Implication

Lecture 4, DSCI 220, 2026W1

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Implication

Implication

Connect operator \(\rightarrow\) to a logically equivalent expression using only and, or, and not.

\(p\)	\(q\)	\(p\rightarrow q\)	\(\underline{\hspace{3em}}\)
F	F	T
F	T	T
T	F	F
T	T	T

• Note: \(p \rightarrow q \equiv\) _______.

• \(p \rightarrow q\) Corresponds to “if \(p\) then \(q\)”

• We’re not promising causation, only a truth condition.

• \(p \rightarrow q\) is False only when \(p\) is True and \(q\) is False.

• Gives us mechanism for reasoning!

• Is this operator commutative?

q and not p is false. so it’s NOT (q^-p) == ??? (apply demorgan)

PL exercises on Logical equivalence involving implication?

More on implication and logical equivalence in tutorial questions

Suppose \(p\equiv\) T . Then \(p\) is a Tautology.

Suppose \(p\equiv\) F . Then \(p\) is a Contradiction.

Converses — check what’s True

Each of the following is \(p\rightarrow q\).   Underline \(p\), dash-underline \(q\).   Then decide whether each direction is true.

#	Statement	p→q	q→p
1	If it’s a kitten, then it’s a cat.	☐	☐
2	If an animal is a mammal, then it is a cat.	☐	☐
3	If I’m an only child, then I have zero siblings.	☐	☐
4	If I eat breakfast, then I ace the quiz.	☐	☐
5	A player is in the NHL only if they are a professional.	☐	☐
6	To be empty, a string must have length 0.	☐	☐
7	No song appears on the Billboard Hot 100 without reaching #1.	☐	☐

Two steps per row, and only the first is logic:

TRANSLATE – what does the sentence assert? EVALUATE – is that true?

Rows 1-4 translate for free (“if” names the antecedent). Rows 5-7 do not:

5 “only if” marks the CONSEQUENT. Students read “p only if q” as q->p because “if” sits next to q. It means p->q. 6 “must” – necessity, so again the consequent. 7 Board work. THE HARD PART IS FINDING p AND q, not the algebra below.

 The sentence carries two negation words -- "No" and "without" -- and
 NEITHER belongs to p or q. Both are structural. p and q are positive:

   p: this song appears on the Hot 100
   q: this song reached #1

 Ask "what two facts about a single song does this sentence relate?"
 That strips the scaffolding and leaves the atoms. Each is plainly
 true-or-false of one song; neither contains a "no" or a "without".
 Students will want q = "does not reach #1", or will attach the "No"
 to p. Underline only the positive spans.

 Once p and q are fixed, *without* = *and not* and the rest is algebra:

   ~(p ^ ~q)  ==  ~p v ~~q     De Morgan        (lecture 3 laws table)
              ==  ~p v q       double negation  (same table)
              ==  p -> q       definition of -> (slide 1 today)

 This is the first CHAIN OF EQUIVALENCES in the course -- lecture 3
 promised two ways to justify an equivalence, and only truth tables have
 been used so far. It is also the worked example for the LE Proof slide
 two slides from now.

 Then the catch: the chain shows the sentence MEANS p->q. It does not
 show p->q is true -- and it isn't. A song that charted but never hit #1
 makes p true, q false, which falsifies BOTH forms. That is what the
 equivalence promised.

 But the CONVERSE is true, and for a different kind of reason:
   p->q  (charted -> hit #1)   FALSE, empirically -- most never do.
   q->p  (hit #1 -> charted)   TRUE, definitionally -- #1 IS a position
                               on the Hot 100. You cannot top a chart
                               you are not on.
 So the sentence asserts the contingent direction, which fails, while the
 direction it does not assert holds by definition. Not a biconditional.

 Same truth pattern as row 2 (mammal -> cat), which they met early and
 easily. Row 7 is that pattern in disguise -- worth pointing back to it.

Row 4 (breakfast/quiz) is the discussion row, not a quick one. Nobody in the room has the facts, so ask what would make it FALSE – one day you ate breakfast and did not ace. That is a counterexample, three lectures early. It is also the only home for vacuous truth: if you never eat breakfast, p is always false and the statement is true, because you never broke the promise.

Rows 3 and 4 are the only propositional ones – about a fixed “I”. The rest range over a domain, which is what predicates are for. Do not say “for all” yet; it becomes the callback when quantifiers arrive.

Vocabulary

Given \(p\rightarrow q\), we define the following 3 terms:

• ________________ is \(\neg q\rightarrow \neg p\)

• ________________ is \(q\rightarrow p\)

• ________________ is \(\neg p\rightarrow \neg q\)

Describe all the logical equivalences among the 4 statements:

we’ll see if English can help us!

Logical Equivalence Proof

Prove that \(p \rightarrow q \equiv \neg q \rightarrow \neg p\):

Biconditionals

We have a special operator for those implications whose converses are also True:

  \((p\rightarrow q) \wedge (q\rightarrow p)\equiv p\leftrightarrow q\)

Some phrases describing this relationship:

• biconditional
• if and only if (iff)
• equivalent

Which of the seven statements were biconditionals?

Words and Ideas

Implication

Converse, Contrapositive, Inverse

Biconditional