Which standard argument forms are valid, which are invalid, and how do you use the counterexample method to tell them apart?
Valid and invalid argument forms: modus ponens, modus tollens, the disjunctive and hypothetical syllogisms; the formal fallacies of affirming the consequent and denying the antecedent; and using the counterexample method to expose an invalid form.
The standard valid argument forms in SQA Higher Philosophy (modus ponens, modus tollens, disjunctive and hypothetical syllogism) and the invalid forms (affirming the consequent, denying the antecedent), with the counterexample method for proving an argument invalid.
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What this dot point is asking
Some argument shapes guarantee a valid inference whatever you slot into them, and some only look as if they do. SQA Higher Philosophy expects you to recognise the standard valid forms (modus ponens, modus tollens, disjunctive syllogism, hypothetical syllogism) and the two classic invalid forms (affirming the consequent, denying the antecedent), and to prove an argument invalid using the counterexample method.
The valid forms
A conditional statement has the shape "if P then Q", where P is the antecedent and Q is the consequent. Three of the four valid forms turn on conditionals.
- Modus ponens. "If the alarm sounds, we evacuate; the alarm sounded; so we evacuate." Affirming the antecedent forces the consequent.
- Modus tollens. "If it is a square, it has four sides; it does not have four sides; so it is not a square." Denying the consequent forces the denial of the antecedent.
- Hypothetical syllogism. "If I study, I pass; if I pass, I graduate; so if I study, I graduate." Conditionals chain together.
- Disjunctive syllogism. "Either the train is late or I missed it; the train is not late; so I missed it." Ruling out one disjunct leaves the other.
The invalid forms
Two shapes mimic modus ponens and modus tollens but are invalid.
- Affirming the consequent. "If it rained, the ground is wet; the ground is wet; so it rained." Invalid, because the ground could be wet for another reason.
- Denying the antecedent. "If it rained, the ground is wet; it did not rain; so the ground is not wet." Invalid, because something else could have wet the ground.
In both, a conditional only tells you that P is sufficient for Q, not that P is the only route to Q, so you cannot reason backwards from Q to P or from not-P to not-Q.
The counterexample method
The method works because validity is about form: if even one argument of a given form has true premises and a false conclusion, the form cannot guarantee truth. You do not need to discuss the original content at all; you just match its skeleton with obviously true premises and an obviously false conclusion.
Examples in context
Suppose you are given: "If a country is democratic, it holds elections; North Korea holds elections; so North Korea is democratic." First identify the form: it is "if P then Q; Q; therefore P", affirming the consequent. To prove it invalid, build a same-form counterexample with clear truth values: "If something is a dog, it is an animal; a cat is an animal; so a cat is a dog." The premises are plainly true, the conclusion plainly false, and the form is identical, so the original argument's form is invalid. That is a complete, full-mark demonstration: name the form, then defeat it with a counterexample.
Try this
Q1. Write modus ponens and modus tollens in symbolic form. [2 marks]
- Cue. Modus ponens: . Modus tollens: .
Q2. Why does a single counterexample prove a form invalid? [2 marks]
- Cue. Validity requires that no argument of the form has true premises and a false conclusion, so one example with true premises and a false conclusion shows the form fails.
Exam-style practice questions
Practice questions written in the style of SQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
SQA Higher (P1)4 marksIdentify the form of the following argument and state whether it is valid: 'If it is raining then the ground is wet; the ground is wet; so it is raining.'Show worked answer →
Marks are awarded for naming the form and correctly judging validity with a reason. This argument affirms the consequent: it has the shape "If P then Q; Q; therefore P", which is an invalid form.
Explain why with a counterexample: the ground could be wet for another reason, such as a burst pipe or a hose, so the premises can both be true while the conclusion (it is raining) is false. Because a situation exists in which the premises hold and the conclusion fails, the form is invalid. Contrast it with the valid modus ponens, "If P then Q; P; therefore Q", which would be "if it is raining the ground is wet; it is raining; so the ground is wet".
SQA Higher (P1)5 marksUse the counterexample method to show that the following argument is invalid.Show worked answer →
The counterexample method earns marks for constructing a new argument with exactly the same form whose premises are clearly true and whose conclusion is clearly false. That single example proves the form cannot be valid.
Take "All dogs are animals; all cats are animals; so all dogs are cats", whose form is "All A are C; all B are C; so all A are B". A counterexample with the same form is "All apples are fruit; all bananas are fruit; so all apples are bananas". The premises are obviously true and the conclusion obviously false, so the shared form is invalid. State explicitly that because the form allows true premises and a false conclusion, every argument of that form is invalid.
Related dot points
- Statements, arguments and standard form: distinguishing arguments from explanations, descriptions and assertions, identifying premises and conclusions using indicator words, and rewriting an argument in standard form.
How to recognise an argument in SQA Higher Philosophy, tell it apart from explanations and descriptions, identify the premises and the conclusion using indicator words, and rewrite the argument in standard form ready for analysis.
- Deductive validity and soundness: the meaning of validity (the conclusion must follow if the premises are true), the meaning of soundness (valid plus all premises actually true), and why a valid argument can have a false conclusion and a true conclusion can come from an invalid argument.
How SQA Higher Philosophy defines deductive validity and soundness: validity as a guarantee that true premises force a true conclusion, soundness as validity plus true premises, and why truth and validity are separate ideas you must not confuse.
- Inductive arguments and reliability: how induction differs from deduction, the main inductive patterns (generalisation from a sample, argument from analogy, causal and predictive inference, appeal to authority), and the criteria that make an inductive argument reliable or weak.
How SQA Higher Philosophy treats inductive reasoning: the difference between induction and deduction, the main inductive forms (generalisation, analogy, causal and predictive inference, appeal to authority), and what makes an inductive argument reliable rather than weak.
- Fallacies: the formal fallacies (affirming the consequent, denying the antecedent) and the common informal fallacies (ad hominem, straw man, false dilemma, slippery slope, appeal to emotion, begging the question, hasty generalisation and others), with how to identify and explain each.
The fallacies SQA Higher Philosophy expects you to recognise: the formal fallacies of affirming the consequent and denying the antecedent, and the common informal fallacies such as ad hominem, straw man, false dilemma, slippery slope, begging the question and hasty generalisation.
- Evaluating arguments: judging the premises for acceptability, judging the premises for relevance to the conclusion, judging whether the premises are sufficient to support the conclusion, and applying the principle of charity when interpreting an argument.
How to evaluate an argument in SQA Higher Philosophy using the criteria of acceptability, relevance and sufficiency, how these criteria connect to validity and soundness, and why the principle of charity matters when interpreting an argument.
- The course assessment: the structure of the externally marked question papers covering Arguments in Action, Knowledge and Doubt and Moral Philosophy, the command words used, and how marks are awarded across short-answer, analysis and extended-response questions.
An overview of how SQA Higher Philosophy is assessed: the externally marked question papers across the three areas of study, the command words (explain, analyse, evaluate), and how marks are awarded across short-answer, analysis and extended-response questions.