Lecture Notes for Logic, Study notes of Logic

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Lecture Notes for Logic

J. Dmitri Gallow

Contents

• 1 Basic Concepts of Logic
  • 1.1 Finding Argumentative Structure
  • 1.2 Conditionals
    • 1.2.1 Necessary and Sufficient Conditions
  • 1.3 Deductive Validity
  • 1.4 Inductive Strength
• 2 Basic Concepts of Logic, Day - 2.0.1 Review
  • 2.1 Proving Invalidity, take
    • 2.1.1 Venn Diagrams
    • 2.1.2 Venn Diagrams, Counterexamples, and Validity
• 3 Basic Concepts of Logic, Day
  • 3.1 Review
  • 3.2 Formal Deductive Validity
  • 3.3 Proving Invalidity, take
  • 3.4 Formal Deductive Invalidity
• 4 Dialectics
  • 4.1 Rules for Dialectics
    • 4.1.1 Example: A Failed Dialectic
• 5 Informal Fallacies
  • 5.1 Fallacies of Irrelevance
    • 5.1.1 Argument Against the Person (Ad Hominem)
    • 5.1.2 Straw Man
    • 5.1.3 Appeal to Force (Ad Baculum)
    • 5.1.4 Appeal to the People (Ad Populum)
    • 5.1.5 Appeal to Ignorance (Ad Ignorantiam)
    • 5.1.6 Red Herring (Ignoratio Elenchi)
• 6 Informal Fallacies, Day
  • 6.1 Fallacies Involving Ambiguity
    • 6.1.1 Equivocation
    • 6.1.2 Amphiboly
    • 6.1.3 Composition/Division
  • 6.2 Fallacies Involving Unwarranted Assumptions
    • 6.2.1 Begging the Question (Petitio Principii)
    • 6.2.2 False Dilemma
    • 6.2.3 False Cause Fallacy
• 7 Propositional Logic
  • 7.1 Syntax for PL
    • 7.1.1 Vocabulary
    • 7.1.2 Grammar
    • 7.1.3 Main Operators and Subformulae
  • 7.2 Semantics for PL
    • 7.2.1 The Meaning of the Statement Letters
    • 7.2.2 The Meaning of ‘’
    • 7.2.3 The Meaning of ‘’
    • 7.2.4 The Meaning of ‘_’
    • 7.2.5 The Meaning of ‘’
    • 7.2.6 The Meaning of ‘’
    • 7.2.7 Determining the Truth-value of a wff of P L
  • 7.3 Translation from PL to English
  • 7.4 Translation from English to PL
    • 7.4.1 Negation
    • 7.4.2 Conjunction
    • 7.4.3 Disjunction
    • 7.4.4 The Material Conditional and Biconditional
• 8 Logical Notions of PL: Validity
  • 8.1 How to Construct a Truth-Table
  • 8.2 What a Truth-Table Represents
  • 8.3 P L-Validity
• 9 Propositional Logic Derivations, day
  • 9.1 The Basics
  • 9.2 Rules of Implication
    • 9.2.1 Modus Ponens
    • 9.2.2 Modus Tollens
    • 9.2.3 Hypothetical Syllogism
    • 9.2.4 Disjunctive Syllogism
    • 9.2.5 Simplification
    • 9.2.6 Conjunction
    • 9.2.7 Addition
    • 9.2.8 Constructive Dilemma
  • 9.3 A Mistake to Avoid
  • 9.4 Rules of Replacement
    • 9.4.1 De Morgan’s
    • 9.4.2 Commutativity
    • 9.4.3 Associativity
    • 9.4.4 Distribution
    • 9.4.5 Double Negation
    • 9.4.6 Transposition
    • 9.4.7 Material Implication
    • 9.4.8 Material Equivalence
    • 9.4.9 Exportation
• 10 Propositional Logic Derivations, Day
  • 10.1 Four Final Rules of Inference
    • 10.1.1 Subderivations
    • 10.1.2 Conditional Proof
    • 10.1.3 Indirect Proof
  • 10.2 P L-Derivability and the Logical Notions of P L
    • 10.2.1 Some New Notation
    • 10.2.2 P L-Validity
    • 10.2.3 P L-Tautologies and P L-Self-Contradictions
    • 10.2.4 P L-Equivalence and P L-Contradiction
    • 10.2.5 P L-Inconsistency
• 11 Categorical Propositions
  • 11.1 Categorical Propositions
    • 11.1.1 The Components of a Categorical Proposition
    • 11.1.2 Quality, Quantity, and Distribution
  • 11.2 The Square of Opposition
  • 11.3 Categorical Syllogisms
    • 11.3.1 Major Terms, Minor Terms, Middle Terms
    • 11.3.2 Mood and Figure
  • 11.4 Venn Diagrams - standing) 11.4.1 Representing Standard Form Categorical Propositions with Venn Diagrams (Modern Under-
    • 11.4.2 Testing the Validity of Categorical Syllogisms with Venn Diagrams (Modern Understanding)
• 12 Quantificational Logic: Syntax and Semantics
  • 12.1 The Language QL
    • 12.1.1 The Syntax of QL
    • 12.1.2 Semantics for QL
  • 12.2 Translations from QL into English
    • 12.2.1 Translating Simple Quantified wffs of QL
    • 12.2.2 Translating More Complicated Quantified wffs of QL
  • 12.3 Translations from English into QL
• 13 Quantificational Logic: Validity, Equivalence, and the Rest
  • 13.1 Notation
  • 13.2 QL-Validity
  • 13.3 QL-tautologies, QL-self-contradictions, & QL-contingencies
  • 13.4 QL-Equivalence & QL-Contradiction
  • 13.5 QL-Consistency & QL-Inconsistency
• 14 Quantificational Logic: Derivations
  • 14.1 Substitution Instances
  • 14.2 QL-Derivations
    • 14.2.1 New Rules of Replacement
    • 14.2.2 New Rules of Implication
    • 14.2.3 Final Thoughts
  • 14.3 QL-Derivability and the Logical Notions of QL
    • 14.3.1 QL-Validity
    • 14.3.2 QL-Tautologies and QL-Self-Contradictions
    • 14.3.3 QL-Equivalence and QL-Contradiction
    • 14.3.4 QL-Inconsistency
• 15 Quantificational Logic with Identity
  • 15.1 The Language QLI
    • 15.1.1 Syntax for QLI
    • 15.1.2 Semantics for QLI
  • 15.2 QLI Derivations
  • 15.3 Translations from English to QLI
    • 15.3.1 Number Claims
  • 15.4 The Only
• 15.5 Definite Descriptions

1. Legalizing gay marriage will lead to the legalization of polygamy.
2. We ought not legalize polygamy.
3. So, we ought not legalize gay marriage.

But perhaps not. Perhaps this passage is best understood in some other way. Perhaps Perkins isn’t making a claim about what would happen if we legalized gay marriage. Perhaps he is making a claim about what follows from the claim that gay marriage ought to be legalized. Perhaps, that is, he is saying that, if we think gay marriage should be legal, then we are committed to thinking that polygamy should be legal as well. That is, perhaps we should understand his argument along the following lines:

1. If we ought to legalize gay marriage, then we ought to legalize polygamy.
2. We ought not legalize polygamy.
3. So, we ought not legalize gay marriage.

Then again, perhaps, rather than providing an argument against gay marriage, Perkins is simply providing an objection to somebody else’s argument for gay marriage. Perhaps he is objecting to another’s premise that all loving relationships deserve the rights of marriage. That is, perhaps his argument is best understood along these lines:

1. If all loving relationships deserve the rights of marriage, then loving polygamous relationships deserve the rights of marriage.
2. Loving polygamous relationships don’t deserve the rights of marriage.
3. So, not all loving relationships deserve the rights of marriage.

As we’ll see later on, good objections to one of these arguments are not necessarily going to be good objections to any of the others. So, what we ought to say about Perkins’ statements here will depend upon how we ought understand them—whether we ought to understand them as implicitly making the first, second, third, or forth argument above (or whether we ought to understand them in some other way).

Logic is the study of arguments. The goal of logic is to give a theory of which arguments are good and which are bad, and to explain what it is that makes arguments good or bad. Since this is our goal, we ought not understand ‘argument’ in such a way that an argument has to be any good. So, in this class, we’ll understand an argument to be any collection of statements, one of which is presented as the conclusion, and the others of which are presented as the premises.

A statement is a sentence which is capable of being true or false. Questions, commands, suggestions, and exclamations are not statements, since they are not capable of being true or false. It doesn’t make sense to say ‘It’s true that Damn it!’ or “It’s false that when did you arrive?’, so ‘Damn it!’ and ‘When did you arrive?’ are not statements. It does make sense to say, e.g., ‘It’s true that the store closes at eleven’, so ‘the store closes at eleven’ is a statement.

a test: given some sentence, P , if ‘It is true that P ’ makes sense, then P is a statement. If ‘It is true that P ’ does not make sense, then P is not a statement.

1.1 Finding Argumentative Structure

As we saw with Tony Perkins above, given a passage, it is not always obvious whether the passage constitutes an argument or not. Given that it is an argument, it is not always obvious which sentences are premises, which are conclusions, and which sentences are extraneous (asides which are not a part of the argument). Some clues are provided by indicator words. For instance, if any of the following words precede a statement which occurs in an argument, then that statement is almost certainly the argument’s conclusion:

therefore, ... hence, ... so, ... thus, ... this entails that... as a result, ... for this reason,... we may conclude ... consequently,... accordingly, ... this implies that... this entails that...

Similarly, if any of the following words precede a statement in an argument, then that statement is almost certainly one of the argument’s premises:

since... for... as... because... given that... may be inferred from... in that... for the reason that... seeing that... seeing as... as is shown by... owing to...

However, often, indicator words are missing, and one must infer from the context and other clues both 1) whether the passage is an argument; and 2) which statements are premises and which are conclusions. For (1), it is important to consider the author’s goal in writing the passage. If their goal is to persuade the reader, then the passage is an argument. If their goal is anything else, then it is not providing an argument. In particular, if the passage is providing an explanation, or providing information, then it is not an argument. Stories may very well contain indicator words like ‘because’ and ‘consequently’, but this does not mean that they are arguments. For instance, if I tell you

Sabeen is visiting New York because her company was hired to do a workshop there.

my goal is not to persuade you that Sabeen is visiting New York. Rather, I’m simply telling you something about why she is there. This is not an argument, even though it contains the indicator word ‘because’. For (2), you should work with a principle of charity—figure out which potential argument the author might be making is the best argument.

principle of charity: When searching for argumentative structure within a passage, attempt to find the argument which is most persuasive.

For instance, the following passage lacks indicator words:

We must give up some privacy in the name of security. If the homeland is not secure, terrorist attacks orders of magnitudes larger than 9/11 will find their way to our shores. No amount of privacy is worth enduring an attack like this.

So, there are a few arguments we could see the author making. They might be making this argument:

1. We must give up some privacy in the name of security.
2. If the homeland is not secure, terrorist attacks orders of magnitude larger than 9/11 will find their way to our shores.
3. So, no amount of privacy is worth enduring an attack like this.

Alternatively, they might be making this argument:

1. We must give up some privacy in the name of security.
2. No amount of privacy is worth enduring an attack orders of magnitude larger than 9/11.
3. So, if the homeland is not secure, terrorist attacks like this will find their way to our shores.

Finally, they might be making this argument:

1. If the homeland is not secure, terrorist attacks orders of magnitude larger than 9/11 will find their way to our shores.
2. No amount of privacy is worth enduring an attack like this.
3. So, we must give up some privacy in the name of security.

Which of these is correct? Well, the first two arguments are just really bad arguments. With respect to the first one, ask yourself: “suppose that there would be a large attack, and suppose, moreover, that we must give up privacy in the name of security. Does this tell me anything about the relative worth of privacy and avoiding such an attack?” Perhaps the first premise (we must give up some privacy in the name of security) does tell us something about the relative worth of privacy and attacks like this, but then the second premise would be entirely unneeded. So there wouldn’t have been any good reason for the arguer to include it. So this argument looks pretty poor.

The second argument is even worse. Ask yourself “suppose that we must give up privacy in the name of security, and suppose, moreover, that no amount of privacy is worth enduring an attack worse than 9/11. Does this tell me anything

N

S

Figure 1.1: : In the diagram, N is necessary for S and S is sufficient for N.

For instance, being French is sufficient for being European. There’s no way to be French without also being European. For another: being square is sufficient for being rectangular. There’s no way to be square without also being rectangular. And the truth of ‘Sabeen is older than 27’ is sufficient for the truth of ‘Sabeen is older than 20.’

We can visualize this with the Venn Diagram shown in figure 1.1. In that diagram, being inside the circle S is sufficient for being inside the circle N —everything inside S is also inside N. And being inside the circle N is necessary for being inside the circle S—everything inside S is also inside N. This diagram also makes it clear that S is a sufficient condition for N if and only if N is a necessary condition for S.

1.3 Deductive Validity

Our goal in Logic is to separate out the good arguments from the bad. Here’s one very good property that an argument can have: it can be deductively valid. An argument is deductively valid if and only if the truth of its premises is sufficient for the truth of its conclusion.

An argument is deductively valid if and only if the truth of its premises is sufficient for the truth of its conclusion.

Equivalently, an argument is deductively valid if and only if there is no way for its premises to all be true while its conclusion is simultaneously false.

An argument is deductively valid if and only if it is impossible for its premises to all be true while its conclusion is simultaneously false.

For instance, each of the following arguments are deductively valid:

1. If Obama is president, then he is the commander in chief.
2. Obama is president.
3. So, Obama is the commander in chief.
4. Gerald is either in Barcelona or in New York.
5. Gerald is not in New York.
6. So, Gerald is in Barcelona.
7. Obama is younger than 30.
8. So, Obama is younger than 40.

(I will often just say that the argument is ‘valid’, rather than ‘deductively valid’.) Just because an argument is deductively valid, it doesn’t follow that the conclusion of the argument is true. The third argument above is deductively valid, but its conclusion is false. Obama is not younger than 40. If, however, a deductively valid argument has all true premises, then its conclusion must be true as well. If a deductively valid argument has all true premises, then we say that the argument is deductively sound.

An argument is deductively sound if and only if it is deductively valid and all of its premises are true.

If an argument is deductively sound, then its conclusion will be true. Of all the good making features of arguments that we will discuss today, none is finer than deductive soundness. Of all the honorifics of arguments that we’ll discuss today, there is no finer compliment to an argument than to say that it’s deductively sound.

1.4 Inductive Strength

Not every good argument is deductively valid. For instance, the following argument is not deductively valid:

1. Every human born before 1880 has died.
2. So, I will die.

However, it is still an excellent argument. Its premise gives us spectacular reason to believe its conclusion. Arguments like these are inductively strong, even though they are not deductively valid. An argument is inductively strong if and only if its conclusion is sufficiently probable given its premises.

An argument is inductively strong to the extent that its conclusion is probable, given the truth of its premises.

This means that inductive strength, unlike deductive validity, is the kind of thing that comes in degrees. Some arguments can be inductively stronger than others. We could, if we like, set some arbitrary threshold and say that an argument is inductively strong—full stop—if and only if its premises probabilify its conclusion above that threshold. For instance, we could say that

An argument is inductively strong if and only if

Pr(conclusion j premises) > 0 : 5

If an argument is inductively strong with all true premises, then it is inductively cogent.

An argument is inductively cogent if and only if it is inductively strong and all of its premises are true.

Important Concepts:

• statement
• argument
• premise
• conclusion
• necessary condition
• sufficient condition
• conditional
• deductive validity
• deductive soundness
• inductive strength
• inductive cogency

D

F G

Figure 2.1: : A Venn diagram

2.1.1 Venn Diagrams

Let’s talk a bit about Venn diagrams. A Venn diagram has 2 components: a box and some number of labeled circles inside of the box. One example is shown in figure 2.1. In order to interpret this diagram, we must say two things: first, what the domain, D, of the diagram is. That is, we must say what the box contains. Secondly, we must say what each of the circles, F and G, represent.

An interpretation of a Venn diagram says

1. what the domain D is; and
2. what each circle represents

In general, a circle will represent a set of things inside the box. An object is represented as belonging to the set if and only if it is inside of the circle. For instance, I could interpret the Venn diagram in figure 2.1 by saying that the domain D is all animals. That is, every animal is located somewhere inside of the box. I could then say that F is the set of all frogs and that G is the set of all green animals. Alternatively, I could interpret this diagram by saying that the domain is the set of all people, F is the set of all fathers, and G is the set of all grandfathers. Thus, either of the following would be an interpretation of the Venn diagram in figure 2.1:

D = the set of all animals D = the set of all people F = the set of all frogs F = the set of all fathers G = the set of all green animals G = the set of all grandfathers

Let’s start with the first interpretation. There are some animals who are neither frogs nor green (zebras). They lie outside of both the circle F and the circle G. There are some animals who are frogs but not green (brown frogs). They lie within the circle F yet outside of the circle G. There are some animals who are both frogs and green (green frogs). They lie inside both the circles F and G. Finally, there are green animals which are not frogs (crocodiles). They lie inside the circle G, but not inside the circle F.

Think now about the second interpretation. There are people who are neither fathers nor grandfathers. There are also people who are fathers but not grandfathers. And there are people who are both fathers and grandfathers. However, there are no people who are grandfathers but not fathers. So there is nobody who is outside of the circle F but still inside of the circle G. Suppose that we want to express the idea that this area is unoccupied. We may do so by crossing out that area of the graph, as shown in figure 2.2. The lines in figure 2.2 make the claim that all Gs are F s. Equivalently: they make the claim that there are no Gs which are not F. Equivalently: they make the claim that being G is sufficient

D

F G

Figure 2.2: : All Gs are F s

for being F. Equivalently: they make the claim that being F is necessary for being G. (Make sure that you understand why all of these claims are equivalent.)

Suppose that we wish to say that some area of the Venn diagram is occupied. Perhaps, that is, we wish to make the claim that there are some fathers who are not grandfathers. That is, we wish to claim that there are some F s that are not Gs. We may indicate this by putting a single ‘’ in the diagram which is inside the circle F yet outside of the circle G, as in figure 2.3. In figure 2.3, the ‘’ makes the claim that some F s are not Gs. Equivalently: it makes the claim that not all

D

F G

x

Figure 2.3: : All Gs are F s and some F s are not Gs

Gs are F s. Equivalently: it makes the claim that being F is not sufficient for being G. Equivalently: it makes the claim that being G is not necessary for being F. (Make sure that you understand why all of these claims are equivalent.)

2.1.2 Venn Diagrams, Counterexamples, and Validity

Suppose that we’ve got an argument from the premises p 1 and p 2 to the conclusion c. This argument is deductively valid if and only if it is impossible for p 1 and p 2 to both be true and yet for c to be simultaneously false. Let’s think about this claim using Venn diagrams. Consider the Venn diagram in figure 2.4. Let us give this diagram the following interpretation. The domain D is the set of all possibilities. If any state of affairs is possible, then that state of affairs is included in D. P1 is the set of possibilities in which p 1 is true. P2 is the set of possibilities in which p 2 is true. And C is the set of possibilities in which c is true.

Figure 2.5: : The argument from p 1 and p 2 to c is deductively invalid

1. The earth moves around the sun.
2. So, the sun does not move.
3. Raising the minimum wage reduces employment.
4. Obama wants to raise the minimum wage.
5. So, Obama wants to reduce employment.
6. We have not discovered life on other planets.
7. So, there is no life on other planets.

Each of these arguments are deductively invalid. And we may demonstrate that they are deductively invalid by providing the following counterexamples. For the first argument, consider the following state of affairs: the earth moves around the sun, and the sun itself moves. In this state of affairs, the premise of the first argument is true, yet the conclusion is false. So, since this state of affairs is possible (it is actual), the argument is invalid. For the second argument, consider the following state of affairs: raising the minimum wage does reduce employment; however, Obama does not know this. Obama wants to raise the minimum wage, but does not want to reduce employment. Since this state of affairs is possible (though perhaps not actual), the argument is invalid. For the third argument, consider the following state of affairs: life on other planets is hidden somewhere we would be unlikely to have yet found it. Though we have not yet found it, it is still out there. In this state of affairs, the premise of the argument is true, yet its conclusion is false. Since this state of affairs is possible (though perhaps not actual), the argument is invalid.

Chapter 3

Basic Concepts of Logic, Day 3

3.1 Review

An argument is deductively valid if and only if the truth of its premises is sufficient for the truth of its conclusion.

An argument is deductively valid if and only if it is impossible for its premises to all be true while its conclusion is simultaneously false.

A counterexample to the validity of an argument is a specification of a possibility in which the premises of the argument are all true, yet the con- clusion of the argument is false.

An argument is deductively valid if and only if it has no counterexample.

3.2 Formal Deductive Validity

Up until this point, both Hurley and I have been defining deductive validity as necessary truth-preservation—that is, a valid argument is one such that, necessarily, if its premises are all true, then its conclusion will be true as well. In §1.5 of Hurley, however, a new idea shows up: that “the validity of a deductive argument is determined by the argument form.”^1 Understanding this definition requires understanding what an argument form is, as well as what it is for a given argument to have a certain form.

Let’s start with the idea of a variable. A variable is just a kind of place-holder for which you can substitute a certain kind of thing—perhaps a number, perhaps a statement, perhaps a name, perhaps something else entirely. Those entities that can take the place of the variable are the variable’s possible values. For instance, we could use ‘x’ as a variable whose possible values are names. We could similarly use ‘p’ as a variable whose possible values are whole statements. Specifying a variable means specifying what its possible values are—those are known as the values over which the variable ranges.

Next, consider the idea of a statement form. A statement form is a string of words containing variables such that, if

(^1) Hurley, §1.5.

Then, we may define a corresponding notion of formal deductive validity. An argument is formally deductively valid if and only if it is a substitution instance of a deductively valid argument form.

An argument is formally deductively valid if and only if it is a substi- tution instance of a deductively valid argument form.

Correlatively, we may define the notion of formal deductive invalidity. An argument is formally deductively invalid if and only if it is not a substitution instance of a deductively valid argument form.

An argument is formally deductively invalid if and only if it is not a substitution instance of a deductively valid argument form.

Here’s the bold and daring and provocative thesis: deductive validity just is formal deductive validity.

bold and daring and provocative thesis: An argument is deductively valid if and only if it is formally deductively valid.

To see some prima facie motivation for this thesis, consider the examples of deductively valid arguments that we en- countered last time.

If Obama is president, then he is the commander in chief. Obama is president. So, Obama is the commander in chief.

Either Gerald is in Barcelon or Gerald is in New York. It is not the case that Gerald is in New York. So, Gerald is in Barcelona.

Each of these arguments has a deductively valid argument form, namely,

If p, then q p So, q

Either p or q It is not the case that q So, p

Despite this strong prima facie motivation, the bold and daring and provocative thesis is still controversial; some philosophers dispute it. Nevertheless, I will assume it in what follows. As it turns out, very little of what we will do in this class will depend upon the thesis.

3.3 Proving Invalidity, take 2

Consider the following arguments:

If Russia invades Ukraine, then there will be war. It is not the case that Russia will invade Ukraine. So, it is not the case that there will be war.

If it’s raining, then (it’s raining and Romney is president). It is not the case that it’s raining. So, it’s not the case that (it’s raining and Romney is president).

Both of these arguments are of the same general form, namely

If p, then q It is not the case that p So, it is not the case that q

(In the first argument, p = ‘Russia invades the Ukraine’ and q = ‘there will be war’. In the second argument, p = ‘it’s raining’ and q = ‘it’s raining and Romney is president’.)

However, this general form is invalid. We can show that the general form is invalid by pointing out that it has a substi- tution instance with true premises and a false conclusion, namely,

If Romney is president, then a man is president. It is not the case that Romney is president. So, it is not the case that a man is president.

(where p = ‘Romney is president’ and q = ‘a man is president’.) In this substitution instance, the premises are true, yet the conclusion is false. Therefore, the argument form ‘if p, then q; it is not the case that p; therefore, it is not the case that q’ is invalid.

We will call a substitution instance of an argument form which has true premises and a false conclusion a formal counterexample to the deductive validity of the argument form.

A formal counterexample to the deductive validity of an argument form is a substation instance of the argument form which has all (actually) true premises and an (actually) false conclusion.

This affords us another (equivalent) definition of the deductive validity of argument forms

An argument form is deductively valid iff it has no formal counterexam- ple.

3.4 Formal Deductive Invalidity

Earlier, I said that

An argument is formally deductively invalid if and only if it is not a substitution instance of a deductively valid argument form.

What I didn’t say, because it is false, was