Lecture 7: Predicate Logic I PDF

LOGIC Lecture 7: Predicate logic I  PREDICATE LOGIC We have learned how to translate English sentences into a formal language called sentential logic. We have learned two different methods for evaluating validity: truth tables natural deduction  PREDICATE LO...

LOGIC Lecture 7: Predicate logic I  PREDICATE LOGIC We have learned how to translate English sentences into a formal language called sentential logic. We have learned two different methods for evaluating validity: truth tables natural deduction  PREDICATE LOGIC Those methods allow us to determine the validity of ANY argument in SL. So if an argument in SL is valid, you can: Show this with a truth table Loading… Prove it with natural deduction Unfortunately, this doesn’t mean that we can prove every argument that is valid in English.  PREDICATE LOGIC As we’ve discussed before, not every valid argument in English translates into a valid argument in SL. Consider this argument: All dogs are mammals. Spot is a dog. Therefore, Spot is a mammal.  PREDICATE LOGIC This is clearly a valid deductive argument. But if we translate it into SL: P Q Therefore, R. Loading… There aren’t any of our SL logical connectives to translate. But as an argument in SL, P,Q, therefore R is clearly INVALID.  PREDICATE LOGIC We are missing something about the first argument when we translate it into SL. The argument is valid, but SL doesn’t show why. We need a more powerful logical language to show why the argument is valid.  PREDICATE LOGIC There is a more powerful system – PREDICATE LOGIC, also called QUANTIFIED LOGIC (QL). Sentential logic breaks arguments up into logical connectives and atomic sentences. But predicate logic breaks arguments up in a different way: into connectives, SINGULAR TERMS, and PREDICATES.  PREDICATE LOGIC A SINGULAR TERM is a word or phrase that ‘picks out’ or REFERS to a single person, place, or object. So “Jenny” is a singular term; so is “Hong Kong”. But not ‘dog’ or ‘green’. We’ll call them NAMES for short.  PREDICATE LOGIC ‘Proper’ names are the most obvious examples of singular terms, but there are other types. DEFINITE DESCRIPTIONS: “The president of the United States” picks out a single person, Joe Biden. The REFERENT of this description changes over time, but at any given time there is only one referent.  PREDICATE LOGIC In predicate logic (QL), we use lower case letters to stand for names. So a, b, c, etc. can all be used for names. We can NOT use x, y, and z – those will be used later. For example, I might translate ‘Hong Kong’ using the lower letter h. Once I use a letter for a name, I can’t use it again for something else in the same translation. So if I have Bill and Bob, only one can be b. (It’s ok to use subscripts though – b1 and b2.) I CAN give the same object two names (e.g. both a and b can name Bill).  PREDICATE LOGIC Names combine with PREDICATES to make the atomic sentences of QL. Consider the sentence “Fluffy is a dog”. “Fluffy” is a name Loading… “is a dog” is a predicate Predicates assign properties or characteristics to objects: “is tall” “is happy” “is a good student”  PREDICATE LOGIC Predicates are represented by capital letters in QL (any – X, Y, Z ok). Again, can’t use the same letter for two predicates. In order to represent a whole atomic sentence, a name and a predicate must be combined. “Fluffy is a dog” = Df. “f” stands for “Fluffy” “D” stands for “is a dog”  PREDICATE LOGIC In QL, we always put the predicate before the name. So Ga is a well-formed formula, but aG is not. aa is not a WFF GG is not a WFF Other WFFs in QL: “Hong Kong is beautiful” = Bh. “Jenny is a professor” = Pj.  PREDICATE LOGIC The sentences we’ve just looked at use MONADIC predicates, or ‘one-place’ predicates. These attach to a single name. But other predicates are POLYADIC: Jane loves Bill. Tuesday is between Monday and Wednesday.  PREDICATE LOGIC These work similarly: Jane loves Bill = Ljb. Tuesday is between Monday and Wednesday = Btmw. The predicate always comes first; the names after. We can even do: Bill loves himself = Lbb.  PREDICATE LOGIC In principle, there is no limit to how many ‘places’ a predicate can have. So Paaaaaaaaaaaaaaaaa is technically grammatical in QL.  PREDICATE LOGIC Technically, WFFs in SL are WFFs in QL (though systems vary on this). A capital letter by itself, without a name, can be treated as a “0-place” predicate. In practice, though, we won’t use these in class. So you will not see things like P, or P&Q, in problems using QL.  PREDICATE LOGIC QL uses all the same connectives that SL does. Sentences in QL can be put together in the same way as in SL. Name-predicate combos are predicate logic’s ‘atomic sentences’, just like sentence letters are in SL. So these are all WFFs in QL: Gaa & Fbc ~Br ~Fa → (Br & Gbb)  PREDICATE LOGIC QL also adds some new logical words that get their own special symbols. There are two new symbols in QL – they are both what are known as QUANTIFIERS.  PREDICATE LOGIC QUANTIFIERS are logical words in natural language that express quantity: “some” “all” “most” “none” “one” “two” In QL, we only symbolize two – ‘some’ and ‘all’.  PREDICATE LOGIC “All” is represented with an upside down A - ∀. We call this the ‘universal quantifier’. “Some” is represented with a backwards E - ∃. We call this the ‘existential quantifier’.  PREDICATE LOGIC ‘Some’ in English is a little different from ∃ in QL. In English, “some dogs are happy” implies that more than one dog is happy. But in QL, it does not. In QL, “some” just means “at least one”.  PREDICATE LOGIC The way in which the quantifiers work in sentences of QL is a bit complicated. We don’t attach names to the quantifiers like this: ∀a We also don’t attach sentences directly to the quantifiers like this: ∀Fa  PREDICATE LOGIC Why not? Well, it wouldn’t make sense in English: “Some Barack Obama”? “All Barack Obama is tall”? We want to use the quantifiers in the way they are used in natural language.  PREDICATE LOGIC Here’s a quantifier sentence that makes sense: “Something is tall” Here, exactly WHO or WHAT is tall is not specified – the sentence just claims that SOMETHING is tall.  PREDICATE LOGIC We can represent the unknown thing that is tall with a VARIABLE – x, y, or z. (Plus subscripts if needed) Variables are not like names in predicate logic – they do not stand for particular individuals or objects. They stand for unknown individuals, like the ‘thing’ in ‘something is tall’.  PREDICATE LOGIC In order to represent the sentence “Something is tall” in predicate logic, we would write ∃xTx. This can be read as “There is an x such that x is tall”. The ∃x means ‘there is an x such that’. It claims that there exists AT LEAST ONE individual/object that has the property being mentioned. In other words – at least one thing that could be substituted for x (one possible value of x) has the property of being tall.  PREDICATE LOGIC “All” in predicate logic works in a similar way. Suppose I want to say “Everything is tall”. “Everything” means the same as “all the things”, and is translated with the symbol for all - ∀. So to say that everything is tall, I would write ∀xTx. The ∀x here can be read as ‘for all x’. So the sentence means “For all x, x is tall”. In other words - every possible thing that could be substituted for x (every possible value of x) has the property of being tall.  PREDICATE LOGIC So sentences like this are grammatical in predicate logic: ∀xTx ∀xGx ∃xBx in: Loading… You can also connect such sentences with connectives, as ∀xTx & ∀yGy ∃xBx v Fa There are some more complexities here (e.g. using x vs y), but we’ll return to them a bit later.  PREDICATE LOGIC When we learned SL, we talked about INTERPRETATIONS. Recall that an interpretation is an assignment of meaning to the non-logical symbols of the language. For SL, that was an assignment of truth values to atomic sentences.  PREDICATE LOGIC Side note: our textbook also uses the term MODEL for interpretations. The term ‘model’ is used in a few different ways in logic, however, so to avoid confusion we’ll stick with INTERPRETATION.  PREDICATE LOGIC Interpretations in QL are more complicated than interpretations in SL. Interpretations in QL include: A specification of the domain/UD (UNIVERSE OF DISCOURSE) An assignment of an element of the UD to the names An assignment of an EXTENSION to the predicates  PREDICATE LOGIC The first element, the universe of discourse (UD), is the ‘domain’ of our problem – it specifies the individuals that our variables can ‘range over’. So, if our UD contains only Mary, John, Mark: ∀xHx is true if all members of the UD are H. Imagine H means happy - so, for this UD, ∀xHx is true if Mary is happy, John is happy, and Mark is happy. ∃xHx is true if at least one member of the UD is H. That is, if Mary is happy, John is happy, OR Mark is happy.  PREDICATE LOGIC The UD can be specified as in these examples: UD = people UD = students UD = {Mary, Bill, Joe} This last uses SET notation – it lists all members of the UD as a set. For QL, the UD must be non-empty – it must contain at least one thing.  PREDICATE LOGIC For homeworks and tests, I will specify the intended universe of discourse for each translation problem. So, for instance, I might tell you to translate “everyone is happy”, assuming the UD = people. In such a case, ‘everyone’ can be translated with ∀x. For non-translation problems, it generally won’t be an issue. I will specify the UD if it is needed; otherwise don’t worry about it.  PREDICATE LOGIC The second element of an interpretation is an assignment of entities in the UD to names: a = Mary b = Joe c = Bill d = Mary Multiple names can pick out the same element of the UD, but a single name cannot pick out multiple elements. The element named by a name is called its REFERENT.  PREDICATE LOGIC The third element is an assignment of extension to the predicates. We can assign an extension using English like this: Hx = is happy Tx = is tall That method is typical when translating from English. But in other cases, we’ll want to use set notation: Extension(H) = {Mary, Bill}  PREDICATE LOGIC Set notation for predicates lists all the members of the UD to which H applies. So it’s only usable when the UD is fairly small. However, when it can be used it is very helpful. Suppose an interpretation specifies that Extension(H) = {Mary, Bill}. If we know that a names Mary, we can immediately tell that Ha is true on this interpretation.  PREDICATE LOGIC Predicates can be empty – e.g., an interpretation can specify that the predicate applies to no members of the UD. We would represent that this way: Extension(P) = { }  PREDICATE LOGIC For polyadic predicates, we specify their extensions in a given interpretation by listing sets of ORDERED N- TUPLES. For the two-place ‘loves’: Extension(L) = {, , , } Notice order matters!  PREDICATE LOGIC It is helpful to think of an interpretation for QL as a little bit like a possible world. The domain/UD tells you what’s in the world (or at least, the part of the possible world we’re looking at). The assignment of names gives you a way to talk about or refer to the individual members of the UD using the language of QL. The assignment of extension to the predicates tells you what the properties of the members of the world are.  PREDICATE LOGIC So for instance, in the actual world I fall under the extension of the predicate ‘is a professor’, and Donald Trump does not. But there are other possible worlds where I became president of the US, and Trump taught this logic class. In a possible world like that, Trump would be in the extension of the predicate ‘is a professor’.  PREDICATE LOGIC Given a QL interpretation, we can determine whether a quantified sentence is true on that interpretation. This is analogous to what we did with SL when we determined, for instance, whether (PvQ)->~Q is true when P = T and Q = F.  PREDICATE LOGIC Let’s practice with this interpretation. UD: {Adam; Bill; Carl} a = Adam; b = Bill; c = Carl Extension(T) = {Adam, Bill} Extension(N) = {Carl} Extension(F) = {Carl, Bill} Extension(L) = {, , , }  PREDICATE LOGIC Sometimes it can help to visualize what the ‘possible world’ corresponding to an interpretation would ‘look like’ – especially when we start evaluating tougher sentences. If the UD is small enough, I like to draw a ‘world’ and the individuals in it like this:  PREDICATE LOGIC Imagine each of these dots as a member of the UD. Using our previous example interpretation, one of these dots is Adam, one is Bill, and one is Carl.  PREDICATE LOGIC Let’s assign them the names this interpretation gives them. a b c  PREDICATE LOGIC And let’s note down which predicates apply to them. Let’s start with the monadic predicates. a T b T,F c N,F  PREDICATE LOGIC We can use arrows to show the dyadic predicate L. L L a L T b L T,F c N,F  PREDICATE LOGIC It’s a little ugly, but you get the idea! Just looking at this diagram of the example interpretation, we can tell that e.g. ∃xLxb is true. We just look if there’s an L-arrow pointing towards the individual named b. L L a L T b L T,F c N,F  PREDICATE LOGIC Note – you won’t ever have to make a diagram like this for class. But if you’re trying to think through a tricky problem, drawing one can sometimes help. L L a L T b L T,F c N,F PREDICATE LOGIC UD: {Adam; Bill; Carl} True or false? a = Adam; b = Bill; c = Carl Ta Extension(T) = {Adam, Bill} Lba Extension(N) = {Carl} ∀xTx Extension(F) = {Carl, Bill} ∃xLxa Extension(L) = {, , , } Lba -> Tb  PREDICATE LOGIC Summary: An interpretation for QL is an assignment of meaning to the non-logical parts of the sentences. For QL, these interpretations are a little like possible worlds. They consist of: a UD/domain which specifies all the entities in the ‘world’ an assignment of names to the entities in the UD an assignment of extensions to the predicates – that is, a specification of which entities in the UD fall under each predicate.  PREDICATE LOGIC Remember that in SL, we could define the various logical properties in terms of interpretations. For instance: A tautology is true on every interpretation – for SL, every assignment of truth values to the sentence letters. So for example, (P&Q)->P is a tautology, because it turns out true for every possible assignment (P=T, Q=T; P=T, Q=F, etc.).  PREDICATE LOGIC We can do the same thing for interpretations in QL. A tautology in QL is true on every interpretation. So, for instance, Fa is not a tautology – because we could specify an interpretation where Fa turns out false. (Just give an interpretation where the entity named by a is not in the extension of F!). Similarly, ∃xFx is not a tautology – remember that extensions for predicates can be empty. But Fa -> ∃xFx is a tautology. There’s no way to specify an interpretation that makes it false.  PREDICATE LOGIC As a more worked-out example, suppose we have the following sentence: ∀xFx → ∃yGy To show this is NOT a tautology, we can simply indicate an interpretation that makes it false: UD = {Mary, Bill, Jane} a= Mary b = Bill c = Jane Extension(F) = {Mary, Bill, Jane} Extension(G) = { }  PREDICATE LOGIC We can also define the other logical properties in terms of interpretations: A contradiction in QL is false on every interpretation. (for instance, Fa & ~ ∃xFx) A set of sentences is consistent in QL if there is an interpretation on which they are all true (for instance, Fa, Ga, and Lab is a consistent set; Fa and ~Fa is not).  PREDICATE LOGIC We can also define the other logical properties in terms of interpretations: An argument is valid in QL if any interpretation that satisfies (makes true) the premises also satisfies the conclusion. This is very much like the original, intuitive notion of ‘no counterexamples’ we initially used to define validity. If an argument is valid in QL, there will be no interpretation (‘possible world’) where the premises are true but the conclusion is false. No counterexample!  PREDICATE LOGIC We can also define the other logical properties in terms of interpretations: One sentence entails another in QL if any interpretation that satisfies the first sentence satisfies the second sentence. Two sentences are logically equivalent in QL if the first entails the second and vice versa.  PREDICATE LOGIC Ichikawa goes into a lot of detail about how notions like truth and satisfaction work in QL. We won’t worry about these details for class. Next time, we’ll apply these basic ideas to more complicated sentences in QL.