0:00:04.640,0:00:11.450
In the last lecture, I said that, we were[br]going to spend this unit learning the

0:00:11.450,0:00:18.152
rules for evaluating deductive arguments.[br]Now this week, we're gonna learn the rules

0:00:18.152,0:00:24.247
for evaluating deductive arguments that[br]involve, what I'm gonna call propositional

0:00:24.247,0:00:30.824
connectives. Now, what are propositional[br]connectives? Well, in order to explain

0:00:30.824,0:00:36.982
what propositional connectives are. I[br]first have to tell you what propositions

0:00:36.982,0:00:42.745
are and then I can talk about connectives.[br]So first, propositions. What's a

0:00:42.745,0:00:49.140
proposition? A proposition, is the kind of[br]thing that can be true, or false and that

0:00:49.140,0:00:54.740
can serve as the premise, or the[br]conclusion of an argument. Here let me

0:00:54.740,0:01:07.236
give you an example. See this book this[br]book is not a proposition. It can't be

0:01:07.236,0:01:13.420
true or false, and it can't serve as the[br]premise or the conclusion of an argument.

0:01:13.700,0:01:21.706
See this hand, this hand is not a[br]proposition it can be true or false, and

0:01:21.706,0:01:30.041
it can serve as the premise or the[br]conclusion of a argument. But now suppose

0:01:30.041,0:01:40.555
I say, the book is in my hand. Now what I[br]just said, that the book is in my hand is

0:01:40.555,0:01:47.721
a proposition, it can be true in fact, it[br]is true, or it can be false and right now

0:01:47.721,0:01:54.446
it is false. It can also serve as the[br]premise of an argument. I could say, the

0:01:54.446,0:02:01.258
book is in my hand, therefore, my hand is[br]not free to shake yours. And it could

0:02:01.258,0:02:08.602
serve as the conclusion of an argument. I[br]could say, you just gave me the book and I

0:02:08.602,0:02:15.030
haven't let go of it, therefore, the book[br]is in my hand. I just told you what

0:02:15.030,0:02:20.609
propositions are. But what's a[br]propositional connective? A propositional

0:02:20.609,0:02:26.502
connective is something that takes[br]propositions and makes new propositions

0:02:26.502,0:02:32.631
out of it. Let me give you an example to[br]illustrate. Consider a proposition, the

0:02:32.631,0:02:39.989
book is under my hand. Now consider the[br]proposition, my foot is under the book. We

0:02:39.989,0:02:47.468
can combine those two propositions using[br]the propositional connective end to make

0:02:47.468,0:02:55.060
the new proposition. The book is under my[br]hand, and by foot is under the book. Now

0:02:55.060,0:03:00.730
what I've just said that the book is under[br]my hand and my foot is under the book,

0:03:00.730,0:03:06.050
that's a proposition. It's the kind of[br]thing that could be true or false. For

0:03:06.050,0:03:11.370
instance, right now it's true and right[br]now it's false. It's also the kind of

0:03:11.370,0:03:16.480
thing that can be premise or the[br]conclusion of the argument. For instance,

0:03:16.480,0:03:22.080
I could say, what you're seeing right now[br]is really happening, therefore, the book

0:03:22.080,0:03:29.279
is under my hand and my foot is under the[br]book. Or I could say, the book is under my

0:03:29.279,0:03:37.700
hand, and my foot is under the book.[br]Therefore, my foot is under my hand. So

0:03:37.420,0:03:41.465
you see that the book is under my hand,[br]and my foot is under the book. That's the

0:03:41.465,0:03:46.016
kinda thing that can be true or false and[br]it's the kind of thing that can be the

0:03:46.016,0:03:51.305
premise, or the conclusion of an argument.[br]So it's a proposition, but it's a

0:03:51.305,0:03:58.158
proposition that we made by combining two[br]other propositions. See how propositional

0:03:58.158,0:04:04.744
connectives work? They're beautiful,[br]aren't they? I just gave you an example of

0:04:04.744,0:04:10.862
a propositional connective. I called it[br]the proposition connective and. But in

0:04:10.862,0:04:16.981
English, the word and can be used in[br]different ways. It's not always used as a

0:04:16.981,0:04:23.418
proposition connective. Let me give you[br]different examples of how and can be used.

0:04:23.418,0:04:29.740
Think about the sentence, Jack and Jill[br]finally talked. Okay now, there are three

0:04:29.740,0:04:36.320
different ways to understand what that[br]sentence is saying. Jack and Jill could be

0:04:36.320,0:04:42.737
the name of a fast food company that[br]serves a special stew that's very popular

0:04:42.737,0:04:49.478
with its patrons. Now, maybe lawyers have[br]been wondering what the ingredients are in

0:04:49.478,0:04:55.895
Jack and Jill's special stew because many[br]of Jack and Jill's patrons have been

0:04:55.895,0:05:02.250
coming down with an unusual disease. So,[br]lawyers have been asking Jack and Jill to

0:05:02.250,0:05:07.968
disclose what's in their stew. Jack and[br]Jill has been refusing to do so. But

0:05:07.968,0:05:13.990
finally, the spokesman for Jack and Jill[br]discloses what's in their stew. I might

0:05:13.990,0:05:20.013
tell you about that situation by saying[br]Jack and Jill finally talked. There, I'm

0:05:20.013,0:05:26.188
telling you there's a particular company,[br]a company called Jack and Jill, and that

0:05:26.188,0:05:32.722
company finally talked through their[br]spokesperson. Here's a second way to

0:05:32.722,0:05:38.155
understand the sentence, Jack and Jill[br]finally talked. Suppose that Jack and Jill

0:05:38.155,0:05:43.520
ar e a couple, and recently they've been[br]going through a tough time. They've been

0:05:43.520,0:05:49.021
angry and resentful towards each other.[br]And they haven't talked about what's been

0:05:49.021,0:05:54.534
bothering them. Now, I might tell you,[br]Jack and Jill finally talked. And what I

0:05:54.534,0:06:00.650
mean by that is, that they finally talked[br]to each other about what's bothering them.

0:06:00.650,0:06:07.375
Now that's different, from the first[br]example in which I said, Jack and Jill

0:06:07.375,0:06:14.554
finally talked. There, I was saying that a[br]particular thing the company, Jack and

0:06:14.554,0:06:21.691
Jill. Finally talked about the ingredients[br]in its special stew. But in this second

0:06:21.691,0:06:27.782
example, I'm saying that two things, Jack[br]and Jill finally talked to each other. Let

0:06:27.782,0:06:33.724
me give you a third way to understand the[br]sentence Jack and Jill finally talked.

0:06:33.724,0:06:39.593
Imagine that Jack, Jill, and Roger are[br]having a silence contest. They're having a

0:06:39.593,0:06:45.610
contest to see who can go for the longest[br]period of time without talking. And I'm

0:06:45.610,0:06:53.129
watching them to see who wins. Well, you[br]call be periodically and ask me as anyone

0:06:53.129,0:06:59.620
of them talked? And for the first few[br]hours I might say to you nope, none of

0:06:59.620,0:07:07.394
them has talked yet. And then at one point[br]you call me and you ask me, has anyone of

0:07:07.394,0:07:14.553
them talked? And I say well Jack and Jill[br]finally talked. Now here I'm not saying

0:07:14.553,0:07:23.373
that Jack or Jill talked to each other.[br]I'm saying that Jack finally talked, and

0:07:23.373,0:07:32.706
Jill finally talked. So, when I say Jack[br]and Jill finally talked. I'm expressing a

0:07:32.706,0:07:40.403
proposition that's made up of two other[br]propositions. The proposition that Jack

0:07:40.403,0:07:48.167
finally talked, and the proposition that[br]Jill finally talked. When I say Jack and

0:07:48.167,0:07:55.454
Jill finally talked, I'm using the word[br]and, as a propositional connective. It

0:07:55.454,0:08:02.087
takes two propositions. First, that Jack[br]finally talked, adn second that Jill

0:08:02.087,0:08:09.153
finally talked. And it combines those two[br]propositions into a third proposition. The

0:08:09.153,0:08:14.803
proposition that, Jack and Jill. Finally[br]talked. Now that's a third way to

0:08:14.803,0:08:20.204
understand the sentence Jack and Jill[br]finally talked and it's different from the

0:08:20.204,0:08:25.471
first two. But that's the only way of[br]understanding the word and, so that it's a

0:08:25.471,0:08:30.271
propositional conn ective in that[br]sentence. So I've just explained to you

0:08:30.271,0:08:35.339
what propositional connectives are. I gave[br]you an example of a propositional

0:08:35.339,0:08:40.539
connective and, and I showed how the[br]English word and, can sometimes be used as

0:08:40.539,0:08:46.176
a propositional connective but sometimes[br]not. Now the English language like every

0:08:46.176,0:08:52.329
other natural language, contains lots and[br]lots of different phrases that can be used

0:08:52.329,0:08:57.603
as propositional connectives. For[br]instance, consider the phrase, I believe

0:08:57.603,0:09:03.868
that. You tell me Jack and Jill finally[br]talked. Well, I might say I believe that

0:09:03.868,0:09:10.008
Jack and Jill finally talked. There, I'm[br]taking one proposition Jack and Jill

0:09:10.008,0:09:16.229
finally talked, and attaching the phrase[br]to it, I believe that to make another

0:09:16.229,0:09:22.606
proposition namely, I believe that Jack[br]and Jill finally talked. Or consider the

0:09:22.606,0:09:28.956
phrase, I hate it when. You might tell me,[br]it's raining, and I might say, I hate it

0:09:28.956,0:09:35.307
when it's raining. See there, I take the[br]proposition it's raining, and I attach a

0:09:35.307,0:09:41.416
phrase to it, I hate it when, to make[br]another proposition, I hate it when it's

0:09:41.416,0:09:47.479
raining. So these are other examples of[br]how phrases in English, I believe that, or

0:09:47.479,0:09:52.749
I hate it when, can be used as[br]propositional connectives. They can take

0:09:52.749,0:09:58.231
other propositions and make new[br]propositions out of them. But and is a

0:09:58.231,0:10:03.703
very different kind of propositional[br]connective from I believe that or I hate

0:10:03.703,0:10:09.385
it when. And it's different in two ways.[br]One way it's different is that it makes a

0:10:09.385,0:10:13.945
new proposition out of two other[br]propositions, not just one other

0:10:13.945,0:10:19.416
proposition. So when you make a new[br]proposition by attaching I hate it when to

0:10:19.416,0:10:24.888
another proposition, what you're doing is[br]turning one proposition into another

0:10:24.888,0:10:30.085
proposition. But when you make a new[br]proposition by combining two other

0:10:30.085,0:10:36.065
propositions using the word and, you're[br]making a new proposition out of two other

0:10:36.065,0:10:41.528
propositions. Jack finally talked, and[br]Jill finally talked. You make the new

0:10:41.528,0:10:47.655
proposition, Jack and Jill finally talked.[br]That's one way in which the propositional

0:10:47.655,0:10:53.340
connective and is different from the[br]propositional connective, I hate it when.

0:10:53.880,0:10:59.295
But there's a second wa y, in which the[br]propositional connective, and, differs

0:10:59.295,0:11:04.999
from the propositional connective, I hate[br]it when. And that is, that when you use

0:11:04.999,0:11:10.630
the propositional connective and, to[br]combine two other propositions into a new

0:11:10.630,0:11:16.623
proposition. Whether that proposition is[br]true or false doesn't depend upon anything

0:11:16.623,0:11:22.038
other than whether the two original[br]propositions that you used to build it

0:11:22.038,0:11:28.708
were true or false, that's all it depends[br]upon. Let me illustrate this point using

0:11:28.708,0:11:35.362
something we call a truth table. A truth[br]table is a way of representing various

0:11:35.362,0:11:42.353
possible situations, and how the truth of[br]a proposition depends upon which of those

0:11:42.353,0:11:49.091
various possible situations is real. For[br]instance, consider the two possibilities

0:11:49.091,0:11:55.830
there are with Jack finally talked. The[br]proposition Jack finally taught, could be

0:11:55.830,0:12:01.913
true. Or it could be false. The[br]proposition Jill finally talked could be

0:12:01.913,0:12:08.499
true or it could be false. So, there are[br]four possibilities we have to consider.

0:12:08.499,0:12:14.859
Either Jack finally talked is true and[br]Jill finally talked is true. Or Jack

0:12:14.859,0:12:21.812
finally talked is true and Jill finally[br]talked is false. Or Jack finally talked is

0:12:21.812,0:12:28.765
false and Jill finally talked is true. Or[br]finally, Jack finally talked is false and

0:12:28.765,0:12:35.334
Jill finally talked is false. Those are[br]the four possible situations. Now, if the

0:12:35.334,0:12:40.911
first of those four situations is the real[br]situation, so it's true that Jack finally

0:12:40.911,0:12:45.824
talked and it's true that Jill finally[br]talked, then is it going to be true or

0:12:45.824,0:12:52.775
false that Jack and Jill finally talked?[br]It's going to be true. All right, cuz Jack

0:12:52.775,0:13:00.370
finally talked, and Jill finally talked,[br]so Jack and Jill finally talked. Now

0:13:00.370,0:13:05.896
suppose it's true that Jack finally[br]talked, but it's false that Jill finally

0:13:05.896,0:13:11.349
talked. Then is it going to be true or[br]false that Jack and Jill finally talked?

0:13:11.349,0:13:18.149
It'll be false, and suppose it's false[br]that Jack finally talked, but it's true

0:13:18.149,0:13:23.836
that Jill finally talked. Then, is it[br]going to be true or false that Jack and

0:13:23.836,0:13:29.810
Jill finally talked, again it's going to[br]be false. And finally, suppose it's false

0:13:29.810,0:13:35.219
that Jack finally talked and it's also[br]false that Jill final ly talked. Then is

0:13:35.219,0:13:40.126
it going to be true or false that Jack and[br]Jill finally talked? Well obviously in

0:13:40.126,0:13:45.677
that situation, it's going to be false[br]that Jack and Jill finally talked. So what

0:13:45.677,0:13:52.843
this truth table demonstrates is that the[br]truth of the proposition Jack and Jill

0:13:52.843,0:13:58.519
finally talked. Just depends, it depends[br]on nothing other than the truth of the

0:13:58.519,0:14:03.835
proposition Jack finally talked, and the[br]truth of the proposition Jill finally

0:14:03.835,0:14:08.667
talked. In other words, the truth of the[br]proposition that we've use the

0:14:08.667,0:14:14.204
propositional connective and to build.[br]Depends on nothing other than the truth of

0:14:14.204,0:14:19.496
the two ingredient propositions that we[br]connected by means of the propositional

0:14:19.496,0:14:24.920
connective and. Because the propositional[br]connective and, works that way because it

0:14:24.920,0:14:30.145
builds new propositions whose truth[br]depends on nothing other than the truth of

0:14:30.145,0:14:35.569
the ingredient propositions that go into[br]building them. That kind of propositional

0:14:35.569,0:14:41.110
connective is one that we're going to call[br]a truth functional connective. Now and,

0:14:41.110,0:14:47.261
the propositional connective and, is a[br]truth-functional connective. But not all

0:14:47.261,0:14:52.230
propositional connectives are[br]truth-functional connectives. For

0:14:52.230,0:14:58.065
instance, suppose we try to construct a[br]truth table for I hate it when, the

0:14:58.065,0:15:04.728
propositional connective I hate it when.[br]Well, so consider the proposition, it's

0:15:04.728,0:15:11.365
raining. Now that proposition could be[br]true or it could be false. Sometimes its

0:15:11.365,0:15:17.748
true, sometimes its false. So lets[br]consider these two possible situations. So

0:15:17.748,0:15:24.555
suppose the proposition Its raining is[br]true. In that situation is it going to be

0:15:24.555,0:15:31.802
true or false that I hated when its[br]raining. Could be either one, it could be

0:15:31.802,0:15:38.770
raining even though I enjoyed the rain or[br]it could be raining even though I hate the

0:15:38.770,0:15:45.040
rain. Or it could be raining even though[br]I'm indifferent to the rain. So the truth,

0:15:45.040,0:15:53.602
of I hate it when it's raining isn't[br]determined by its raining. So, if it's

0:15:53.602,0:16:00.012
true that it's raining. It's unclear[br]whether, I hate it when it's raining.

0:16:00.012,0:16:06.466
Could be true, could be false. Suppose[br]it's false, that it's raining. Then, is it

0:16:06.466,0:16:12.327
going to be true or false that I hate it[br]when it's raining. Again, could be either

0:16:12.327,0:16:18.264
one. The truth of I hate it when it's[br]raining isn't determined by the falsehood

0:16:18.264,0:16:23.900
of it's raining. So, even if it's not[br]raining, that doesn't mean anything one

0:16:23.900,0:16:29.836
way or the other for whether I hate it[br]when it's raining. So once again, if it's

0:16:29.836,0:16:37.377
false that it's raining, I hate it when[br]it's raining, could be true or could be

0:16:37.377,0:16:44.211
false. So the proposition, I hated when[br]it's raining. Whether that proposition is

0:16:44.211,0:16:51.390
true or false doesn't just depend on the[br]truth or the falsehood of the proposition

0:16:51.390,0:16:57.964
it's raining, that you built this[br]proposition out of using the propositional

0:16:57.964,0:17:04.690
connective, I hate it when. Because of[br]that, the propositional connective I hate

0:17:04.690,0:17:10.303
it when is not a truth-functional[br]connective. It's different from the

0:17:10.303,0:17:16.893
propositional connective and which is a[br]truth-functional connective. A moment ago,

0:17:16.893,0:17:23.482
we built a truth table for a proposition[br]that was built using a truth-functional

0:17:23.482,0:17:29.014
connective, specifically the[br]truth-functional connective and. But I'd

0:17:29.014,0:17:35.583
like us to notice something about that[br]truth table. Notice that if we replace the

0:17:35.583,0:17:40.633
particular propositions that we are[br]putting together using the

0:17:40.633,0:17:47.208
truth-functional connective and, to make a[br]different resultant proposition. Even if

0:17:47.208,0:17:53.617
we change the ingredient propositions the[br]truth table looks the same. Let me show

0:17:53.617,0:17:59.757
you what I mean. Suppose instead of having[br]Jack and Jill finally talked. We have Jack

0:17:59.757,0:18:05.386
finally walked and Jill finally talked. So[br]now, we're connecting two different

0:18:05.386,0:18:10.942
propositions using the truth-functional[br]connective and. There's Jack finally

0:18:10.942,0:18:16.863
walked, there's Jill finally talked. And[br]then we connect them up into Jack finally

0:18:16.863,0:18:22.709
walked and Jill finally talked. Okay, now[br]the truth of that resultant proposition,

0:18:22.709,0:18:28.261
Jack finally walked and Jill finally[br]talked. How does that depend on the truth

0:18:28.261,0:18:32.936
or false sort of the ingredient[br]propositions?Jack Jack finally walked and

0:18:32.936,0:18:38.629
Jill finally talked. Well it's the same[br]patent we saw earlier. If it's true that

0:18:38.629,0:18:44.520
Jack finally walked, and it's also true[br]that Jill finally talked. and it's also

0:18:44.520,0:18:49.393
true that Jill finally tal ked.[br]Then it's going to be true that Jack

0:18:49.393,0:18:55.006
finally walked and Jill finally talked. If[br]it's true that Jack finally walked, but

0:18:55.006,0:19:00.227
it's false that Jill finally talked, then[br]it's going to be false that Jack finally

0:19:00.227,0:19:05.449
walked and Jill finally talked. If it's[br]false that Jack finally walked but it's

0:19:05.449,0:19:10.736
true that Jill finally talked, then it's[br]going to be false that Jack finally walked

0:19:10.736,0:19:16.156
and Jill finally talked. And if it's false[br]that Jack finally walked and it's false

0:19:16.156,0:19:21.378
that Jill finally talked, then it's going[br]to be false that Jack finally walked and

0:19:21.378,0:19:28.182
Jill finally talked. So even if we change[br]one of the ingredient propositions as long

0:19:28.182,0:19:33.882
as we're combining propositions using the[br]truth functional connective and, the

0:19:33.882,0:19:41.607
overall truth table looks the same. We[br]could change them some more to illustrate

0:19:41.607,0:19:47.520
this point. I suppose if we changed Jill[br]finally talked to the zebra escaped.

0:19:59.693,0:20:10.322
Change it here, so notice what we have[br]here. We take two propositions, the

0:20:11.950,0:20:20.190
proposition Jack finally walked, and the[br]proposition the zebra escaped. And we put

0:20:20.190,0:20:28.023
them together with the truth functional[br]connective, and to create a resultant

0:20:28.023,0:20:35.246
proposition Jack finally walked and the[br]zebra escaped. Now, when is that

0:20:35.246,0:20:43.012
proposition gonna be true? Well, again it[br]depends just on, when these propositions

0:20:43.012,0:20:48.244
are true. So if it's true, that Jack[br]finally walked, and it's also true that

0:20:48.244,0:20:53.760
the zebra escaped, then it's going to be[br]true that Jack finally walked and the

0:20:53.760,0:20:58.523
zebra escaped. If it's true that Jack[br]finally walked, but it's false that the

0:20:58.523,0:21:03.269
zebra escaped, then it's going to be false[br]that Jack finally walked and the zebra

0:21:03.269,0:21:07.895
escaped. If it's false that Jack finally[br]walked and it's true that the zebra

0:21:07.895,0:21:12.763
escaped, then it's going to be false that[br]Jack finally walked and the zebra escaped.

0:21:12.763,0:21:17.327
And finally, if it's false that Jack[br]finally walked and it's false that the

0:21:17.327,0:21:22.074
zebra escaped, then of course it's going[br]to be false that Jack finally walked and

0:21:22.074,0:21:27.047
the zebra escaped. So once again, same[br]truth table even if we change the

0:21:27.047,0:21:33.412
ingredient propositions that we're putting[br]together with the functional connective an

0:21:33.412,0:21:40.459
d to make the resultive proposition. Now,[br]since the truth table stays the same even

0:21:40.459,0:21:47.530
when we change these propositions up on[br]top. We could represent that fact, by

0:21:47.530,0:21:55.242
replacing these propositions altogether[br]with variables that can range over any

0:21:55.242,0:22:03.052
proposition. So, for instance, instead of[br]saying Jack finally walked, we could just

0:22:03.052,0:22:11.057
have a variable here call it P1 Our first[br]proposition. Instead of saying the zebra

0:22:11.057,0:22:17.988
escaped we can have a variable there, call[br]it P2 our second proposition. And finally,

0:22:17.988,0:22:26.852
when we put those two propositions[br]together using the truth functional

0:22:26.852,0:22:36.019
connective and. We'll have P1 one and. P2.[br]So that's going to be our resultant

0:22:36.019,0:22:43.264
proposition, P1 and P2. And whatever[br]exactly that is, is going to depend of

0:22:43.264,0:22:51.262
course on what P1 is and what P2 is. But[br]whether this third proposition is true or

0:22:51.262,0:22:59.284
false again is only going to depend on the[br]truth or falsehood of P1 and of P2, when

0:22:59.284,0:23:07.507
P1 whatever exactly that is, is true and[br]P2 is true then. The proposition p1 and p2

0:23:07.507,0:23:15.729
is going to be true. Whatever proposition[br]that is, is going to be true. And in every

0:23:15.729,0:23:23.852
other possible situation, that proposition[br]is going to be false. So no matter what

0:23:23.852,0:23:32.075
proposition we have for P1 and P2 their[br]conjunction P1 and P2 is going to be true,

0:23:32.075,0:23:39.619
just in those situations when P1 and P2[br]are both true. That's the truth table for

0:23:39.619,0:23:46.053
the truth functional connective and, which[br]we'll also call conjunction. In the next

0:23:46.053,0:23:52.143
lecture, we're going to see how we can use[br]the truth table for the truth functional

0:23:52.143,0:23:58.156
connective and. To figure out the rules[br]for evaluating deductive arguments that

0:23:58.156,0:24:04.246
rely on the truth functional connected[br]and. And in the following three lectures,

0:24:04.246,0:24:10.640
we'll see how we can use the truth tables[br]for other truth functional connectives, to

0:24:10.640,0:24:16.502
figure out the rules for evaluating[br]deductive arguments that use those other

0:24:16.502,0:24:18.938
connectives. See you in next lecture.