Formal Logic Introduction Flashcards
Implications
There’s one very important thing to remember: The fact that a deductive argument is valid doesn’t necessarily mean that its conclusion holds. That may seem confusing, but it’s because of the slightly counter-intuitive nature of how implication works. Obviously you can build a valid argument out of true propositions. But you can also build a completely valid argument using only false propositions. For example: All insects have wings (premise) Termites are insects (premise) Therefore termites have wings (conclusion) The conclusion isn’t true because the argument’s premises are false. If the argument’s premises were true, however, the conclusion would be true. So the argument is entirely valid. More subtly, you can reach a true conclusion from false premises – even ludicrously false ones: All fish live in the ocean (premise) Sea otters are fish (premise) Therefore sea otters live in the ocean (conclusion) However, there’s one thing you can’t do: start with true premises, go through a valid deductive argument, and arrive at a false conclusion. If the premises are false and the inference valid, the conclusion can be true or false. (Lines 1 and 2.) If the premises are true and the conclusion false, the inference must be invalid. (Line 3.) If the premises are true and the inference valid, the conclusion must be true. (Line 4.) A sound argument is a valid argument whose premises are true. A sound argument therefore arrives at a true conclusion. Be careful not to confuse sound arguments with valid arguments. Ultimately, the conclusion of a valid logical argument is only as compelling as the basic premises it is derived from. Logic in itself does not solve the problem of verifying the basic assertions which support arguments. The only way to verify basic assertions is by scientific enquiry.
Recognizing an argument
Sometimes arguments won’t follow the order described above. For instance, the conclusions might be stated first, and the premises stated afterwards in support of the conclusion. This is perfectly valid, if sometimes a little confusing. Arguments are harder to recognize than premises or conclusions. Lots of people shower their writing with assertions, without ever producing anything you might reasonably call an argument. To make the situation worse, some statements look like arguments but are not. For example: “If the Bible is accurate, Jesus must either have been insane, an evil liar, or the Son of God.” The statement above isn’t an argument; it’s a conditional statement. It doesn’t assert the premises which are needed to support what looks like its conclusion. (Even if you add those assertions, it still suffers from a number of other logical flaws.) Here’s another example: “God created you; therefore obey and worship God.” The phrase “obey and worship God” is neither true nor false. Therefore it isn’t a proposition, and the sentence isn’t an argument. Causality is important as well. Suppose we’re trying to argue that there’s something wrong with the engine of a car. Let’s look at two statements of the form “A because B”. Here’s the first: “The car won’t start because there’s something wrong with the engine.” That’s not an argument for there being something wrong with the engine; it’s an explanation of why the car won’t start. We’re explaining A, using B as the explanation. Now consider a second statement: “There must be something wrong with the engine of the car, because it won’t start.” Here we’re arguing for A, giving B as evidence. The statement “A because B” is an argument. The difference between the two cases might not be completely clear. So, remember that “A because B” is equivalent to “B therefore A”. The two statements then become: “There’s something wrong with the engine, therefore the car won’t start.” And: “The car won’t start, therefore there’s something wrong with the engine.” We’re supposed to be arguing that there’s something wrong with the engine, but now it’s obvious that the first statement doesn’t do that at all. Only the second statement is arguing that there’s something wrong with the engine.
Types of Argument
There are two traditional types of logical argument: deductive and inductive. 1. A deductive argument is one which provides conclusive proof of its conclusions. It is either valid or invalid. A valid deductive argument is defined as one where if the premises are true, then the conclusion must also be true. 2. An inductive argument is one where the premises provide some evidence for the truth of the conclusion. Inductive arguments are not valid or invalid, but we can talk about whether they are better or worse than other arguments. We can also discuss how likely their premises are. There are forms of argument in ordinary language which are neither deductive nor inductive. However, we’ll concentrate on deductive arguments, as they are often viewed as the most rigorous and convincing. Here is an example of a deductive argument: Premise: Every event has a cause Premise: The universe has a beginning Premise: All beginnings involve an event Inference: This implies that the beginning of the universe involved an event Inference: Therefore the beginning of the universe had a cause Conclusion: The universe had a cause Note that the conclusion of one argument might be a premise in another argument. A proposition can only be a premise or a conclusion of a particular argument; the terms don’t make sense in isolation.
What is an argument?
There are three stages to an argument: premises, inference, and conclusion. Stage 1: Premises For the argument to get anywhere, you need one or more initial propositions. These initial statements are called the premises of the argument, and must be stated explicitly. You can think of the premises as the reasons for accepting the argument, or the evidence it’s built on. Premises are often indicated by phrases such as “because”, “since”, “let’s assume”, and so on. Stage 2: Inference Next the argument continues step by step, in a process called inference. In inference, you start with one or more propositions which have been accepted. You then use those propositions to arrive at a new proposition. The new proposition can, of course, be used in later stages of inference. There are various kinds of valid inference – and also some invalid kinds, but we’ll get to those later. Inference is often denoted by phrases such as “implies that” or “therefore”. Stage three: Conclusion Finally, you arrive at the conclusion of the argument, another proposition. The conclusion is often stated as the final stage of inference. The conclusion is affirmed on the basis the original premises, and the inference from them. Conclusions are often indicated by phrases such as “therefore”, “it follows that”, “we conclude” and so on. Note that the phrase “obviously” is often viewed with suspicion, as it gets used to intimidate people into accepting things which aren’t true at all. If something doesn’t seem obvious to you, don’t be afraid to question it. You can always say “Oh, yes, you’re right, it is obvious” when you’ve heard the explanation.
Basic Concepts
The building blocks of a logical argument are propositions, also called statements. A proposition is a statement which is either true or false. For example: “The first Holden car was built in 1948.” “Ginger cats are always male.” “Canberra is the capital of Australia.” Propositions may be either asserted (said to be true) or denied (said to be false). Note: This is a technical meaning of the word “deny”, not the everyday meaning. When a proposition has been asserted based on some argument, we usually say that it has been affirmed. The proposition is generally viewed as the meaning of the statement, and not the particular arrangement of words used. So “An even prime number greater than two exists” and “There exists an even prime number greater than two” both express the same (false) proposition. Sometimes, however, it is better to consider the wording of the proposition as significant, and use linguistic rules to derive equivalent statements if necessary.
Introduction
There is a lot of very poor argument in modern Biblical studies (and other fields as well no doubt) even from very well known and high profile scholars. This document attempts to provide a basic introduction to logic in order to guide readers in constructing a careful logical argument that can stand up to scrutiny as well as helping readers to spot poor and invalid arguments. Logic is the science of reasoning, proof, thinking, or inference [Concise OED]. Logic will let you analyze an argument or a piece of reasoning, and work out whether it is correct or not. To use the technical terms, logic lets you work out whether the reasoning is valid or invalid. Note also that this document deals only with simple boolean logic. Other sorts of mathematical logic, such as fuzzy logic, obey different rules. When people talk about logical arguments, though, they usually mean the type being described here. One problem with boolean logic is that people don’t have to be consistent in their goals and desires. People use fuzzy logic and non-logical reasoning to handle their conflicting goals; boolean logic isn’t good enough. For example: “John wishes to speak to the person in charge. The person in charge is Steve. Therefore John wishes to speak to Steve.” Logically, that’s a totally valid argument. However, John may have a conflicting goal of avoiding Steve, meaning that the answer obtained by logical reasoning may be inapplicable to real life. Garlic tastes good, strawberry ice cream tastes good, but strawberry garlic ice cream is only logically a good idea. Sometimes, principles of valid reasoning which were thought to be universal have turned out to be false. For example, for a long time the principles of Euclidean geometry were thought to be universal laws. However, keeping those caveats and limitations in mind, let’s go on to consider the basics of boolean logic.
Formal systems of logic
Sentential logic (SL) is a formal system of logic. It is a very simple system of logic. When people study formal logic this is usually the first thing that they would study. Other more complicated systems include for example predicate logic (PL), and modal logic.
So what is a system of logic? Basically, it is a set of rules that tell us how to make use of special symbols to construct sentences and do proofs. To define a particular system of logic, we need to specify :
The formal language of the system
The semantic rules for the formal language
The rules of proof for the language
A formal language in a system of logic is a language with precisely specified rules that tell us how to construct grammatical sentences. Such rules are called syntactic rules. They are equivalent to the rules of grammar you find in English or Cantonese.
The semantic rules are rules for interpreting the sentences in the language. They tell us what the sentences mean and the conditions under which the sentences are true or false.
The rules of proof are rules that specify how logical proofs are to be constucted. They tell us what conclusions can be derived given certain initial assumptions.
Why study formal systems of logic?
There are many reasons for creating and studying such formal systems of logic:
Systems of logic can be used to formalize arguments in natural languages. A natural language is a language that is used for normal everyday communication in a human society. So languages such as Japanese, Irish, and French are all natural languages. By formalization we refer to the process of translating arguments or sentences in natural languages into the notations of formal logic. The reason for carrying out formalization is that very often they can help us understand the logical structure of arguments better, by identifying patterns of valid arguments. Also, the rules of proof in a formal system of logic are precisely specified. By formalizing an argument we can use the rules of proof to check whether the argument can indeed be proved to be valid.
Because the rules of formal systems of logic are defined very clearly, we can program them into a computer and get a computer to construct and evaluate proofs quickly and automatically. This is particularly important in areas such as Artificial Intelligence, where many researchers teach computers to use formal logic in reasoning.
Linguists are scientists who study natural languages. Many linguists also study formal languages and use them to compare and contrast with natural languages.
Many philosophers are also interested in formal systems of logic. One reason is that natural languages are sometimes not precise enough to express certain ideas clearly. So sometimes they turn to formal systems of logic instead.
Formal systems of logic are also interesting in their own right. Logicians and mathematicians are interested in finding out what they can or cannot prove, and also their many other logical properties. Formal systems of logic also play an important role in understanding the foundations of set theory and mathematics.
Characterizing Levels of Truth
If a statement is not true, then it is either false or possible.
If a statement is not false, then it is either true or possible
**possible statements can be expressed as could be true and could be false**
If a statement does not qualify as could be true, then it must be false
If a statement does not qualify as could be false, then it must be true
