Logic is the study of the formation of arguments and their structure. It explains how arguments can be formed so as to be valid, sound and convincing. As humans, we are capable of understanding and applying rules of logic in our arguments. However, this does not often happen for two main reasons.
First, one would think that knowing how to construct good arguments under a system of logic should be a basic component of the education system. However, logic is worryingly absent from the curricula. Too much importance is put on remembering and memorising arguments, while the way in which arguments are formed is wholly disregarded, resulting in a logically-handicapped population.
Second, our very own nature does not always help us approach each problem with rationality and critical thinking, form logical arguments, or spot invalidities or fallacies in arguments we face. We often do not have the time or the resources to rationally approach arguments or situations, and we thus employ heuristics, mental shortcuts we use when we want to make quick decisions or reach quick solutions.
However, this does not even slightly diminish the importance of logic. The rules of logic are considered true a priori; they are fundamental laws. There are three fundamental laws of logic, as identified by Bertrand Russell, and previously by Aristotle. The law of identity states that a thing is necessarily identical with itself. The law of contradiction states that, for any proposition p, it is impossible that p is true and not true at the same time. It is impossible that a statement p (e.g. ‘My coat is wet’) is both true and not true at the same time. The law of excluded middle states that at most one of two contradicting statements must be true. It must be true that either ‘My coat is wet’ or ‘My coat is not wet’. Here are some fundamental axioms on which logic is built:
Deductive logic
Constructing arguments on the fundamental laws of logic, can help us understand validity in arguments. An argument is formed by a number of premises which lead to a conclusion. A deductive argument is valid when it is impossible for it to have true premises and a false conclusion. For example, the following argument is valid:
All men are humans.
Plato is a man.
Therefore, Plato is a human.
Given that the premises are true, the conclusion that ‘Plato is a human’ must also be true, if the argument is to be valid.
However, validity does not necessarily deal with the actual truth of the premises in question. The following argument is also valid:
All giraffes are aliens.
Aristotle is a giraffe.
Therefore, Aristotle is an alien.
Hence, what matters in deductive logic is the structure of the sentences. The above sentences follow the same structure, and are therefore both valid, despite their actual claims. What matters, then, is that, given that the premises were true – whatever they might claim – the conclusion must also be true.
The structure that the above arguments form is this:
All P are Q.
R is a P.
Therefore, R is a Q.
Whatever one chooses to substitute for P, Q, and R, they will be able to form a valid argument.
What deals with the actual truth of sentences, however, is soundness. An argument is sound if and only if it’s a valid argument and all its premises are actually true. Therefore, the first argument above is both valid and sound, but the second is only valid, as all its premises and conclusion are false.
Two further rules of logic which are very often and easily confused, leading to thinking errors in deductive validity are modus ponens and modus tollens.
Modus ponens arguments take the following form:
If P, then Q
P
Therefore, Q
For example,
If it’s raining, I will buy an umbrella.
It’s raining.
Therefore, I will buy an umbrella.
It’s a logically valid argument, though not necessarily sound. What it says is that, if the argument is to be logically valid, and its premises are true, the conclusion must also be true.
Modus tollens arguments take the following form:
If P, then Q
Not Q
Therefore, not P
For example:
If it’s raining, the sky is grey.
The sky is not grey.
Therefore, it’s not raining.
All modus tollens arguments tell us about is the structure of the sentence and its validity. The actual content is irrelevant.
Again, whatever one chooses to substitute for P and Q in the above argument, they will always get a valid argument. An argument that is valid, however, is not always a good argument. A good argument has to be sound and valid at the same time.
Nevertheless, the above argumentative structures are very often confused and incorrectly applied, through their invalid variations: affirming the consequent anddenying the antecedent. These two forms of argument are invalid and should be avoided.
Affirming the consequent takes this form:
If P, then Q
Q
Therefore, P
The first premise, however, does not exhaust all the possibilities under which Q arises. P is not the only cause of Q, therefore the fact that Q is true does not necessitate that P has to also be true.
Denying the antecedent takes this form:
If P, then Q
Not P
Therefore, not Q
Again, this argument tells us nothing about all the conditions under which Q arises.
Taking, for example, the argument ‘If it rains, then you will get wet’.
Saying ‘You are wet, therefore it rained’, is fallacious, as there are other ways of getting wet. This is the fallacy of affirming the consequent.
Saying ‘It did not rain, therefore you are not wet’, is also fallacious, as there are other ways of getting wet. This is the fallacy of denying the antecedent.
As with their valid counterparts, what matters with ‘affirming the consequent’ and ‘denying the antecedent’ arguments is their structure, not the actual truth of the sentences one may substitute for P and Q. One may be able to form an argument with true premises and conclusion which is, nevertheless, invalid.
All the above deal with deductive logic, that is, logic dealing with the logical form of the sentences. The actual content does not matter.
Inductive logic
Inductive logic is the branch of logic which deals with the content of the sentences, not their structure.
Unlike valid arguments in deductive logic, true premises in inductive arguments do not guarantee the truth of their conclusion; they only make it increasingly possible. Thus, inductive arguments can vary in strength. The general idea is that deductive logic infers the specific from the general, while inductive logic infers the general from the specific.
For example, the statements ‘The Sun will rise tomorrow’, and ‘The next swan I will see will be white’, are both inductive statements, as they are based on assertions about the future based on past experience. Nothing, however, absolutely guarantees that the laws which cause the phenomenon of sunrise will apply tomorrow, or that the next swan I will see will not have a mutation which caused it to be black.
Inductive logic is important, as a large part of science is based on it. We base our best scientific theories on observations and assertions about the future; theories that we expect to be true under all identical circumstances.
Inductive logic, however, is also what we mostly use in our everyday lives. When crossing an empty street in a rural area, we don’t think of the possibility that a Ferrari will appear speeding at 250 km/h and killing us. When we are going to pay for a meal in London using British Pounds, we don’t think of the possibility that the UK currency changed to Euro overnight. These scenarios are not improbable, but the probability of them happening is extremely small. We base this on our experiences and our current knowledge of the matters in question, and not necessarily by using any deductive logic.
Logical Fallacies
Inductive logic’s common and everyday usage, however, brings with it a great variety of language misuses within inductive arguments, in the form of logical fallacies. Inductive fallacies are weak argumentative techniques that do not provide any support for the conclusion they present. People may commit logical fallacies unwittingly, but fallacies may also be used intentionally to manipulate an argument and make it look more convincing, especially when presented towards people who are unaware of the rules of logic or do not know how to spot logical fallacies in arguments. In reality, however, fallacies take away an argument’s strength and make it less convincing. Fallacies in inductive logic are called informal fallacies.
Some informal fallacies are stated and explained below:
Begging the Question
When this fallacy is committed, the truth of the conclusion is assumed in one of the premises. The conclusion is taken for granted, when it itself requires proof. It’s a form of circular reasoning, where the premises do not provide additional evidence for the conclusion, but pose the same statement as the conclusion.
Example:
‘Minority interests should not be considered as much as majority interests, for the majority is always more important than the minority.’
No independent argumentation is given as to why minority interests should not be considered. Saying that ‘the majority is always more important’ is just a restatement of what was previously said.
Post Hoc Ergo Propter Hoc
Under this fallacy, when an event B often happens after an event A, it is falsely claimed that event A caused event B, when they are completely unrelated.
Example:
‘I always pray to God before each football match. That’s why we win every game.’
The fact that an action was followed by the desired ‘outcome’ does not signify that the former event caused the latter. Causation has to be proven and not just asserted. Also, it is often the case that people forget cases where the latter event (Winning every game) happened in the absence of the former event (Praying to God), or did not happen despite the fact that the former event happened.
In statistics and science, this is often brought up by the statement ‘Correlation does not imply causation’, although this sentence can also be applied with events that happen at the same time, and not only when one comes after the other.
False Dichotomy
This occurs when an argument only presents two possible options, ignoring other alternatives that may exist. It’s an ‘either-or’ fallacy, in which it is argued that ‘Either X or Y is true’ when there could be an option Z worth considering.
Example:
‘Creationism is true because the theory of evolution has many flaws.’
The creationist fails to see that there might be other options to consider if they believe that the theory of evolution is false. What is more, proving one argument wrong, does not make the opposing argument stronger.
Slippery Slope
Slippery slope fallacies occur when it’s assumed that the acceptance of a statement will bring about bad and undesired future consequences in a chain of, possibly unrelated, events.
Example:
‘Legalising same-sex marriage will increase divorce rates between heterosexual couples and will lead society in accepting paedophilia and bestiality’
Legalising same-sex marriage is not, however, related in any way in divorce rates between heterosexual couples and acceptance of pedophilia and bestiality. If concrete causal evidence is not presented, slippery slope arguments fail to provide a reason to refuse the initial argument.
No True Scotsman
Occurs when the person making an argument redefines a group in a way which the new group definition excludes members which have some undesirable qualities.
Example:
‘The man speaking for the Christians’ side at the debate, said that the Bible is a work of fiction. I do not believe him though; this proves he is not a real Christian.’
The person here re-defined, ad hoc, who can qualify as a ‘real Christian’, i.e. the debater who would not say that the Bible is a work of fiction.
Had the debater said that we should accept the Bible as describing true events, the person would accept the debater’s words as true. However, when the words of the debater do not coincide with the person’s worldview, the debater is not considered a ‘real Christian’, and their views are wholly disregarded.
In conclusion, it is of utmost importance to understand how to construct logically valid arguments, but also how to be convincing by formulating arguments which are coherent and sound.
Understanding basic logic allows us to better judge situations in which we find ourselves in, and also being able to tell when others use fallacious arguments, on purpose or not, essentially making the world a more understandable place where we more effectively communicate with each other.