (ꓱx)({[(Rx Dxp) (ꓯy)[(Ry Dyp) y=x]} Cxg) A 2 (2) Rj Djp A 3 (3) {[(Rx Dxp) (ꓯy)[(Ry Dyp) y=x]} Cxg Ass. (ꓱE) 3 (4) (Rx Dxp) (ꓯy)[(Ry Dyp) y=x] 3, E 3 (5) (ꓯy)[(Ry Dyp) y=x] 4, E??? 1, 2 (n) Cjg? Now we can perform ꓯE, replacing y with j since we know that j is in the domain: 1 (1) (ꓱx)({[(Rx Dxp) (ꓯy)[(Ry Dyp) y=x]} Cxg) A 2 (2) Rj Djp A 3 (3) {[(Rx Dxp) (ꓯy)[(Ry Dyp) y=x]} Cxg Ass. (ꓱE) 3 (4) (Rx Dxp) (ꓯy)[(Ry Dyp) y=x] 3, E 3 (5) (ꓯy)[(Ry Dyp) y=x] 4, E 3 (6) (Rj Djp) j=x 5, ꓯE 2, 3 (7) j=x 2, 6, E 3 (8) Cxg 3, E 2, 3 (9) Cjg 7, 8, =E 2 (10) ({[(Rx Dxp) (ꓯy)[(Ry Dyp) y=x]} Cxg) Cjg 3, 9, I 1, 2 (11) Cjg 1, 10, ꓱE We did it! 8