Proofs

What is this Tutorial About?

The tutorial Compound Indication dealt with the realization of assents through speech and covered the various aspects of sentences. The tutorial Indicated Assent then dealt with the statements that sentences represent and covered aspects such as contradiction and inverse. This tutorial now uses what was taught therein to discuss the theory of proofs. In order to fully appreciate this discussion, it is recommended that the reader have also read Modern Logic, although this is not a strict prerequisite.

Proof of Compound Statements

∃h (S(h) ∧ R(h)) → ◇ (∀p G(p))

If we are required to prove any complicated claim like the above, we must first divide it into its most atomic propositions and deal with each separately. Recall from Modern Logic that propositions (and their negations) can be connected with one another using various connectives, as the above statement is a combination of two propositions connected through implication (IF-THEN). If we separate these two, we get the following.

∃h (S(h) ∧ R(h)) → ◇ (∀p G(p))

We have now divided the complicated statement into two slightly more manageable components. How we go about dealing with each component is determined by the types of operations used to connect them. Recall that all statements can be expressed as a combination of conjunction (AND) and disjunction (OR). In particular, implication (P → Q) can be expressed as disjunction (¬P ∨ Q). Rewriting the above statement we get the following.

¬∃h (S(h) ∧ R(h)) ∨ ◇ (∀p G(p))

We must apply this process recursively at all levels of the statement so, in the end, all we have left is a series of conjunctions and in each conjunction we have a series of disjunctions, or we have a series of disjunctions and in each disjunction we have a series of conjunctions. Any statement can be reduced to this form – this is called Normal Form (NF). An example follows.

Notice that, in the end, all we have is conjunctions, disjunctions, and atomic propositions or their negations. If a set of propositions are connected by conjunctions (AND), we must prove all propositions. If, however, they are connected by disjunctions (OR), we need only show one of the propositions to be true. Consider the following statement.

The way we prove this is by noticing that the top-level components of this statement are (A(a) ∨ ¬B(b) ∨ C(c)) and (A(b) ∨ B(c) ∨ C(a)) and that they are connected by conjunction. This means that we need to prove both components separately. In proving the first component, we need to show that either A(a) holds, ¬B(b) holds, or that C(c) holds. And in proving the second component we apply a similar algorithm. Notice the recursive nature.

Once we are done with this recursive process, we will eventually need to prove the atomic propositions or their negations. In other words, we need to actually prove at some point A(a), A(b), and so forth. We may need to do this with basic propositions, quantified propositions, and/or moderated propositions. There are various methods to this which are detailed in the rest of this tutorial.

Proof of Atomic Propositions

Generalization & Induction

There are three major forms of proof, one of which is of interest and the other two of which are given less attention due to their lack of decisiveness. One of the two is generalization (تمثيل) which is the process of applying the ruling of one thing onto another merely by virtue of the two sharing something in common. For instance, one may read the Bible and deem it to be a long, religious text which is complicated. The reader may subsequently deem that the Qur’an, because it too is a long, religious text, is also complicated.

The other type of indecisive proof is induction (استقراء). This is the process of extrapolating a ruling over an entire class based on having witnessed the ruling hold for a subset of the individuals of that class. For example, one may witness several dozen occurrences of a mosquito biting someone and causing an allergic reaction in the victim. One may then generalize over the entire class of mosquitoes and deem all to cause allergic reactions upon biting. Notice that this is significantly dissimilar to generalization in two ways:

·         the process of induction involves generalization over individuals of a genus by virtue of their being in that genus, whereas generalization (تمثيل) involves generalization based on some property or attribute

·         induction requires testing the ruling over a sufficiently large subset of individuals before applying it to the entire class, whereas generalization is often a one-to-one application

Deduction

Deduction (قياس) is an entire class of proofs. What’s common between them is that the method of arriving at a result, given some prerequisite information, is decisive and airtight. Whereas generalization, induction, etc may yield incorrect results, deduction always yields decisive results, assuming that the information used to derive the result is itself valid.

One of the forms of deductive proof is Proof by Contradiction (دليل الخلف). This is a deduction whereby a proposition’s contradiction is proven true (or false), thereby proving the proposition itself false (or true). Since the very definition of Contradiction implies that a proposition is false whenever its contradiction is true and vice versa, this is indeed an airtight consequence and hence a form of deduction.

For example, we can prove the proposition “all humans are animals” by opting to disprove its contradiction, which is “some humans are not animals.” If we can disprove this, we have proven the original statement.

Another type of deduction is the syllogism. A syllogism is a type of deduction where the conclusion is supported by two premises from which it results. For example, “if all A is B, and all B is C, then all A is C.” The subject of the conclusion (A in the example) is called the minor term (أصغر) and the premise which contains it is called the minor premise (صغرى). The predicate of the conclusion (C in the example) is called the major term (أكبر) and the premise that contains it is called the major premise (كبرى). The major and minor premises then contain a shared middle term (حد أوسط) that binds them together (B in the example).

Syllogisms come in four flavours depending on where in the major and minor premises the common term falls. Each form of the syllogism is called a figure (شكل) as named by Aristotle. The table below gives these figures and the syllogisms in each.

The syllogisms in the first flavour are quite obviously valid. The syllogisms in the other three flavours, however, are not so obviously correct. They are, in fact, achieved by applying inversion to one or both of the two premises.

Types of Deduction Based on Composition

A deductive proof, as mentioned, is one in which the process of deriving a consequence is completely sound. For example, it is not logically cogent to assume that all mosquitoes cause an allergic reaction when they bite merely by witnessing this happen sufficiently many times. It is, however, perfectly cogent to deduce that all men are animals given that all men are humans and that all humans are animals. The principle that allowed us to reach the conclusion in the latter example is a sound principle, unlike the one in the former example; it is therefore a type of deduction.

Now, the principle may be sound, but that doesn’t entail that the premises used in the deduction are also sound. We can say that everything in the air is a bird because anything in the air is flying and anything that is flying is a bird. Although the principle in this deduction was correct, the premises were faulty and erroneous.

Given this, deduction (قياس) is divided into five categories:

So the برهان is a deduction in which the premises are axiomatic or are already proven theorems. This is the most cogent type of proof and the hallmark of classical logic. Axioms, then, are categorized:

1.       أوليات – axioms which the mind accepts immediately without explanation, like “the whole is greater than the part”

2.       فطريات – axioms which the mind accepts after some simple analysis, like “4 is even” which merely requires conceptualizing “4” and the definition of “even”

3.       حدسيات – axioms which, not immediately, but quickly lend themselves to what is proposed; whereas actual thought requires acquiring parts of a concept then organizing them, these require merely acquiring the parts without organizing them

4.       تجربيات – axioms which the mind accepts based on experience

5.       متواترات – axioms accepted because their opposites are deemed impossible

6.       مشاهدات – axioms accepted by virtue of the senses

*Perception & The Five Senses

We have now seen the most powerful form of proof; the برهان. Because it is a sub-type of deduction, the mechanism used to derive a consequence from its premises is sound. Moreover, it is a sub-type of deduction where the premises themselves are either axioms or proven claims.

We then categorized axioms based on how the mind accepts them and one of the ways was through the five senses. This section of the tutorial is an addendum that builds on the concept of the five senses and presents the Classical Logic and Classical Philosophy thought on this issue. Although the concepts we have studied so far in our pursuit of Classical Logic have all been valid, this particular quondam philosophy has been abrogated by modern neuroscience. By the same token, the translations of certain Arabic terms we use hereafter are merely approximations from modern neuroscience. The reason we add this addendum is because this philosophy is a vital prerequisite for many topics on Arabic Rhetoric.

The mind will either perceive universals (كلي) or non-universals (جزئي). In order to perceive universals – in other words, abstract concepts – the mind will use the Cerebral Cortex (عقل). When the Cerebral Cortex has completed its conceptualization, it will store its information in the Hippocampus (العقل الفعال).

In order to perceive non-universals – in other words, tangible entities – the body will first use one of the five senses (الحواس الخمسة). These are as follows.

1.       vision (القوة الباصرة); things that are seen (مبصرات)

2.       audition (القوة السامعة); things that are heard (مسموعات)

3.       olfaction (القوة الشامة); things that are smelled (مشمومات)

4.       gustation (القوة الذائقة); things that are tasted (مذوقات)

5.       somato-sensation (القوة اللامسة); things that are felt (ملموسات)

These senses will then carry information (the صورة) to a the Thalamus (الحس المشترك). When the Thalamus has finished processing the information, it will store it or send it for further processing to the Neocortex (خيال).

If a non-universal entity cannot be carried by the five senses and processed by the Thalamus, such as a particular instance of anger, it is called معنى in Arabic as opposed to صورة and it is processed by the Frontal Lobe (الوهم). When the Frontal Lobe has finished processing it, it stores it in memory (حافظة).

Finally, the Association Area (متصرفة) is the faculty that gathers various memories and associates or separates some to or from others.