PHIL102: Introduction to Critical Thinking and Logic

3a. Contrast formal logic and informal logic

• What is the difference between formal and informal logic?

Formal logic is concerned with deductive logical systems. In other words, a logical system consisting of formal language, rules for determining truth and falsity, and rules for deriving the conclusion of a valid argument is considered "formal logic". Informal logic is the name given to patterns of reasoning that do not exhibit those structures found in formal logic.

To review, see the section on Formal and Informal Logic.

3b. Identify declarative, interrogative, and imperative sentences

• What is a declarative sentence?
• What is an interrogative sentence?
• What is an imperative sentence?
• What is the difference between a sentence and a statement?

A sentence is a complete, grammatical construction. Sentences can be declarative, interrogative, or imperative. (A sentence can also be exclamatory, but we do not cover that type here.) 

A complete grammatical declarative sentence makes a claim, e.g., "Today is Tuesday". A complete grammatical interrogative sentence asks a question, e.g., "How are you?" Interrogatives can also be rhetorical, e.g., "Did you really finish the exam in 20 minutes?" A rhetorical interrogative is effectively a declarative sentence with different punctuation. Finally, an imperative sentence is a command or request, e.g., "Shut the door".

The difference between a declarative sentence and the other types shows us how we use sentences in an argument. More specifically, a declarative sentence has a truth value. In other words, a declarative sentence is either true or false. We cannot respond to the interrogative or imperative sentence types with "True", or "False". They are, in other words, not statements. For our purposes, the difference between a sentence and a statement is determined by the concept of a truth value. A statement is a sentence that is true or false.

To review, see Logical Statements, Connectives, and Relations. 

3c. Define and identify several kinds of logical statements: negations, conjunctions, disjunctions, conditionals, and biconditionals  

• What is a logical connective?
• What is the function of the negation?
• What is the function of the conjunction?
• What is the function of the disjunction?
• What is the function of the conditional?
• What is the function of the biconditional?
• What are the common symbolic logic symbols to notate connectives?

The structures, or forms, of sentences with which we are concerned in critical thinking, are logically simple or complex. Any statement in one of these forms will be a statement we can evaluate in terms of its truth-value.

The more complicated the statement, the more complicated the logical relationship between its parts. For example, "The dog is barking, but there is no intruder", connects the independent clause, "The dog is barking", with the independent clause, "There is no intruder", with the coordinating conjunction, "but". Notice also that the second clause, "There is no intruder", is the negation of the more simple statement, "There is an intruder".

What we've just seen, and will continue to study in more detail below, is called sentential, statement, or propositional logic. If you take a formal, or symbolic logic course, you will learn quite a bit more about this system of logic. In this course, we want to develop a working understanding of what types of sentences are candidates for arguments. We will leverage that understanding when we identify, evaluate, and construct arguments.

Breaking down complex statements to their most simple forms means identifying and separating out the coordinating conjunctions, which we will call connectives. We do this in logic to evaluate the truth-value of the statement. A statement's truth-value is its truth or falsity. Because every statement is either true or false, we can say that every statement has a truth-value. The truth-value of the statement, "The dog is barking, but there's no intruder", is determined by the simple statements that constitute it: "The dog is barking", and "There is an intruder", and their connectives: "but" and "not". The reason for evaluating truth-values of statements is to help us determine the truth-values of any statement governed by a particular connective, but also to determine the quality of the argument in which such statements appear.

As a reminder, an affirmative subject-predicate statement is a simple statement, and consequently does not have any connectives. As such, it is either true or false. The truth or falsity is altered when connectives are added to the statement. We generally use five connectives in logic:

The Five Logical Connectives and Their Ordinary Language Words and Phrases:

A negation denies a statement. It is generally understood to mean, "It is not the case that". So, the negation of "There is an intruder", is "It is not the case that there is an intruder", or "There is no intruder". Other words you can substitute for "It is not the case that", include "Not", "It is false that", "It is not true that", and the contraction of "either-or", which is "neither-nor". Basic grammar does not distinguish between a simple statement and a negated simple statement. Because this distinction has logical importance, we will isolate it from the instances in which it appears, and designate any negated statement compound. Moreover, statements with other connectives are negated, so in a way, a negated statement that's already logically complex is doubly, triply (and so on) complex. Here are some examples:

Conjunction joins together two statements, so that both are asserted to be the case. "The dog is barking but there is no intruder", is the conjunction of the simple statement, "The dog is barking", with the compound statement, "There is no intruder". The conjunction in this case is made with the word "and". Alternative words include "But", "However", "Still", "Nevertheless", "In addition", "Furthermore", "Moreover", "Also", and "Although". If some of these words seem like odd conjunctions, take a moment to see how a few of them work as replacements for our dog barking-intruder statement:

Disjunction separates two statements, so that at least one or the other is the case. Notice the phrase, "at least one", in the last sentence. It is a clue to how to understand a logical disjunction: It is inclusive, not exclusive, so that, when we assert "one or the other", both is also asserted to be the case. Typically, we think of disjunctions in terms of exclusivity, such as when you are presented with a choice of soup or salad at a restaurant. The intended sense is that you may have soup, or you may have salad, but you may not have both. In logic, however, we use the inclusive sense of the disjunction. In terms of a disjunctive sentence's truth-value, this means that one or the other "side" of the disjunct can be true, and in fact, both may be true. The two simple sentences that make up the statement, "That dog looks like a Beagle or a Lab", could both be true; the dog could be a mix of Beagle and Lab, so it could look like both breeds. Other words you can use to express a disjunction are "either-or", "or", and "unless". Here are some examples:

Notice with these examples that the sense of the disjunct is not inclusive but exclusive. After all, we might initially think Monique cannot both be at home and at work. Logically, however (and certainly if Monique works at home!), each disjunct could be true at the same time. There are plenty of disjuncts that are exclusive, and we should be aware of the difference between the two. When we want to communicate exclusivity in logic, we explicitly assert it: A or B, but not both A and B. Here are some more examples of disjunctive statements that can be thought of as inclusive:

In ordinary language, the parts of a conditional statement are dependent on each other, and that makes it difficult to understand just how those parts relate to each other. "If" is a concept that makes asserting truth or falsity troublesome. The tricky thing about conditional statements in logic is that, unlike statements governed by other connectives, the conditional statement is always only hypothetical. It does not say that something is the case, only that something might be the case, will be the case, could have been the case, or would have been the case. The problem with this type of statement is it does not assert that something is the case, and so also something about which you can assert truth or falsity, but that something might be (or have been) the case.

Moreover, we tend to think that denying the antecedent will result in a denial of the consequent. We tend to make this mistake because we think of the antecedent as being a necessary condition, when, in fact, it is not. Take the following reasoning: “If you eat spinach, you’ll be healthy and strong. But you don’t eat spinach, so you won’t be healthy and strong.” The denial of the consequent does not follow from the denial of the antecedent. This will become clearer as we learn more about logically correct reasoning.

The last connective is the biconditional. If you take apart the word, you'll notice that you have "bi" and "conditional". We've already seen what a conditional statement is, but what does "bi" mean? Simply put, it means "two-way". So, the biconditional connective expresses a two-way relationship. It says, "If this, then that, and, if that, then this". The biconditional tells us that both parts of the conditional expression are required in order for the statement to be true. The English expression of the biconditional is "if and only if". Notice the biconditional is made up of "if", "and", and "only if". These are all connectives we already know. It's the way they're put together that expresses a new logical relationship. Here's an example:

She wants to go to college if and only if she is accepted to her first choice.

The sense of the biconditional is equality: going to college is equal to being accepted by a first choice. This means, then, that not going to college will result from not being accepted by the first choice. At the same time, not being accepted by the first choice can be inferred from not going to college. Here are the ways we can express a biconditional relationship:

Here are some of the common symbolic logic symbols used to notate each connective:

To review, see Statements, Logical Connectives, and Logical Relations.

3d. Identify the scope and main connective for a well-formed formula

•     What is a well-formed formula?
•     What is the scope of a connective?
•     What is a sentence's main connective?

A well-formed formula is a correctly structured statement in symbolic logic notation. Suppose we want to notate the sentence, "Today is not Tuesday", in logical notation. We need a symbol for the sentence, "Today is Tuesday", and for the negation of that sentence. Let's stipulate the capital letter, "T", stands for "Today is Tuesday", and we decide to use the tilde for the negation symbol: ~. Both the original simple sentence, T, and the compound sentence, ~T are well-formed.

The scope of a connective is the range of sentences it covers. For example, the scope of the negation in the sentence, ~T is the simple sentence, T. Now suppose we have the sentence, "It's not Tuesday, but it is Wednesday", where we translate the ordinary English word, "but" with the conjunction symbol – we'll use the ampersand: &. The simple sentence, "Today is Wednesday", is translated with W. So, we have ~T&W for the complete sentence. Notice here that the scope of the negation is still restricted to T, while the scope of the & covers both ~T and W – the conjunction conjoins the two simpler sentences (the first of which is itself compound).

The discussion leads us to the idea of a compound sentence's main connective. When a sentence involves two or more connectives, only one can be the governing or main connective. Think about the ordinary English sentence above, "It's not Tuesday, but it is Wednesday", and its translation: ~T&W. Is this a negation or a conjunction sentence? You can tell which is the main connective by looking at the scope.

Now suppose we say, "It's not the case that today isn't Tuesday but it is Wednesday. In other words, here, we deny the entire original sentence: ~(~T&W). The parentheses group together the original sentence's elements to show that the negation covers the whole.

To review, see Sentential Logic and Well-Formed Formulas.

3e. Create truth-tables for several kinds of statements, sentences and arguments, such as negations, conjunctions, disjunctions, conditionals, and biconditionals

• How are truth-values determined in truth-functional (sentences governed by a connective) logic?
• What are the truth definitions for the negation, conjunction, disjunction, conditional, and biconditional?

Determining the truth or falsity of a simple statement can be tricky business. Consider assertions such as, "God exists", and "God doesn't exist". In truth-functional logic, the focus is on possible truth values, so that the actual determination issue does not arise. In other words, rather than focusing on determining whether or not it is true to claim, "This tree is a conifer", in formal logic, the focus is on the fact that there are two possible truth values for the claim: True or False.

When dealing with a compound sentence, that is a sentence involving one or more connectives, the possible truth-values are a function of the original simple statement, and the meaning of the connective(s).

Here are some examples of the concept of possible truth values and truth-functionality:

The logic of natural deduction is a logic involving the assembling and disassembling of statements at their connective "joints". The assembling and disassembling, in turn, is based on each connective's possible truth values – it's truth-functionality. Consider the following statement from the example set above: George W. Bush was not the 44th president of the United States, but Barack Obama was. The compound statement, governed by the connective, "but", is made from two other statements, the first of which is itself compound: George W. Bush was not the 44th president of the United States, and Barack Obama was the 44th president of the United States. Now consider what makes the conjunction true. When we make assertions such as, "Both…and", "and", "yet", and "but", for example, the claim is that each element of the "both", or each side of the "and", is true. So, for the compound sentence, George W. Bush was not the 44th president of the United States, but Barack Obama was, each of the conjuncts must be true: the sentence, George W. Bush was not the 44th president of the United States, must be true, and the sentence, Barack Obama was the 44th president of the United States, must also be true.

Digging a bit deeper, we can also see that for the sentence, George W. Bush was not the 44th president of the United States, to be true, the simple sentence, George W. Bush was the 44th president of the United States, must be false. The negation, "not" denies the original truth-value. So, when we want to assemble (the rule, Conjunction) or disassemble (the rule, Simplification) a conjunction, we do so based on its truth definition, namely that a conjunction is true when, and only when, each of the conjuncts is true.

Here is an overview of each of the connectives we are studying, and the introduction of each's truth definition:

Here is another chart depicting the shorthand for each connective's truth definitions, where P and Q stand for any statement – they are the logical equivalent of variables. Note that the ordinary language phrase for the biconditional, "if and only if", is abbreviated to "iff":

Remember that the reference columns always lay out the possible truth values in the same way, and that the number of rows of a truth table increases exponentially with the addition of each simple sentence.

Reading left-to-right, the first simple sentence's values are half true and half false. The second simple sentence's truth values are half of the first sentence's values – again, half true and half false.

One simple sentence:

Two simple sentences:

Three simple sentences:

The formula for determining the number of rows in a truth table: R = 2^{number of simple sentences}

Here are the truth definitions of each connective:

Here are some examples of possible truth values of compound sentences, where the reference column values are directly beneath each simple sentence:

To review, see:

• Truth Tables
• Complex Truth Tables

3f. Translate ordinary statements into logical language

• How do we translate ordinary language statements into symbolic logic notation?

In the version of sentential logic we're studying, we translate ordinary English statements into symbolic logic notation using the symbols discussed in 3c. We also use parentheses to signify that elements in a compound statement are to be read as a unit. So, capital letters A-Z are used to notate simple sentences. (Reminder: A simple sentence is the smallest linguistic entity that has a truth value, e.g., "Socrates sits". A simple sentence does not contain a connective, i.e., it does not include a negation, conjunction, disjunction, condition, or biconditional).

The connectives are symbolized as follows:

Here are some examples of translating into symbolic logic notation:

To review, see Formalization. 

3g. Explain the limitations of truth-tables as assessment tools

• What are some of the limitations of truth-tables?

The truth table is a powerful tool for determining the possible values of truth-functional sentences and the validity of truth functional arguments. As we know, however, not every statement is truth-functional, and not every argument is constituted by at least one truth-functional statement.

Consider the following valid arguments:

Notice there are no connectives in any of the statements. If we put them into a truth table, we'll end up with at least one row on which the argument fails, which shouldn't happen! This means that the truth table is effective only for those arguments involving connectives.

In addition, consider the fact that the conditional claim (the material conditional) is false in one and one case only, namely when the antecedent is true and the consequent is false. This comports with the concept of validity: An argument is invalid when the premises are true and the conclusion is false. At the same time, however, there are times when the truth table doesn't seem to comport with our intuitions about the conditional claim. Consider the following sentence: "If apples are fruits, then Los Angeles, California is in the United States". On a truth-functional interpretation, the statement is true, but we don't think it's true in any meaningful way. Or consider this sentence: "If fish have legs, then fish take hikes in the woods". From the standpoint of the truth table, where both the antecedent and consequent are false, the statement is true. But this is counterintuitive. As long as we remember how the truth-table works and what its restrictions are, we will not misuse it, which can lead to erroneous outcomes.

To review, see Material Conditional. 

Unit 3 Vocabulary

This vocabulary list includes the terms listed above that you will need to know to successfully complete the final exam.

• biconditional
  
• conditional
• conjunction
• conjuncts
• connectives
• declarative sentence
• disjunction
• formal logic
• function
• imperative sentence
• informal logic
• interrogative sentence
• main connective
• negation
• scope
• statement
• truth-value
• truth definition
• truth table
• well-formed formula