Formal fallacies1 are fallacies that relate to the form of the argument or arrangement of the terms rather than their meaning. They are fallacies of the form of the arguments as represented in logical form. Logical form is a special way of representing an argument using precise language or mathematical symbolism. When argument from everyday language are rearranged into logical form, these can be represented mathematically by variables. a mathematical type logical argument can be made using these variables in place of everyday words. Error in the arrangement of these variables or their relationships to each other are called formal fallacies. Formal arguments include conditional and unconditional (categorical) arguments.
Hypothetical syllogisms - Conditional arguments
A hypothetical syllogism is a conditional argument with two premises and a conclusion. The first premise is an if...then statement. The 'If' part is the antecedent, and the 'then' part is the consequent. The second premise is a confirmation or denial of either the antecedent or the consequent. There are four possible forms of this type of argument. The argument are structured like this:
If A, Then B
Either A or Not A or B or Not B depending on argument form
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Either A or Not A or B or Not B
This argument form is common in the Scriptures. Of the four possible forms of this argument, two of the forms are valid and the other two are invalid.
Valid forms
The two valid forms are modus pollens - confirming the antecedent and modus tollens - denying the consequent.
Modus ponens or confirming the antecedent2 takes the form:
1. If A, Then B
2. A
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Therefore B
In Scripture we see the modus pollens form in 1 John 1:9 " If we confess our sins, he is faithful and just to forgive us our sins, and to cleanse us from all unrighteousness. " It can be expressed this way.
1. IF we confess our sins, THEN our sins are forgiven.
2. We have confessed our sins
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Therefore, our sins are forgiven. (paraphrase)
Modus Tollens or denying the consequent is an argument form where the second premise denies the 'then' part of the if..then statement. It takes the form.
1. If A, Then B
2. Not B
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Therefore Not A
In Scripture, we see the modus tollens form in 1 Corinthian 15
1. If Christ is not raised, THEN Preaching and faith worthless, Christians still in sins, and apostles false witness.
2. However, Preaching and faith with power (not worthless) & Christian freed from sin (not still in sins) & apostles true witnesses (not false witnesses )
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Therefore Jesus is risen from the dead.
Invalid forms
There are two invalid forms. They are denying the antecedent and confirming the consequent. These involve that do not follow from the premises because there are other possible conditions.
Denying the antecedent occurs when you draw conclusions about the consequent (the then part) based on a denial of the antecedent. Denying the consequent takes this form.
1. If A, Then B
2. Not A
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Therefore Not B
This fails because there are other conditions that may account for B besides A. Consider this example.
1. If a student makes straight A's, THEN he will get a scholarship
2. The student did not make straight A's
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The student did not get a scholarship.
This conclusion fails because there may be other ways that a student might get a scholarship. Students get scholarships for financial need, musical and athletic ability, and a host of other reasons.
When some interpreters read 1 John 1:9, they draw the wrong inference. When they read that "IF we confess our sins, THEN He will forgive our sins" this way.
1. IF we confess our sins, THEN He will forgive our sins.
2. We have not confessed our sins
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He has not forgiven our sins.
This fails because we are justified by faith. Instead of seeing confession as a promise of affirmative action by Our Lord and Savior, people turn it into a legalistic requirement by committing the fallacy of denying the antecedent. Instead of being a legalistic requirement, confession is a tool to exercise faith and release power from God.
The fallacy of affirming the consequent occurs when people draw conclusions about the antecedent (the if part) by affirming the consequent. Affirming the consequent takes the form.
1. If A, Then B
2. B
----------
Therefore A
Because the consequent might be true for reasons other than the antecedent, this fork fails. A real world example of this form may look something like this.
1. If a student makes straight A's, THEN he will get a scholarship
2. The student got a scholarship
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Therefore student made straight A's
Again, this fails because a student could have got the scholarship through other causes.
Disjunctive argument.
Disjunctive arguments are exclusive OR argument that show a contradictory relationship between two ideas. These two ideas are called disjuncts. This type of argument is extremely common in the Scripture. It takes the following form
A OR B
NOT A
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THEREFORE B
This argument form is used by Jesus in Matthew 6:24. It can be expressed this way.
Since " No man can serve two masters ; Ye cannot serve God and mammon."
he will serve God OR he will serve mammon (money)
He serves God
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Therefore he does not serve money.
This argument form has three invalid forms. He cannot deny both disjuncts, he cannot affirm both disjuncts, and he cannot deny and affirm the same disjunct.
Categorical Syllogisms
A categorical syllogism is an argument type that, when expressed in logical form, has exactly two premises followed by a conclusion. When expressed in logical form, these arguments have three terms. The validity of this type argument depends solely on the form of the argument; it is agnostic concerning the content of the argument. This argument form is not as common in the Scripture.2
There are four types of premises in a categorical syllogism. These types of statement are labeled A, E, I, and O type statements as defined in the square of opposition:
A type: All A are B
E type: No A are B
I type: Some A are B
O type: Some A are not B
Categorical syllogisms can be evaluated by using three rules of inference. The first two rules relate to whether a term is distributed to all members of the class designated by that term, and the third relates to whether a conclusion affirm or denies a truth.
Distribution refers to universal application. For example, in the statement 'All men are mortal,' 'men' is distributed because it applies to each and every man. 'Mortal' is not distributed as the statement does not apply to every mortal. There may be mortals who are not men. There are three ways to determine if a term id distributed.
1. Every term preceded by the word 'All' is distributed
2. Every term preceded by the word 'Not' is distributed.
3. In very sentence in which the subject is preceded by the word 'No,' both the subject and predicate are distributed
Here are the two rules on distribution:
The middle term must be distributed. It is impossible to deduce any link between the other two terms if the statement that contain the middle term do not absolutely apply to every member of the class referenced by the middle term.
If a term is distributed in the conclusion, it must be distributed in one of the premises. If there are no statements that absolutely apply to all members of a class in the premises, then it is impossible to deduce an absolutely applicable statement in the conclusion.
The third rule relates to how statement affirm or negate. Affirmation occurs in statements containing 'all' and 'some.' Negation occurs in statements containing 'no' and 'not.'
In order to draw any conclusions at least one statement must be affirmative, and the sign value of the conclusion must be a valid product of the multiplied sign values of the premises. Two affirmative statement in the premises must produce an affirmative conclusion (think +1 * +1 = +1). An affirmative and a negative statement (think -1 * +1 = -1) produce a negative conclusion. Two negative statements produce no conclusion, as it denies a link between the statements in the premises.
References
1 "Formal Fallacy", The Fallacy Files
2 The categorical syllogism, at least in its formal application, is extremely uncommon in the Scripture. As a formal application, it is primarily connected to Aristotelian logic. It is included for completeness, but I will not be asking questions on this section. Of the 64 possible forms, only a few appear in Scripture. This one is so obviously valid that little formal training in logic is needed to understand its validity
1 What are formal fallacies?
2 What is logical form?
3 What are two main types of arguments?
4 What is a conditional argument?
5 What are the two valid conditional argument forms?
6 What are the two invalid conditional argument forms?
7 What is a disjunctive syllogism?
8 What are the two valid disjunctive syllogisms.
9 What are three invalid disjunctive syllogisms.
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