Example: The first theorem Euclid’s Elements provides a good example of the kind of reasoning that people admire.

Suppose we construct a triangle in the following way: 1. Draw a circle centered at point A. Mark a point B on the circumference and draw a line from A to B. Draw a second circle centered at B that passed through A. Mark one of the points at which the circles intersect as B and draw lines from C to A and from C to B.

Theorem: All the sides of the triangle ABC are of equal length.

Proof: Let |AB| denote the length of the line segments AB, and so on.

Definition: An argument is a list of statements, one of which is the conclusion and the rest of which are the premises.

The conclusion states the point being argued for and the premises state the reasons being advanced in support the conclusion. They may not be good reasons. There are good and bad arguments.

Tip: To identify arguments look for words that introduce conclusions, like "therefore", "consequently", "it follows that". These are called conclusion indicators. Also look for premise indicators like "because" and "since".

Remark: Each of the five steps in the proof to Euclid’s first theorem is an argument. The conclusions in steps 1 to 4 are called intermediate conclusions, while the conclusion in step 5 is the main conclusion.

Question: All arguments, or sequences of arguments, are examples of reasoning, but is every piece of reasoning an argument? A perceptual judgment such as "I see a blue square", or the conclusions of scientific experts reading in X-rays, or looking through a microscope, may be examples of reasoning that are not arguments. They are derived from what Kuhn called tacit knowledge, acquired through training and experience (e.g., knowing how to ride a bicycle). It is not easily articulated, and is not stated in any language.

In logic, we assume that any reasoning is represented as an argument, and the evaluation of an argument involves two questions:

1. Are the premises true?
2. Supposing that the premises are true, what sort of support do they give the conclusion?

Answers to question 2: Compare the following arguments.

1. All planets move on ellipses. Pluto is a planet. Therefore, Pluto moves on an ellipse.
2. Mercury moves on an ellipse. Venus moves on an ellipse. Earth moves on an ellipse. Mars moves on an ellipse. Jupiter moves on an ellipse. Saturn moves on an ellipse. Uranus moves on an ellipse. Neptune moves on an ellipse. Therefore, Pluto moves on an ellipse.

Definition: An argument is deductively valid if and only if it is impossible that its conclusion is false while its premises are true.

Examples: Argument 1 is deductively valid, while argument 2 is not.

Remark on terminology: The notion of deductively validity is such a central and important concept in philosophy, that is goes by several names. When an argument is deductively valid, we say that the conclusion follows from the premises, or the conclusion is deduced from, or inferred from, or proved from the premises. Or we may say that the premises imply, or entail, or prove the conclusion. We also talk of deductively valid arguments as being demonstrative. All these different terms mean exactly the same thing, so the situation is far simpler than it appears.

What’s possible? The sense of "impossible" needs clarification. Consider the example:

3. George is a human being. George is 100 years old. George has arthritis. Therefore, George will not run a four-minute mile tomorrow.

Suppose that the premises are true. In logic, it is possible that George will run a four-minute mile tomorrow. It is not physically possible. But logicians have a far more liberal sense of what is "possible" in mind in their definition of deductive validity. Argument 3 is not deductively valid on their definition. So, argument 3 is invalid.

Key idea: In any deductively valid argument, there is a sense in which the conclusion is contained in premises. Deductive reasoning serves the purpose of extracting information from the premises. In a non-deductive argument, the conclusion ‘goes beyond’ the premises. Inferences in which the conclusion amplifies the premises is sometimes called ampliative inference.

Therefore, whether an argument is deductively valid or not, depends on what the premises are.

‘Missing’ premises?: We can always add a premise to turn an invalid argument into a valid argument. For example, if we add the premise "No 100-year-old human being with arthritis will run a four-minute mile tomorrow" to argument 3, then the new argument is deductively valid. (The original argument, of course, is still invalid).

Definition: An argument is inductively strong if and only if it is improbable that its conclusion is false while its premises are true.

Remember: This definition is the same as the definition of "deductively valid" except that "impossible" is replaced by "improbable."

The degree of strength of an inductive argument may be measured by the probability of that the conclusion is true given that all the premises are true.

The probability of the conclusion of a deductively valid argument given the premises is one, so deductively valid arguments may be thought of as the limiting case of a strong inductive arguments. Ampliative arguments have an inductive strength less than one.

The probability of the conclusion given the premises can change from person to person, as it depends on the stock of relevant knowledge possessed by a given person at a given time.

Summary: In response to question 2, we may give answers like "the argument is valid", "the arguments is inductively strong" or "the argument is inductively weak."

Exercise: Discuss the following examples (all statements are understood to refer to the year 1998):

Nevertheless, in logic, it is assumed that the answer to question 1 is relevant to the evaluation of an argument. But it is a question that needs to be asked in addition to question 2. So, if the premises of an inductively strong argument are false, then logicians are forced to say that the argument is not a good one. It is confusing to say that an inductively strong argument is a weak argument, but this is how the terms are defined.

Tip: Defined terms must be used as defined. You can’t use the term differently just because you don’t agree with the definition.

Definition: Any argument that is not deductively valid, or deductively invalid, is called an ampliative argument. The term refers to the fact that the conclusion of such argument goes beyond, or amplifies upon, the premises.

Remark on terminology: Again the notion of ‘invalid’ is so common and central, that it goes by many names. Other terms commonly used are inductive and non-demonstrative. I prefer ‘ampliative’ because it reminds us that the conclusion ‘goes beyond’ the premises, and it does not have the bad reputation that sometimes goes along with the word ‘induction.’

Simple enumerative induction goes from a list of observations of the form "this A is a B" to the conclusion "All A’s are B’s". The example Hume made famous is like this:

Some ampliative arguments go from general statements to general statements:

Others go from general statements to specific statements:

Conclusion: To understand empirical science we need to understand ampliative inference.

Two Kinds of Science? A Priori and Empirical?

1. A priori science, like Euclid’s geometry, is where the conclusions are deduced from premises that appear to be self-evidently true.
2. In empirical science, like physics, conclusions are based on observational data.

• This is similar to the distinction between pure mathematics and applied mathematics. The distinction is not always sharp.

Ever since Einstein rejected the use of Euclidean geometry in his new physics at the turn of the 20^th century, it seems that a priori sciences cannot tell us anything about the real world. The focus of recent philosophy of science is on the empirical sciences.

• A priori sciences contain the strongest form of reasoning, at the expense of telling us less about the real world.

Definition: In philosophy of science, we refer to what we already know directly through observation as the empirical evidence (we are open-minded about the possibility that some of these ‘facts’ are mistaken). See Exercise 1.

All of empirical science uses ampliative arguments. Hume made the same point in a different way. He pointed that in example 8, it is possible that the premises are true and the conclusion is false. No matter how many instances of a generalization we observe, it does not prove that the generalization is true.

What is the difference between science and pseudoscience? You often hear that science is based on the ‘facts’ while pseudoscience is not. Or you say that religious belief is based on faith, whereas scientific belief is not. Unfortunately, both scientific and non-scientific reasoning go beyond the facts. So, can we tell them apart?

Argument:

1. The demarcation between science and pseudoscience depends only the nature of the reasoning used.
2. Genuine science involves ampliative inference.
3. Pseudoscience involves ampliative inference.
4. Therefore, there is no demarcation between science and pseudoscience.

The problem of demarcation is to say what is wrong with this argument. (Question: what are the two things that can be wrong with an argument?)

Definition: An argument is deductively valid if and only if it is impossible that its conclusion is false while its premises are true.

Remark: The notion of deductively validity is such a central and important concept in philosophy, that is goes by several names. When an argument is deductively valid, we say that the conclusion follows from the premises, or the conclusion is deduced from, or inferred from, or proved from the premises. Or we may say that the premises imply, or entail, or prove the conclusion. We also talk of deductively valid arguments as being demonstrative. All these different terms mean exactly the same thing, so the situation is far simpler than it appears.

Definition: Any argument that is not deductively valid, or deductively invalid, is an ampliative argument. The term refers to the fact that the conclusion of such argument goes beyond, or amplifies upon, the premises.

Remark: Again the notion of ‘invalid’ is so common and central, that it goes by many names. Other terms commonly used are inductive and non-demonstrative. I prefer ‘ampliative’ because it reminds us that the conclusion ‘goes beyond’ the premises, and it does not have the bad reputation that sometimes goes along with the word ‘induction.’