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Two of the four things a second premise can do to a rule are valid, and each of the other two is mistaken for one of them.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will distinguish MP and MT from affirming the consequent and denying the antecedent, justify valid cases, and construct complete countermodels for invalid cases without inventing probability claims.
You know that a conditional forbids one combination, that the contrapositive is free and the converse is not. This lesson turns those facts into four named moves, because a move with a name is one you can catch somebody making.
| Term | What it means |
|---|---|
| Modus ponens | P -> Q and P entail Q. |
| Modus tollens | P -> Q and ~Q entail ~P. |
| Affirming the consequent | P -> Q and Q do not generally entail P. |
| Denying the antecedent | P -> Q and ~P do not generally entail ~Q. |
| Countermodel | All premises true and conclusion false on one assignment. |
Start from the rule P → Q. A second premise can do four things to it, and each gives a different argument.
| Second premise | Conclusion drawn | Name | Verdict |
|---|---|---|---|
| P | Q | affirming the antecedent | valid |
| Q | P | affirming the consequent | invalid |
| not Q | not P | denying the consequent | valid |
| not P | not Q | denying the antecedent | invalid |
The two valid ones run along the arrow: forwards from the if-part, backwards from the failure of the then-part. The two invalid ones try to run the other way, and they are the converse and the inverse from lessons 9 and 10 turned into arguments.
Each invalid move is set beside the valid one it is mistaken for, on purpose. Affirming the consequent means nothing on its own — it sounds like a technical insult you could throw at any argument. It means something when it sits next to affirming the antecedent, which is valid, and the only difference between them is which half the second premise picks up.
Another way: picture
One-way street. The arrow is the direction of travel: you may drive from P to Q, and you may work out that a car which never reached Q cannot have set off from P. What you may not do is drive back up it — arriving at Q tells you nothing about where the car came from.
Another way: steps
To judge a move:
An invalid conditional inference is not necessarily unlikely, and it is not necessarily usually correct. Its probability depends on evidence that the logical form alone does not provide. If a wet surface has many possible causes, seeing wetness may or may not make rain likely. The countermodel test asks a different question: can the stated premises hold while the conclusion fails? Do not invent frequencies to explain why someone found a pattern tempting.
For P -> Q and Q, with conclusion P, the assignment P false and Q true answers that logical question. It keeps the conditional true because its antecedent is false, keeps Q true as explicitly required, and falsifies P. No numerical probability is needed. The row proves that the two premises do not guarantee the proposed conclusion.
For P -> Q and ~P, with conclusion ~Q, the same assignment works. The denied antecedent is true and the conditional holds, while ~Q is false. Denying one sufficient condition leaves the result possible through another route. A practical explanation should name a plausible alternate route when the context supplies one, but the formal demonstration is the complete assignment and premise audit.
Do not call an invalid inference worthless in every context. It may suggest a hypothesis worth testing, or its conclusion may have separate support. The defect is specifically that it fails as a deductive guarantee from these premises. A revised argument might add a justified reverse conditional, use probabilistic evidence, or lower its conclusion from certainty to a tentative proposal. State which change is being made.
Modus ponens has two inputs: a conditional and its complete antecedent. Suppose both are true. The true antecedent rules out the rows where P is false. The true conditional then rules out P true with Q false. The only remaining possibility has Q true. This explains why the result follows without consulting whether the particular story is familiar or believable.
Modus tollens uses the same conditional but a denial of its complete consequent. Suppose P -> Q and ~Q are true. Q is false. If P were true, the conditional would be false, so P must be false. Hence ~P follows. This is a truth-preserving inference from two premises, not merely a restatement of the conditional alone. The observation ~Q has an independent role.
The rules apply to whole formulas. If the antecedent is P & Q, modus ponens needs that conjunction, not merely P. If the consequent is Q | R, modus tollens needs ~(Q | R), not merely ~Q. A supplied denial of one disjunct leaves the other potentially true and therefore does not deny the whole consequent. Scope errors can make an invalid step look like a correct named rule.
Conversely, a negative consequent does not automatically mean the inference is modus tollens. From P -> ~Q and P, the result ~Q follows by modus ponens: the antecedent was affirmed. Identify the rule by the relationship between the complete inputs, not by counting negation signs. This habit becomes especially useful once the statements contain conjunctions, disjunctions, or nested denials.
An objection that a premise is false differs from an objection that the conclusion does not follow. Both can matter, but they require different evidence. To test validity, temporarily grant the complete stated premises and search for a false conclusion. Rejecting the premise instead may challenge the argument's soundness without establishing that its inference form is invalid.
Imagine a stated rule that every accepted item has a tracking label. If the argument also asserts that this item was accepted, a conclusion that it has a label is valid. Finding an actual accepted item without a label challenges the rule's truth in the application. It does not create a countermodel to the formal MP argument, because that assignment would make the conditional premise false.
Now consider the reverse inference from a label to acceptance. A labeled item that was never accepted can preserve the original one-way rule and falsify the conclusion. This is a genuine countermodel to that reversed argument. The difference lies in which premises the case preserves, not in whether the story sounds critical of the same organization.
When presenting a counterexample, list the truth value of every premise and the target. If one premise is false, explain that the case challenges a factual input instead. If all premises are true and the target false, explain that it challenges the inference. Keeping those objections distinct makes an argument review more useful than attaching a fallacy label to any disagreement.
After identifying a valid pattern, ask whether the actual inputs have been established. A rule about final records cannot be applied using an observation from an unfinished draft. A denial inferred from an incomplete search may not supply the negative premise MT needs. A premise about one object cannot be combined with an observation about a similar object unless the key explicitly connects them.
After identifying an invalid pattern, ask what would close the particular gap. For affirming the consequent, a separately justified Q -> P would permit the desired result by MP. For denying the antecedent, that same reverse conditional together with ~P would support ~Q by MT. This repair is not free: the reverse conditional is exactly the new evidence commitment that the original argument lacked.
A final review can therefore record the pattern, the countermodel or truth-preservation explanation, and the factual evidence still needed. The named forms help organize the work, but the explanation earns the verdict. A learner who can write a complete witness or rule justification has demonstrated more than the ability to recognize a Latin name.
From P | Q and ~P, Q follows: the disjunction requires at least one true part, and the denial removes P. But from P | Q and P, ~Q does not follow, because both parts may be true. The assignment P=T,Q=T preserves those premises and falsifies the proposed denial. This comparison reinforces the conditional lesson's method: first identify the connective's complete truth condition, then ask what the extra premise actually excludes. Do not transfer a pattern from an exclusive everyday choice into an exercise that explicitly uses inclusive or. The supplied semantics determines which alternatives remain available.
A workshop's declared rule is that every accepted item has a tracking label at the time of acceptance. Let A mean accepted and L mean label present at that same time. Four reviewers propose different inferences. The first has evidence A and concludes L; this is MP and valid. The second has a verified absence ~L and concludes ~A; this is MT and valid, provided the rule accurately covers that time.
The third has a label L and concludes acceptance A. A pre-labeled item waiting for acceptance would make both given premises true and the conclusion false. The fourth has evidence ~A and concludes ~L. The same pre-labeled waiting item refutes that move too. The label could exist before acceptance because the original rule never prohibited that order.
Suppose the audit record contains six accepted labeled items, three waiting labeled items, and two waiting unlabelled items. There are no accepted unlabelled items in this complete declared record. The original rule has no violation here. Yet the three waiting labeled items defeat both reverse inferences. Counting all eleven records is useful for illustrating the situation, but one of the three counterexamples already suffices for invalidity.
If the workshop wants a label to certify acceptance, it needs an additional rule governing when labels may be attached, plus evidence that the rule is followed. The existing one-way guarantee cannot do that job by itself. The logical review identifies this missing commitment without prescribing how the workshop should redesign its process.
MP and MT are valid forms. Calling an instance sound additionally requires actually true premises, which the pattern alone does not establish. Do not claim that reverse inferences usually work without data about the context. Their deductive invalidity does not determine a probability. Finally, a failed first search for a countermodel is not a proof of validity: a positive validity verdict requires checking all possibilities or giving a complete truth-preserving argument.
State the argument.
A -> L,A; therefore L
The second premise affirms the antecedent.
Demand a false conclusion.
L=F
A countermodel would need this value.
Preserve the atomic premise.
A=T
Acceptance is explicitly asserted.
Check the conditional.
A -> L=F
The attempted row fails a premise.
Conclude that the inference is valid.
Valid MP
No target-false row can preserve both premises.
State the reverse inference.
A -> L,L; therefore A
The result is being used to infer its sufficient condition.
Falsify the target.
A=F
The item is not accepted.
Keep the observed label.
L=T
This is required by the second premise.
Check the arrow.
A -> L=T
A false antecedent satisfies the conditional.
Audit the full row.
Premises T,T; conclusion F
Both input claims survive on the same assignment.
State the verdict.
Invalid; A=F,L=T
A pre-labeled waiting item illustrates the countermodel.
State the available premises.
P -> (Q | R),~Q
The observation denies only Q.
Identify the actual consequent.
Q | R
It is a complete disjunction.
Write the denial MT would require.
~(Q | R)
The rule needs the negation of the whole consequent.
Choose a surviving alternate disjunct.
R=T
The observation ~Q leaves R unconstrained.
Complete a countermodel to ~P.
P=T,Q=F,R=T
The target denial is false.
Check every premise.
P -> (Q | R)=T; ~Q=T
Both premises hold despite P being true.
Reject the proposed MT application.
Invalid inference to ~P
A denial of one disjunct does not deny the consequent.
Identify the premises.
P -> ~Q,P
The antecedent P is affirmed.
Match the complete consequent.
~Q
This is what the conditional guarantees.
Name and justify the move.
Test P -> ~Q, P; therefore ~Q. Decide validity and supply a countermodel if invalid.
P -> ~Q
P
∴ ~Q
valid invalid — countermodel:
Audit P=F,Q=T for P -> Q,Q; therefore P. Count true premises and false conclusions.
Evaluate the supplied premises.
P -> Q=T,Q=T; premises true premises
The arrow's antecedent is false and Q is directly true.
Evaluate the target.
P=F; conclusions false conclusion
The proposed antecedent does not hold.
State the formal verdict.
invalid
All premises survive while the conclusion fails.
Test P -> (Q | R), ~(Q | R); therefore ~P. Decide validity and supply a countermodel if invalid.
P -> (Q | R)
~(Q | R)
∴ ~P
valid invalid — countermodel:
Test P -> (Q & R), Q & R; therefore P. Decide validity and supply a countermodel if invalid.
P -> (Q & R)
Q & R
∴ P
valid invalid — countermodel:
A complete record has 7 accepted labeled items, 0 accepted unlabelled, 4 waiting labeled, and 3 waiting unlabelled. A means accepted, L labeled. Count original A -> L violations, counterexamples to L therefore A, and counterexamples to ~A therefore ~L when the original rule is retained.
Rule violations: b0
Reverse counterexamples: b1
Denial counterexamples: b2
Test (P | Q) -> R, ~(P | Q); therefore ~R. Decide validity and supply a countermodel if invalid.
(P | Q) -> R
~(P | Q)
∴ ~R
valid invalid — countermodel:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Test ~P -> Q, ~Q; therefore P. Decide validity and supply a countermodel if invalid.
~P -> Q
~Q
∴ P
valid invalid — countermodel:
You can name a move from an if-then, decide whether it is valid, and produce the countermodel when it is not. Tell someone why the pitch is wet, so it rained is invalid without assuming anything about how probable rain is. Next: the difference between an argument being valid and its conclusion being true.
15. A negative result can still be MP, step 3
MP gives ~Q
The negative conclusion does not turn an antecedent-affirming step into MT.