1 Part II: How to Evaluate Deductive Arguments Week 4: Propositional Logic and Truth Tables Lecture 4.1: Introduction to deductive logic Deductive arguments = presented as being valid, and successful only if they are valid. By comparison: inductive arguments are presented as being strong (not necc. valid) and are successful to the extent they are strong. Remember this example from 3.1: Mary has a child who is pregnant. Only daughters can become pregnant. Mary has at least one daughter Other arguments may use the same form but differ in subject matter: Terry has a job in which she arrests people. Only police officers arrest people. At least one of Terry s jobs is as a police officer. Both arguments are valid it s impossible to imagine that both premises are true and the conclusion false. Forms of valid argument Quantifiers: only, at least, some, all, none, etc. we ll go into these in next week s categorical logic lessons Propositional connectives (partic. truth functional connectives), which we ll discuss this week. Lecture 4.2: Propositions and propositional connectives propositional logic the rules that det. the validity of an argument based on the propositional connectives used within it. What is a proposition? Something that can be either true or false, and that can be the premise or conclusion of an argument. Proposition: The binoculars are in my hand. Not propositions: Binoculars, a hand What is a propositional connective? Something that takes propositions and makes new propositions out of them. Example: and I am holding the binoculars and looking through them.
2 Lecture 4.3: And and truth-functional connectives And doesn t always function as a propositional connective. Jack and Jill finally talked and could serve three functions, giving sentence 3 meanings: Joins two names to make a complex name ( The fast food company Jack and Jill finally talked. as in, its spokesman finally held a news conference.) Propositional connective that joins two propositions to make a complex proposition made of just those two propositions. ( Each of two contestants, Jack and Jill, finally talked. they re in a silence contest with a third party and I m relaying results: Jack finally talked and Jill finally talked.) Expresses a proposition that involves something over and above Jack s finally talking and JIll s finally talking not just a propositional connective. ( Two people, Jack and Jill, finally talked to each other. ) Here, we ll focus on and as propositional connective we ll look at other words used that way such as or, not, but, only, if, etc. later on. And can be used in diff. ways, even as a propositional connective. I took a shower and got dressed conveys sense of temporal order; propositional connective. I am holding the binoculars and looking through them does not convey sense of temporal order; here, it s not just a PC but also a truth functional connective. Truth functional connective propositional connective that makes a new proposition whose truth or falsity depends solely on the truth or falsity of its propositional ingredients. conjunction a truth functional connective expressed with the use of and Examples of truth functional connectives: As long as it s true that Jack finally talked and Jill finally talked, it s true that Jack and Jill finally talked. Jack finally talked Jill finally talked Jack and Jill finally talked True True True True False False False True False False False False Some arguments are valid because of the truth functional connectives they use. Lecture 4.4: Using truth tables to show validity What is a truth table?
3 Device that can be used to explain how a truth functional connector or a truth functional operator works. (For example, at the end of the last lesson.) Conjunction introduction inference introduces a conjunction that wasn t present in either of the inference s two premises: P Q P & Q Truth table tells you that in any situation in which both premises are true, the conclusion must be true. That means conjunction introduction inferences are all going to be valid. P Q P & Q True True True True False False False True False False False False Conjunction elimination conclusion of argument eliminates a conjunction in its premise P&Q P&Q P Q Truth table for conjunctions (above) still applies therefore, all conjunction elimination arguments are valid, regardless of what they re about. Lecture 4.5: Rules, variables and generality So, why do we use variables when we state rules? Rules are general Designed to apply to many different possible cases RULE: You should never under any circumstances hit another person. Use of pronoun you and never under any circumstances indicate this rule applies to a multitude of possible situations. RULE: You should never hit your brother. NOT A RULE: Stop hitting your brother right now! Use of right now indicates that this command applies to a particular situation. Rule could get even more general never hit another person never... do violence to another person never... do something unkind to another person Another example: Walter should not force his dog to kill his cat some generality to it
4 More general: People should not force creatures to kill innocent creatures. Last sentence is unclear: how should creatures be understood? Plural understanding: People should not force a whole bunch of creatures to kill innocent creatures. Distributive understanding: For any creature you pick, people should not force that creature to kill an innocent creature. So we clarify: Where x is any creature, and y is any innocent creature: People should not force x to kill y. Using variables here helps us express the distributive rule precisely and clearly. Variables are equally useful in propositional logic. I m holding my binoculars and looking through them. I m holding my binoculars. P & Q P For any argument where the premise is two propositions, and the conclusion is one of those conjoined propositions, that argument is also going to be valid. Notation Variables are arbitrary we ve used Roman letters, but anything else that s easy to produce and recognize can be used. (Symbols may not be easy to recognize as variables because they re commonly used to express other things.) Notation for truth functions (also arbitrary, but here s what we ll use) Conjunction: & Disjunction: (not entirely unlike a letter V) Negation: or or ~ Lecture 4.6: Disjunction Disjunction truth functional connective usually expressed with the inclusive use of or Either Manchester won it or Barcelona won it. exclusive use; here, they cannot both win. This is breakfast or lunch. inclusive use; here, it could be either or both. Truth table for disjunctions P Q P V Q true true true true false true false true true false false false
5 If P and/or Q is true, then their disjunction (P V Q) is true. If both P and Q are false, then their disjunction is false. Disjunction introduction argument: P Q P V Q P P V Q Any argument of either form is valid. Disjunction elimination argument: P V Q P I m going to tickle you with my right or left hand. I m going to tickle you with my right hand. An argument of this form need not be valid. (For one thing, I could be tickling you with my left hand.) Lecture 4.7: Combining conjunctions and disjunctions They can be used together to string a bunch of propositions into a larger proposition. This example strings together three propositions: I m going to tickle you with hand #1 & I m either going to tickle you with hand #2 or hand #3. (Each hand is a proposition) Truth table for conjoining a disjunction I m going to tickle you with hand #1 I m going to tickle you with hand #2 I m going to tickle you with hand #3 I m going to tickle you with either hand #2 or hand #3 I m going to tickle you with hand #1, and with either hand #2 or hand #3 true true true true true true true false true true true false true true true true false false false false false true true true false false true false true false
6 false false true true false false false false false false The disjunction can only be true if one part is true (either hand #2 or #3). The full proposition can only be true if the hand #1 part is true and the disjunction is true. Conjunction and disjunction are not associative When we use truth functional connectives to build new propositions, the truth table for the new proposition depends on the order in which the connectives are applied. Truth table for (P & Q) V R (below) is different from that for P & (Q V R). Truth table for disjoining a conjunction I m going to tickle you with hand #1 I m going to tickle you with hand #2 I m going to tickle you with hand #3 I m going to tickle you with hand #1 and hand #2 I m going to tickle you with hand #1 and hand #2, or with hand #3 true true true true true true true false true true true false true false true true false false false false false true true false true false true false false false false false true false true false false false false false Lecture 4.8: Negation and truth-functional operators propositional operator a propositional device that operates on a proposition to convert it to another proposition (rather than connecting two to make a larger one). Ex: I believe that truth functional operator propositional operator that creates a proposition whose truth depends solely on the truth of the proposition to which the operator is applied. I believe that is not a truth functional operator the truth or falsity of I believe that it s raining today doesn t depend solely on the truth or falsity of it s raining today. Negation truth functional operator that takes a proposition and creates a new proposition whose truth value is the opposite of the original. Can be expressed with it is not the case that or simply not.
7 P P true false false true Does not express negation? Sometimes, but not always. Walter has stopped beating his dogs. vs. Walter has not stopped beating his dogs. The former can be false even when the latter is not true. (Walter may still be beating dogs, may have never beaten dogs, may not even have dogs to beat, or flat out may not exist.) But Walter has stopped beating his dogs NOT! is true. Lecture 4.9: Negating conjunctions and disjunctions Negating a conjunction says only that it s not the case that both of the two conjuncts are true one or none could be true. Negating a disjunction says that neither conjunct is the case. Truth table for negating a conjunction I m going to eat the plug I m going to eat the cylinder I m going to eat the plug & the cylinder It s not the case that I m going to eat the plug & the cylinder true true true false true false false true false true false true false false false true Truth table for negating a disjunction I m going to eat the plug I m going to eat the cylinder I m going to eat the plug or the cylinder It s not the case that (I m going to eat the plug or the cylinder) true true true false true false true false false true true false false false false true Lecture 4.10: Commutativity and associativity Two properties we find in conjunction and disjunction, but not in other truth functions. Commutativity A function of two things is commutative when it delivers the same result regardless of the order in which it operates on those things.
8 Consider arithmetic: Addition is commutative (in general: x + y = y + x) Multiplication is commutative (in general: xy = yx) But subtraction is not commutative (in general: x y y x) Division is also not commutative. Commutativity in propositional logic: Conjunction is commutative: I m standing and waving = I m waving and standing (in general: p & q = q & p) Disjunction is commutative: I m standing or waving = I m waving or standing (in general: p V q = q V p) The conditional (which we ll learn about later this week), is not commutative the order determines the result you get. Associativity A function of three or more things is associative when it delivers the same result regardless of the order in which it operates on those things. In arithmetic: Addition is associative [in general: x + (y + z) = (x + y) + z] Multiplication is associative [in general: x(yz) = (xy)z] Subtraction is not associative [in general: x (y z) (x y) z]. Nor is division. In propositional logic: Conjunction is associative: I m standing and he s sitting and waving = I m standing and he s sitting, and he s also waving [in general: p & (q + r) = (p & q) + r] Disjunction is associative: I m standing or he s sitting and waving = I m standing or he s sitting, or he s waving [in general: p V (q V r) = (p V q) V r] Lecture 4.11: The conditional conditional a truth functional connective that is especially important in understanding the rules by which we assess validity of deductive arguments. If Walter is eating lunch at New Havana, then the private investigator is eating lunch at New Havana. If then... is a propositional connective How to argue that it s a truth functional connective as well? Consider this construction: (P & Q) is true. (negation of the conjunction of P and negation of Q) P & Q is false. If P is true, then Q is false If P is true, then Q is true That is to say: If P, then Q So: If P, then Q follows from (P & Q)
10 Truth table for (P & Q) P Q Q P & Q (P & Q) true true false false true true false true true false false true false false true false false true false true Now let s make that argument starting with If P, then Q : If P, then Q. If P is true, then Q is true. If P is true, then Q is false. (P & Q) is not true. (P & Q) is true. So: (P & Q) follows from If P, then Q. The truth table for the conditional If P, then Q is the same as that of (P & Q): P Q If P, then Q true true true true false false false true true false false true Since If P then Q has a truth table, that proves it s a truth functional connective. Some good rules governing our use of the conditional: Modus ponens: From the premises P and If P, then Q, infer the conclusion Q. (In other words: In any situation where P is true and If P, then Q is true, Q has got to be true.) Conditional truth table shows this is a good rule of inference for P to be true, it must be scenario 1 or 2; for If P, then Q to be true, it rules out scenario 2. Modus tollens: From the premises Q and If P, then Q, infer the conclusion P (In any situation where the negation of Q is true and If P, then Q is true, the negation of P has got to be true.) Truth table: Q means it s scenario 2 or 4; If P, then Q rules out 2. Why is the conditional an important truth functional connective? Can use it to express, in the form of a single proposition, the validity of an argument from premises P to conclusion C whatever the argument s about.
11 Lecture 4.12: Conditionals in ordinary language Use of If then often but not always signifies a material conditional or any other truth functional connective: For example, propositions within it cannot be in the subjunctive mood: If I had been 4 feet tall, then I would have been in the Guinness Book of World Records. Both propositions are false, but statement may or may not be true. And phrases other than if then may be used, such as: if : The private investigator is eating lunch at New Havana if Walter is. only if : Walter is eating lunch at New Havana only if the P.I. is. Lecture 4.13: Biconditionals if and only if conjoins conditional if P, then Q and conditional If Q, then P biconditional propositional connective connecting two propositions into a larger proposition, and the larger proposition is true just in case the two propositions that are part of it have the same truth value (both true or both false). P if and only if Q is true just in case P and Q are both true, or both false. Bob was born in the U.S. if, and only if, George was. equivalent to saying both Bob was born in the U.S. if George was. AND Bob was born in the U.S. only if George was. Truth table for biconditional: P Q P = Q true true true true false false false true false false false true Some examples would seem to counter claim that conditional and biconditional are truth functional connectives: If = 4, [then] Pierre is the capital of South Dakota. first proposition after if is antecedent. second proposition after then is consequent. Isn t this a baffling thing to say? Yes, but that doesn t mean it s not true.
12 Week 5: Categorical Logic and Syllogisms Lecture 5.1: Categorical logic This week: looking at arguments that are valid for a reason other than because of the propositional connectives they use some of these have no propositional connectives. Propositional logic: 1. Jill is riding her bicycle if, and only if, John is walking to the park 2. John is walking to the park if, and only if, premise 1 is true. 3. Jill is riding her bicycle We have a tricky deductive argument whose validity is not obvious it takes a while to check: Jill is riding her bike. John is walking to the park. Jill is riding her bike if, and only if, John is walking to the park. John is walking to the park if, and only if, the statement to the left is true. true true true true true false false true false true false false false false true false We don t just use truth tables to determine when or if arguments are valid we can also use truth tables to explain why they re valid, even in cases where it s obvious (as the ones we looked at last week were). This week: We can t use truth tables on inferences/arguments that don t use truth functional connectives, so we need to find another method to discover whether and why an argument is valid. No fish have wings All birds have wings All animals with gills are fish No birds have gills This is a valid argument we ll learn how to suss it out via the central method of categorical logic. Mary has a child who is pregnant Only daughters can become pregnant. Therefore, Mary has at least one daughter. What is it about the use of the terms only and at least that makes this valid? The concept of the quantifier is our central concept this week. Lecture 5.2: Categories and quantifiers Categories in arguments
13 Brazilians speak Portuguese. Portuguese speakers understand Spanish. Brazilians understand Spanish This argument brings 3 categories into relation with each other: Brazilians, speakers of Portuguese, and people who understand Spanish. Using some as quantifier on premises and conclusion = not a valid argument: Possible scenario: all the Portuguese speakers who understand Spanish live outside Brazil. Using most as quantifier = also not valid. Possible scenario: most Portuguese speakers live outside Brazil, and only those outside Brazil understand Spanish. Using all as quantifier = valid, but not sound. It s almost certainly not the case that both premises are true. So we started with 3 categories, then saw that by use of certain modifiers, we could make the original argument more precise and thus test its validity. To test arguments that use categories and quantifiers, we use a Venn diagram. The one at right covers just one premise; to cover all the premises and the conclusion, we d use one like the diagram below, for the some argument: At the right is one representing the all argument. The unshaded portion shows
14 Brazilians who ve got to understand Spanish; the Venn diagram shows the argument to be valid. Lecture 5.3: How quantifiers modify categories Types of quantifiers (F and G represent any category at all.) A: All Fs are Gs. All things in the first category also fall into the second category. E: No Fs are Gs. No things in the first category also fall into the second. I: Some Fs are Gs. Some things in the first category also fall into the second. O: Some Fs are not Gs. Some things in the first category don t fall into the second. As these Venn diagrams show, propositions of the A and O forms are negations of one another; the same is true between forms E and I. Lecture 5.4: Immediate categorical inference Immediate categorical inference: an inference with just one premise, in which both the premise and the conclusion are of form A, E, I or O. (They don t both have to be the same form.) Subject term and predicate term Each proposition of form A, E, I or O has a subject term and a predicate term. subject term the one directly modified by the quantifier (in our examples, labeled F) predicate term the other term, not directly modified by quantifier (labeled G here)
15 Conversion Inference in which the conclusion switches the subject and predicate term that appear in the premise. The most common example of immediate categorical inference. No Fs are Gs. All Fs are Gs. No Gs are Fs. All Gs are Fs. The first example is plausibly valid; the second is not. Conversion inferences are valid for E and I propositions (No Fs are Gs/Some Fs are Gs), but not A and O propositions. Venn diagrams illustrate why. A All Fs are Gs/All Gs are Fs : Non overlapping F circle is excluded, which could mean all the Gs there are are in overlapping section but it leaves open that there are plenty of Gs in the rest of that circle E No Fs are Gs/No Fs are Gs : Overlapping section is excluded, showing that there is the possibility of Fs and Gs, but not both at the same time. I Some Fs are Gs/Some Gs are Fs : X in overlapping section can be read as something being in both circles. O Some Fs are not Gs/Some Gs are not Fs : X in F circle outside the G circle. Does not imply that some Gs are not Fs there may be no Gs at all, or they may all be in overlapping section. Lecture 5.5: Syllogisms Syllogism: an argument with two premises and a conclusion, where all three propositions are of the A, E, I or O form. subject term of the syllogism = subject term of the conclusion (modified by quantifier). One of the premises the minor premise must also contain the subject term. predicate term of the syllogism = predicate term of the conclusion (not modified). One of the premises the major premise must also contain the predicate term. Example 1: Example 2: All Duke students are humans. Some Duke students are humans. All humans are animals. All humans are animals. All Duke students are animals. Some Duke students are animals.
16 Example 3: No Duke students are humans. All humans are animals. No Duke students are animals. Lecture 5.6: Categories, individuals and language How ordinary language can mislead us: Many statements that seem to be about individuals really are about categories. Example: Mary owns a Ferrari. Doesn t appear to involve categories, does it? A particular person owns a particular car. But compare that to: Some of Mary s possessions are Ferrari cars. It s a statement of the I form ( Some Fs are Gs. ) The two statements seem to amount to the same thing. Mary owns a Ferrari is true if and only if some of Mary s possessions are Ferrari cars. Thus, our categorical logic and Venn diagrams can be used to study the validity of such arguments. Not every statement is of the A, E, I or O forms, but lots of ordinary statements we make are. Lecture 5.7: Venn diagrams and validity Let s try to diagram some of our deductive arguments from last week: Mary has a child who is pregnant Only daughters can become pregnant. Mary has at least one daughter Terry has a job in which she arrests people. Only police officers arrest people. At least one of Terry s jobs is as a police officer.
17 Robert has a pet who is canine. Only mammals are canine. At least one of Robert s pets is mammal. Labels aside, the diagrams for all three of these arguments look the same, which tells us the arguments have the same form, and are valid for the same reason. Lecture 5.8: Other ways of expressing A, E, I or O propositions Propositions of the form not all : Not all Fs are Gs. (= Some Fs are not Gs O form) Not all geniuses take Coursera courses. Not everything that Pat does is intended to annoy Chris. Propositions of the form All Fs are not Gs (=No Fs are Gs E form) All Nobel Prize winners are not alcoholics. Everything that Pat does is not designed to achieve victory. Propositions of the form Some Fs are both Gs and Hs : Not the same as Some Fs are Gs and some Fs are Hs. Some philosophers are both robotic and monotone. Some of the things that Pat does are both intended to amuse and to provoke. Week 6: Representing Information Lecture 6.1: Reasoning from Venn diagrams or truth tables alone All the arguments we ve looked at over the last few weeks are given in ordinary language we can understand. But not all arguments are like that they might be given in technical jargon or in a foreign language. So, could you predict and explain those arguments validity based just on diagrams or truth tables? Yes. Example 1: You re an anthropologist studying another culture and trying to translate its untranslated language the word SPooG has eluded you. By observing their behavior, you devise a hypothesis: SPooG is used as a truth functional connective with this truth table:
18 P Q P SPooG Q True True True True False True False True False False False True You overhear an argument you translate as: John is riding his bicycle SPooG Jill is walking to the park. Jill is walking to the park. Therefore, John is riding his bicycle. Is this argument valid? Use the truth table (with headings John is riding his bicycle / JIll is walking to the park / John is riding his bike SPooG Jill is walking to the park ). The only scenario that works is the first one, in which all three are true. Example 2: John is riding his bicycle SPooG (Jill is walking to the park or Frank is sick) Frank is not sick. John is not riding his bicycle. Therefore, Jill is walking to the park. John is riding Jill is walking Frank is sick Jill is walking or Frank is sick John is riding SPooG (Jill is walking or Frank is sick) True True True True True True True False True True True False True True True True False False False True False True True True False False True False True False False False True True False False False False False True Premise 3 (John is not riding) narrows it down to the last four scenarios. Premise 2 (Frank is not sick) narrows it down to scenarios 1, 3, 5 and 7. Premise 1 (John is riding SPooG (Jill is walking or Frank is sick)) narrows it to first 4 scenarios or #8
19 This leaves us with scenario #8 but that states Jill is not walking to the park, which doesn t match the conclusion that she is. Thus, this argument is not valid. Quantifiers Another word in this language Jid! is always uttered in a high pitched, excited tone. Your hypothesis is that this word is a quantifier which works like this: Jid! (F or G) = There are no Fs outside G, but there are Gs outside F. (See Venn diagram at right.) You hear this argument: Jid! giraffes are herbivores. Jid! herbivores are mammals. There are some giraffes, and all of them are mammals. From premise 1, we rule out all of giraffes that s not under herbivores. We know there s an X in the overlap, but not where yet. From premise 2, we rule out all of herbivores that doesn t overlap with mammals, and an X in the overlap somewhere. We wouldn t know where if it weren t for premise 1. So it goes in the central overlap. Our diagram matches the conclusion, so the argument is valid. How might you translate Jid! into English? All Fs are Gs, but not all Gs are Fs. Lecture 6.2: Different ways of representing information What s the point of using truth tables or Venn diagrams to represent information? Unlike sentence representations, they allow us to clearly see which deductive inferences involving that information are valid. We can see relations of deductive validity that we can t with sentences.
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