Logic Part 3: Deductive Argument
The use of the word ‘argument’, in the context of logic, is a process of reasoning. The deductive argument is merely one type of reasoning.
For example, Mary could argue:
General Premise 1: Tigers are cats that have stripes
General Premise 2: At the present time (in biological evolution) tigers are the only very large cats with stripes
General Premise 3: Leopards are the only very large cats with spots
Specific Premise 4: That animal is a cat, it is very large, and it has stripes
Conclusion: That animal is a tiger, not a leopard
The general premises above are to some extent definitions and might be regarded as true for that reason. They are also categorical in nature because of the take individual examples and make named categories, in this case of cat types. Categorical assertions found within arguments are creating identifiable or quantifiable sets (in the normal mathematical sense).
The point about the formal argument is that we are seeing some form of rationalised justification. In the example above, it is clear that rational justification is implicit in the communally recognised meaning and use of terms. In a more general sense, we cannot separate the concept of logical truth (or some conclusive equivalent) from the concepts of meaning, linguistic use, communally agreed definitions and implication.
Almost bizarrely, formal logic, as presently conceived, is not concerned with the content of the assertions that form the premises, merely the nature of their relationship to the conclusion. Deductive structures, by themselves, are hollow, in the sense of a computer program lacking required input variables needed for the production of a meaningful output. A program might function superbly well but can only function given input data that has meaning (or truth value) on its own account.
Truth, Validity and Soundness in Deduction
When considering the use and meaning of the word ‘truth’ in a logical sense, we should be very clear about what part of the argument requires this property. When people can be heard to commonly argue that something is “logically true”, they are possibly referring to an attribute in logic that is more technically referred to as ‘soundness’. A logical argument requires that we start with opening thoughts, known as ‘premises’. The premises then need to be drawn together in an instinctively or culturally dependent way that we refer to as ‘logically valid’. If the premises are ‘true’ and the relationship of the premises to the conclusion is ‘valid’, then we will arrive at a ‘sound’ conclusion to the argument. The type of ‘logical truth’ often being referred to in everyday speech is more formally called ‘soundness of logical deduction’, since most people are concerned most of the time, not about the structure of arguments but on the utility of the conclusion.
We, of course, might use the ‘garbage-in probably garbage-out’ principle. If we start with a hypothetically untrue premise, we cannot rationally expect to produce a ‘sound’ conclusion, although the structure of the argument or relationship of the premises might be described as instinctively or intuitively valid.
From the stop-motion clay animation film ‘A Grand Day Out’ – Landing on the Moon – Wallace and Gromit. Even Wallace is not sure about the moon being made of cheese. Of course, he does the wise thing and looks for another sample to test his hypothesis. The ever-wise Gromit is clearly adopting the strategy of using his senses and moderating his opinion with due scepticism. Clearly, it is very important that in the conduct of our everyday affairs we reach sound conclusions.
An extreme example of ‘garbage-in-garbage-out’ inspired by the behaviour of Wallace above can be written as follows:
Premise 1. Blue-veined cheese is edible
Premise 2. The moon is made of blue-veined cheese.
Conclusion: The moon is therefore edible.
Another way to phrase this argument is known as Modus Ponens
Premise 1. If the moon is made of green cheese then the Moon is edible
(This statement has the general symbolic form symbolically expressed as if P then Q)
Premise 2. The Moon is made of cheese
( This has the general symbolic form P)
Conclusion: The Moon is edible
(Symbolically written as Q)
(The dictionary definition of Modus Ponens is “the rule of logic which states that if a conditional statement (‘if p then q ’) is accepted, and the antecedent (p) holds, then the consequent (q) may be inferred”)
The conclusion above is very obviously ‘unsound’ because premise number 2 is a fantasy, even although the structure of the argument, i.e., the relationship of the premises to the conclusion is valid. Notice also that the conclusion is also just a rearrangement of the information contained in the premises. Notice also that although Modus Ponens is normally said to be a ‘rule of inference’ it really has nothing to say about whether we should be accepting or revising either of the premises. This rule is also doing something much more subtle. It is telling the user that in the event that he or she cannot hold the beliefs simultaneously or synchronously, if for some reason we feel that must reject something in the argument. Logic, it has been argued, applies to synchronously held beliefs rather than function diachronically in the development of ideas. Modus Ponens is, therefore, better thought of as a rule of substitution in logic. Quoting Wikipedia, “Modus ponens allows one to eliminate a conditional statement from a logical proof or argument“.
Of course, by sheer chance, we could reach an acceptable explanation concerning some feature of the world but for the ‘wrong’ or ‘unsound’ reasons. These instances have become eponymously known in philosophy as Gettier Cases, although they are usually framed in terms of knowledge.
Where Does The Truth of Deductive Premises Originate?
A deductive argument is described as ‘sound’ when the relationship of the premises to the conclusions is valid and the premises are assumed to be true! Deductive argument, by its very nature, requires truth as a starting condition or a substitute for truth such a ‘probable truth’, ‘socially contextualised plausibility’ or ‘assertability’. We hope to achieve pragmatically useful ideas by the use of validly structured arguments and true premises or a property substituted for an absolute conception of truth. If we are not so concerned with an idealised notion of truth, then we might want our starting assumptions for deductive arguments to be reliable enough to make their use worthwhile.
For those who wish to apply a Coherence Theory of Truth (discussed later in section 3), the idea that propositions about the world ‘cohere’ together because of supportive deductive arguments, is virtuous in itself. In other words, if we have a web of interrelated sound or possibly (or probably) true arguments that cohere together through reasoned conclusions, that might be all we need to create a pragmatically useful view of the world. You might even insist on calling that a (partial) basis for truth.
The question then arises as to where the truth of the premises in any logical argument originate? In some instances, inductive conclusions resulting from repeated observations are used to form the premises of deductive arguments. For example in medicine, the results of clinical research studies, which yield probable truths on groups of people, could be used as the basis for an argument about why a particular patient should be treated in one way but not another.
In other instances the premises of a deductive argument might be true by definition:
All doctors of medicine have received a medical education and have passed the relevant exams.
Helen was a doctor of medicine.
Conclusion: Helen must have had a medical education and passed the relevant exams
The truth of the conclusion is merely a logical consequence of the definition of a medical doctor. The first sentence is simply defining the properties of members of a set and is also said to be a categorical proposition. Another way of looking at this example is to say that “Helen is a member of the set of humans that have received a medical education, and have passed the relevant exams, and are called doctors”. The apparent conclusion has then been reduced to a series of logical conjunctions by the use of the logical connective AND the declaration of set membership.
Yet another way of looking at the origins and authority of premises is to assert that the premises are themselves the conclusion of other logical arguments. For example, we can rationally argue by using more than 1 deduction, referring to the video above, that the moon is not made of cheese:
1. Cheese is a dairy product that it is manufactured from cow’s milk by human cheesemakers in small quantities.
2. The large size of the moon has been calculated from optical measurements made here on earth and so is known to be larger than anything humans can produce.
Conclusion 1: As cheese is a relatively small-scale human-manufactured product the moon cannot be made of cheese
We did not need humans to visit the moon on the Apollo space missions to be certain that the moon is not made of cheese. Deduction has done this job for us. However, it is clear that we can only have reached premise 2 inductively using empirical observation.
Before we become too self-congratulatory about the use of deduction we should bear in mind that deductive conclusions may also result from probable truths taken as the premises, rather than any idealisation of truth.
A Weakness of Deduction
A potential weakness of deduction is the presumption of truth as a starting point in arguments in which there is possibly some degree of doubt or conditionality. In other words, if the premises are true the conclusion then applies. When we need to go beyond mere assumptions of truth, we can still apply logic and admit the possibility of conditionality.
We can, for example, take a syllogism, and apply conditionality.
Premise 1: Roses are flowers
Premise 2: All flowers are pretty
Conclusion: Roses are pretty
If it is true that (Roses are flowers)
If it is true that (All flowers are pretty)
Assuming those conditions then it is true that (Roses are pretty)
Alternatively, we could write If (Roses are flowers AND all flowers are pretty) then roses are pretty.
The injection of conditionality above has not disturbed our view of the world. It has only injected a useful degree of pragmatism. If we do allow ourselves conditionality, we can then be more open to counterfactual thinking and so potentially become more original and more creative.