Philosophy and Logic The Process of Correct Reasoning October 23, 2006; updated Sept. 7, 2015

Logic is the art & science of correct reasoning Analysis Clarification Evaluation Words Statements Arguments of

What is a "word"??? According to Webster's New World Dictionary (2nd College Edition), a word is (a) a speech sound, or series of them, serving to communicate meaning and consisting of at least one base morpheme with or without prefixes or suffixes but with a superfix; [a] unit of language between the morpheme and the complete utterance; (b) a letter or group of letters representing such a unit of language, written or printed usually in solid or hyphenated form . . . . [italics added] OK, what's a morpheme, a base morpheme, a prefix, a suffix, a superfix; and what is meant by "the complete utterance"? To find out, do some research on this.)

Statements: an OFFICIAL definition A statement (also known as a "proposition") is a verbal expression* that is either true or false and that may therefore be either affirmed or denied. *A verbal expression is an expression in words, either spoken or written.

Problem: Are statements (propositions) really verbal expressions? Can't I believe, for example, that "it is raining" without saying or writing it?

In other words, can a "propositional mental state" (i.e., a belief) be considered a "statement" (in a sense)???

Another problem There may be various verbal expressions of the same statement or proposition. "It is raining." "Es regnet." "Il pleut." "Esta lloviendo."

Still another problem Must a statement or proposition be either true or false? What about "The present King of France is bald" & "This sentence is false"???

"The present King of France is bald." If there is no present King of France, how can it be true that he is bald? But is it false that he is bald? In that case, it would be true that he is "haired." But how can that be since there is no King of France at present?

Louie the Bald "The present King of France is bald" is a sentence and therefore a verbal expression. Is it a statement? If so, then some statements are (APPARENTLY) neither true nor false.

However . . . ,

the great 20th century philosopher, Bertrand Russell (1872-1970), thought that "The present King of France is bald" and "The present King of France is not bald" should be interpreted as follows:

1. "There is a present King of France, and he is bald." France, and he is not bald." According to Russell, both of these statements are false.

What makes conjunctions true as opposed to false? P & Q T T T T F F F F T F F F

What do you think and why? Is it possible that "The present King of France is bald" is NOT a statement? What do you think and why? And how about . . . .

"This sentence is false" ? ? ?

If "This sentence is false" is true . . . . then it is false (because it truly states that it is false); and if it is false, then it's true (again because it truly states that it is false). Is the sentence both true and false at the same time??? How can that be???

Is the set of all sets that are not members of themselves Very Hard Question: Is the set of all sets that are not members of themselves a member of itself or not???

And what follows if it isn't?

Sets that are and sets that aren't members of themselves: Most sets are NOT members of them-selves. E.g., the set of all cats is not a cat; the set of all tables is not a table; the set of all human beings is not a human being; & so on. But there ARE sets that ARE members of themselves. E.g., the set of all countable things is a countable thing; the set of all conceivable things is a conceive-able thing; & so on.

But, once again, is the set of all sets that are not members of themselves a member of itself or not???

Well, anyway . . . . let's move on.

For most or all PRACTICAL purposes, we can assume that a STATEMENT is a verbal expression that is EITHER TRUE OR FALSE and that may therefore be either affirmed or denied. Now,

having discussed words and statements, let's talk about ARGUMENTS. The OFFICIAL DEFINITION of an ARGUMENT is as follows:

An ARGUMENT is a group, series, or set of STATEMENTS in which one of the statements, known as the CONCLUSION, is claimed by the arguer to follow logically (by way of INFERENCE) from the other statements in the argument, which are known as PREMISES (and which the arguer claims to be TRUE).

All arguments have the same basic structure or format: 1. Premise 2. Premise 3. Premise 4. Conclusion Factual Claim (premises are true) Inference Inferential Claim - that the truth of the conclusion follows logically (by way of inference) from the ASSUMED truth of the premises

Factual Claim & Inferential Claim The factual claim in an argument is the claim, made by the arguer, that all of the premises in the argument are true (as opposed to false or unconvinc-ing). The inferential claim in an argument is the claim, made by the arguer, that the conclusion of the argument follows logically from its premises, assuming that the premises are true.

How to (1) analyze and (2) evaluate an argument

First, we need to find an argument to analyze and evaluate.

Suppose someone were to argue something really silly, like

"All bats have two heads because all bats are kangaroos, and all kangaroos have two heads."

The argument must be subjected to a 6-step analysis & evaluation. Step 1. Identify the conclusion. Step 2. Identify the premises. Step 3. Set the argument up in "standard form." These three steps constitute an "argument analysis." An "argument evaluation" consists of the next three steps, which are:

Step 4. Evaluate the factual claim. Are. the premises true, false, or Step 4. Evaluate the factual claim. Are the premises true, false, or unconvincing? Step 5. Evaluate the inferential claim. Does the conclusion follow logically from the premises (assuming that they are true)? Step 6. Evaluate the argument as a whole. Is it sound or unsound?

Let's apply the six-step method to our sample argument about kangaroos and bats. (1) All bats have two heads because (2) all bats are kangaroos and (3) all kangaroos have two heads. Step 1. What's the conclusion?

Step 1: Can you see that the conclusion of the argument is "All bats have two heads"

Step 2: and that the premises are and "All kangaroos have two heads"? "All bats are kangaroos" and "All kangaroos have two heads"?

Thus, the logical (or "standard") form of the argument is Step 3: Thus, the logical (or "standard") form of the argument is 1. All kangaroos have two heads. 2. All bats are kangaroos. 3. All bats have two heads.

Or to put it more abstractly, 1. All K is T. 2. All B is K. 3. All B is T.

and even more abstractly, Two- headed things Kangaroos Bats

We have now That is what is meant by an ARGUMENT ANALYSIS. (1) identified the conclusion of the argument, (2) identified the premises of the argument, and (3) represented the argument in STANDARD FORM. That is what is meant by an ARGUMENT ANALYSIS.

ARGUMENT EVALUATION. Now we need an Is the argument successful ("sound")?

For the argument to be "sound," the premises of the argument must be true (as opposed to false or unconvincing) and the conclusion of the argument must follow logically from the premises (assuming that they are true).

Step 4: Are the premises of the argument true, or false, or unconvincing? Premise 1: Is it true or are you convinced that "all kangaroos have two heads"? Premise 2: Is it true or are you convinced that "all bats are kangaroos"?

This step is easy (in this case). It is obvious to anyone in her (or his) right mind that both premises in this argument are FALSE.

Another point about Step 4: We need to explain WHY we think the premises are true, false, or unconvincing.

Step 5: But what about the INFERENCE (or INFERENTIAL CLAIM) in this argument? Does the conclusion follow logically from the premises (assuming that they are true)? In other words, IF all kangaroos were two-headed, and IF all bats were kangaroos, would it follow logically that all bats have two heads?

It would, wouldn't it? The inference (reasoning) in the argument is "good." The conclusion does follow logically from the premises (on the assumption that the premises are true, which is an assumption we always make at Step 5).

Step 6: Is the argument as a whole "sound"? Well, at Step 5 we saw that the inference (reasoning) in the argument is good, but at Step 4 we found that both premises in the argument are false.

For an argument to be sound, all of its premises must be true (i.e., the "factual claim" in the argument must be justified) (Step 4), and the inference in the argument must be good (i.e., the "inferential claim" in the argument must be justified) (Step 5).

The argument we have been considering is UNSOUND because, although it contains good reasoning, at least one of its premises is not true.

For an argument to be sound as opposed to unsound, both the factual claim and the inferential claim in the argument must be justified. If the factual claim is not justified (i.e., if at least one premise is false or unconvincing), then the argument is unsound. If the inferential claim is not justified (i.e., if the conclusion does not follow logically from the premises, assuming that they are true), then the argument is unsound.

then the argument is unsound. And, of course, if NEITHER the factual claim NOR the inferential claim is justified (i.e., if the argument fails on both counts), then the argument is unsound.

Let's now apply the six-step method of argument analysis and evaluation to a few simple (and unrealistic) arguments, beginning with this one: All cats are animals, and all tigers are cats. Therefore, all tigers must be animals. The argument contains three statements, right? Which one of them is the conclusion (Step 1)?

All cats are animals, and all tigers are cats All cats are animals, and all tigers are cats. Therefore, all tigers must be animals. Step 2: What are the premises of this argument? Step 3: What is the logical ("standard") form of the argument? (See next slide)

This is it, right? 1. All cats are animals. 2. All tigers are cats. 3. All tigers must be (are) animals.

Argument Evaluation 1. All cats are animals. 2. All tigers are cats. 3. All tigers are animals. Step 4: Is the factual claim justified? That is, are both premises true (as opposed to false or unconvincing)?

Step 5: Does the conclusion follow logically from the premises? 1. All cats are animals. 2. All tigers are cats. 3. All tigers are animals. That is, if all cats are animals, and if all tigers are cats, does it follow that all tigers are animals?

If all cats are animals, Animals

and if all tigers are cats, Animals it looks like all tigers must be animals, right? Cats Tigers

Step 6: Is the argument as a whole sound or unsound? That is, are the factual claim and the inferential claim both justified? Are the premises true (Step 4), and does the conclusion follow logically from the premises (assuming that they are true) (Step 5)? 1. All cats are animals. 2. All tigers are cats. 3. All tigers are animals.

What about the following argument? Tigers must be cats because all cats are animals and all tigers are also animals. Class Participation Exercise: Write a six-step analysis & evaluation of this argument.

1. All cats are animals. 2. All tigers are animals. Here's the STANDARD FORM of the argument: 1. All cats are animals. 2. All tigers are animals. 3. All tigers are cats.

A possible misconception at Step 6 To prove that an argument is unsound is not to prove that its conclusion is false.

Enthymemes, i.e., incompletely expressed arguments

Deductive vs. nondeductive ("inductive") arguments A deductive argument is one that contains a deductive inferential claim. A nondeductive ("inductive") argument is one that contains a nondeductive inferential claim.

Deductive vs. nondeductive inferential claims A deductive inferential claim is the claim, made by the arguer, that the truth of the conclusion follows with the force of absolute logical necessity from the assumed truth of the premises. A nondeductive inferential claim . . . .

. . . is the claim, made by the arguer, that the truth of the conclusion follows with some significant degree of probability from the assumed truth of the premises.

Deductive inferential claims are either "valid" or "invalid." Nondeductive inferential claims are either "strong" or "weak."

A deductive inferential claim (or argument) is valid when the truth of its conclusion follows necessarily from the assumed truth of its premises. 1. If Polly is a cat, then Polly is an animal. 2. Polly is a cat. 3. Polly is an animal. is valid.

A deductive inferential claim (or argument) is invalid when the truth of its conclusion DOES NOT follow necessarily from the assumed truth of its premises. 1. If Polly is a cat, then Polly is an animal. 2. Polly is an animal. 3. Polly is a cat. is invalid.

A nondeductive inferential claim (or argument) is strong when the truth of its conclusion follows with some significant degree of probability from the assumed truth of its premises.

is strong. 1. Millions of crows have been observed. 2. All of them have been black. 3. All crows are black (probably). is strong.

A nondeductive inferential claim (or argument) is weak when the truth of its conclusion DOES NOT follow with any significant degree of probability from the assumed truth of its premises.

1. The great majority of college professors are politically liberal. 2. Patricia Quinn is a college professor. 3. Patricia Quinn is (probably) politically liberal. is weak.

Necessary & Contingent Statements

Statements (e.g., premises in an argument) are either true or false. However, some statements are necessarily true or false, while others are contingently true or false. Thus, there is a distinction between necessary and contingent statements (or propositions).

Necessary statements Necessarily true -- formal and informal tautologies Necessarily false -- formal and informal contradictions (re: the law of non-contradiction) A priori verification & falsification (i.e., verification or falsification by logical analysis alone (no empirical appeal)

Tautologies & contradictions The negation of a tautology is a contradiction, and the negation of a contradiction is a tautology.

Some tautologies & their negations Either angels exist, or they don't. All triangles have three sides. Every effect has a cause. If God would not permit the existence of pain and pain exists, then God does not exist. Angels both exist and do not exist. Some triangles do not have three sides. Some effects are uncaused. If God would not permit the existence of pain and pain exists, then the existence of God is still possible.

Some contradictions & their negations It is raining, and it is not raining. Polly is a cat, but she is not an animal. John's siblings are all males, but Mary is John's sister. A perfectly good being is partly evil Either it is raining, or it is not raining (not both). All cats are animals If John's siblings are all males, then John has no sisters. A perfectly good being is not evil at all.

Contingent statements Neither necessarily true (tautology) nor necessarily false (contradiction) True under some conditions; false under others A posteriori verification & falsification (i.e., verification or falsification on empirical grounds -- not by logical analysis alone -- & verification/falsification not always possible)

Some Contingent Statements There are rocks (or rocks exist). Washington DC is the capital of the US. Oranges are not grown in Antarctica. Unicorns exist. Abraham Lincoln is [now] President of the US. There are palm trees growing on the moon.

Tautologies, contradictions, & contingent statements The negation of a tautology is a contradiction; the negation of a contradiction is a tautology; and the negation of a contingent statement is neither a tautology nor a contradiction, but another contingent statement.

A Final Point Necessary statements can be proved (tautologies) or disproved (contradictions) via logical (a priori) analysis of their form or meaning. Many contingent statements can be proved or disproved a posteriori (empirically), but some are (at least at present) beyond verification and falsification because the needed evidence is unavailable. In some cases, the evidence will become available in time; in other cases, the evidence will never be available.

To Be Continued . . . .