Logic and Conditional Reasoning: When Things Are Iffy

Foundations and Structure of the Conditional Proposition

• Definition and Role in Logic:

  • The conditional proposition, traditionally structured around the connective words "if… then…", is recognized as the foundational "heart of logic."

  • Unlike simple Aristotelian propositions (categorical claims that link terms via quantifiers such as all, no, some, and some are not), the conditional connective links entire propositions together into compound statements.

  • In symbolic logic, the basic conditional form is expressed as "If $p$ then $q$", where:

  • $p$ represents the antecedent (the premise or triggering condition).

  • $q$ represents the consequent (the proposition that necessarily follows).

  • Example: In the statement "If Fifi is a poodle, then Fifi is a dog," the proposition "Fifi is a poodle" serves as the antecedent ($p$), and "Fifi is a dog" serves as the consequent ($q$).

• Core Logical Properties:

  • Sufficiency: The antecedent $p$ is a sufficient condition for $q$. If $p$ is true or occurs, $q$ must necessarily occur or be true.

  • Necessity: The consequent $q$ is a necessary condition for $p$. In the absence of $q$, $p$ cannot have occurred (e.g., if Fifi is not a dog, Fifi cannot possibly be a poodle).

  • Implicit Formulations: In everyday language, the term "then" is frequently omitted or implied (e.g., "If wishes were horses, beggars would ride").

• Functional Varieties of Everyday Conditionals:

  • Causality: Temporal linking of events where the antecedent produces the consequent (e.g., "If you press the button, the computer will come on"—meaning the computer activated because the button was pressed).

  • Promises: Conditional commitments (e.g., "If I get paid on Friday, then I will pay you the money I owe you").

  • Threats: Conditional punishments or restrictions (e.g., "If you don't do your homework, then you can't watch TV").

  • Entailment: Definitional or geometrical necessity (e.g., "If the figure is a square, then it has four sides").

  • Evidential Inference: Providing grounds or evidence for a deduced conclusion (e.g., "If you earned an A in the course, then you must have worked hard").

• The Paradox of Protagoras and Euathlus:

  • Recorded by Aulus Gellius, the classical dilemma highlights the inherent complexity of conditional contracts:

  • The Sophist teacher Protagoras contracted to teach law and rhetoric to Euathlus for a large fee. Half was paid upfront, and the remaining half was due when Euathlus won his first legal case.

  • When Euathlus delayed entering legal practice, Protagoras sued him for the balance, arguing:

    • "If you lose this case, then you must pay me for the judges and the law would have found in my favor. If you win this case, then you must pay me because according to our bargain, you must pay me when you win your first case."

  • Euathlus countered using the identical conditional premises reversed:

    • "If you lose this case, then I owe you nothing by virtue of the decision of the judges and the law. If you win this case, then according to our bargain I shall not pay you because I have not yet won my first case."

  • Confounded by the contradictory, self-referential conditionals, the judges postponed the verdict indefinitely.

• Information Asymmetry and Strategic Conditionals:

  • The Four-Shape Cognitive Experiment: In a problem designed by Peter C. Wason and Philip Johnson-Laird, subjects are shown a blue diamond, a yellow diamond, a blue circle, and a yellow circle. The experimenter conceives of an undisclosed color and shape, accepting any object containing either property. When the blue diamond is accepted, observers deduce that one of three conditionals must hold:

  1. If the target is blue and diamond, then the yellow circle is rejected.

  2. If the target is blue and circle, then the yellow diamond is rejected.

  3. If the target is yellow and diamond, then the blue circle is rejected.

  • Deduction halts because without knowing which antecedent ($p$) is true, one cannot determine which consequent ($q$) must follow.

  • Winkler's Cryptologic Bridge Bidding: Mathematician Peter Winkler devised an encrypted bidding system in contract bridge that exploited this property. Players bid using rules such as: "If I hold the ace and king of the suit you named, then I hold the ace of clubs; if I hold the king and queen, then I hold the ace of diamonds; if I hold the ace and queen, then I hold the ace of hearts." Although the rule definitions were fully disclosed, opponents could never identify which antecedent held, preventing them from inferring the consequent. Because this gave partners an unassailable informational advantage, the system was officially banned from North American tournament play.

The Wason Selection Task and Cognitive Biases

• The Original Selection Task (Wason, 1966):

  • Designed by Peter C. Wason to evaluate formal hypothesis testing and deductive capacity, it remains one of the most replicated experiments in cognitive psychology.

  • Experimental Setup: A subject is presented with four cards laid flat on a table. Each card has a letter on one side and an integer on the other. The visible faces display:

  • Card 1: A (vowel)

  • Card 2: D (consonant)

  • Card 3: 4 (even integer)

  • Card 4: 7 (odd integer)

  • Test Rule: "If a card has a vowel on one side then it has an even number on the other."

  • Instructions: Select only those cards that must be turned over to determine whether the rule is true or false.

• Logical Breakdown of the Cards:

  • Card A (Antecedent $p$): Must be selected. If the reverse is an even number, the rule is corroborated; if odd, the rule is definitively falsified via counterexample ($p ∧ ¬q$).

  • Card 7 (Negation of Consequent $¬q$): Must be selected. If the reverse is a vowel, the rule is falsified ($p ∧ ¬q$). If the reverse is a consonant, the rule remains unaffected.

  • Card D (Negation of Antecedent $¬p$): Irrelevant. The conditional states nothing about consonants. A consonant paired with an even or odd number does not violate the rule.

  • Card 4 (Consequent $q$): Irrelevant. A vowel on the other side satisfies the rule, but a consonant on the other side does not invalidate it, as the rule does not assert that only vowels have even numbers.

  • Correct Solution: Only cards A and 7.

• Observed Performance and Systematic Errors:

  • Fewer than $10\%$ of adult subjects identify the correct cards; adults perform as poorly as children.

  • The dominant incorrect choices are selecting A and 4, or selecting A alone.

  • Upper-Face Misinterpretation: An early hypothesis suggested subjects misread the task as applying strictly from upper face to lower face ("Cards with a vowel on the upper side have an even number on the lower side"). However, the mixture of numbers and letters on the exposed faces makes this explanation untenable.

  • Confirmation Bias: Subjects display a marked preference for seeking confirming instances ($p$ and $q$) while actively ignoring potential falsifiers ($¬q$).

  • Matching Bias: Formulated to describe how linguistic saliency overrides formal logic. The explicit mention of lexical items ("vowel", "even number") in the rule increases their psychological prominence. Subjects select the matching cards (A and 4) while failing to recognize the relevance of unnamed alternatives (odd numbers like 7).

  • Cognitive Inertia and Rationalization: When exposed to their errors and shown that turning the 7 card can falsify the conditional, subjects frequently concede the point intellectually but immediately rationalize away their mistake, maintaining unwarranted confidence in their incorrect initial answers.

• The Cube Task:

  • A related spatial-deductive experiment places a square, triangle, or circle on each face of a cube.

  • Rule: "If one side of a cube has a triangle, then the opposite side has a circle."

  • Prompt: Subjects are asked if it is possible for the opposite side of a visible triangle to display a square, a circle, or a triangle.

  • Findings: Between $30\%$ and $50\%$ of subjects state that it is possible to have a triangle on the opposite side—which represents the single impossible outcome under the rule.

Linguistic Formulations, Content Effects, and Contextual Reasoning

• Linguistic Equivalence vs. Psychological Reality:

  • In formal logic, the conditional $p \rightarrow q$ can be expressed in numerous equivalent phrasings, all sharing identical truth tables (falsified if and only if $p$ is true and $q$ is false):

  • "If $p$ then $q$"

  • "$q$ if $p$"

  • "Wearing my mittens implies I have my coat on"

  • "I never wear my mittens without wearing my coat" ($p$ never without $q$)

  • "I do not wear my mittens unless I have my coat on" ($p$ not unless $q$)

  • "I have my mittens on only if I have my coat on" ($p$ only if $q$)

  • "Whenever I have my mittens on, I have my coat on" ($q$ whenever $p$)

  • "Only with my coat on do I wear my mittens"

  • "$p$ is a sufficient condition for $q$"

  • "$q$ is a necessary condition for $p$"

  • Despite identical logical values, human comprehension differs markedly across these forms due to shifts in syntax, emphasis, and pragmatic connotation.

• Content Facets in Sentence Comprehension:

  • Understanding conditional statements relies on three interacting linguistic tiers:

  1. Denotation: Explicit, literal dictionary definitions.

  2. Linguistic Structure: Syntactical arrangement and semantic dependencies.

  3. Connotation: Real-world knowledge, factual memory, and past experiences brought to bear on the terms filling the propositional variables.

• Facilitation Through Familiar and Experiential Content:

  • The Travel Journey Task (Wason and Diana Shapiro):

  • Abstract alphanumeric symbols were replaced with journeys featuring destinations and travel modes.

  • Four cards displayed: Manchester, Leeds, Train, Car.

  • Rule: "Every time I go to Manchester I travel by train."

  • Performance: $10$ out of $16$ subjects ($62.5\%$) selected the correct cards (Manchester and Car), compared to only $2$ out of $16$ subjects ($12.5\%$) in the abstract control group.

  • The Florida Drinking Age Experiment (Richard Griggs and James Cox):

  • Presented an everyday regulatory rule: "If a person is drinking beer, then the person must be over 19."

  • Four displayed cards: Drinking beer, Drinking coke, 16 years of age, 22 years of age.

  • Result: $74\%$ of subjects correctly picked Drinking beer ($p$) and 16 years of age ($¬q$).

  • Cognitive mechanism: To verify the rule, reasoners must access allowable alternatives from long-term memory. When evaluating the 22 years of age card, subjects readily recall that a legal adult may drink soda or alcohol; hence, they omit it. Conversely, an underage individual (16 years of age) consuming beer immediately triggers an accessible, salient violation.

• Constraints on Concrete Content:

  • Concreteness alone does not ensure deductive accuracy:

  • Arbitrary and Counter-Intuitive Rules: When subjects evaluated the rule "If a person is over 19 then the person must be drinking beer," accuracy dropped back to abstract baseline levels because the premise contradicted real-world experience without offering a sensible rationale.

  • Rationale Requirement: Performance improves dramatically only when the rule makes intuitive sense or when subjects are explicitly supplied with the rationale behind the regulation.

• Herman Staudenmayer's Processing Model:

  • Inferences are not generated in isolation; they depend on an individual's internal processing strategies and interpretive frameworks.

  • Tested dimensions:

  • Connective form: "If $p$ then $q$" versus "$p$ causes $q$".

  • Material type: Abstract propositions ("If $X$ occurs then $Y$ occurs"), meaningful concrete relations ("If I turn on the switch then the light will go on"), and anomalous concrete relations ("If she waters the tropical plant then the light will go on").

  • Core Conclusions:

  • Overwhelmingly, subjects misinterpret conditional statements.

  • Abstract Consistency: Subjects made more total errors with abstract material than with concrete material, but their reasoning systems were applied far more consistently. Because abstract symbols lack distracting connotations, reasoners apply an internal logic systematically.

  • Anomalous Processing: When encountering bizarre statements, subjects attempt to rationalize them by constructing hypothetical contexts (e.g., imagining specialized circuits connecting potted plants to lighting fixtures).

  • Two-Stage Deduction Theory: Theories of logic must separate the interpretation process (how premises are encoded via bias, context, and linguistics) from the evaluation process (how logical operations are executed on the interpreted premises).

• Domain Specificity and Pragmatic Inferences (David O'Brien):

  • Different domains trigger distinct underlying assumptions about causality and persistence:

  • Medical Domain: A doctor states: "If a particular drug is taken to reduce inflammation, then the pain will go away." If the patient foregoes the drug and the pain subsides anyway, the doctor is not deemed wrong or deceitful because biological organisms have innate self-healing capabilities.

  • Mechanical Domain: A mechanic states: "If the thermostat is replaced, the car's overheating problem will stop." If the owner leaves the thermostat untouched and the overheating stops, the mechanic's competence and diagnostic accuracy are immediately questioned because mechanical systems do not self-heal.

• Emotional Distortion:

  • Emotionally charged statements undermine deductive discipline.

  • Example: Assessing "If a person convicted of a crime has paid his full debt to society, he should be able to live a private life" becomes emotionally compromised if the subject discovers the individual in question is a convicted child molester.

Structural Transformations: Converse, Inverse, and Contrapositive

• Definitions and Formal Relations:

  • Original Conditional: "If $p$, then $q$" ($p \rightarrow q$)

  • Converse: "If $q$, then $p$" ($q \rightarrow p$)

  • Truth value: Does not share the truth value of the conditional. Confusing the conditional with its converse is a formal logical fallacy (fallacy of conversion).

  • Contrapositive: "If not $q$, then not $p$" ($\neg q \rightarrow \neg p$)

  • Truth value: Logically equivalent to the original conditional. If the conditional is true, the contrapositive is necessarily true; if the conditional is false, the contrapositive is false.

  • Aristotelian Principle of Contraposition: "If it is necessary that $B$ should be when $A$ is, it is necessary that $A$ should not be when $B$ is not."

• Inverse: "If not $p$, then not $q$" ($\neg p \rightarrow \neg q$)

  • Truth value: Equivalent to the converse ($q \rightarrow p$), but not equivalent to the original conditional. Inferring the inverse from the conditional is an illicit inversion.

• Logical Transformations of Concrete Statements:

  • Conditional: "If you exceed the speed limit, then you are breaking the law." (True)

  • Converse: "If you are breaking the law, then you are exceeding the speed limit." (Indeterminate/False; other laws exist).

  • Contrapositive: "If you are not breaking the law, then you are not exceeding the speed limit." (True)

  • Inverse: "If you are not exceeding the speed limit, then you are not breaking the law." (Indeterminate/False).

• The Commuter Train Illustration:

  • Premise: "If the train is traveling from Washington, DC, to Boston, then it stops in New York." (True)

  • Train travels DC to Boston $\rightarrow$ Does it stop in NY? Yes.

  • Train stops in NY $\rightarrow$ Is it traveling from DC to Boston? No. (Converse fallacy).

  • Train stops in NY en route to Boston $\rightarrow$ Did it originate in DC? No.

  • Train originates in DC and stops in NY $\rightarrow$ Is it traveling to Boston? No.

• Linguistic Misuse and "Invited Inferences":

  • Everyday communication routinely blurs conditionals with converses:

  • Voting eligibility: "If you are over 18, then you are eligible to vote" reverses logical reality. The proper statement is: "If you are eligible to vote, then you are over 18" (as non-citizens or felons over 18 may be ineligible).

  • Invited Inferences (M. Geis and A. M. Zwicky):

  • Conditional promises and threats naturally prompt the inference of their inverse:

    • "If you eat your dinner, you may have dessert" routinely invites "If you don't eat your dinner, you won't get dessert."

    • Jonathan Baron's threat example: "If you don't shut up, I'll scream." If the listener complies and shuts up, yet the speaker screams anyway, the listener feels deceived because pragmatic speech assumes: "If you don't shut up, I'll scream, and if you do shut up, I won't scream."

  • Negative Antecedents and Biconditionals:

  • Statements with negated antecedents strongly bias individuals toward biconditional reading ($p \leftrightarrow q$).

  • Example: "If you don't see a trash can, then you put the litter in your pocket" is almost universally interpreted to mean: "If you don't see a trash can, you pocket litter, AND if you pocket litter, you didn't see a trash can."

• Conditional Probabilities and the Prosecutor's Fallacy:

  • Equating a conditional with its converse produces severe errors in probabilistic evaluation:

  • Medical Screening (Sensitivity vs. Predictive Value):

  • Sensitivity: Probability of a positive test given the presence of disease: $P(\text{Positive} \mid \text{Disease})$.

  • Specificity: Probability of a negative test given the absence of disease: $P(\text{Negative} \mid \text{No Disease})$.

  • Misconception: If a diagnostic test has a sensitivity of $0.90$ ($90\%$), physicians often report that a patient testing positive has a $90\%$ probability of having the disease ($P(\text{Disease} \mid \text{Positive})$).

  • A 1978 New England Journal of Medicine study and analyses by David Eddy showed that both practicing physicians and clinical literature routinely conflate $P(\text{Positive} \mid \text{Disease})$ with its converse $P(\text{Disease} \mid \text{Positive})$.

  • Forensic Testimony and Juror Bias:

  • Expert witnesses may testify that if a defendant were not the source of a DNA profile, the likelihood of an accidental match is 1 in 20 million ($P(\text{Match} \mid \text{Innocent}) = \frac{1}{20\,000\,000}$.

  • Jurors routinely invert this conditional, concluding that given a match, the probability of innocence is 1 in 20 million ($P(\text{Innocent} \mid \text{Match}) = \frac{1}{20\,000\,000}$.

  • This error dramatically inflates conviction rates among mock jurors.

• Mitigating Factors: Necessity, Visuals, and Contradiction:

  • Articulating Consequent Necessity: In a four-box experiment matching stuffed animals and fruits, changing the baseline prompt "If there is an apple in the box then there is a horse" to "If there is an apple in the box then there has to be a dog" caused fifth graders' correct logical decisions to rise from $15\%$ to $70\%$, and adults' from $75\%$ to $100\%$.

  • Visual and Pictorial Cues:

  • In Guy Politzer's visual tasks ("If I wear my dress, I wear my hat"), visual representation inhibited illicit conversion.

  • Susan Argent demonstrated that given the statement "If it's a diamond, then it's green," $20$ of $24$ subjects inferred the converse ("All green shapes are diamonds"). When provided with physical drawings or visual depictions, illicit conversions dropped to $6$ of $24$.

  • Contradiction Training: Actively confronting reasoners with counterexamples suppresses fallacious deductions in subsequent tasks.

Causality, Material Implication, and Universal Categoricals

• The Causal Bias:

  • Humans instinctively view the antecedent $p$ not merely as a logical premise, but as the sole operative cause of the consequent $q$.

  • In "If it rains, then she will spoil her new shoes," individuals treat rain as the unique cause of shoe damage, overlooking alternative explanations (e.g., mud, spilled paint).

  • Temporal Asymmetry: The sequential presentation of antecedent followed by consequent produces a cognitive cause-and-effect mind frame.

• "If $p$ then $q$" vs. "$p$ only if $q$":

  • Wason and Johnson-Laird observed that although "If the merger is announced, then the stock goes up" and "The merger is announced only if the stock goes up" are strictly equivalent in classical logic, their psychological connotations differ:

  • "If $p$ then $q$" strongly implies that $p$ causes $q$ or precedes $q$ in time.

  • "$p$ only if $q$" obscures the causal and temporal linkage, reading as a structural restriction.

  • Forensic/Chemical Testing: A forensic test shows: "If a solution contains salt, adding silver nitrate yields a white precipitate." Treating salt as the unique cause is erroneous, as other chemical compounds precipitate similarly with silver nitrate.

  • Logical conditionals do not require causal connection. The formula $p \rightarrow q$ states nothing about the mechanism linking $p$ and $q$.

• Practical vs. Pure Logic:

  • Wason and Johnson-Laird observed that human reasoning naturally abandons formal logic in favor of establishing a coherent "causal matrix" that makes the world predictable.

  • Abstract notions of deductions valid purely by virtue of formal structure—wholly independent of empirical truth—run counter to intuitive everyday cognition.

• Historical Perspectives on the Conditional:

  • Stoic Logic: The formal treatment of conditional propositions originated with the ancient Stoics, who defined valid conditionals as those where the consequent was necessitated or caused by the antecedent.

  • William and Martha Kneale's Critique: Historians Kneale and Kneale observed that while the Greek tradition treated conditionals as expressions of logical entailment, equating "if… then…" exclusively with entailment has introduced pervasive confusion into modern formal logic.

• Truth Values of Ex Falso Quodlibet:

  • In classical logic, any conditional statement with a false antecedent is deemed vacuously true, regardless of the truth value of the consequent:

  • "If the moon is made of green cheese, then pigs can fly" is logically true.

  • Justification: Its contrapositive, "If pigs cannot fly, then the moon is not made of green cheese," possesses both a true antecedent and a true consequent.

  • Numerical Example: "If $3 + 1 = 5$, then $3 + 2 = 6$" is logically true because its contrapositive, "If $3 + 2 \neq 6$, then $3 + 1 \neq 5$", is self-evidently true.

• Transformation of Universal Aristotelian Categoricals:

  • All Aristotelian Type-A universal affirmative statements ("All $S$ is $P$") can be systematically mapped into hypothetical conditionals ("If a thing is an $S$, then it is a $P$"):

  • "All mothers are women" $\rightarrow$ "If a person is a mother, then the person is a woman."

  • "All taxicabs are yellow" $\rightarrow$ "If a car is a taxicab, then the car is yellow."

  • Converting categorical assertions into propositional conditionals establishes the formal groundwork for linking propositional connectives (if, and, or) into expanded deductive syllogisms.