A Basic Introduction to Logic

The primary tool of philosophers is logic, essential to construct arguments and analyze concepts. The field of logic is the systematic study of the principle of valid reasoning. If philosophy is going to arrive at the truth of anything, it requires methods to arrive at the truth. There is nothing more fundamental.

An argument in philosophy is a series of statements or claims which are presented in support of a particular conclusion. A premise is a statement in the argument that leads to the conclusion. There are two basic kinds of arguments, deductive arguments and inductive arguments. The most common example of a valid argument is:

1. All men are mortal.

2. Socrates is a man.

C. Socrates is mortal.

This is an example of a deductive argument. Deductive arguments are arguments which are intended to reason from general principles to specific conclusions. In a valid deductive argument, if the premises are true, then the conclusion must also be true. An argument is sound if and only if, it has true premises and a valid form. It is possible for an argument to be valid and yet unsound.

It is impossible for the premises of a deductive argument to be true, yet for the conclusion to be false. If the first two premises of this example argument are true then the conclusion is entailed by logical necessity. If all men are mortal and Socrates is a man, then it cannot be the case that “Socrates is not mortal.”

This example argument is also a kind of Categorical Syllogism. A syllogism is a formal argument with numbered or otherwise ordered steps and a conclusion. It is called a categorical syllogism because it reasons about categorical claims, i.e. about the category “men.”

Inductive arguments differ in that they argue from specific observations to a more general conclusion. They include probabilistic arguments, and they seek to make probable conclusions about the world from some set of data which seems to favor the conclusion.

1. About 99% of all species that have ever lived on earth are now extinct.

2. Canis familiaris is a species that has lived on Earth.

C. Therefore, the species canis familiaris is extinct.

The conclusion of this inductive argument is factually wrong, canis familiaris refers to dogs, which are quite common nowadays. Hence, this is an unsound argument. But, it is not an invalid argument. Both of the premises are true and yet the conclusion is false. This illustrates that inductive arguments do not lead to certain conclusions. It is possible for all the premises of an inductive argument to be true and yet for the conclusion to be false—this is unlike deductive arguments, where, if the premises are true then the conclusion cannot be false.

Abductive arguments are sometimes set as a third class of argument (some authors include them as a subset of inductive argument). These are inferences to the best explanation. Most of our daily reasoning may be considered abductive. Consider, when someone sits down in their favorite chair it is because they trust that the chair will be stable when they sit in it. Very rarely do people first test the stability of the chair. It is often possible that someone filed down the legs of the chair or otherwise rigged the chair to collapse when anyone attempted to sit in it. But the chair was stable in the past, and the chair looked fine upon cursory glance, so people tend to sit in their chair and often are right in their belief that the chair will be stable when they sit in it. They make an abductive inference that the chair will not collapse once they sit in it. An example of an abductive argument would be:

1. There are cell phones in the world.

2. Humans tend to make cell phones.

C. All cell phones are made by humans.

It is possible that there is a cell phone made by an artificial intelligence, or an alien or by some unknown process. But the best explanation for all the cell phones in the world is that humans have made them. This is the simplest explanation, consistent with what is usually true about cell phones, as stated in premise 2. Many would even think it “crazy” to deny the conclusion of the argument without some strong evidence of non-human cell phone makers. Why is this the “best explanation”? What constitutes a “best explanation?” This depends upon the standards of explanation, justification, and other criteria that the reasoner has in place. But generally, and in our everyday life, explanations are good insofar as they are simpler than rival explanations, explain the data under consideration, and are consistent with our background knowledge.

Symbolic logic is a way of talking about logical expressions and structures without using traditional language. It allows for more precise expressions of logical formulas than vernacular speech. Consider again the following example:

1. All men are mortal.

2. Socrates is a man.

C. Socrates is mortal.

It is possible to symbolize this argument in the following way, if we let M(x) = “x is mortal” and H(x) = “x is a man” and s = “Socrates”

1. ∀x(H(x) ⊃ M(x))

2. H(s)

C. M(s)

The subjects are substituted with variables or letters and operators or adjectival expressions are substituted for symbols. For example, ⊃ means “then” or “entails.” The symbol ∀ means “all” or “for every.” To say “∀x” is then, to make a claim about all x’s i.e. about anything which is an x. The parentheses are used to divide up the expressions and organize them, so we know the full range of applications for the expressions given i.e. it groups things together for us.