0:00:04.440,0:00:09.654
In the last lecture. We talked about the[br]truth functional connective, conjunction.

0:00:09.654,0:00:14.934
We gave the truth table for conjunction.[br]And we showed how we could use the truth

0:00:14.934,0:00:20.409
table for conjunction to figure out which[br]inferences that use conjunction are valid

0:00:20.409,0:00:25.883
and which inferences are not. Today, we're[br]going to talk about the truth functional

0:00:25.883,0:00:31.157
connective, disjunction. We're going to[br]give the truth table for dis-junction, and

0:00:31.157,0:00:36.634
we're going to show how we can use that[br]truth table to figure out which inferences

0:00:36.634,0:00:41.862
that use dis-junction are valid and which[br]are not. Now in English, we usually

0:00:41.862,0:00:47.991
express disjunction by using the word or[br]but the word or can be used in a couple

0:00:47.991,0:00:53.590
different ways in English. For instance,[br]suppose that Manchester is playing

0:00:53.590,0:00:59.189
Barcelona tonight and you ask me, who's[br]going to win? And I say, well, I have no

0:00:59.189,0:01:04.713
idea who's going to win but I can tell you[br]this, it's going to be Manchester or

0:01:04.713,0:01:10.471
Barcelona. Now, what I'm suggesting when I[br]say, it's going to be Manchester or

0:01:10.471,0:01:16.186
Barcelona, is that it's not going to be[br]both. Manchester might win, Barcelona

0:01:16.186,0:01:21.743
might win. But there's no possible way[br]that both of them are going to win.

0:01:21.743,0:01:27.697
Sometimes, in English, when you want to[br]say that it's going to be one thing or the

0:01:27.697,0:01:33.334
other, but not both, you say, either or[br]either Manchester is going to win, or

0:01:33.334,0:01:40.467
Barcelona is going to win. But sometimes[br]when we use the word or, we mean it could

0:01:40.467,0:01:46.776
be one, or the other, or both. So for[br]instance, suppose you ask me what we

0:01:46.776,0:01:54.002
should have for dinner tonight and I say[br]well we could have chicken or fish. Well

0:01:54.002,0:02:01.088
there's no suggestion that we couldn't[br]have both maybe we could have a little bit

0:02:01.088,0:02:08.930
of chicken and a little of fish. So it has[br]to be chicken or fish or both. When I say

0:02:08.930,0:02:16.489
chicken or fish, I'm not suggesting it[br]can't be both. Sometimes in English we use

0:02:16.489,0:02:23.839
the phrase and, or to express that it[br]could be one or the other or both. I'll

0:02:23.839,0:02:31.350
say, we could have the chicken and, or the[br]fish. The truth functional connective

0:02:31.350,0:02:37.563
disjunction is expressed by the second[br]meaning of or. It's expressed by the

0:02:37.563,0:02:44.272
English phrase and, or where you me an it[br]could be one or the other or both. That's

0:02:44.272,0:02:50.567
what we're going to call disjunction in[br]this class. Now let's look at the truth

0:02:50.567,0:02:58.413
table for disjunction. So lets look at the[br]truth table for dis-junction. Suppose

0:02:58.413,0:03:06.866
you're using disjunction to combine the[br]propositions We eat chicken and we eat

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fish into the disjunctive proposition We[br]eat chicken or fish. Well when is that

0:03:15.214,0:03:23.347
disjunctive proposition going to be true?[br]If it's true that we eat chicken, and it's

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true that we eat fish, then it's going to[br]be true that we eat chicken or fish cuz

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remember, when we use or here, we don't[br]mean either or, but not both. We mean and

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or. Could be one, could be the other, or[br]could be both. So if it's true that we eat

0:03:45.622,0:03:50.866
chicken and it's true that we eat fish,[br]it's going to be true that we eat chicken

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or fish. Now supposed it's true that we[br]eat chicken, but its false that we eat

0:03:56.043,0:04:03.232
fish. Well. Then, it's still going to be[br]true that we eat chicken or fish. Suppose

0:04:03.232,0:04:10.301
it's false that we eat chicken, but true[br]that we eat fish. Then, it's still going

0:04:09.764,0:04:17.369
to be true that we eat chicken or fish.[br]But suppose it's false that we eat chicken

0:04:17.369,0:04:24.675
and it's also false that we eat fish.[br]Then, is it going to be true that we eat

0:04:24.675,0:04:31.626
chicken or fish? No! Because we won't be[br]eating either. So then it'll be false that

0:04:31.626,0:04:38.362
we eat chicken or fish. This is the truth[br]table for disjunction. And, like the truth

0:04:38.362,0:04:44.657
table that we saw for conjunction, it's[br]going to work no matter what propositions

0:04:44.657,0:04:50.953
we put into here, or here, or here. So, no[br]matter what proposition you have right

0:04:50.953,0:04:57.007
here, call it P1. And, no matter what[br]proposition you have right here, call it

0:04:57.007,0:05:03.480
P2. When you use the truth functional[br]connective disjunction. To create a new

0:05:03.480,0:05:09.594
proposition out of those two[br]proposition's, so you got a new

0:05:09.594,0:05:16.930
proposition P one or P two. That new[br]disjunctive proposition is going to be

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true. Whenever P1 is true, and it's also[br]going to be true whenever P2 is true. So

0:05:23.552,0:05:31.023
unlike conjunction. Where you need both of[br]the two ingredient propositions to be true

0:05:31.023,0:05:36.833
in order for the conjunctive proposition[br]to be true. In disjunc tion, you only need

0:05:36.833,0:05:42.428
for one of the of the two ingredient[br]propositions to be true in order for the

0:05:42.428,0:05:47.951
disjunctive proposition to be true. The[br]disjunctive proposition is false only

0:05:47.951,0:05:53.191
when. Both of the two ingredient[br]propositions are false. That's the only

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time a disjunction is false. So now, let[br]me give you an example, of how you can use

0:05:59.183,0:06:04.970
the truth table for disjunction. Just show[br]that a particular kind of argument is

0:06:04.970,0:06:10.101
valid. We're going to discuss, a kind of[br]argument that is sometimes known. As

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process of elimination. Here's how it[br]goes. Suppose, that you have to solve. A

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murder mystery. Mister Jones, has been[br]stabbed in his living room. With a knife

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in the back. Now, you figured out that[br]there were only two people in the house at

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the time of his stabbing, the butler and[br]the accountant. You also know that the

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knife is positioned in Mr. Johnson's back[br]in such a way that he couldn't possibly

0:06:41.635,0:06:47.034
have stabbed himself. So it had to be[br]someone else. And whoever else it was it

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had to be someone who's in the house at[br]the time of the stabbing. So it could only

0:06:52.783,0:06:59.793
have been, the butler or the accountant,[br]or maybe both. So you know that the butler

0:06:59.793,0:07:06.293
did it, or the accountant did it. Now you[br]find out that the accountant is a

0:07:06.293,0:07:13.247
quadriplegic, so the accountant couldn't[br]have stabbed Mr. Jones in the back. So now

0:07:13.247,0:07:20.410
you know that the account didn't do it.[br]And so, from the two premises, the butler

0:07:20.410,0:07:27.520
did it, or the accountant did it. And the[br]accountant didn't do it. You can conclude,

0:07:27.520,0:07:33.660
the butler did it. Now, why is that[br]argument valid? Here's why. Think about

0:07:33.660,0:07:39.777
the truth table for disjunction again. So[br]remember the first premise, the butler did

0:07:39.777,0:07:44.495
it or the accountant did it is a[br]disjunction. It's going to be true

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whenever one of it's disjuncts is true,[br]one of it's ingredient propositions is

0:07:50.244,0:07:55.551
true. So it's going to be true whenever[br]the butler did it, and it's going to be

0:07:55.551,0:08:00.711
true whenever the butler did it. The[br]second premise tells you that the

0:08:00.711,0:08:06.857
accountant didn't do it. So the only way[br]for the first premise to be true, given

0:08:06.857,0:08:12.828
that the accountant didn't do it, is fo r[br]the butler to have done it. And so you

0:08:12.828,0:08:18.000
know, since the accountant couldn't have[br]done it. That the only way for the

0:08:18.000,0:08:23.451
dis-junction, the butler did it or the[br]accountant did it to be true, is for the

0:08:23.451,0:08:29.182
butler to have done it and that's why you[br]can conclude the butler did it and your

0:08:29.182,0:08:35.228
argument is valid. That's one example of a[br]process of elimination argument. Of course

0:08:35.228,0:08:40.650
there are lots of others, but with all of[br]those others you can see why they are

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valid by looking at the truth table for[br]dis-junction. Remember how you can use the

0:08:46.276,0:08:51.972
truth functional connective conjunction to[br]build a new proposition out of not just

0:08:51.972,0:08:57.599
two other propositions but sometimes three[br]other propositions. You can conjoin one

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proposition with a second and with a[br]third. Well, you can do the same thing

0:09:03.476,0:09:10.838
with disjunction. You can disjoin one[br]proposition with a second and a third, to

0:09:10.838,0:09:19.187
create the proposition. Either this, or[br]that or the other or any combination of

0:09:19.187,0:09:27.764
the three. What does the truth table for[br]that look like? Here it is. The

0:09:27.764,0:09:36.852
disjunctive proposition, P1 or P2 or P3,[br]is going to be true. Whenever P1 is true,

0:09:36.852,0:09:43.123
it's also going to be true whenever P2 is[br]true. And it's also going to be true

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whenever P3 is true. In fact, the only[br]time that P1 or P2 or P3, the only time

0:09:49.743,0:09:55.928
that, that disjunctive proposition is[br]going to be false is when all these

0:09:55.928,0:10:03.402
ingredient propositions are false. So[br]here's what the truth table for P1, or P2,

0:10:03.402,0:10:10.585
or P3 looks. Now let's use the truth table[br]for our triple disjunction to show how a

0:10:10.585,0:10:16.336
particular process of elimination argument[br]can be valid. Let's go back to our murder

0:10:16.336,0:10:21.470
mystery in order to do that. Now suppose[br]that you find out contrary to what you had

0:10:21.881,0:10:27.563
previously believed, that Butler and the[br]accountant were not the only people in the

0:10:27.563,0:10:33.383
house, at the time of Mr. Jonathan's[br]death. In addition, the maid was in the

0:10:33.383,0:10:38.255
house and the cook was in the house.[br]Alright. Well, now, you know, that the

0:10:38.255,0:10:44.035
butler or the maid or the cook did it. We[br]don't yet know which of them did it, but

0:10:44.035,0:10:49.589
we know that the butler or the maid or the[br]cook did it. Now suppose that yo u find

0:10:49.589,0:10:54.353
out that the maid and the cook, at the[br]time of the stabbing we're off in the

0:10:54.353,0:10:58.866
opposite corner of the house doing[br]something else together. Well now you

0:10:58.866,0:11:03.969
know, that the maid didn't do it. And you[br]know that the cook didn't do it. So what

0:11:03.969,0:11:09.348
can you conclude from those three[br]premises? Premise one, the butler or the

0:11:09.348,0:11:15.243
maid or the cook did it. Premise two, the[br]maid didn't do it. And premise three: the

0:11:15.243,0:11:21.455
cook didn't do it. Well, lets use the[br]truth table to figure this out. Premise

0:11:21.455,0:11:28.463
one of the truth table tells you that the[br]butler or the maid or the cook did it. So

0:11:28.463,0:11:35.556
the situation in which it falls that the[br]butler or the maid or the cook did it that

0:11:35.556,0:11:42.649
situation is ruled out by premise one. So[br]premise one tells you at that situation is

0:11:42.649,0:11:51.507
not the actual situation. Premise two[br]tells you that the maid did not do it. So

0:11:51.507,0:11:56.768
any situation in which its true that the[br]maid did it is also not the actual

0:11:56.768,0:12:02.098
situation. So this situation is one in[br]which its true that the maid did it so

0:12:02.098,0:12:07.774
that's not the actual situation according[br]to premise two. This situation is one in

0:12:07.774,0:12:13.381
which its true that the maid did it. So[br]that's not the actual situation according

0:12:13.381,0:12:18.535
to premise two. This situation is one in[br]which its true that the maid did it. So

0:12:18.535,0:12:23.873
that's not the actual situation according[br]to premise two, and this situation is one

0:12:23.873,0:12:28.632
in which it is true that the maid did it.[br]So that's not the actual situation

0:12:28.632,0:12:35.021
according to premise two. Premise three[br]tells you that the cook didn't do it. So,

0:12:35.021,0:12:40.886
that rules out any situation in which it's[br]true that the cook did it. Well, here's a

0:12:40.886,0:12:46.821
situation in which it's true that the cook[br]did it. So, that situation is ruled out by

0:12:46.821,0:12:52.474
premise three. And, here's a situation in[br]which it's true that the cook did it. So,

0:12:52.474,0:12:57.773
that situation is ruled out by premise[br]three. So, premise one rules out this

0:12:57.773,0:13:05.377
situation. Premise two, rules out this,[br]this, this and this situation. And premise

0:13:05.377,0:13:14.791
three, rules out this, this, this and this[br]situation. Well, whats left? The only

0:13:14.791,0:13:25.018
situation left that could be the actual[br]situation is this one. See cause in this

0:13:25.018,0:13:30.595
situation, it's t rue that the butler or[br]the maid or the cook did it just as

0:13:30.595,0:13:36.692
premise one tells us. Its false that the[br]maid did just as premise two tells us, and

0:13:36.692,0:13:43.031
its false that the cook did just as[br]premise three tells us. But, that's the

0:13:43.031,0:13:48.812
situation in which it's true that the[br]butler did it. So, the conclusion that we

0:13:48.812,0:13:54.889
can draw, based on the situations that are[br]ruled out by premises one, two, and three,

0:13:54.889,0:14:00.818
is that the actual situation is this one,[br]and in that actual situation, it's true

0:14:00.818,0:14:08.579
that the butler did it. So, the butler did[br]it That's why the process of elimination

0:14:08.579,0:14:15.174
reasoning that we just considered is[br]valid. If premise one says, the butler or

0:14:15.174,0:14:20.879
the maid or the cook did it. Premise two[br]says the maid didn't do it, and premise

0:14:20.879,0:14:26.727
three says that the cook didn't do it.[br]Then by process of elimination we can draw

0:14:26.727,0:14:32.289
the valid conclusion that the butler did[br]it and this is why. Let me give you

0:14:32.289,0:14:38.362
another example of how you can use the[br]truth table for disjunction in order to

0:14:38.362,0:14:44.203
show whether or not the process of[br]elimination argument is valid. Suppose we

0:14:44.203,0:14:49.968
know that Walter is a professional[br]football player. Well, that means that he

0:14:49.968,0:14:55.425
plays either American football, U.S.[br]Football, or European football, which

0:14:55.425,0:15:01.882
Americans call soccer, or Australian rules[br]football. But now suppose we find out that

0:15:01.882,0:15:08.618
Walter does not play American football.[br]And you conclude from that, that he must

0:15:08.618,0:15:14.998
play European football. So you argue as[br]follows. Premise 1- Walter plays either

0:15:14.998,0:15:21.019
American football or European football or[br]Australian Rules football, premise 2- he

0:15:21.019,0:15:26.673
does not play American football and[br]therefore you conclude he plays European

0:15:26.673,0:15:32.107
football. Well, that argument is invalid[br]and we can use the truth table for

0:15:32.107,0:15:38.241
disjunction to show why it's invalid. Look[br]at this truth table. Premise one, recall,

0:15:38.241,0:15:45.068
is that Walter plays American or European[br]or Australian rules football. So premise

0:15:45.068,0:15:51.562
one rules out the situation in which it's[br]false that Walter plays American or

0:15:51.562,0:15:58.555
European or Australian rules football. And[br]that's all it rules out. It just rules out

0:15:58.555,0:16:05.938
the situation in which it's false that[br]Walter plays an y of those. Premise two,

0:16:05.938,0:16:12.546
Walter doesn't play American Football.[br]That rules out the situation in which it's

0:16:12.546,0:16:21.191
true that Walter plays American football.[br]So it rules out this situation. And rules

0:16:21.191,0:16:33.058
out this situation. And it rules out this[br]situation. And it rules out this

0:16:33.058,0:16:41.390
situation. So, premise one rules out the[br]situation represented at the bottom.

0:16:41.390,0:16:48.171
Premise two rules out the situations[br]represented by these four columns at the

0:16:48.171,0:16:57.486
top. So, can we conclude that Walter plays[br]European football? No. He might play

0:16:57.486,0:17:04.519
European football but he might also play[br]Australian Rules football. He's looked all

0:17:04.519,0:17:11.213
the premise one and premise two together[br]rule out is these five situations. But

0:17:11.213,0:17:17.992
there is no three situations that are[br]possible. In one of them Walter plays both

0:17:17.992,0:17:24.420
European football and Australian rules[br]football. In another one of them, Walter

0:17:24.420,0:17:29.851
plays European football, but not[br]Australian rules football, and in the

0:17:29.851,0:17:35.438
third situation, this left open by[br]premises one and two, it's false that

0:17:35.438,0:17:41.734
Walter plays European football but true[br]that he plays Australian rules football.

0:17:41.734,0:17:48.109
So based on the information that premises[br]one and two give us, we cannot conclude

0:17:48.109,0:17:54.090
that Walter plays European football. He[br]might play Australian rules football

0:17:54.090,0:17:59.989
instead. So the argument that you made is[br]invalid. In the next lecture, we're going

0:17:59.556,0:18:05.547
to consider a truth functional connective[br]that's different from conjunction and

0:18:05.547,0:18:10.757
disjunction in the following way. While[br]conjunction and disjunction are

0:18:10.757,0:18:16.412
connectives that can be used to build[br]propositions out of two or more other

0:18:16.412,0:18:22.068
propositions. Negation, the connective[br]that we'll talk about next time, is the

0:18:22.068,0:18:27.946
connective that is used to build new[br]propositions out of just one single other

0:18:27.946,0:18:33.825
proposition. Negation, in other words, is[br]a connective that you apply to one

0:18:33.825,0:18:39.777
proposition to build a second proposition.[br]And that's what we'll talk about next

0:18:39.777,0:18:44.750
time. Now, there's some exercise's for you[br]to do. These exercises test your

0:18:44.750,0:18:50.151
understanding of the truth table for[br]disjunction and of how the truth table for

0:18:50.151,0:18:55.352
dis-junction can be used to determine[br]whether a part icular argument that uses

0:18:55.352,0:18:58.020
disjunction is a valid argument or not.