Hi, I'm David Feddes. And this talk is about conjunction based on Matthew van Cleves book  on logic and critical thinking. conjunction is an aspect of propositional logic. So before we go  any further about conjunction, let's just say a few words about propositional logic. It's an area  of formal logic that deals with the logical relationships between propositions. Quite  remarkable. propositional logic is about propositions. It's also sometimes called sentential.  Logic because it's dealing with sentences. A proposition is just a statement. A statement is  something that asserts something to be a fact not every sound you make, like hello or wow, is a proposition or a statement. But a proposition is a statement that is asserting something  goes snow is white. James is a pastor, the floor has been mopped, the dishes have been  washed, or you can put propositions together into one bigger proposition containing two or  more, the floor has been mopped, and the dishes have been washed. So again, a proposition  is just saying something making a statement. An atomic proposition is not a proposition that  goes Kapooie and as a huge explosion, or has anything sensational about it. The word atomic here is just referring to atom, it's the smallest unit. It's a proposition that makes just one claim  a statement that is saying just one thing. And there aren't any connectives No, we'll see in a  moment what a truth functional connective is. But examples are and or not, or if then, an  atomic proposition doesn't have connectives or conjunctions like that. It's just making one  single claim the floor has been mopped. The dishes have been washed. It would not be an  atomic proposition if you said the floor has been mopped, and the dishes have been washed.  A complex proposition is one that puts them together. A combination of propositions linked by  truth functional connective words like and or not, or if then, so the floor has been mopped and the dishes have been washed is a complex proposition and it has two atomic propositions, the floor has been mopped. The second atomic proposition is the dishes have been washed and  the truth functional connective is and another complex proposition the team won, or the team  lost. The atomic proposition is the team won the second atomic proposition is the team lost  and the truth functional connective is, or so it's not all that complicated, even though we're  using the word complex, complex proposition just puts more than one proposition together  using truth functional connectives. Now, what is the truth functional connectives. It's a way of  connecting propositions or statements such that the truth value of the resulting complex  proposition can be determined by the truth value of the propositions that compose it. So let's  say that you have two different propositions or statements. And if you know that both of them  are true, and you know what the connective is, then you'll know that the complex proposition  is true, you might be able to determine the truth value, whether it's true or false, of the whole  statement, once you know the truth value of each atomic statement or proposition within it,  and what those connectives are. If that sounds kind of complicated, forget it. We'll just move  forward and soon you'll get the hang of it by using these things. A conjunction. Suppose the  floor has not been mopped, but the dishes have been washed. Okay. In that case, if I assert  the whole conjunction, the the complex proposition, the floor has been mopped and the  dishes have been washed, then I have asserted something false. If only one of those things is true, then it's false to say the conjunction of both of those things is true. That's all we mean by conjunction is the two propositions are connected and you're saying that both of them are  true. A conjunction is a statement combining multiple statements P and Q it is sunny and it is  hot. And the conjunct is simply the individual statement within the combined statement. So  when we're talking about a conjunction, these are just terms we use the conjunction is the  whole statement that combines more than one the conjunct is each atomic proposition within  those statements. So P would be a conjunct Q would be a conjunct P and Q is the conjunction of those two conjuncts it is sunny is a conjunct, it is hot is a conjunct, it is sunny and it is hot is a conjunction. Not all that complicated even though the words are kind of long. So you have a conjunction and in logic a conjunction is going to be two or more things put together P and Q. 

And in that case, if both conjuncts P and Q, if both of those are true, then the conjunction is  true. If either conjunct is false if P is false, or if Q is false, then the conjunction P and Q is  false. If either one of them is false, then the statement P and Q is false. A conjunction is true  as a whole if and only if both conjuncts are true. There's an old proverb, a half truth is a whole lie. That proverb is reminding people to be transparent night not to fool people say well, I told  part of the truth. Now, if you apply that to logic, if you have a conjunct of two statements, and  one is false, then you might say, well, that's a half truth. Now it's a whole lie. In logic, the  conjunction of something that's true with something that's false, is itself false. It is sunny and it is hot, is false as a whole conjunction if either conjunct is false, it is sunny. But it happens to  be a wintry day. Even though the sun is shining. It is hot, but it's a cloudy day, even though it's very hot and humid, then you wouldn't say it's sunny and it is hot. It's true. You see what I  mean? The conjunction is false. If only one conjunct one member of the statements is true.  Now we're going to talk in logic about four truth functional connectives. Just this talk, we're  gonna be focusing on conjunction, but just to give you a little preview, conjunction in Symbolic Logic is symbolized by a dot a raised dot so If you have P with a raised dot and Q, you would  call that P and Q. negation is symbolized by the Tildy sign, which is kind of up on the upper  left hand side of your keyboard on a computer, which you may never have used, but negation  is symbolized by the tilde. And so if you see tilde P, it means not P conjunction is symbolized  by a wedge, which is also just a lowercase V. And if you have P wedge Q, you'd say that's P  or Q. That's the meaning of that. And then if you see a horseshoe, that means it's a  conditional. And so if you saw P horseshoe Q, it would mean P entails Q, or another way of  saying that is if P, then Q, let's not worry too much about the tilde the wedge of the horseshoe  just yet. But we are going to worry about that raised dot the conjunction and how it symbolizes P and Q. Now, a truth table is a way of evaluating the truth of propositions a truth table  represents how the truth value of a complex proposition depends on the truth values of the  propositions that compose it. So you take the different elements of a complex proposition, find out what the truth values of those are, or the potential truth values, and then you'll get the  whether the whole statement is true or false. I'll show you what I mean. A truth table looks like this. The heading shows your two atomic propositions the two individual claims P is one claim  you're making this, you know the sun is hot, Q, the sun is bright, whatever proposition you  have, but P is one proposition, the other one is Q and then for the conjunction P dot Q is how  you would assert P and Q. And the truth table will show you whether the conjunction is true  based on whether either of the atomic propositions is true. So let's look at that. If P and Q are  both true, then P and Q is true. Now what if P is true, and Q is false, the second line down  there the table well, then the conjunction of P and Q is false because remember, a half truth is a whole lie. The both statements have to be true for a conjunction to be true. Then we look at  the third line of the truth table. If P is false, and Q is true, that still means that the statement as a whole P and Q is false. And then to get the fourth line. You see that if P is false and Q is  false, well then obviously P and Q is false. on that table there's only one situation in in which  P and Q is true, there's four possibilities of combine. There's four possibilities for P and Q,  where you give them different truth values, either true or false. But there's only one set of truth values, which makes the conjunction true. That's if P is true, and Q is true. Now, truth  functional conjunctions are common, we've got some common words that often function in  that way, we've already used the word and, but there's other ways that we express  conjunctions in English sentences besides, and, but in logic, they're used the same way and  but yet, also, although still, however, moreover, nevertheless, now these words have different  shades of meaning, but in logic, they're used to hook two propositions together, and so they're conjunctions. Here's some truth functional conjunctions. Jesus is God, and the Spirit is God.  God is good. But Satan is bad. Although Jesus died, he lives those are two propositions and 

although is a truth functional conjunction, sin provokes God. Nevertheless, God loves sinners. So each of those complex propositions is made up of two atomic propositions. And then  there's a truth functional conjunction, you see, the and is a conjunction but's a conjunction,  although nevertheless, those all function as conjunctions, which are tying two different  propositions together into a complex proposition. Now, we can point out that English  conjunctions often carry more information than the symbolic connective dot here, here's an  example Bob voted, but Carolyn didn't. Now you've got two independent propositions Bob  voted, Carolyn didn't B symbolized Bob voted, C symbolizes Carolyn didn't. And so you  symbolize the conjunction of those two as B dot C. But you'll notice the dot is There  connecting B and C, but there you don't get a sense of contrast, because normally, we just  read B dot C as B and C. And there's a sense in which it's true. It's true, Bob voted, and  Carolyn didn't. But in a normal English sentence, you'll say Bob voted but Carolyn didn't  because you're giving a contrast, and the dot doesn't capture that nuance of contrast. So  sometimes logic, something gets left out when you try to sort it and, and make it into a  symbolic logic proposition. Here's another example. Bob brushed his teeth and went to bed.  That is logically the same as Bob went to Bed and brushed his teeth. But if you heard  somebody say that, it's that's kind of odd, most people don't go to bed and brush their teeth.  But in logic, with conjunction, you're not saying something about the order. And the  conjunction and use in English gives you a sense of the timing. So if you read Bob brushed  his teeth and went to bed, you take that as the order in which those two things happened. But  if you have B dot C, they could have happened in either order. And the dot just means that the two statements are connected, and both are being asserted. Now not all conjunctions are  truth functional. There are conjunctions where and or something else put together. And yet it's not truth functional, you can't determine that whether it's true or false. Whether the total  proposition is true or false based on trying to analyze either one, you can't identify two  independent propositions. In the sentence James and Alice are married. So it's not a true  functional statement. Because we'll I'll go into a little detail why that's hard to do with this  particular statement. When you hear James and Alice are married, what's the first thing that  comes to mind? Well, James and Alice are married to each other is what most of us would  have come to mind. But if you break that down into its atomic propositions, you would say  James is married. And your other atomic proposition is Alice is married. Now, if you take those two atomic propositions, they don't mean that James is married to Alice, do they? James is  married, Alice is married, but they don't mean that they're married to each other, even though  that's what the English sentence seems to indicate. So that's one reason for thinking this isn't  really a truth functional. Conjunction. Another possibility. James and Alice are married. So you might say, well, we're going to use our two atomic propositions to be James is married to each other. Alice is married to each other and then conjoined those two with a conjunction but  those two statements make no sense. James is married to each other. What's that about?  You can say James and Alice are married and break that down into James is married to Alice.  And the other statement would be Alice is married to James. Both statements are true. And if  you join them together, it will get you at what you're trying to say James is married Alice and  Alice is married to James. But the problem is the truth value of those two propositions. The  truth values are not independent of each other and the truth values of propositions in logic  have to be independent of each other if you make a conjunction out of them. So there's the  there's a difference, we have to know that some propositions are not truth functional, and the  conjunctions are not. Here's an example of a truth functional conjunction James and Alice are  persons James is a person, Alice is a person, those are your two atomic propositions. You  can bind them together into James is a person and Alice is a person or even shorten it up  James and Alice are persons, those two propositions have independent truth values, and they

can be expressed as the conjunction J dot A if you want to express it that way. So the way to  determine whether or not a conjunction is truth functional is to ask this question is it formed  from two propositions whose truth is independent of each other? When you finished this talk,  and when you've read carefully through Dr. Van Cleves, article about conjunction then you  can work through some exercises and help this to sink in 



Остання зміна: середа 9 березня 2022 08:50 AM