Guide to Negating Formulas
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- Rebecca Pearson
- 6 years ago
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Transcription
1 Guide to Negating Formulas
2 Hi everybody!
3 We spent a little bit of time in class talking about how to negate formulas in propositional or frst-order logic.
4 This is a really valuable skill! If you ever need to write a proof by contradiction or a proof by contrapositive, you'll need to know how to negate formulas.
5 While this might seem a bit tricky at frst, the good news is that there's a nice, mechanical way that you can negate formulas!
6 There's still a bit of art to it, but by learning a few simple rules and how to apply them, you can negate just about anything!
7 For example.
8 p (q r) Let's imagine that you want to negate this formula to the left.
9 ( p (q r)) To do so, we're going to begin by surrounding the formula in parentheses.
10 ( p (q r)) nd putting a negation symbol in front.
11 ( p (q r)) Technically speaking, this formula is the negation of the original formula, though it's hard to see exactly what this formula says.
12 ( p (q r)) Most of the time, when you need to fnd the negation of a formula, you're going to want to simplify it by pushing the negations inward.
13 ( p (q r)) The good news is that there are a number of rules we can use to do this.
14 ( p (q r)) If you remember from our lecture on propositional logic, we saw a series of rules for simplifying negations.
15 ( p (q r)) For example, we saw de Morgan's laws, which say that ( B)
16 ( B) ( p (q r)) I'm going to write this rules up at the top of the screen.
17 ( B) ( p (q r)) We haven't talked about this notation before, but the good news is that it's not too bad. Let's take a look at this.
18 ( B) Generally speaking, these rules look like this: Before fter ( p (q r))
19 ( B) rule like this means if you see Before, replace it with fter. Before fter ( p (q r))
20 ( B) So let's look at this rule on the right. ( p (q r))
21 ( B) This rule says if you see something of the form replace it with. ( p (q r))
22 ( B) ( p (q r)) Take a look at our original formula. If you look at it closely, you'll see that it matches this particular pattern.
23 ( B) ( p (q r)) That means that we can apply this rule and rewrite it.
24 ( B) p (q r).so that it looks like this.
25 ( B) p (q r) Notice that we've pushed the negation deeper into the formula.
26 ( B) p (q r) Basically, we just need to keep applying these rule templates over and over until the formula is as simple as it can get.
27 ( B) p (q r) So let's focus on this part of the formula for now.
28 ( B) p (q r) Notice that it matches this particular template.
29 ( B) p (q r) So let's apply this rule to simplify the formula!
30 ( B) p ( q r) Now we've got this, which has the negation pushed deeper into the formula! We're making progress!
31 ( B) p ( q r) So what about the other part of this formula?
32 ( B) p ( q r) Well, this is a double-negation. Intuitively, we'd expect these two negations to cancel out.
33 ( B) p ( q r) Indeed, that's what happens. We can write this down as the rule
34 ( B) p ( q r) So let's go and apply this rule!
35 ( B) p ( q r) Tada!
36 ( B) p ( q r) t this point we can't push the negations any deeper into the formula. They're directly applied to propositions, which can't be simplifed.
37 ( B) p ( q r) So we're done!
38 ( B) p ( q r) Of course, before moving on, we should be sure to check our work!
39 ( B) p ( q r) p (q r) Here's the original formula. (Think of this as the before in the before and after shots.)
40 ( B) p ( q r) p (q r) Is there some way we can check whether we negated this formula properly?
41 ( B) p ( q r) p (q r) Yes, actually! There are several.
42 ( B) p ( q r) p (q r) Let's begin with an intuitive check. What do each of these formulas actually mean?
43 ( B) p ( q r) p (q r) This formula means p is true, or both q and r are false.
44 ( B) p ( q r) p (q r) This formula means p is false and at least one of q and r is true.
45 ( B) p ( q r) p (q r) Intuitively, these statements seem to check out as opposites.
46 ( B) p ( q r) p (q r) If the frst one is true, then either p is true (so p is false), or q and r are false (so q r is false.)
47 ( B) p ( q r) p (q r) Similarly, if the second one is true, then p is true (so p is false), and at least one of q and r are true (so q r is false).
48 ( B) p ( q r) p (q r) So in that sense, intuitively, everything checks out.
49 ( B) p ( q r) p (q r) Of course, a lot of things that make intuitive sense aren't right, so perhaps we should check this in a diferent way.
50 ( B) p ( q r) p (q r) When we're dealing with propositional logic, we can always check the truth tables for these formulas and see how they compare!
51 ( B) p ( q r) p (q r) If you punch in these two formulas, here's the truth tables that you'll get back.
52 ( B) p ( q r) p (q r) Notice that, going row by row, the truth values are opposites of one another. That means that they're negations of one another!
53 ( B) p ( q r) p (q r) There's actually a diferent way to use the truth table tool to check if two formulas are negations of one another.
54 ( B) p ( q r) p (q r) Imagine that you have two formulas and B. If is a negation of B, then any time is true, B is false and vice-versa.
55 ( B) p ( q r) p (q r) That means that the formula B should always evaluate to false.
56 ( B) p ( q r) p (q r) So in our case, we can try punching in this formula into the truth table tool. (p (p ( q ( q r)) r)) ( p ( p (q (q r)) r))
57 ( B) p ( q r) p (q r) If we do, here's what we get back. (p (p ( q ( q r)) r)) ( p ( p (q (q r)) r))
58 ( B) p ( q r) p (q r) Since the formula is always false, we know that the two formulas must be negations of one another. Nifty! (p (p ( q ( q r)) r)) ( p ( p (q (q r)) r))
59 ( B) That wasn't so bad! Let's do another one.
60 ( B) (p q) ( p q) Let's try taking the negation of this formula.
61 ( B) (p q) ( p q) ctually, before we do this together, why don't you try doing it yourself frst!
62 ( B) (p q) ( p q) Take a minute or two to walk through this one on your own. See what you come up with!
63 ( B) (p q) ( p q) So did you do it? If not, you really should. It's good practice!
64 ( B) (p q) ( p q) So you did it? You've got an answer? Great! Let's work through this one together.
65 ( B) ((p q) ( p q)) s before, we begin by surrounding it in parentheses.
66 ( B) ((p q) ( p q)).and putting a negation at the front.
67 ( B) ((p q) ( p q)) Now, we're going to keep applying rules to push the negation deeper and deeper into the expression.
68 ( B) ((p q) ( p q)) Looking over the structure of this formula, we can see that it's a negation of two things NDed together.
69 ( B) ((p q) ( p q)) That means that we want to use this rule.
70 ( B) ((p q) ( p q)) Color-coding things to make it easier to see which part is which, we can now apply the template.
71 ( B) (p q) ( p q).which gives us this initial simplifcation.
72 ( B) (p q) ( p q) Now, we can keep repeating this process on each smaller part of the expression.
73 ( B) (p q) ( p q) Just because we can, let's start with this part.
74 ( B) (p q) ( p q) That matches the same rule as before!
75 ( B) (p q) ( p q) So we can color-code things.
76 ( B) (p q) ( p q).and apply the template.
77 ( B) (p q) ( p q) Progress!
78 ( B) (p q) ( p q) reasonable next step would be to look at these two subexpressions, each of which is a double-negation.
79 ( B) (p q) ( p q) Here's the template.
80 ( B) (p q) (p q).and here's the result!
81 ( B) (p q) (p q) Cool! We're almost there.
82 ( B) (p q) (p q) Our last step is to deal with this part of the expression.
83 ( B) (p q) (p q) That matches this template.
84 ( B) (p q) (p q) So we see how it matches.
85 ( B) ( p q) (p q).and apply the rule!
86 ( B) ( p q) (p q) That's as deep as the negations are going to go. We're done!
87 ( B) ( p q) (p q) Of course, we have to check our work.
88 ( B) ( p q) (p q) (p q) ( p q) Here's where we started.
89 ( B) ( p q) (p q) (p q) ( p q) Why don't you go use the truth table tool to check whether these statements actually are negations of one another?
90 ( B) Okay! So let's recap where we are right now.
91 ( B) We have these nice and handy templates available for simplifying expressions with negations in them.
92 ( B) We've talked about three ways that you can check your work.
93 ( B) nd we've gone through two examples!
94 ( B) However, the rules above don't cover all possible formulas.
95 ( B) p q r For example, how would we negate this formula?
96 ( B) p q r First, let's refresh how to negate an implication.
97 ( B) p q r ctually, before we review that one directly, why don't you review your notes and see how you negate an implication?
98 ( B) p q r What goes below this line? ( B)??
99 ( B) p q r Here's the answer: ( B) B
100 ( B) ( B) B p q r Remember, the only way for an implication to be false is if the antecedent () is true and the consequent (B) is false!
101 ( B) ( B) B p q r One of the most common mistakes we see people make when negating formulas is to get this wrong, so commit this one to memory!
102 ( B) ( B) B p q r Okay! Now that we have our rule, let's see how to negate this formula.
103 ( B) ( B) B (p q r) s before, we parenthesize it.
104 ( B) ( B) B (p q r).and put a negation out front.
105 ( B) ( B) B (p q r) Now, we need to see what template to apply.
106 ( B) ( B) B (p q r) Here, we're going to run into an operator precedence issue, which we haven't seen yet.
107 ( B) ( B) B (p q r) Do we interpret the formula like this? ((p q) r) (?)
108 ( B) ( B) B (p q r) Or like this? ((p q) r) (?) (p (q r)) (?)
109 ( B) ( B) B (p q r) Before we tell you, why don't you look over your notes and see what you fnd? ((p q) r) (?) (p (q r)) (?)
110 ( B) ( B) B (p q r) Okay. you've got a guess? ((p q) r) (?) (p (q r)) (?)
111 ( B) ( B) B (p q r) The top one has the right operator precedence. ((p q) r) (?) (p (q r)) (?)
112 ( B) ( B) B ((p q) r) ((p q) r) (?) (p (q r)) (?) Just to make it easier for us to remember that, let's go and add those parentheses into our top-level formula.
113 ( B) ( B) B ((p q) r) So now we can look at our templates and see which one applies.
114 ( B) ( B) B ((p q) r) Here, we're negating an implication, so this rule applies.
115 ( B) ( B) B ((p q) r) So we fnd the correspondence.
116 ( B) ( B) B (p q) r.and apply the template!
117 ( B) ( B) B (p q) r That gives us this expression.
118 ( B) ( B) B p q r We can actually drop the parentheses here, since everything is getting NDed together and ND is associative.
119 ( B) ( B) B p q r t this point, we can't simplify things any more, so we're done!
120 ( B) ( B) B p q r s before, we should check our work.
121 ( B) ( B) B p q r p q r Here's the original formula.
122 ( B) ( B) B (p q r) (p q r) If we plug this expression into the truth table tool, we can see that these formulas are indeed negations of one another. (Try it!)
123 ( B) ( B) B p q r p q r But, intuitively, why is that?
124 ( B) ( B) B p q r Notice that the original formula p q r (the one on the bottom) says p and q implies r.
125 ( B) ( B) B p q r That statement is going to be p q r true unless the antecedent (p and q) is true and the consequent (r) is false.
126 ( B) ( B) B p q r So take a look at what our p q r negation (the top formula) says: it says p and q are true, but r is false. It checks out!
127 ( B) ( B) B p q r Before we do another example, p q r one detail that's worth noting is that we didn't even touch the antecedent.
128 ( B) ( B) B p q r Notice that it's the same in both p q r the original formula and the negation.
129 ( B) ( B) B p q r When you're negating statements, p q r you don't necessarily negate every part of them.
130 ( B) ( B) B p q r This is why these rules matter: p q r they show you what parts get negated and what parts stay the same.
131 ( B) ( B) B p q r It's important to keep that in mind one of the more common p q r mistakes we see people make is negating way more than they should.
132 ( B) ( B) B p q r p q r So, when in doubt, just keep applying the templates!
133 ( B) ( B) B With that said, let's do another example.
134 ( B) ( B) B p q r How might we negate this statement?
135 ( B) ( B) B (p q r) You know the drill! frst we parenthesize.
136 ( B) ( B) B (p q r).then we stick a negation out front.
137 ( B) ( B) B (p q r) Now we just need to apply templates.
138 ( B) ( B) B (p q r) It should be clear that we need to use this template for implications, but how do we apply it?
139 ( B) ( B) B (p q r) Specifcally, which of these interpretations of the expression are correct? ((p q) r) (?) (p (q r)) (?)
140 ( B) ( B) B (p q r) ((p q) r) (?) (p (q r)) (?) Take a minute to review your notes and make a guess. Seriously, try it! It's a good exercise.
141 ( B) ( B) B (p q r) So you have a guess? If not, go make one before going on. ((p q) r) (?) (p (q r)) (?)
142 ( B) ( B) B (p q r) So this is the correct interpretation. The operator is right-associative. ((p q) r) (?) (p (q r)) (?)
143 ( B) ( B) B (p (q r)) Let's go put those parentheses in up here. ((p q) r) (?) (p (q r)) (?)
144 ( B) ( B) B (p (q r)) Okay! Now let's start applying templates.
145 ( B) ( B) B (p (q r)) Color-coding the antecedent and consequent lets us see the structure.
146 ( B) ( B) B p (q r).and makes it clear how we push the negation inward.
147 ( B) ( B) B p (q r) Progress!
148 ( B) ( B) B p (q r) Now, we can apply this template another time.
149 ( B) ( B) B p (q r) Before.
150 ( B) ( B) B p q r.and after!
151 ( B) ( B) B p q r So we're done!
152 ( B) ( B) B p q r Here's the original formula and the ultimate negation. You can p q r (and should!) go check whether this is right using the truth table tool.
153 ( B) ( B) B p q r But hold on a second. p q r does this negation seem familiar?
154 ( B) ( B) B p q r Hopefully it does we saw p q r earlier that it's the negation of p q r!
155 ( B) ( B) B p q r p q r This is exciting, because it means we just discovered something new!
156 ( B) ( B) B p q r Since these two formulas have the p q r same negation, they must be equivalent to one another! p q r p q r
157 ( B) ( B) B p q r This particular identity shows up p q r in a lot of places, actually. Curious? Come ask us about it! p q r p q r
158 ( B) ( B) B Okay, so now we have a rule for negating implications. Great!
159 ( B) ( B) B Before we move on, there's something important I'd like to address.
160 ( B) ( B) B If you'll remember, in class we saw this equivalence: B
161 ( B) ( B) B We can treat this as a new template for negating ND: B
162 ( B) ( B) B Let's add it to the list. B
163 ( B) ( B) B B We now have two diferent rules for negating NDs.
164 ( B) ( B) B B () ( B) We can use the truth table tool to check that both of these rules are correct. Just punch in the formulas to the left and see what you get!
165 ( B) ( B) B B But what does it mean to have two diferent rules lying around for negating ND?
166 ( B) ( B) B B This means that if we want to negate an ND, we have two diferent ways of doing it.
167 ( B) ( B) B B This template uses de Morgan's laws. It tends to be more useful when working in propositional logic.
168 ( B) ( B) B B This template is based on the negation rule for implications. It's more useful in frst-order logic.
169 ( B) ( B) B B Now that we have several diferent negation rules available, we can talk about a key property of negations we've skipped up to this point.
170 ( B) ( B) B B p q r Let's look at how to negate this formula.
171 ( B) ( B) B B p q r Before we go over this one together, take a few minutes to negate this one on your own.
172 ( B) ( B) B B p q r Really, it's a good exercise. Don't continue onward until you've done that.
173 ( B) ( B) B B p q r So you negated it? Good! Great! Let's do it together.
174 ( B) ( B) B B (p q r) The frst few steps should be pretty routine by this point - parenthesize and negate!
175 ( B) ( B) B B (p q r) Now, we have a lot of diferent options here. We have two diferent templates for negating ND.
176 ( B) ( B) B B (p q r).and at the same time, there are ((p q) r) (p (q r)) two equally correct ways of parenthesizing this expression.
177 ( B) ( B) B B (p q r) Depending on which templates you use, and how you group ((p q) r) (p (q r)) things, you may end up with totally diferent answers!
178 ( B) ( B) B B (p q r) ((p q) r) (p (q r)) Let's see what they look like.
179 ( B) ( B) B B (p (q r)) For starters, let's parenthesize things like this.
180 ( B) ( B) B B (p (q r)) Let's use this frst template. We can be old-school.
181 ( B) ( B) B B (p (q r)) Here's the before picture.
182 ( B) ( B) B B p (q r). and the after picture.
183 ( B) ( B) B B p (q r) So now we have to simplify the second part of this expression.
184 ( B) ( B) B B p (q r) Let's use this template.
185 ( B) ( B) B B p (q r) Before.
186 ( B) ( B) B B p (q r). and after!
187 ( B) ( B) B B p (q r) So this is one possible formula we could arrive at as the negation of the original formula.
188 ( B) ( B) B B p (q r) This isn't all that easy to read. Let's try negating it a diferent way.
189 ( B) ( B) B B p (q r) (p q r) Resetting back to where we began.
190 ( B) ( B) B B p (q r) ((p q) r) Let's parenthesize it like this.
191 ( B) ( B) B B p (q r) ((p q) r) Now, let's use this template.
192 ( B) ( B) B B p (q r) ((p q) r) That groups things like this.
193 ( B) ( B) B B p (q r) p q r.giving us this result!
194 ( B) ( B) B B p (q r) p q r nd we're done! That was pretty fast. nd we now have a very diferent formula than the frst one!
195 ( B) ( B) B B p (q r) p q r Of course, this isn't the only way you could do this.
196 ( B) ( B) B B p (q r) p q r (p q r) Let's see another one.
197 ( B) ( B) B B p (q r) p q r (p (q r)) We'll group things this way.
198 ( B) ( B) B B p (q r) p q r (p (q r)) Now, we'll apply this template.
199 ( B) ( B) B B p (q r) p q r (p (q r)) This groups things in a pretty diferent way from before.
200 ( B) ( B) B B p (q r) p q r p (q r).so we end up with something pretty diferent than before!
201 ( B) ( B) B B p (q r) p q r p (q r) We still need to simplify that second part.
202 ( B) ( B) B B p (q r) p q r p (q r) Let's apply this template again.
203 ( B) ( B) B B p (q r) p q r p (q r) That takes us from here.
204 ( B) ( B) B B p (q r) p q r p q r.to here.
205 ( B) ( B) B B p (q r) p q r p q r nd we're done! nother totally valid negation.
206 ( B) ( B) B B p (q r) p q r p q r Let's do one more.
207 ( B) ( B) B B p (q r) p q r p q r (p q r) We'll start here.
208 ( B) ( B) B B p (q r) p q r p q r ((p q) r). and group things like this.
209 ( B) ( B) B B p (q r) p q r p q r ((p q) r) Let's use the frst template, for a change.
210 ( B) ( B) B B p (q r) p q r p q r ((p q) r) Here's where we start.
211 ( B) ( B) B B p (q r) p q r p q r (p q) r.and where we end.
212 ( B) ( B) B B p (q r) p q r p q r (p q) r Now, we need to fx up that frst part.
213 ( B) ( B) B B p (q r) p q r p q r (p q) r We'll use this template again.
214 ( B) ( B) B B p (q r) p q r p q r (p q) r.taking us from here.
215 ( B) ( B) B B p (q r) p q r p q r p q r.to here.
216 ( B) ( B) B B p (q r) p q r p q r p q r nd we're done! totally diferent, totally valid negation.
217 ( B) ( B) B B p (q r) p q r p q r p q r p ( q r) (p q) r For reference, here are all the statements you can get to using the above templates. Is your answer on that list?
218 ( B) ( B) B B p (q r) p q r p q r p q r p ( q r) (p q) r So why did we go through this (admittedly lengthy) diversion?
219 ( B) ( B) B B p (q r) p q r p q r p q r p ( q r) (p q) r Well, any of these formulas are valid negations of the original formula.
220 ( B) ( B) B B p (q r) p q r p q r p q r p ( q r) (p q) r However, as you can see, not all of them are very easy to read.
221 ( B) ( B) B B p (q r) p q r p q r p q r p ( q r) (p q) r We personally think that these two are the cleanest of the negations.
222 ( B) ( B) B B p (q r) p q r p q r p q r p ( q r) (p q) r lthough we've stressed the importance of working through these negations using templates, there still is a bit of an art to it.
223 ( B) ( B) B B p (q r) p q r p q r p q r p ( q r) (p q) r You'll sometimes have a choice about which templates to use, and sometimes one choice will be much better than the others.
224 ( B) ( B) B B p (q r) p q r p q r p q r p ( q r) (p q) r Learning which rules to use takes some practice, but once you get the hang of it, it's really not too bad!
225 ( B) ( B) B B p (q r) p q r p q r p q r p ( q r) (p q) r If you ever make an inelegant choice, just back up and choose something else!
226 ( B) ( B) B B To see this in action, let's talk about how to negate biconditionals.
227 ( B) ( B) B B lthough we didn't talk about it in class, there are two nice rules you can use to negate a biconditional.
228 ( B) ( B) ( B) B B ( B) B B They're shown above. Basically, you negate one side, leave the other alone, and leave the connective unchanged.
229 ( B) ( B) ( B) B B ( B) B B (s an aside, the biconditional is the only connective that doesn't change when negated. Nifty!)
230 ( B) ( B) ( B) B B ( B) B B ((p q) r) ((q r) p) So let's put these new rules to practice by negating the statement to the left.
231 ( B) ( B) ( B) B B ( B) B B ((p q) r) ((q r) p) This one looks tricky! Don't worry it's not as bad as it looks.
232 ( B) ( B) ( B) B B ( B) B B (((p q) r) ((q r) p)) We start of the same way that we always do parenthesize and negate.
233 ( B) ( B) ( B) B B ( B) B B (((p q) r) ((q r) p)) Now, we need to apply some template here, so we need to pick one of the two biconditional templates.
234 ( B) ( B) ( B) B B ( B) B B (((p q) r) ((q r) p)) Now, we have a choice. We can either negate the left-hand side or the right-hand side.
235 ( B) ( B) ( B) B B ( B) B B (((p q) r) ((q r) p)) Either way will work, but one way is a lot easier than the other.
236 ( B) ( B) ( B) B B ( B) B B (((p q) r) ((q r) p)) Let's think a step ahead.
237 ( B) ( B) ( B) B B ( B) B B (((p q) r) ((q r) p)) If we negate this side, we're going to have to negate the OR, and that's going to require us to then negate the ND.
238 ( B) ( B) ( B) B B ( B) B B (((p q) r) ((q r) p)) On the other hand, if we negate this side, we just negate the implication, which is going to leave the OR unchanged.
239 ( B) ( B) ( B) B B ( B) B B (((p q) r) ((q r) p)) s a result, let's try applying this template, which negates the left-hand side.
240 ( B) ( B) ( B) B B ( B) B B (((p q) r) ((q r) p)) Here's the before shot.
241 ( B) ( B) ( B) B B ( B) B B ((p q) r) ((q r) p). and the after.
242 ( B) ( B) ( B) B B ( B) B B ((p q) r) ((q r) p) So now we need to push this negation deeper.
243 ( B) ( B) ( B) B B ( B) B B ((p q) r) ((q r) p) That's a relatively straightforward application of this template.
244 ( B) ( B) ( B) B B ( B) B B ((p q) r) ((q r) p) So we color-code for convenience.
245 ( B) ( B) ( B) B B ( B) B B ((p q) r) ((q r) p).and simplify.
246 ( B) ( B) ( B) B B ( B) B B ((p q) r) ((q r) p) nd bam! We're done.
247 ( B) ( B) ( B) B B ( B) B B ((p q) r) ((q r) p) ((p q) r) ((q r) p) You can and should check that this formula is correct by using the truth table tool. Here's the original as a reminder.
248 ( B) ( B) ( B) B B ( B) B B So wow! We've covered a lot of ground. Let's recap.
249 ( B) ( B) ( B) B B ( B) B B We've recently introduced new templates for negating implications and biconditionals.
250 ( B) ( B) ( B) B B ( B) B B You've seen that there's also a diferent way to negate NDs using implications.
251 ( B) ( B) ( B) B B ( B) B B nd we touched a bit on the idea that there's an art to choosing these templates wisely.
252 ( B) ( B) ( B) B B ( B) B B That's basically everything we'd like to say about negating formulas in propositional logic.
253 ( B) ( B) ( B) B B ( B) B B To wrap things up, let's talk about how to negate frst-order logic formulas. Don't worry! They follow the same basic set of rules.
254 ( B) ( B) ( B) B B ( B) B B Fundamentally, frst-order logic is quite diferent from propositional logic. Structurally, though, it's still propositions joined by connectives.
255 ( B) ( B) ( B) B B ( B) B B The only major diference is the introduction of quantifers. Once you know how to negate those, you can negate frst-order formulas with ease!
256 ( B) ( B) ( B) B B ( B) B B Let's see this with an example.
257 ( B) ( B) ( B) B B ( B) B B x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Here's a formula from class. Do you remember what this says?
258 ( B) ( B) ( B) B B ( B) B B x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Let's see how to negate it.
259 ( B) ( B) ( B) B B ( B) B B x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) s before, we put a negation at the front. (Since this whole formula is controlled by the quantifer, we don't need to parenthesize it.)
260 ( B) ( B) ( B) B B ( B) B B x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Now, we need some kind of template for simplifying this expression.
261 ( B) ( B) ( B) B B ( B) B B x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) If you'll remember from lecture, we saw how to negate quantifers: push the negation across the quantifer, then fip the quantifer.
262 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Those rules look like this.
263 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) So let's go apply them!
264 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Here, we can apply this template.
265 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) That looks like this.
266 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ).so we end up here.
267 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) That's progress!
268 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Now, how do we continue?
269 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Well, notice that we have a negation applied to an implication.
270 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Well, notice that we have a negation applied to an implication.
271 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) That suggests we should use this template.
272 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Here's the antecedent and consequent, in case that makes things clearer.
273 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) pplying the rule.
274 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ).gives us this.
275 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Cool!
276 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) It's just a matter of repeating this process until we're done.
277 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Here we have a negated existential statement.
278 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ).which simplifes like this.
279 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Great! Now, we have a negated ND. It's a three-way ND, actually, but fortunately we've seen lots of ways to handle this!
280 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) While we can use any template we'd like here, when dealing with frst-order logic, it's often useful to use this rule, which turns NDs into implications.
281 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) We'll group things like this.
282 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ). which gives us this as a result.
283 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) nd now we're done!
284 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) So how can we check whether this is right?
285 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) When dealing with propositional logic, we could just plug our result into the truth table tool.
286 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Unfortunately, frst-order logic doesn't have truth tables, so that particular approach isn't going to work.
287 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) It turns out, more generally, that it is objectively hard to check whether one frst-order formula is the negation of another.
288 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) You know how we've been talking about problems so hard that they can't be solved by a computer? This is one of them!
289 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) We're not going to talk about that in CS103. Take Phil 152 or CS154 for details!
290 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) So if there's no simple way to confrm that we have the right answer, how do we know that we've gotten it right?
291 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) t some level, we have to trust that we applied the templates correctly.
292 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) However, we still can do a few quick checks to make sure we didn't mess anything up.
293 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) For example, we know that is (usually) paired with and that is (usually) paired with.
294 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) great way to check your work is to make sure that you didn't break that rule. If you broke the rule, chances are you made a mistake somewhere.
295 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Here, notice that we do indeed obey the rules.
296 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) This is a good sign that we didn't make any major mistakes.
297 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) To check whether we really have it, it's often useful to translate the original and negated statements into English and to check those.
298 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) This statement says there is someone that doesn't love anyone else.
299 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Here's our original formula.
300 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) This says everyone loves someone else.
301 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) (Let's reset back to the negated formula.)
302 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) The two translations are indeed negations of one another.
303 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) If everyone loves someone else, there's no way that there could be a person who doesn't love anyone else.
304 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Similarly, if there's someone who doesn't love anyone else, then it's defnitely not the case that everyone loves someone else.
305 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) So overall we should be pretty confdent that we've properly negated our formula.
306 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) It passes our basic structural tests, and also intuitively syncs up with what the formulas mean.
307 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Before we move on, I wanted to touch on one question: why did we use the ND-to-implies template to negate the NDs here?
308 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) To answer that, let's roll back the clock to the point where we made that choice.
309 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Here we are!
310 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Now, we saw a number of ways that we can negate a three-way ND. (It's almost like we did that example intentionally to set this discussion up!)
311 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. (Person(y) y x Loves(x, y) ) ) Let's apply de Morgan's laws here, using the rule that ( B C) C
312 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. ( Person(y) y x Loves(x, y) ) ) That gives us this result.
313 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. ( Person(y) y x Loves(x, y) ) ) So this result is correct we know this because we applied valid rules the whole way through.
314 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. ( Person(y) y x Loves(x, y) ) ).but it's a lot harder to read this statement. That quantifer reads any y either isn't a person, or is equal to x, or x doesn't love y.
315 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. ( Person(y) y x Loves(x, y) ) ) That's quite a mouthful.
316 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. ( Person(y) y x Loves(x, y) ) ) Plus, this breaks the usual pattern of getting paired with, which makes it harder to check.
317 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. ( Person(y) y x Loves(x, y) ) ) One of the reasons why the NDto-implies rule is so useful is that it preserves which connectives pair with which quantifers.
318 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x. (Person(x) y. ( Person(y) y x Loves(x, y) ) ) This is why we recommend that you not use de Morgan's laws with ND and instead go with the other route. It's just a lot easier!
319 ( B) ( B) ( B) B B ( B) B B x. x. x. x. Having said that, let's go and do another, more complex example.
320 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) Here's a formula from set theory. Before we go on, can you translate this into English?
321 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) This says any two sets are equal if and only if they have the same elements. If you're not sure why, dive into this and see if you can convince yourself!
322 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) Let's see how to negate this.
323 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) ctually, let's not! Let's begin by having you take a stab at it. Try negating this one!
324 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) Seriously, go try to negate it. Don't keep reading until you do.
325 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) So you negated it? Like, really? Because once you see us do it you'll never get to experience the thrill of discovering it yourself.
326 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) Okay! So let's do this!
327 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) Let's begin by putting a negation at the front. gain, since the whole statement is quantifed, we can pass on the parentheses.
328 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) We now have something matching this template, so let's go apply it!
329 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) Before.
330 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ). and after!
331 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) We can now apply this template a second time to this next quantifer.
332 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) Before.
333 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ). and after!
334 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) We now have a negation lined up against an implication, so let's go and apply this template.
335 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) Before.
336 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ). and after!
337 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) Now our negation is sitting in front of a biconditional.
338 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) That means we have two choices of which template to apply.
339 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) But hey! Didn't we do some practice with that one earlier?
340 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) One of the lessons we took from that was that it's often useful to look at which half would be easier to negate.
341 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) This half would be pretty easy to negate we just change the = to a.
342 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) This half would be a lot harder to negate we'd have to push the negation across the quantifer, then handle another biconditional.
343 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) Based on that, it makes a lot more sense to negate the frst half of this biconditional, not the second.
344 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S = T x. (x S x T)) ) So we'll apply this template.
345 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S T x. (x S x T)) ).giving us this result.
346 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S T x. (x S x T)) ) nd at this point, we're done! Does this match what you had?
347 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S T x. (x S x T)) ) You might have negated the second half of the biconditional rather than the frst.
348 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S T x. (x S x T)) ) If you did, you'd get something like this instead. S. T. (Set(S) Set(T) (S = T x. (x S x T)) )
349 ( B) ( B) ( B) B B ( B) B B x. x. x. x. S. T. (Set(S) Set(T) (S T x. (x S x T)) ) If you did this, no worries! You're still correct. We just think this is a harder route to go down. S. T. (Set(S) Set(T) (S = T x. (x S x T)) )
350 ( B) ( B) ( B) B B ( B) B B x. x. x. x. Let's do one fnal example before moving on to our last topic.
351 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) How might you negate this formula?
352 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) s before, take a few minutes to work through this one on your own time. See what you come up with!
353 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) You know the drill. This slide is here to make sure that you actually did that work and didn't just skip by it.
354 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) Okay! So let's go negate this.
355 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) This formula is quite diferent from the previous two in a subtle way.
356 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) The two previous formulas we looked at were formulas that were quantifed at the top level.
357 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) By that, we mean that those formulas were of the form x. or x. B.
358 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) This formula, on the other hand, is an implication where both the antecedent and consequent use quantifers. However, the whole formula is not quantifed.
359 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) This means that, when we start of, we'll fully parenthesize the expression and negate it, rather than just tacking a negation up-front.
360 ( B) ( B) ( B) B B ( B) B B x. x. x. x. (( x. Happy(x)) ( y. Happy(y))) Here's what this now looks like.
361 ( B) ( B) ( B) B B ( B) B B x. x. x. x. (( x. Happy(x)) ( y. Happy(y))) Notice that we now have a negation applied to an implication.
362 ( B) ( B) ( B) B B ( B) B B x. x. x. x. (( x. Happy(x)) ( y. Happy(y))) That means that we'll use this template, just as we normally do.
363 ( B) ( B) ( B) B B ( B) B B x. x. x. x. (( x. Happy(x)) ( y. Happy(y))) Here are the antecedent and consequent each of which is a bit complicated. If we apply the template.
364 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)). we end up here.
365 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) ll that's left to do is push the negation across the quantifer.
366 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) That means we apply this rule.
367 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)).like this!
368 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) nd now we're done. Nifty!
369 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) We included this example so that you'd see that, when dealing with frst-order logic, you still have to follow all the regular rules for negating statements.
370 ( B) ( B) ( B) B B ( B) B B x. x. x. x. ( x. Happy(x)) ( y. Happy(y)) You have to be careful to make sure that you don't negate everything, and that you fgure out what the top-level structure is.
371 ( B) ( B) ( B) B B ( B) B B x. x. x. x. So we're almost done! We're going to do one last example.
372 ( B) ( B) ( B) B B ( B) B B x. x. x. x. s you saw in lecture, sometimes we allow for quantifers over sets.
373 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x S. y S. x y This means that you might see formulas like this one.
374 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x S. y S. x y How do you negate formulas like these?
375 ( B) ( B) ( B) B B ( B) B B x. x. x. x. x S. y S. x y The good news is that the rules for negating these set quantifers are basically the same as the rules for negating the regular quantifers.
376 ( B) ( B) ( B) B B ( B) B B x. x. x S. x S. x. x. x S. x S. x S. y S. x y I've included them here. You just push the negation across the quantifer and fip which quantifer you're using.
377 ( B) ( B) ( B) B B ( B) B B x. x. x S. x S. x. x. x S. x S. x S. y S. x y Based on that, let's fnish this one of and go home. ^_^
378 ( B) ( B) ( B) B B ( B) B B x. x. x S. x S. x. x. x S. x S. x S. y S. x y We begin as we are wont to do by putting a negation in front.
379 ( B) ( B) ( B) B B ( B) B B x. x. x S. x S. x. x. x S. x S. x S. y S. x y We match this template, so let's go apply it!
380 ( B) ( B) ( B) B B ( B) B B x. x. x S. x S. x. x. x S. x S. x S. y S. x y Tada!
381 ( B) ( B) ( B) B B ( B) B B x. x. x S. x S. x. x. x S. x S. x S. y S. x y Now we match this one.
382 ( B) ( B) ( B) B B ( B) B B x. x. x S. x S. x. x. x S. x S. x S. y S. x = y nd we're done!
383 ( B) ( B) ( B) B B ( B) B B x. x. x S. x S. x. x. x S. x S. So there you have it. We have a collection of templates that we can apply to negate formulas.
384 Hope this helps! Please feel free to ask questions if you have them.
385 Did you fnd this useful? If so, let us know! We can go and make more guides like these.
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