[Music] hey welcome back to critical thinking last time we did an introduction to deductive logic and this time we're going to be focusing on our very first topic within the study which is going to be categorical logic and that's actually a really good place to start because categorical logic was one of the first things developed within the discipline of logic going all the way back to the ancient greeks at least to the time of aristotle so why don't we go ahead and take a look and dive right in we're going to be spending several episodes on categorical logic and since our emphasis here is on the basics we're going to recap briefly some of what we covered last time like math deductive logic requires we build from the ground up you can't move on to algebra until you know how to add and multiply and you can't test the validity of a categorical syllogism until you know all of its building blocks categorical propositions are the building blocks of the categorical syllogism and as you recall these are statements that can be analyzed as being about relationships among classes or categories of things they affirm or deny that one class is wholly or partially included in another class or we could say they include or exclude all or some of one category x in or from another category y using the same examples i gave you last time all dogs are mammals and some dogs are not german shepherds you may also recall that because we can either affirm or deny wholly or partially we end up with four subtypes universal affirmatives affirming something about an entire category universal negatives denying something about an entire category particular affirmatives affirming something about part of a category and particular negatives denying something about part of a category and since categorical propositions must affirm or deny something medieval logicians came up with a convention of giving each type a single letter abbreviation based on the latin words effermo i affirm and nego i deny the a and i from a pheromone are used of affirmative propositions and the e and o from nego are used of negative propositions the a is a universal affirmative and e is a universal negative i stands for particular affirmative and o stands for the particular negative here are examples of each type written in standard form a all dogs are mammals e no dogs are cats i some dogs are wolves and o some dogs are not wolves yet categorical propositions are not the most fundamental pieces that we're going to be working with they too are made of more fundamental building blocks and there are four of these as well the most important of which in a way are the two terms since they relate what it is we're actually talking about the first is the subject term the thing or thought about which an assertion is being made in the proposition all dogs are mammals dogs is my subject the second is the predicate term that which is asserted about the subject here the category mammals third is the copula that which joins the subject to the predicate and in logic it is always the verb to be is or is not are or are not was or was not present tense past tense it doesn't matter in all dogs or mammals the coppola is are rather than are not because this is an affirmative claim not a negative the copula determines what we call the quality and i'll get to this in just a minute the fourth and final part is the quantifier that which indicates extent or number being discussed what we call the quantity in a categorical proposition it's either the whole category or some part of it so we use words all or some in the universal negative the word no stands in for both the quantifier all and the negative aspect of the copula and among our particulars the quantifier sum can indicate any amount of a category in our example here our quantifier is all so all categorical propositions have both quality and quantity when we speak about quality we're referring to whether the proposition affirms or deny something it has to do one or the other as we said it's always indicated by the copula there are two possible qualities and only two possible qualities in a categorical proposition affirmative affirming something saying it is the case and negative denying something saying it is not the case a and i have an affirmative quality e and o have a negative as their names indicate all dogs are mammals which is affirmative no dogs are cats in other words dogs are not cats quantity specifies whether proposition refers to all or some of the members of the subject category has to do again one or the other and it's indicated by the quantifier all or some and there are only two possible quantities universal which refers to the whole class all of something and particular which refers to a portion which is any amount of a class some of the class so all dogs are mammals universal no dogs are cats in other words all dogs are not cats that's also universal and we never write a universal negative this way because the construction all are not can be ambiguous as we'll see in just a minute and some dogs are wolves that would be particular if we want to visually represent the relationship between subject and predicate terms we could use euler circles named after the 18th century mathematician leonard euler and there are only five possible relationships that can be expressed by each of the four proposition types a all a is b may be illustrated in two ways where a and b completely overlap like all circles are geometric figures with all points on their circumference equidistant from one central point or where a is entirely contained within b like all circles are geometric figures i some a is b may be illustrated in four ways the two we just looked at and two additional ways where a and b slightly overlap which looks much like a venn diagram another system for illustrating categorical propositions that we might take a look at down the road and where b is entirely contained within a like some dogs are wolves e no a is b can be illustrated with a fifth diagram two completely separate circles like no dogs are cats o some a is not b has already been illustrated with diagrams three through five so we don't need to add anything further beyond these five drawings notice the quantifier sum can apply to any amount of a class up to and including the entire class which is why diagrams one and two illustrate an i proposition and diagram 5 illustrates an o so if all dogs are mammals then it's also true that some dogs are mammals and that may sound strange but if you think about it for a bit it makes perfect sense now that we know the four parts of a categorical proposition let's spend the rest of our time looking at how we might translate ordinary sentences into proper categorical form so that they'll formally resemble one of our four types of propositions with all four parts clearly distinguished this might be tricky but it's a necessary skill to master because formal logic demands an analysis of an argument's form and if we can't clearly render the argument into a simplified form it makes our job much more difficult but unfortunately there are no hard and fast rules the key to successfully translating a non-standard proposition into standard categorical form is to clearly understand what is being stated and to translate the meaning of the original proposition not simply the words let's start with propositions dealing with singular subjects sometimes we assert something about a single object in these cases it's important to remember that the singular object can form a class or category of one or a class occupied by only one member and if there is only one member then we'll be speaking about the entire class and this is the situation generally when we talk about proper nouns people's names for example bill went to the store bill is the subject and he's also a class that includes only one thing bill himself so here it's appropriate to add the proper quantifier to the term bill and since he is the entire class it'll be the quantifier all all bill went to the store believe me when we're done it won't sound pretty but we're after clarity not poetry of course we're not done yet we only have two of our four parts in place what do we do when we have propositions lacking the proper copula and or including a prepositional or agitable phrase standard categorical form requires the copula to be no other verb is going to work if the statement includes some other verb it'll need to be rewritten using a form of the verb to be and if the subject or predicate term contains a prepositional phrase or adjective rather than a noun a generic noun of the intended class can be added let's keep working with bill here all bill went to the store this statement includes the verb to go not the verb to be it also contains a prepositional phrase to the store in the predicate position we can start by adding the verb to be all bill is next we'll add our generic noun to what we were given initially all bill is a person who went to the store and to make sure it's as clear as it can be it's good to compartmentalize each element quantifier subject copula predicate so all bill is a person who went to the store and i like to stick parentheses around my subject and predicate and often i'll replace long and complicated terms with single letter placeholders to make it even simpler sometimes sentences already include words or clues indicating quantity in these cases we may need to remove words and replace them with the only quantifiers acceptable for standard categorical form all some and no which again is not technically a quantifier but we're going to treat it as one some words that indicate quantity could be things like anyone everyone whoever the article a the all not etc the important thing is to stop and make sure you know what's being stated find the meaning here's an example anyone without hair is bald we could translate it like this all people without hair are people who are bald it completely captures the meaning has all four parts required of the standard categorical form and it's logically clear and precise here are some other examples the dolphin is an aquatic mammal translation may require a judgment call most likely we're not talking about one specific dolphin out of the class of dolphins but the entire class as a whole so we might translate it this way all dolphins are aquatic mammals the morning paper was soaked by the rain this probably doesn't refer to every morning paper people still get those the neighbor's paper could be fine so it might be okay to translate it as some morning papers were paper soaked by the rain now the tricky one all dogs are not black now be careful i mentioned that the construction all are not can be ambiguous it sounds like a universal all and it definitely sounds negative so it would be tempting to translate it as an e type of proposition no dogs are black but this probably isn't what the speaker intended to communicate so even though it uses the quantifier all it doesn't mean all it doesn't mean there are no black dogs rather it's expressing the idea that not all dogs are black and this we can render as some dogs are not black it's a particular negative and note all s is not p often means sum s is not p what about exclusive propositions propositions including exclusive terms like only or none but they'll translate into universal propositions a types but it requires two steps to translate number one we add the universal quantifier all when we remove the only or the none but and two we reverse the order of the subject in the predicate for example none but animals are veterinary patients this does not mean all animals are veterinary patients the squirrel that ran out in front of my car isn't on his way to the hospital rather all veterinary patients are animals now here's an important exception with the word only and it happens when only is preceded by the definite article v the key as always is understanding the meaning and asking questions such as is my translation logically equivalent to the original sentence in other words is it possible that one can be true while the other is false if it is possible the sentences are not equivalent here's an example the only person in the pool is tom which means that the entire class of people in the pool is contained in a single individual tom but it won't translate like we saw with singular subjects and proper nouns and we don't need to swap the subject in predicate because it does not communicate merely that all tom is a person in the pool that's true for tom that he's in the pool but this proposition doesn't rule out the possibility that others are in there with him which is completely ruled out by the original proposition so if our new sentence can still be true if another person is in the water and our original sentence would be false in that same situation they can't be logically equivalent here's what we want all persons in the pool are persons identical with tom remember it doesn't have to be pretty now what about negative propositions propositions making a denial of an entire class s is not p the entire class s is not part of class p and the denial of an entire class is a universal negative proposition an e type so these should be relatively straightforward we know the form that they take although the negative quality is actually the result of the copula is not we can separate the negation from the verb and place it in the quantifier position as universal so dogs are not humans becomes no dogs are humans like we said no is technically part of the negative copula and it's not a quantifier itself the actual quantifier here is all because we're talking about all dogs not being something but universal negatives are never written this way and we've already seen why and lastly we have acceptive propositions which assert that all members of a class accept a portion of a subclass are members of another class these actually include two claims in one one the claim that all members of the class not in the subclass are in the predicate class a universal affirmative and two the claim that all members of the subclass are not in the predicate class which is a universal negative the key word to look for is accept let's look at this example all police officers except those wearing gas masks passed out from the gas this must be translated into two separate categorical propositions but before we translate we need to establish the class and the subclass here we have the class police officers and they can be divided into those wearing gas masks and those not wearing gas masks and believe me when i say it's best to distinguish the two groups by the use of the prefix non gas mask wearing officers and non-guest mask wearing officers we'll discuss term compliments and how we can play with them in our next episode but for now just trust me our universal affirmative proposition is going to be all non-gas mask wearing officers are officers who passed out from the gas and our universal negative will be no gas mask wearing officers are officers who passed out from the gas now of course there are many other situations that are going to require translation and there may not always be simple rules to refer to so when it comes down to it the key to all translation is determine what the statement means then translate the meaning not the words you probably weren't expecting to be doing translation when you're doing logic it's not an english course it's not a foreign language course but logic is a language of its own and it's a skill like i said that is essential that you master next time we're going to be continuing on with categorical logic specifically we'll be looking at the square of opposition and immediate inference so until then take care and i'll see you in the next video [Music]