Transcription
If you are a complete beginner when it comes to logic, you're in the right place. This is a small course that I've put together during the past month or so. In this video I will go through these topics. Aristotle's Laws of Thought, Simple Truth Tables, Deductive Reasoning, Redundancy and Complex Truth Tables.
If you watched this whole episode, I think you will have gained some understanding of the basics of logic and how it works. But as it is with most things, practice is needed. You can't expect to become a master without practicing the craft. But if you do, I believe it can do wonders to all your rational inquiries. So, let's begin.
Logic is a huge part of intelligence and is arguably even the foundation for thought itself. If you think about it, if you're not thinking logically or rationally, what are you doing? Well, whatever it is, it would be irrational, which means that it is either false or meaningless. Well yeah, I mean sure it can be true as well, but the chance is significantly reduced. I believe that the fastest way to improve one's own intelligence is through the study of logic. Some people might find logic to be so fundamental that it is easy and simple so one doesn't have to study it. But it is an incredibly deep topic within itself. But one doesn't have to become a logician in order to improve one's intelligence. Just a little bit of studying might yield fantastic results in all other domains of study since logic is so fundamental. Indeed, more fundamental than mathematics if you ask me.
Probably the first real logician was Aristotle. He laid down what is commonly referred to as the laws of thought. These rules simply cannot be broken if you are to think rationally. The laws are 1. The law of identity. 2. The law of non-contradiction. And 3. The law of the excluded middle. The law of identity simply states that A equals A. If I have decided to call something a chair, for example, then that object is a chair and not something else. In other words, everything is identical with itself. Whatever is, is.
Something cannot be A and not be A at the same time. Which is indeed the second law. The law of non-contradiction. Something cannot both be a chair and not be a chair at the same time. And then we have the third one. The law of the excluded middle. This one states that for any proposition, it or its negation has to be true. A or not A, for example. For instance, if we have a scale from, let's say, hot to cold, then it's not possible for something to be both hot and cold at the same time. Either it's hot or it's cold. Some might dispute this and say, hey, something can be lukewarm, which is sorta both hot and cold at the same time. But Aristotle might reply by saying that you have only introduced a new scale here, which means that something cannot be both hot and lukewarm at the same time.
These three laws are fundamental to logic. If you don't follow them, you're indeed thinking irrationally. And if you think about it, almost everything within any domain of study is dependent upon these three laws. For instance, if not everything was identical with itself, i.e. A equals A, then we would never be able to talk about anything really. Words would lose their meaning. Hence, the study of logic is the foundation for thought itself.
Another important thing that was systematized by Aristotle are the syllogisms, which is a form of deductive reasoning. Take this argument for an example. All men are mortal, Socrates is a man, therefore Socrates is mortal. This is an Aristotelian syllogism, and it is both valid and sound, which means that both the premises here are true, and they necessarily make the conclusion true as well. Deductive reasoning is a big subject in itself.
Another very important subject within logic is truth tables. They are a key in understanding logic. So if truth tables are the foundations of logic, and logic is the foundation of thought, then that's a proper place to begin. If you don't know anything about truth tables, you're in the right place. This will be a comprehensive beginner's guide. In this episode, I will go through the truth tables for the Boolean connectives and or a negation. So here are the operators. Logical, not, and or.
To begin with, a truth table is a tool that displays all possible combinations of truth values for a given sentence in an organized manner. It helps us to determine what is true or false in different situations. Before we begin drawing our tables, we need to know how many variables we have since the number of rows is calculated as such, 2 to the power of the number of variables we're dealing with. So let's say that we have a and b, that is two variables, which means that we need four rows. But if we have, let's say, a, b, and c, it would be eight rows, since 2 to the power of 3 equals 8. Just a quick tip, whenever I see a logical sign, I sometimes whisper the name of it in my head. That way it all makes more sense, like not a, for an example.
Now let's begin with the simplest one, and that would be negation. A negation is simply a denial of a proposition, in other words, claiming that something is not the case. Notice that the negation sign is not a minus sign. One might think it is the same thing, but they are very different. So let's begin by drawing up our tables. This one is simple, since we only have one variable, a. And what is 2 to the power of 1? Well, 2. So we only need two rows. So let's fill out the different possible truth values here. If we only have one variable, it can only be either true or false, right? So let's fill out those possibilities here. True and false. Then, if we have the negation, not a, in other words, they will always have the opposite truth value, right? Let's say that a means that I have a hamburger. Then, if it is the case that I have a hamburger, as it is in the case here, the negation of it, it is not the case that I have a hamburger, is false, right? And if it was the case that I didn't have a hamburger, then the negation is true. Does it make sense? In other words, the negation always reverses the truth value. So here we have the truth table for negation. Simple enough? Let's move on.
Now to logical and, or as it's sometimes called, conjunction. A quick tip. To remember this simple, I sometimes think that it looks like an a missing a line for and. So here we have a sentence a and b, but let's make it more easy and concrete by switching out the letters. Let's say that a means that my friend will give me a hamburger and b means that my friend will give me fries. So before we begin, we have two variables here, right? So that means 2 to the power of 2, which is 4. So we need 4 rows. Then what we want to do is a fully exhaustive truth evaluation. In other words, test out each possible situation. So we simply begin by filling out these possible truth values. So here we just write down true, true, false, false. And here under b we have to write it down in a different order so we can then test each possible scenario. So here instead we write down true, false, true, false.
Now the fun begins. Let's say that a friend of mine told me, "When we get to my place, I will give you a hamburger and fries." Okay, here we have 4 different hypothetical scenarios. First one, we arrive at my friend's place and he gives me both a hamburger and fries. Then he told me the truth, right? He gave me exactly what he told me before. So then we write down true here. But then in scenario 2 he gives me only a hamburger. But then he lied to me, right? Because I would complain and say, "Hey man, you said that I would get a burger and fries, not just a burger." So if a is true but b is false, that would make the conjunction of them false as well. And then in scenario 3 my friend gives me only fries. Same thing here, I would complain since he said that I would get both. So the conjunction is false. He lied to me. And lastly in scenario 4 we arrive at my friend's place and he doesn't give me anything. What kind of a friend is that? Well obviously then, the sentence is false. I would have felt that he lied to me. So here we have the truth table for a and b. A conjunction is only true when all of the variables are true.
So now let's move on to disjunction or logical or as it's called. Here is the logical symbol for or. And the way I remember it is that it looks like a v. V for versus. If that helps. So let's stick to the hamburger example. But this time around my friend tells me that I will get a hamburger or fries. Same thing here. We have four possible different scenarios. In the first scenario my friend gives me both a hamburger and fries. Some might find this one odd. But in this case it is in fact true. In standard logic we commonly have an inclusive disjunction rule. So in other words if both the disjuncts are true then the whole sentence is true as well. But I'm curious. Would you say that my friend lied to me when he said that I would get burger or fries but then brought me both? Please leave a comment down below. I would love to hear your thoughts. Then in scenario two he brings me only a burger. That makes the disjunction true. Right? Burger or fries. One is enough to make the whole sentence true. In scenario three he brings me only fries. Same thing here. That also makes the disjunction true. And lastly in scenario four he gives me nothing. But then that is false right? I mean he said that I would get something burger or fries. So this scenario is false. So here we have the truth table for disjunction. And it is true in all cases except when all of the variables are false.
So here we have all the truth tables that we have discussed. Make sure to subscribe next episode I'll go through the truth tables for conditionals and biconditionals. Or I mean logic and philosophy. In the So here are the operators that I will go through in today's episode. Logical, material, conditional and material biconditional. As we discussed in the last episode the number of rows is calculated by taking 2 to the power of the number of variables we're dealing with. And in today's example we only have two variables a and b. So we only need four rows. So now let's begin.
Here is the logical symbol for material conditional. What signifies a conditional sentence is every if then sentence. If a then b for example. When we discuss conditionals it's even more important to whisper the name of the symbols or in other words translating them into English. That way it makes way more sense. At least in the beginning. So I will recommend you to say if a then b rather than saying a arrow b. Also it's good to know that the proposition that comes before the symbol is sometimes referred to as the antecedent. And the proposition that comes after is referred to as the consequent. And just like before we want to do a fully exhaustive truth evaluation. I.e. test out each possible situation. So here we fill out the different possible truth values. True true false false. And then under b true false true false.
So now let's make this example a bit more concrete. Let's say that a means come over to my place and b means I will give you a hamburger. So let's say that my friend told me if you come over to my place I will give you a hamburger. In the first scenario here both of them are true. That will make the whole sentence true right? If it is true that I went to my friend's place and that he gave me a hamburger that will make the whole sentence true. In the second scenario I come over to my friend's place but then he didn't give me a hamburger. But wouldn't you say that my friend lied to me then? He said that I would get a hamburger if I came over to his place. So here the whole sentence is false. But then in scenario three let's say that I stayed at home instead but my friend missed me so he came over to me. With a hamburger. Some might find this one odd but in this scenario the sentence is true. Because we didn't really test out the whole statement did we? So if the antecedent is false but the consequent is true that makes the whole sentence true. This is sometimes referred to as a vacuous truth and this will be important for other reasons that I will explain later. In scenario four both are false. I didn't go to my friend's place and I didn't get a hamburger either. This one is true as well oddly enough since we didn't test out the statement did we? I wouldn't know if my friend told me the truth or not since I never went to his place and I didn't get a hamburger. So here we have the truth table for material conditional and it is false only when the antecedent is true and the consequent is false.
Now let's move on to bi-conditional. Here is a symbol for bi-conditionals. This one is sometimes referred to as equivalence as well but I do not appreciate calling it that for sometimes equivalence can mean something else which I will discuss in a future episode. But bi-conditionals is very important to know if you're into analytic philosophy since it's commonly used for stipulating definitions. Let's stick to the last example but this time around my friend says "if and only if you come over to my place I will give you a hamburger". Same thing here we have four possible different scenarios. In the first example both are true and that would make the whole sentence true as well right? I came over to my friend's place and he gave me a hamburger. In scenario two it is true that I went to my friend's place but he didn't give me a hamburger. Then the sentence is false right? He lied to me. If and only if I come by his place will I get a hamburger. And here in scenario three it is false that I went to my friend's place but true that he gave me a hamburger. But then that is false. He said if and only if I came over to his place would he give me a hamburger. So here my friend lied to me. So in this scenario the whole sentence is false. And lastly in scenario four it is false that I went to my friend's place and false that he gave me a hamburger. This makes actually the whole sentence true since we didn't really perform a test to see if my friend lied to me or not did we? In other words if both are false it doesn't contradict the whole sentence. So here we write down true. Like I said before, biconditional is very important for analytic philosophers. For an example some philosophers claim that someone possesses propositional knowledge if and only if he or she has a justified true belief. So k if and only if j and t and b. Now if I would say instead if k then j and t and b and we look at the truth table here in the third scenario that will make the whole sentence k is false that I have knowledge but true that I have a justified true belief true. But that's not what we want right? Because here what is said is that you can have a jtb without it being knowledge at the same time. So in other words something more is needed for knowledge. But what we want is a definition of what knowledge is. So whenever you have knowledge you also have a justified true belief and vice versa. This is why the biconditional is so important for analytic philosophers. So here it should be instead if and only if. A lot of information there. But if you learned something new then consider subscribing.
What is Deduction? A deductive argument is a type of inference where the conclusion logically follows from the premises. In other words, it is impossible for the premises to be true and the conclusion to be false at the same time. For instance, all men are mortal, Socrates is a man, hence Socrates is mortal. This argument is banal and simple and we can see instantly that it is valid. But how do we go about when we have a more complex argument and want to figure out whether it is valid or not, in a more systematic way? Simply through truth tables. We have gone through the basics of truth tables, but now let's use them in a more practical way. There are three simple and classical variants of deductive arguments that I want to go through in today's episode. Firstly, modus ponens. Secondly, modus tollens. And thirdly, disjunctive syllogism. Let's begin with the first one.
This is the structure of a modus ponens argument. If P then Q, P therefore Q. But as always, let's make it a bit more concrete. Let's say that P means there is sugar in the cookie and Q means that the cookie is sweet. So here the whole argument can be translated to, if there is sugar in the cookie, then the cookie is sweet. There is sugar in the cookie, therefore the cookie is sweet. Is this valid? Well, yes, but let's look at the truth table to see if it is. So let's begin to check the first premise. It's true here, here and here. And then we check the second premise. It's true here and here. But as you can see, the only place where both the premises are true is here and on the very same row the conclusion is true as well. Hence the argument is valid. If the conclusion here would have been false, the argument would have been invalid. But let's take another argument instead which looks very similar. If P then Q, Q therefore P. And same thing here, let's translate it. If there is sugar in the cookie, then the cookie is sweet. The cookie is sweet, therefore there is sugar in the cookie. This one is a bit fishy, right? There is something wrong with this argument. Let's check out the truth table once more. Our first premise is true here, here and here, just like before. But then the second premise is true here and here. So we need to look at these two rows here. Here in scenario one, the conclusion together with the premises are true. So far so good. But here in scenario three, both the premises are true, but the conclusion is false. What makes an argument valid is when the premises necessarily makes the conclusion true. Hence the argument is fallacious since there is a possibility for both the premises to be true at the same time as the conclusion is false. In other words, there could be something else rather than the sugar that makes the cookie sweet, right? This fallacy is sometimes referred to as affirming the consequence.
So now let's move on to modus tollens. If P then Q, not Q, therefore not P. So if we translate this argument, it would sound something like this. If there is sugar in the cookie, then the cookie is sweet. The cookie is not sweet. Therefore, there is no sugar in the cookie. That sounds like a valid argument, right? Let's check the table. Here is a slightly bigger table where I included the negations. Then we begin by checking the first premise. Same thing here. The first premise is true in these three places. And then the second premise is true here and here. But here both of the premises are true at the same time as the conclusion and we can't find any other row where both the premises are true but the conclusion falls. So the argument is valid. Let's also take this argument that looks like a modus tollens. If P then Q, not P, therefore not Q. If there is sugar in the cookie, then the cookie is sweet. There is no sugar in the cookie, therefore the cookie is not sweet. This is a fallacy and it's commonly called denying the antecedent. So if we look at the truth table, we have the row here where both the premises are true but the conclusion is false. Hence it is a fallacy.
Now on to the last one. Disjunctive syllogism. Let's take this argument. P or Q, not P, therefore Q. Here I think we should switch the variables meaning to make it sound more reasonable. So I will give you a hamburger or I will give you fries. I didn't give you a hamburger, therefore I gave you fries. This one might sound a bit fishy but let's look at the truth table to see if it's valid or not. P1 is true here, here and here. P2 is true here and here. But here both of the premises are true at the same time as the conclusion and here the conclusion is true as well. Hence the argument is valid.
Now you know the basics of utilizing truth tables to analyze arguments. The logical implication that we have used a lot in today's episode is actually redundant. In other words, we don't really need it. Logical, not and or. Conditional, bi-conditional. These operators are more or less based in how our language works. But some of you maybe had some objections to a few of the truth tables, especially when it comes to logical or. Let's look at the truth table and let's say that A means I will give you a hamburger and B means I will give you fries. Here in the first scenario both of them are true and in this case the disjunction of them is true. If I said to you that I will give a hamburger or fries and then gave you both, did I lie to you? Some might say yes, but in standard logic we have an inclusive disjunction rule for really good reasons that I will discuss in a future episode, but there is a logical sign for an exclusive disjunction. In this case the truth table looks like this instead and this is also sometimes referred to as XOR for exclusive disjunction. But this logical sign is redundant. In other words, it is not needed for we can express its meaning with the operators we already have discussed A or B and it is not the case that A and B. So here we're saying for an example that you will get a hamburger or fries and that is not the case that you will get a hamburger and fries. Thus we have an exclusive disjunction. This formulation is logically equivalent as the exclusive disjunction operator. In other words they are interchangeable or means the same thing. And to show why this is the case we simply need to look at their respective truth tables. As you can see the truth tables are identical with each other, hence they are logically equivalent. And to express this is through the symbol of logical equivalence. This symbol is different from biconditional since logical equivalence is not a logical operator. That was why I didn't like calling material biconditional for equivalence. So we don't really need the exclusive disjunction and there are actually more operators that are redundant in the same manner and we can show this by looking at the truth tables. If we start with the biconditional, as the logical symbol sort of shows us it is simply a double material conditional, an arrow that goes both ways. So a biconditional can be expressed as such if p then q and if q then p. And to show this we simply need to look at their respective truth tables. As you can see they are true and false in exactly the same places, hence they are logically equivalent. And in matters of fact logical material conditional is itself also redundant. Take the statement if there is sugar in the cookie then the cookie is sweet. This statement is logically equivalent to there is no sugar in the cookie or the cookie is sweet. Why? Well same thing here if we look at the truth table they are true and false in exactly the same scenarios. Hence every logical implication can be expressed through a disjunction and a negation. Therefore every biconditional can also be expressed through a disjunction and a negation right? So we don't really need any of these. In other words all logical operators can be expressed by only using these three operators, logical and or and negation. And the funny thing is it can actually be reduced even further, but then we need two completely new operators one called noned not and and nor neither nor. Here is the truth table for those operators and here is the thing all logical operators can be expressed through these two which means that we can create a fully complete logical system with only two operators. So logic can indeed be reduced to binary functions and what else is binary? Well ones and zeros well binary code. I'm not a coding expert but this is certainly related. But just because we can reduce everything to binary code doesn't mean that we have to. It can become incredibly complex with only two operators. It is way simpler to use all the standard operators for they are more grounded in common language.
In today's episode I would like to go through some more complex arguments. So now let's take this argument for an example. P if and only if Q. P or R not Q therefore R. Just looking at it like this is tough but as always let's make it a bit more concrete. Let's say that P means come over to my place Q means I will give you a hamburger and R means I will give you fries. So this argument can be translated to this. If and only if you come over to my place will I give you a hamburger. You come over to my place or I will give you fries. I did not give you a hamburger therefore I gave you fries. Is this argument valid? It sounds like it right but let's check our truth table to be sure. In this argument we have three variables right? So here we need eight rows. Remember the number of rows is calculated by two to the power of the number of variables we're dealing with. And as always we want to do a fully exhaustive truth evaluation. We begin by filling out underneath P true true true true false false false false. Then under Q true true false false true true false false. And lastly under R true false true false true false etc. There is a way to be more systematic when filling out these initial values by counting in binary but I believe one can make an entire video about that subject but I'll leave a link down below if you want to know more about that. Good so now we then fill out the individual premises of the argument. Let's begin with the simplest one not Q. This one is simple since it's only the reverse value of Q. Then we have P if and only if Q. Now here it can be a little bit tricky since we need to remember the truth table for biconditional while at the same time compare the already existing rows with each other and the biconditional is true only when all of the variables have the same truth value. So here we see that P and Q have the same hence the biconditional is true. And then here true as well but then here they differ so false then false false false true right and lastly true. Cool then we move on to P or R. Same thing here we need to remember the truth table for disjunction while we compare the rows underneath P and R and a disjunction is only false when all of the disjuncts are false. So here it should be true true true true true then false here right then true and false. Nice now we have filled out all the truth values. Let's figure out whether it is valid or not. So here we need to find one row where all the premises are true and where the conclusion is true as well. And there cannot also be a row where the premises are true and the conclusion is false at the same time. So let's begin with not Q it's true in these rows then P if and only if Q is true in these places and then we have P or R and it is true everywhere except for here and here. And as you can see here on this row all the premises are true at the same time as the conclusion is true and we cannot find a row where the premises are true and the conclusion is false at the same time. Sweet now we know that the argument is valid right.
Let's do another one. So here we have an argument if it is not the case that P and Q then R or Q not P not Q therefore R. The translation of this argument could be something like this if it is not the case that you came over to my place and that I gave you a hamburger then it is the case that I gave you fries or a hamburger. You did not come over to my place I did not give you a hamburger therefore I gave you fries. Same thing here three variables so we have already done the groundwork. So if we begin with the simple premises here they are only negations so we can simply reverse their truth value. But the first premise here is a bit more complex. How do we go about in writing the truth table here? Where should we even begin? Well whenever we have a complex premise like this one we always begin with the operator that possesses the smallest scope. In other words the operator that affects the least amount of variables and operators and then we work ourselves up to the operator with the second to smallest scope and then the third etc etc until we've reached the main operator. So first we take the two operators that affects two variables let's begin with this one. A conjunction which is only true when all of the variables are true. So here we compare it with the truth value below p and q. So here we write true true false false false false false false and false. Then here we have a disjunction between r and q. So here it is true true true false true true true and false. Good now let's take the negation. As always the negation reverses the truth value and it is the truth value below the conjunction in this case. Good now we move on to the main operator in this premise which is the material conditional for it affects the whole premise and it is false only when the antecedent is true and the consequence is false. So here we need to compare the truth value below the negation sign with the disjunction sign. I know a bit tedious but we're almost done. So this one is true true true false true true true and false. Now just like before we need to find a row where all the premises are true and where the conclusion is true as well. And as you can see here in scenario 7 the premises are all true and the conclusion is true as well. We cannot find a row where all the premises are true while the conclusion is false at the same time. Hence the argument is valid.
As you can see writing complex truth tables with more than two variables already that point where truth tables become insufficient in my opinion and one should instead utilize natural deduction or logical proofs. In my opinion when you're doing truth tables you're not really doing logic or thinking logically. In a way you are just doing a mechanized work comparing values with each other. So if you really want to improve your logical thinking I would suggest to learn and practice natural deduction but I would say that formal proofs like that is beyond the basics of logic. That is big brain logic and it takes a lot of practice. At least it did so for me. But I would love to make a series on natural deduction here on this channel but that would require a lot of time for me. So if you would like to see that please leave a comment down below. But anyways make sure to subscribe if you want to know more about logic and philosophy in the future. Take care until the next one.