Propositional Logic: Operators, Truth Tables and Syllogisms

Formalise statements using the basic logical operators (not, and, or, if-then), build truth tables to work out when compound statements are true, and recognise the common valid syllogisms — and the fallacies that imitate them.

1 Introduction

Once you can reconstruct an argument into premises and a conclusion (see Constructing and Reconstructing Arguments), the next step is to analyse its form. Propositional logic is the study of how the meaning of logical operators — words like not, and, or, and if … then — determines the truth conditions of compound statements, and which patterns of inference are reliable.

The payoff is practical: a small set of symbols and tables lets you see at a glance whether a piece of reasoning is built on a valid pattern or a tempting imitation of one.

2 Notation: Writing Logic

A few conventions make reasoning easier to handle:

• Use symbols (letters such as p, q, s, …) to stand for whole statements (“propositions”).
• Separate the conclusion from the premises with a line, or with the symbol ∴ (“therefore”).
• Use logical operators to combine statements.
• Assign truth values (T, F) to the component statements and work out the truth value of the whole — this is what a truth table does.

For example, the argument

Premise 1: If it rains, the street is wet. Premise 2: It rains. Conclusion: The street is wet.

becomes, with r = “it rains” and w = “the street is wet”:

P1: r → w P2: r ∴ w

3 The Basic Logical Operators

3.1 Negation (¬)

¬p (“not p”) reverses the truth value of p.

p ⇒ “you shall pass” ¬p ⇒ “you shall not pass”

For ¬p to be true, p must be false; for ¬p to be false, p must be true.

3.2 Conjunction (∧)

p ∧ q (“p and q”) is true only when both p and q are true. Order does not matter: p ∧ q ≡ q ∧ p.

3.3 Disjunction (∨)

p ∨ q (“p or q”) is true when at least one of p, q is true. (This is the inclusive or — it is also true when both hold.) Order does not matter: p ∨ q ≡ q ∨ p.

3.4 The conditional (→)

p → q (“if p, then q”) links a condition p to a consequence q. Unlike the others, order matters: p → q is not the same as q → p.

The conditional has one feature that surprises newcomers:

• If p is true, then p → q takes the truth value of q.
• If p is false, then p → q is true by definition (the promise was never triggered, so it cannot have been broken).

4 Truth Tables

A truth table lists every combination of truth values for the components and gives the resulting value of the compound statement.

Negation

p	¬p
T	F
F	T

Conjunction and disjunction

p	q	p ∧ q	p ∨ q
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	F

The conditional

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

The only way to make p → q false is to have a true condition p with a false consequence q — the second row.

5 Logical Truths: Contradictions and Tautologies

Two compound statements are worth memorising because their truth value never depends on the facts:

• p ∧ ¬p is a contradiction — always false. (For a conjunction to be true we need both parts true, but p and ¬p can never both be true.) Example: “g is green and g is not green.”
• p ∨ ¬p is a tautology — always true. (For a disjunction we need at least one part true, and exactly one of p, ¬p always is.) Example: “the door is open or it is not open.”

6 Common Syllogisms

A syllogism is a valid deductive argument with (typically) two premises and a conclusion. If the premises are true and the syllogism is valid, the conclusion must be true. Four patterns recur constantly.

Modus ponens — affirm the condition:

P → Q P ∴ Q

If it rains, the street is wet. It rains. ∴ The street is wet.

Modus tollens — deny the consequence:

P → Q ¬Q ∴ ¬P

If our pricing strategy is competitive, sales will increase. Sales have not increased. ∴ Our pricing strategy is not competitive.

Hypothetical syllogism — chain two conditionals:

P → Q Q → R ∴ P → R

If I don’t wake up, I can’t go to work. If I can’t go to work, I won’t get paid. ∴ If I don’t wake up, I won’t get paid.

Disjunctive syllogism — eliminate one option:

P ∨ Q ¬P ∴ Q

I eat soup today or I eat salad today. I do not eat soup today. ∴ I eat salad today.

7 Two Tempting Fallacies

Two invalid patterns look like modus ponens and modus tollens but are not valid. Watch for them.

Valid		Invalid
Modus ponens		Fallacy of the converse
P → Q; P; ∴ Q		P → Q; Q; ∴ P
Modus tollens		Fallacy of the inverse
P → Q; ¬Q; ∴ ¬P		P → Q; ¬P; ∴ ¬Q

Using “If it rains, the street is wet”:

• Fallacy of the converse: “The street is wet, ∴ it rains.” Invalid — the street could be wet for another reason.
• Fallacy of the inverse: “It is not raining, ∴ the street is not wet.” Invalid for the same reason.

Note🎓 MA / PhD

Propositional logic is the floor, not the ceiling. It treats statements as indivisible atoms; predicate (first-order) logic adds quantifiers (“for all”, “there exists”) and lets you reason about relationships within statements, which is needed for most genuinely scientific generalisations. The valid argument forms above also reappear as rules of inference in formal proof systems. If you work with formal models, the connection between truth tables and logical validity (a conclusion is valid iff there is no row in which all premises are true and the conclusion false) is worth internalising.

8 Summary

• The operators ¬, ∧, ∨, → combine statements; their truth tables define exactly when the compound is true.
• p ∧ ¬p is always false (contradiction); p ∨ ¬p is always true (tautology).
• Modus ponens, modus tollens, hypothetical and disjunctive syllogisms are valid patterns.
• The fallacy of the converse and the fallacy of the inverse imitate the valid conditional patterns but are invalid.

9 What to Read Next

• Evaluating Arguments: Validity, Soundness and Fallacies — moving from form to a full evaluation of whether an argument is good.

10 Further Reading

• Gensler, H. J. (2010). Introduction to Logic (2nd ed.). Routledge.
• Hurley, P. J. (2000). A Concise Introduction to Logic (7th ed.). Wadsworth.