Arguments, premises and inference
Logic is the study of correct reasoning. An argument is a group of statements where some (the premises) are given as reasons for another (the conclusion). Inference is the mental step from premises to conclusion.
Indicator words help: because, since, given that introduce premises; so, therefore, hence, thus introduce a conclusion. A statement (proposition) is a sentence that is either true or false; questions and commands are not statements.
Language and clarity
Names refer to things; sentences make claims. Many errors come from unclear language: ambiguity (a word with two meanings, like 'bank'), vagueness ('tall', 'rich' with no limit) and loaded words. Clear terms come first, then good reasoning.
Deductive and inductive arguments
A deductive argument aims for certainty: if the premises are true, the conclusion cannot be false. Example: All metals conduct electricity; copper is a metal; so copper conducts electricity.
An inductive argument aims for probability: the premises make the conclusion likely, not certain. Example: every crow I have seen is black; so all crows are probably black.
Deduction often goes from general to particular, induction from particular to general, but the real test is: does the conclusion follow necessarily (deductive) or only probably (inductive)?
Validity, soundness and counterexamples
A deductive argument is valid if its form guarantees that true premises give a true conclusion. Validity is about form, not about whether the premises are actually true.
It is sound if it is valid and all its premises are true. Only a sound argument proves its conclusion.
Common forms
- Modus ponens (valid): If P then Q. P. So Q.
- Modus tollens (valid): If P then Q. Not Q. So not P.
- Affirming the consequent (invalid): If P then Q. Q. So P.
- Denying the antecedent (invalid): If P then Q. Not P. So not Q.
A counterexample to a form is an argument with the same form whose premises are true and conclusion false. Finding one proves the form invalid.
Kinds and strength of inductive arguments
Inductive arguments are strong or weak. If a strong argument also has true premises it is cogent.
- Generalisation: from a sample to a whole group. Stronger with a large, random, varied sample. Weak if hasty (too few cases) or biased.
- Statistical syllogism: 90% of A are B; x is an A; so x is probably B. Strength depends on the percentage and on whether x is typical.
- Analogy: two things are alike in many ways, so probably alike in one more.
- Causal reasoning: from repeated patterns to a cause, as in science experiments.
New evidence can weaken an inductive conclusion; it cannot weaken a valid deduction with true premises.
Categorical propositions and predicate logic
A term names a class, like 'humans' or 'mortal things'. A categorical proposition links a subject term S and a predicate term P:
- A: All S are P. — ∀x (Sx → Px)
- E: No S are P. — ∀x (Sx → ¬Px)
- I: Some S are P. — ∃x (Sx ∧ Px)
- O: Some S are not P. — ∃x (Sx ∧ ¬Px)
∀ means 'for all', ∃ means 'there exists', → 'if…then', ∧ 'and', ¬ 'not'. This is monadic predicate logic (each predicate takes one thing). A and O contradict each other, as do E and I. Circle diagrams (like the 3D) test syllogisms: shade or place dots and see whether the conclusion is forced.
Try it: argument detective
In the last 3D step test all four forms. For each invalid one, invent your own counterexample with true premises and a false conclusion. At home: find one advert or social media post that makes an argument. Write its premises and conclusion. Is it deductive or inductive? Is it valid or strong?
Key formulas and definitions
- Argument = premises + conclusion
- Valid: true premises cannot give a false conclusion
- Sound = valid + all premises true
- Modus ponens: P → Q, P ⊢ Q; Modus tollens: P → Q, ¬Q ⊢ ¬P
- A: ∀x(Sx→Px) E: ∀x(Sx→¬Px) I: ∃x(Sx∧Px) O: ∃x(Sx∧¬Px)
- Strong inductive argument + true premises = cogent
Worked examples
1. Is this valid? Is it sound? 'All fish live in water. A dolphin lives in water. So a dolphin is a fish.'
Step 1: Form: All F are W; d is W; so d is F. Step 2: Counterexample with the same form: All cats are animals; a dog is an animal; so a dog is a cat. True premises, false conclusion. Step 3: So the form is invalid, and an invalid argument cannot be sound.
2. 'If the battery is dead, the phone will not switch on. The phone switches on. So the battery is not dead.' Name the form and judge it.
P = battery dead, Q = phone will not switch on. Premises: P → Q and not Q. Conclusion: not P. This is modus tollens, which is valid. If both premises are true, it is also sound.
3. '92% of students at this school pass maths. Meera is a student here. So Meera will probably pass.' What kind of argument is this, and what could weaken it?
It is a statistical syllogism (inductive). It is fairly strong because 92% is high. It is weakened if we learn Meera is not typical, for example she missed half the year, or if the 92% figure comes from a small or old sample.
Common mistakes
- Thinking valid means true. A valid argument can have a false conclusion if a premise is false.
- Calling an inductive argument 'valid' or 'invalid'. Inductive arguments are strong or weak.
- Affirming the consequent: 'If it rains the road is wet; the road is wet; so it rained.' Other causes are possible.
- Generalising from too few cases (hasty generalisation), like judging a whole city by one visit.