Deductive Reasoning and Logical Connectives

Where does the certainty in mathematics actually come from? Not from the size of your example set, and not from how confident the author of the textbook sounds. It comes from deductive reasoning: a chain of statements where each link is forced by the ones before it. If the premises are true and the steps follow the rules, the conclusion has to be true — there is no escape. This section unpacks the machinery: how to spot premises and conclusions, how to combine simple statements with the connectives $\wedge$, $\vee$, $\neg$, and how to recognize the two argument patterns (modus ponens and modus tollens) that show up in nearly every proof you will ever write.

Premises, conclusion, validity

A deductive argument is a list of premises followed by a conclusion. The argument is valid when the conclusion must be true whenever every premise is true. Notice what validity does not require: it does not require the premises to actually be true. "All cats can fly. Whiskers is a cat. Therefore Whiskers can fly." is a valid argument with a false premise — the form is correct, the inputs are wrong. Validity is about logical shape, not about the world.

An argument is invalid if there exists a counterexample — a single scenario where all premises are true but the conclusion is false. One counterexample is enough to refute a claim of validity forever.

The three basic connectives

Conjunction, written $P \wedge Q$ and read "$P$ and $Q$," is true exactly when both $P$ and $Q$ are true. Disjunction, written $P \vee Q$ and read "$P$ or $Q$," is true when at least one of $P$, $Q$ is true — this is the inclusive or, so "$P \vee Q$" is still true when both hold. Negation, written $\neg P$ and read "not $P$," flips the truth value: $\neg P$ is true exactly when $P$ is false.

A statement built from atomic propositions and these connectives, with parentheses to group, is called a well-formed formula (wff). For example $(P \wedge Q) \vee \neg R$ is well-formed; the string $P \wedge \vee Q$ is not (two binary connectives back to back).

Two valid forms you will use forever

Two reusable argument templates appear in almost every proof:

Modus ponens. From $P \Rightarrow Q$ and $P$, conclude $Q$. (If rain implies wet, and it is raining, the ground is wet.)

Modus tollens. From $P \Rightarrow Q$ and $\neg Q$, conclude $\neg P$. (If rain implies wet, and the ground is dry, then it did not rain.)

Both forms are valid by the truth table of the conditional — the only way $P \Rightarrow Q$ can fail is the row $P=T, Q=F$, and each form rules that row out.

Pause and think: Suppose you know $P \Rightarrow Q$ and $Q$. Can you conclude $P$? If your gut says yes, ask which truth-table row would be a counterexample. (Hint: $P=F, Q=T$ makes both premises true but $P$ false — this is the fallacy of affirming the consequent.)

Try it

• Before writing anything: identify the premise(s) and the conclusion in "Every prime greater than 2 is odd. The number 17 is a prime greater than 2. So 17 is odd." Which argument form is this?
• Predict first: is the argument "If $n$ is divisible by 6, then $n$ is even. $n$ is even. Therefore $n$ is divisible by 6." valid? Find a counterexample using small integers.
• Translate into symbols using $C$ = "it is cold" and $R$ = "it is raining": "It is cold but not raining." Then "It is neither cold nor raining."
• Open the truth-table widget. Build the column for $(\neg P) \vee Q$. Compare it row by row with $P \Rightarrow Q$. What do you notice?

A trap to watch for

The most common beginner error is confusing modus tollens with its evil twin, the fallacy of denying the antecedent: from $P \Rightarrow Q$ and $\neg P$, mistakenly concluding $\neg Q$. The mistake feels natural — "if rain means wet, and it is not raining, then the ground should not be wet" — but the ground could be wet for any number of other reasons (sprinklers, snow, dew). The conditional $P \Rightarrow Q$ only forbids the row $P=T, Q=F$; it says nothing about what happens when $P$ is false. Train yourself to ask, every time, "which row of the truth table am I ruling out?" before you draw a conclusion.

What you now know

You can break an English argument into premises and conclusion, translate the connectives into symbols, and tell whether the form is valid by running through truth-table rows. In the next section we make this enumeration systematic: the truth table itself becomes the basic verification tool of propositional logic, and you will learn to certify any compound formula as tautology, contradiction, or contingency.

References

• Velleman, D. J. (2019). How to Prove It: A Structured Approach (3rd ed.). Cambridge University Press, ch. 1.
• Enderton, H. B. (2001). A Mathematical Introduction to Logic (2nd ed.). Academic Press, §1.1.
• Smullyan, R. M. (1995). First-Order Logic. Dover Publications, ch. 1.
• Mendelson, E. (2015). Introduction to Mathematical Logic (6th ed.). Chapman and Hall/CRC, ch. 1.