Basic Concepts

Gig Φ Philosophy
(at-a-glance overviews of philosophical concepts)

Published September, 2022

Symbolic Logic : PREVIOUS

Introduction: Logic & Arguments

LOGIC

Logic, traditionally understood, is centered around the analysis and study of argument forms and patterns. In other words, logic is the study of proper rules of reasoning and their application to arguments. Arguments come in many forms but, as we shall see, we will find it helpful to develop and refine a system of rules and methods that help us deal with language and arguments. More specifically, we want to be able to identify good patterns of reasoning, and crucially, be able to separate them from bad forms of reasoning. This is one of the many ways logic can help. It can give us, in varying degrees of success, methods for improving and evaluating not only our own reasoning, but that of others as well. Given that we are constantly faced and confronted with claims, arguments, and pieces of reasoning, the usefulness of logic can seem all the more appealing.

See here for more about the relationship between logic and philosophy

ARGUMENTS

So if logic is the study of argument forms and patterns, what, then is an argument exactly? An argument is a set of sentences, one or more of which we call the premise or premises, which are intended to provide support for or reasons to believe another sentence, the conclusion. In other words, to present an argument is to give reason(s) for thinking that some conclusion is true. What we do not mean by argument is a quarrel or verbal fight (though arguments can certainly lead to that). Here is an example of an elementary argument:

EXAMPLE ARGUMENT

All states have capitals. Washington is a state.

Therefore, Washington has a capital.

As we can see, there are two sentences in the passage here that are intended to provide support for, or reasons to believe, one of the other sentences. The first two sentences give us reason to believe the statement “Washington has a capital”. Both the premises and the conclusion are composed of what philosophers call propositions or statements. An argument can have one or more premises but only one conclusion. 

What is a Proposition / Statement?

STATEMENTS / PROPOSITIONS

There is much philosophical discussion about the distinction between propositions, statements, and claims. While this is an important topic in philosophical logic, we will set aside those discussions here and use the terms interchangeably, but settle by using the word statement throughout. For more discussion on this topic, see Grayling (2001).

By statement we mean simply a sentence that is either true or false. To say that a statement is either true or false is just to say that it has a truth value. Bringing this together, an argument then, consists of at least two statements: a statement to be supported (the conclusion) and the statement (premise) meant to support it. Another way of demarcating between statements and nonstatements is to think of them as being synonymous with propositions.

In symbolic logic, any proposition can be translated using a surprisingly limited pattern of statements. 

Most Common Statements in Logic

Conditional Statements = If P, then Q

P = Antecedent

Q = Consequent

Disjunctive Statements = Either P, or Q

P & Q = Disjuncts

However, most statements are not constructed this way. See the following examples:

EX: STATEMENTS

The first two statements are in fact true while the third statement is actually false. Interestingly, to the fourth statement, while many people may claim that they know the answer, most people understandably will say that they do not know if this statement is true or false. Even in these cases, we still want to say that statements, on our definition, are either true or false. 

EX: NONSTATEMENTS

To see the difference between statements and nonstatements, let's look at the following examples of nonstatements:

In these examples, there is nothing claimed by the sentences that is either true or false in the same way as the examples of statements, so we say, for example, that questions do not have a truth value. 

What is an Argument?

INDICATOR WORDS

Now that we have introduced the idea of arguments being composed of statements, we want to successfully identify the premise(s) and conclusion of an argument. To help us do so often involves indicator words. Conclusion indicators are words used (in our case, English) that lead us to believe that what follows is the argument's conclusion. 

EX: PREMISE
Indicator Words

Conversely, we can often identify premises by using indicator words. Common premise indicators are: 

since
because
given that
for
as indicated by
implied by
due to the fact that

EX: CONCLUSION
Indicator Words

Here are some of the most commonly used conclusion indicator words:
thus
therefore
accordingly
consequently
hence
it follows that
we may conclude that
as a result

EX: Identifying an Argument

To see how these words can help us identify respective premise(s) and the conclusion, let's look at the following argument:

EXAMPLE ARGUMENT

Because Sophie was born in August, it follows that she is a Leo.

Notice that we have two indicator words that tip us o to what the premise and conclusion are. It is worth noting, though, that indicator words may not always be present. Sometimes, we need to assess the inferential relationship between statements in order to determine if an argument is present [i.e., if some statement(s) is meant to support another]. 

PATTERNS OF REASONING

Inference is the process of reasoning wherein we recognize patterns and relationships between propositions / states of affair. Even if one has never heard the infamous syllogism (three-lined argument) about Socrates; you can probably guess what the conclusion is from its premises. See if you can infer what follows from this variation on the classic:

EXAMPLE ARGUMENT

All women are mortal. Hypatia is a woman. Therefore, [...?]

 These premises imply that “Hypatia is mortal”.

STANDARD FORM

One practice that helps us focus in on an argument's content is called putting an argument into standard form. This refers to the process of taking an argument in passage form, identifying and organizing the premise(s) (typically abbreviated as “P1.”, “P2.”, etc.) such that they lead inferentially to the conclusion (typically abbreviated as “C.” with “∴” to indicate “therefore”) making the argument as a whole clearer to the reader. 

Let's say that we have an argument contained in the following passage:

EXAMPLE ARGUMENT

Public schools deserve increased financial assistance. The amount of money spent per student has been decreasing for years in this state. At current funding levels, the state cannot fulfill its constitutional obligation to provide public education to all of its citizens.

Although there are no indicator words in the above example, upon analysis, the second and third statements seem to provide reason to believe the first. By putting this argument into standard form we can better appreciate the flow of the argument:

EXAMPLE ARGUMENT IN STANDARD FORM

P1. The amount of money spent per student has been decreasing for years in this state.

P2. At current funding levels, the state cannot fulfill its constitutional obligation to provide public education to all of its citizens.

C. ∴ Public schools deserve increased financial assistance.

Notice importantly that the first sentence in our original passage was the conclusion itself, which should always be listed last in standard form. It should be pointed out that while the conclusion may appear at the end of a passage, it can often appear as the first sentence, as in our case above. Additionally, many simple arguments, like the example above, can be combined to form more complex arguments; in which case the conclusion of the simpler argument becomes a premise of the larger, more complex, argument.

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