12A: Problem Set 13 Due date: Friday May 1, 12pm. Upload completed problem set to bCourses in PDF format. Reminder: late problem sets do not receive credit. 1. For each of the following arguments, provide an fol Fitch proof which shows the argument to be valid. Don’t forget to number your lines and cite the rules of inference as appropriate. (a) ` (¬(a = a) (a = b)) (b) 9x8zHxz ` 8x9zHzx (c) (Rcd 8xRxx) ` ¬9y¬(Rcd Ryy) (d) (9xF x _ 9xGb) ` 9x(F x _ Gb) (This one is a bit tricky. -2 if you get it wrong, but +5 bonus if you get it right.) 2. Specify a model where 8x9zHzx is true but 9x8zHxz is false. 1 Fitch-style natural deduction Handout 6, Philosophy 12A, Berkeley Contents 1 Introducing Fitch-style proof 1 2 Proof rules for the connectives 2.1 Conjunction Introduction and Conjunction Elimination . 2.2 Disjunction Introduction and Disjunction Elimination . . 2.3 Negation Introduction and Negation Elimination . . . . . 2.4 Conditional Introduction and Conditional Elimination . . 2.5 Biconditional Introduction and Biconditional Elimination 2.6 Reiteration . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 5 6 8 9 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Definitions refresher: quantifiers, scope, binding, substitution instances 11 4 Proof rules for the quantifiers 4.1 Existential Introduction . . . 4.2 Universal Elimination . . . . 4.3 Universal Introduction . . . . 4.4 Existential Elimination . . . . . . . . 12 12 12 13 14 5 Rules for identity 5.1 Identity Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Identity Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 15 15 6 Some closing definitions 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introducing Fitch-style proof Now we want to define a formal notion of proof. A formal proof is a syntactic object. Think of it as made of symbols. Obviously, not just any collection of symbols from fol counts as a proof. So we need some rules that tell us what exactly counts as a proof in fol. That’s what a proof system, such as the system described below, gives us. There are various perfectly good ways of defining a formal notion of proof for fol—that is, there are various proof systems for fol. What do they all have in common, which makes them all proof systems for fol? Here’s the answer: The central thing any proof system for fol must do is supply a notion of proof such that a proof exists for an argument if and only if the argument is valid in fol. (Recall that an argument is valid in fol just in case any model where the premises are all true is a model where the conclusion is true.) 1 The proof system introduced below is called a natural deduction system, because it mimics (in a highly idealized way) the structure of (good) argumentation in ordinary language. This particular style of natural deduction is called Fitch-style. Proofs constructed in this style are called Fitch proofs or Fitch diagrams. We are first going to restrict our attention to Fitch proofs in pl. Once we get the hang of proof in pl, it will be straightforward to extend our proof system to fol. Here is an example of a Fitch proof: 1 2 ((A ^ B) C) A Premise Assumption 3 B Assumption 4 (A ^ B) ^I 2,3 5 C 6 7 (B (A E 1,4 C) (B I 3-5 C)) I 2-6 A Fitch-proof starts with a vertical list of zero or more premises arranged to the immediate right of a vertical line, called a scope line. In the above example we have just one premise. A short horizontal line is drawn between the premises of a Fitch proof and the lines beneath them; this marks o↵ where the reasoning from the premises begins. The goal is to build a proof step-by-step which ends with the conclusion of the argument we are trying to prove. Each step is justified by rule of inference (these are explained below), or else serves as a temporary assumption. A new scope line within the original scope line appears whenever a provisional assumption is made, with a short horizontal line marking o↵ the assumption. The new scope line corresponds to a subproof environment. Certain proof rules allow you to discharge the assumption of a subproof environment and return to the previous scope line. ( I does this on lines 6 and 7 of the example above—this is explained more below.) Each line of the proof must be explicitly permitted by the proof system rules, as described below. As the example above illustrates, each line of a Fitch proof requires some annotation which indicates the justification for that line. Each time you extend your proof by an additional line, you need to indicate the justification for that extension, by citing the rule you used to together with the lines that allowed you to apply the rule. The point of everything below is to explain how these rules work, and how you can use them to build proofs. 2 Proof rules for the connectives We introduce a standard set of Fitch-style natural deduction rules below. In the statement of each rule, the annotated lines (towards the bottom) show how a proof is permitted to 2 continue, given what kind of things occur in the proof above it. (If there is more than one annotated line displayed for a given rule (e.g., ^E), you can derive either of the lines displayed.) Here’s a useful notion for understanding and explaining the rules. Let’s say that if appears on line i of a proof, then is available at a line j below line i if and only if every scope line that is within extends to line j. To illustrate by using our example above: the w↵ on line 1 is available on every line below it. It is along the main scope line of the proof, and the main scope line of course extends to every numbered line. (Thus every premise in any proof is available on any line beneath it.) By contrast, the w↵ on line 3 (viz., B) is available on lines 4 and 5, but not on lines 6 or 7, because there is a scope line that line 3 is within which does not extend to lines 6 or 7. Now let’s look at the rules of inference—our proof rules. Each connective comes with an introduction rule and an elimination rule. We’ll start with those. 2.1 Conjunction Introduction and Conjunction Elimination These abstract pictures describe how these rules work: ^I: ^E: i i .. . j ^E i .. . ( ^ ) ( ^ ) .. . ^E i ^I i, j Conjunction Introduction (^I) says that you can write a conjunction on a line if each conjunct is available on that line. Note: we won’t interpret this rule (or the other rules) as saying that line i literally has to be above line j in the proof. The order doesn’t matter. Let’s see a simple example of this rule in action. Suppose our premises are A and B, and our conclusion is (A ^ B). Now clearly that’s a valid argument: {A, B} ✏ (A ^ B). Is it provable in our Fitch proof system—that is, {A, B} ` (A ^ B)? Indeed it is. The proof is very simple and direct: 1 A Premise 2 B Premise 3 (A ^ B) ^I 1,2 The annotation “^I 1, 2” on line 3 supplies the justification for this line. It is a key part of the proof. It means that you derived line 3 from lines 1 and 2 using the ^I proof rule. 3 Note that we also could have used the ^I proof rule to prove (B ^ A): 1 A Premise 2 B Premise 3 (B ^ A) ^I 1,2 That’s to say, when you derive a conjunction, the order of the conjuncts is up to you—either order is permitted by the ^I rule. In your annotations, you generally have to state the rule you used and the lines you applied the rules to, but those don’t have to be written in a particular order. We could have supplied the annotation like this: 1 A Premise 2 B Premise 3 (B ^ A) 1, 2 ^I A point of clarification: in the abstract picture of how ^I works above, it look as if the two w↵s used to derive a conjunction must occur along exactly the same scope line as the conjunction derived from them. But we’re going to interpret the rule in a more flexible manner than this. Again, the intended understanding of the ^I rule above is that the w↵s on lines i and j must be available at the line the where the conjunction is derived. Observe that this more flexible understanding is already in play in our example proof from earlier. Let’s see it again: 1 2 ((A ^ B) C) A Premise Assumption 3 B Assumption 4 (A ^ B) ^I 2,3 5 C 6 7 (B (A E 1,4 C) (B I 3-5 C)) I 2-6 We haven’t explained some of the rules used here yet, but don’t worry about that for now. The relevant thing to notice is that the w↵s on line 2 and 3 are available at line 4, and hence we can form the conjunction of these and write the conjunction down on this line, citing ^I as justification. Conjunction Elimination (^E) says that from a conjunction you can derive either of the conjuncts. A simple example: 4 1 (A ^ B) Premise 2 B 1 ^E This is a Fitch proof that B follows from (A ^ B). Here is another example: 1 ((A 2 (A B) ^ C) Premise 1 ^E B) Putting our two rules together, we can prove that (A ^ B) entails (B ^ A): 1 (A ^ B) Premise 2 A 1 ^E 3 B 1 ^E 4 (B ^ A) 2,3 ^I 2.2 Disjunction Introduction and Disjunction Elimination Disjunction Introduction (_I) says that if is available at a line you can derive the disjunction of with any other w↵ at all, and you can arrange the resulting disjunction in either order. Disjunction Elimination (_E) says that if a disjunction is a available at a line, and the negation of one of the disjuncts is also available at a line, you can derive the other disjunct. _I: _E: i i ( _ ) [or: ( _ )] .. . j ¬ .. . .. . ( _ ) _I i ( _ ) _I i _E i, j A simple example of _I: 1 P Premise 2 (P _ Q) 1 _I Note that the introduced disjunct can be simple or complex: 5 1 P Premise 2 (P _ (Q ^ (¬P _ R))) 1 _I ... and you can make it appear first or second in the disjunction: 1 P Premise 2 ((Q ^ (¬P _ R)) _ P ) 1 _I Here is a simple example of _E: 1 (A _ B) Premise 2 ¬A Premise 3 B 1,2 _E Another: 1 ((A Q) _ ¬B) Premise 2 ¬(A Q) Premise 3 ¬B 1,2 _E Let’s now see all of our rules for conjunction and disjunction in action in a single proof: 1 ((P _ Q) ^ ¬Q) Premise 2 ¬Q 1, ^E 3 (P _ Q) 1, ^E 4 P 1,3 _E 5 (P ^ ¬Q) 2,4 ^I 6 ((P ^ ¬Q) _ R) 5 _I 2.3 Negation Introduction and Negation Elimination Negation Introduction (¬I) encodes the idea of reductio ad absurdum, or proof by contradiction. It allows you to derive a sentence of the form ¬ by provisionally supposing is true and deriving a contradiction within the scope of that supposition. Negation Elimination (¬E) encodes the idea that a pair of negation symbols cancel out. 6 ¬I: ¬E: i Assumption .. . i .. . ¬E i ¬ j ¬ ¬¬ .. . ¬I i-j Let’s see a simple example of ¬I in action: 1 (P ^ Q) 2 ¬P Assumption 3 P 1 ^E 4 ¬¬P Premise 2-3 ¬I We’re showing here that ¬¬P can be proved from (P ^ Q). Taking (P ^ Q) as a premise, we immediately temporary suppose ¬P . This is what happens on line 2, where a subproof environment is created. Next, within the subproof environment, we derive P from our premise using ^E. Now notice that within our subproof environment, we can find contradictory lines: line 2 has ¬P , whereas line 3 has P . This means that from our assumption of ¬P , we can derive a contradiction. And this fact means our temporary assumption must be false, given our premises. This is what the ¬I rule encodes. Once it is evident a contradiction can be derived, we can return to the earlier scope line as we do on line 4, and conclude the negation of the thing we temporarily supposed. When we leave the subproof and justify this with ¬I, we cite the lines corresponding to the whole subproof environment. Note that in this example of ¬I, one of our two contradictory sentences was actually the very assumption introduced on line 2. That is unusual, but permissible. As long as you can find two sentences along a subproof scope line such that one is the negation of the other, you can apply ¬I to conclude the negation of the thing you supposed right outside that scope line. The two sentence may both occur beneath the assumption, or one of them may be the assumption itself. The rule for ¬E is straightforward. It just lets us drop double negations. For example: 1 ¬¬(P ^ Q) Premise 2 (P ^ Q) 1, ¬E 7 2.4 Conditional Introduction and Conditional Elimination I: E: i i .. . Assumption .. . j .. . j I i-j E i,j Conditional Introduction ( I) lets you introduce a conditional by using a subproof environment. If you provisionally suppose the antecedent of a conditional and are able to derive the consequent of the conditional within the scope of this supposition, then you can “discharge” the supposition and derive the conditional immediately outside the subproof environment. Here is a simple example: (P ^ Q) 1 Premise 2 P Assumption 3 Q 1, ^E 4 (P Q) 2-3, I Another: 1 (P ^ Q) Premise 2 P Assumption 3 Q 1, ^E 4 (Q _ A) 3, _I 5 (P (Q _ A)) 2-4, I Conditional Elimination ( E) is also called Modus Ponens. If a conditional together with its antecedent are available at a line, you can derive the consequent of the conditional at that line. Example: 1 ((P _ R) (Q ^ R)) 2 (P _ R) Premise 3 (Q ^ R) 1,2 Premise E Our example from earlier provides an illustration of both rules: 8 1 ((A ^ B) 2 C) A Premise Assumption 3 B Assumption 4 (A ^ B) ^I 2,3 5 C 6 7 (B (A E 1,4 C) (B I 3-5 C)) I 2-6 On line 2, we make the supposition of A. On line 3, we make another supposition within that supposition, of B. (You can do that!) From these two suppositions, (A^B) is easily derived on line 4. E is then applied to derive C on line 5 within the sub-subproof environment. Then the innermost supposition (of B) is discharged, and the conditional (B C) is derived on line 6 with I. Finally the original supposition A is discharged on line 7, using I once again. 2.5 Biconditional Introduction and Biconditional Elimination $I: $E: i ( .. . ) j ( .. . ) ( $ ) i $I i, j ( $ ) .. . ( ) ^E i ( ) ^E i Biconditional Introduction ($I) and Biconditional Elimination ($E) are reminiscent of the corresponding conjunction rules. That is not surprising, since of course a biconditional just is equivalent to a conjunction of conditionals. 2.6 Reiteration This rule doesn’t have to do with any particular connective, but it’s useful to have. 9 Reit: Reiteration lets you repeat a w↵ occurring on a line above it, as long as that w↵ is along the same scope line or along a scope line to the left of the reiterated w↵. In particular it allows you to “move formulas inward”. i .. . Reit i Here is a simple example of a case where Reit is handy: 1 P Premise 2 ¬P Assumption 3 P 1, Reit 4 ¬¬P 2-3, ¬I Here is another: 1 ((¬P ^ W ) _ ¬R) Premise 2 P Assumption 3 ((¬P ^ W ) _ ¬R) 1, Reit 4 (P ((¬P ^ W ) _ ¬R)) 10 2-3, I 3 Definitions refresher: quantifiers, scope, binding, substitution instances To extend our Fitch proof system to fol, we need to state introduction and elimination rules for the quantifiers and for identity. Before we get to those, however, here are some definitions that will be helpful to keep in mind. Def. A quantifier is an expression of the form Q⌫, where Q is a quantifier symbol (i.e., 9 or 8) and ⌫ is a variable. Def. The scope of a quantifier is the w↵ that it attaches to. More precisely: If Q⌫ is a quantifier and is a w↵, then the scope of the occurrence of Q⌫ in the expression Q⌫ is . Examples: • The scope of 9x in 9x8y(F x • The scope of 8y in 9x8y(F x • The scope of 9x in (9x8y(F x Gy) is 8y(F x Gy) is (F x Gy). Gy). Gy) ^ F x) is 8y(F x Gy). Def. Any occurrence of the variable ⌫ in the scope of an occurrence of 9⌫ or 8⌫ is bound by that occurrence of the quantifier, as long as that occurrence of ⌫ is not also in the scope of some closer occurrence of 9⌫ or 8⌫. Examples: • In (9x8y(F x ables. Gx y) ^ F x), the quantifier 9x binds the highlighted vari- • In 9x8x(F x Gx y), the quantifier 8x binds the highlighted variables, and the quantifier 9x binds no variables. Def. A substitution instance of a closed quantified formula is the result of deleting the quantifier, then replacing every variable bound by the quantifier with the same individual constant. If we have a quantified sentence of the form Q⌫ and ↵ is a constant, then ↵/⌫ denotes the substitution instance of Q⌫ which results from deleting Q⌫ and replacing every occurrence of ⌫ in which was previously bound by Q⌫ with ↵. Examples: • F a is a substitution instance of 9xF x, 8xF x, 9yF y, and so on. • F aa and F bb are substitution instances of 9xF xx, but F ab is not. (The same constant has to replace each variable that was bound by the quantifier removed.) • 9xF xb is a substitution instance of 8y9xF xy, and also of 9z9xF xz. • 9xF x is a substitution instance of 8y9xF x—and indeed, its only substitution instance. 11 4 Proof rules for the quantifiers 4.1 Existential Introduction Existential Introduction (9I) lets you derive an existentially quantified formula from any substitution instance of that formula. The form of reasoning this rules encodes is also called existential generalization. It says that if you have established on some line of the proof, and is a substitution instance of a w↵ of the form 9⌫ , then you can derive any w↵ of that form: 9I: Where ↵/⌫ is a substitution instance of 9⌫ , i .. . 9⌫ 9I i ↵/⌫ Here are a number of di↵erent ways of applying the rule: 1 F ab Premise 2 9xF xb 9I 1 a/x 3 9xF ax 9I 1 b/x 4 9yF yb 9I 1 a/y 5 9yF ay 9I 1 b/y 6 9x9yF xy 9I 5 a/x 4.2 Universal Elimination Universal Elimination (8E) says that you can derive any substitution instance of a universally quantified formula: 8E: Where i ↵/⌫ is a substitution instance of 8⌫ , 8⌫ .. . 8E i ↵/⌫ For example, from 8xF bx, one can use 8E to derive F ba, or F bb, or F bc: 12 1 8xF bx Premise 2 F ba 8E 1 a/x 3 F bb 8E 1 b/x 4 F bc 8E 1 c/x That’s because these are all substitution instances of 8xF bx. 4.3 Universal Introduction The rule for 8I involves a subproof environment. But it is a subproof environment of a special sort. This subproof environment serves to mark the fact that a constant symbol ↵ is temporarily being used to stand for an arbitrary object. We represent this by opening the subproof environment with a numbered line that just contains that constant symbol surrounded by a box. This is a “flagging step”: we are flagging that the constant is going to be used to reason about an arbitrary thing. The constant selected must not appear in a w↵ outside the subproof environment. The flagging step is like saying “Consider an arbitrary thing, call it a.” (Or whatever constant it is that you choose.) The normal course is then derive to some w↵s containing a within the subproof context introduced by the flagging step. Since the choice of a was arbitrary, whatever conclusions were drawn could have been drawn about anything; and thus a universal claim is derivable immediately outside the subproof environment. Here is a formal way to state the rule: 8I: Where the constant ↵ does not occur outside lines i-j, i ↵ .. . j 8⌫ ⌫/↵ 8I i-j ⌫/↵ By ⌫/↵ , we mean the result of replacing any occurrences of the constant ↵ in with the variable ⌫ in such a way that the resulting sentence, when prefixed with 8⌫, yields a universally quantified sentence of which is a substitution instance. For example, if is F ab, then 8x x/a is just 8xF xb, and 8x x/b is just 8xF ax. Here is an example of the rule in action: 13 1 8x(F x Gx) Premise 2 8x(Gx Hx) Premise 3 a 4 (F a Ga) 8 E 1 a/x 5 (Ga Ha) 8 E 2 a/x 6 Fa Assumption 7 Ga E 4,6 8 Ha E 5,7 9 (F a Ha) 10 8x(F x Hx) I 6-8 8I 3-9 x/a 4.4 Existential Elimination If you have an existentially quantified formula to work with from your position in a proof, the rule of Existential Elimination lets you provisionally reason as though you have a substitution instance of that formula. The idea is then to derive something using that substitution instance that doesn’t depend on the particular constant you chose to instantiate the existential claim. Here’s how it works. Given an existential sentence, you may start a new subproof whose first line is a substitution instance of it. This line will also contain, to the right of the substitution instance, a box that flags the constant you chose to instantiate the existential sentence. You have to choose a fresh constant, not one appearing elsewhere in the proof. You can close the subproof after any line which does not contain the flagged constant, and reiterate the final w↵ of the subproof immediately outside of the subproof. On this line you will cite 9E, the line containing the existentially quantified formula, and also the whole subproof. 9E: Where the constant ↵ does not occur outside lines j-k, i 9⌫ .. . ↵/⌫ j ↵ Instantiating i .. . k 9E i, j-k 14 This rule is a bit involved, so let’s have an example: 1 9x(F x _ Ga) Premise 2 8x¬F x Premise 3 (F b _ Ga) b Instantiating 1 4 ¬F b 8E 2 5 Ga _E 3,4 6 9E 1, 3-5 Ga Remember, the constant you instantiate with has to be “fresh”: we couldn’t have used a to instantiate the existential w↵ on line 1, because it already appears in the proof (indeed, in the very existential w↵ we are instantiating). Remember also that you can’t use 9E to “export” w↵s that contain the constant you flagged. So for instance we could not have exited the subproof environment on line 5 and concluded ¬F b along the main scope line, because that w↵ contains the flagged constant. Notice that in the annotation for existential elimination, you cite the line you are instantiating, and also the whole subproof environment. 5 Rules for identity 5.1 Identity Introduction This rule is simple: any proof can be extended, along any available scope line, with (↵ = ↵), where ↵ is any constant, citing as justification: =I. We don’t need to cite any previous lines of the proof here. Each constant picks out some thing or other, and it is part of the logic of identity that each thing is identical to itself. To illustrate, here’s how we can use this rule to prove that everything is self-identical: 1 a Flagging step 2 (a = a) 3 8x(x = x) =I 8I 1-2 x/a 5.2 Identity Elimination If a = b, then if F a, we know it must be that F b. Identity Elimination captures the relevant generalization here: 15 =E: Where ↵ is formula that contains the constant ↵, and where ↵/ is a result of replacing one or more occurrences of ↵ in ↵ with the constant , i (↵ = ) [or: ( = ↵)] .. . ↵ j .. . ↵/ =E i,j The following example illustrates several di↵erent ways of applying the rule: 1 (Raa ^ F a) Premise 2 (a = b) Premise 3 (Rab ^ F a) =E 1,2 4 (Rbb ^ F a) =E 1,2 5 (Rbb ^ F b) =E 1,2 6 Some closing definitions Def. A Fitch proof is any sequence of assumptions and applications of the rules above such that the last line of the proof has only a single scope line to its left (i.e., all assumptions aside from the premises have been discharged). Def. A w↵ of L is deducible from a set of w↵s (written as ‘ ` ’) if and only if there is a Fitch proof starting with all the w↵s of as assumptions and ending with . As a special case of the above, A w↵ of L is provable (written as ‘` ’) if and only if there is a Fitch proof which has no starting assumptions and ends in . 16
Purchase answer to see full attachment