The structure of deductive arguments reveals that there is a necessary implication between the premises and the conclusion.
The conclusion necessarily follows from the premises. So the reasoning here is conclusive. Further the conclusion cannot have more generality than the premises.
What is already assumed in the premise can only be brought out in the conclusion. As there is a relation of formal entailment between the premise and the conclusion the conclusion cannot go beyond the premise. Let us take two examples to explain it:
1. All teachers are literate. Some literate people are teachers.
2. All philosophers are wise.
3. Russell is a Philosopher. Russell is wise.
The first argument comes under immediate inference and the second one under mediate inference i.e. syllogism. In each case the conclusion necessarily follows from the respective premise or premises.
To accept the premise and deny the conclusion would land us in contradiction. Further the conclusion in a deductive argument will always have equal or less generality than the premise or premises from which it is derived. Again one important characteristic of deductive reasoning is that the conclusion cannot have any novelty.
It cannot give any new information that is not inherent in the premise. Thus in a valid deductive argument the conclusion can never extend our knowledge beyond the content of the premise. It only brings out or makes explicit what is implicitly inherent in the premise.
In a valid deductive argument since the conclusion necessarily follows from the premises, if the premises are true as a matter of fact, the conclusion must also be true.
A false conclusion cannot follow from true premises in a valid argument. But if the premises are false, the conclusion will be false even if the argument is valid. Let us take an example-
All men are immortal.
All philosophers are men.
All philosophers are immortal.
Here no consistency is violated since the premises do necessarily imply the conclusion. It is an argument in form of Barbara of the first figure. But here the conclusion is false as a matter of fact.
The falsity of the conclusion is due to the fact that the premise “All men are immortal” is false as a matter of fact. So a conclusion can be false even in a valid argument if the premise is false.
Similarly in other cases the premises and the conclusion may be true. But die argument may be invalid. Consider this example:-
All men are mortal.
All students are mortal.
All students are men.
In this argument the premises as well as the conclusion are ture as a matter of fact but the argument is invalid. It is invalid because the conclusion does not necessarily follow from the premises for it involves the fallacy of undistributed middle.
Thus truth and validity are different concepts. A proposition is true or false depending upon the situation it describes. If it tallies with the actual state of affairs, it is true otherwise it is not. But on the other hand an argument is valid or invalid.
An argument is valid if its conclusion is being justified by the premises. Validity, therefore, is a rule guiding concept. If some rules of reasoning are satisfied with regard to an argument then the argument is valid. For then is complete justification or consistency between the premises and the conclusion.
In an invalid argument the premises do not justify the conclusion. But it should be noted that once an argument is’ valid and the premises are true the conclusion must be true and can; never be otherwise.
From this analysis it appears that in case of deductive reasoning the thrust is on formal consistency. Logic is primarily concerned with the forms of valid arguments. Logic has explored different forms of valid argumentation.
Once a valid form is made explicit any subject matter filled into it will make the argument a valid one. Since the content of the argument may be anything a logician is not interested what the subject matter of an argument is. But on the contrary he is very much concerned with the form of the argument. For example, consider these two instances:-
In the first example, (a), S, P and M stand for terms of a proposition and in (b), P and Q stand for propositions. In (a) whatever term will be substituted uniformly in place of S. P and M, will result in a valid argument. So too in (b) any proposition substituted in place of P and Q will make a real valid argument. For these are valid forms of arguments.
Once the forms of valid argument are ascertained, any real argument can be symbolized into its formal structure. If the formal structure is in accordance with valid reasoning, then the argument, irrespective of its subject matter, will be treated as valid. Logic, therefore, is very much concerned with the forms of validity like mathematics.
Deductive logic and mathematics are thus considered as formal sciences. A formal science has universal application. The procedure or reasoning in formal science is based on self-consistency. In mathematics, particularly in Euclid’s geometry, the deductive procedure is evident. From a limited axiom hundreds of theorems are brought out by applying the deductive procedure.
Great logicians and mathematicians have presented frameworks to show that from a very limited axiom with the help of a few definitions and rules of syntax any tautology, which is a necessarily true expression, can be derived. All these show that deductive logic is primarily concerned with the forms of valid reasoning.