Suppressing valid inferences with conditionals



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Suppressing valid inferences with conditionals

  • Ruth M.J. Byrne, MRC Applied Psychology Unit, Cambridge
  • (1987, 1988, 1989)

Four Conditional Inferences

  • Modus Ponens (MP)
  • Modus Tollens (MT)
  • Denial of the Antecedent (DA)
  • Affirmation of the Consequent (AC)

Modus Ponens

  • Premise 1 (conditional)
    • If P then Q
  • Premise 2 (categorical sentence)
    • P
  • Valid conclusion
    • Q

Modus Tollens

  • Premise 1 (conditional)
    • If P then Q
  • Premise 2 (categorical sentence)
    • ~Q
  • Valid conclusion
    • ~P

Correct inferences

  • MP is generally self-evident for validity and the conclusion is deduced easily.
    • Some tricky cases ($20 in pocket, warnings)
  • MT is more difficult to infer, but generally judged to be valid.

Denial of the Antecedent

  • Premise 1 (conditional)
    • If P then Q
  • Premise 2 (categorical sentence)
    • ~P
  • Invalid conclusion
    • ~Q

Affirmation of the Consequent

  • Premise 1 (conditional)
    • If P then Q
  • Premise 2 (categorical sentence)
    • Q
  • Invalid conclusion
    • P

Incorrect inferences (fallacies)

  • DA and AC are the most common errors among test groups.
    • If she has an essay to write then she will study late in the library.
      • She does not have an essay to write, so…
        • She will not study in the library. [Nothing follows.]
      • She will study late in the library, so…
        • She has an essay to write. [Nothing follows.]

Formal theories

  • Fallacies are difficult for formal (rule-based) theories to explain.
  • They are usually attributed to comprehension processes by which the premises are decoded into incorrect representations used by the rules (e.g. obverse of conditional).

Suppressing invalid inferences

  • When accompanied by alternative antecedents, people systematically reject invalid DA and AC conclusions.
  • (Markovits, 1985; Rumain et al., 1983)

Example (suppressing invalid inferences)

  • If she has an essay to write then she will study late in the library.
  • If she has some textbooks to read then she will study late in the library.
    • She does not have an essay to write, so…
      • Nothing follows – what if she has some textbooks to read?
    • She will study late in the library, so…
      • Nothing follows – maybe she has an essay to write, maybe she has some textbooks to read, or maybe something else.

What if…

  • The opposite is true, and we can suppress valid inferences.

Example (suppressing valid inferences)

  • If she has an essay to write then she will study late in the library.
  • If the library stays open then she will study late in the library.
    • She has an essay to write, so…
      • Nothing follows – what if the library doesn’t stay open? [She will study late in the library.]
    • She will not study late in the library, so…
      • Nothing follows – what if the library doesn’t stay open? [She doesn’t have an essay to write.]

Suppressing valid inferences

Experiment 1 setup

  • 24 subjects, 3 groups (no logic tuition)
  • 3 argument types
    • Simple conditional, conditional + alternative, conditional + additional
  • 4 inference types
    • MP, MT, DA, AC
  • Given 3 conclusions – which one follows?

Experiment 1 results

  • Inference Type
  • Argument Type
  • MP
  • MT
  • DA
  • AC
  • Simple arguments
  • 96
  • 92
  • 46
  • 71
  • Alternative arguments
  • 96
  • 96
  • 4
  • 13
  • Additional arguments
  • 38
  • 33
  • 63
  • 54

Experiment 1 summary

  • Additional antecedents suppressed MP and MT inferences, while, as proven before, alternative antecedents suppressed DA and AC inferences.
  • Alternative or additional antecedents in the second conditional must alter the interpretation of the first conditional.

But why?

  • Perhaps the relevant formal rule no longer applies to the interpretation.
  • More plausible: suppression depends on the categorical information supplied in the premises.

What if…

  • We supply more categorical information.

Example (not suppressing valid inferences)

  • If she meets her friend then she will go to a play.
  • If she has enough money then she will go to a play.
    • She meets her friend and she has enough money, so…
      • She will go to a play.
    • She will not go to a play, so…
      • She doesn’t meet her friend.

Example (not suppressing invalid inferences)

  • If she meets her friend then she will go to a play.
  • If she meets her family then she will go to a play.
    • She does not meet her friend and she does not meet her family, so…
      • She will not go to a play. [Nothing follows – what if some other reason compelled her to go?]
    • She will go to a play, so…
      • She meets her friend. [Nothing follows – what if some other reason compelled her to go?]

Not suppressing valid inferences

  • Hypothesis 2: when accompanied by additional antecedents together with the conjunction of both antecedents, people will systematically affirm valid MP and MT inferences (and invalid DA and AC inferences), as in the simple argument.

Experiment 2 setup

  • 24 subjects, 3 groups (no logic tuition)
  • 3 argument types
    • Simple conditional, conditional + alternative + conjunction/consequent, conditional + additional + conjunction/consequent
  • 4 inference types
    • MP, MT, DA, AC
  • Given 3 conclusions – which one follows?

Experiment 2 results

  • Inference Type
  • Argument Type
  • MP
  • MT
  • DA
  • AC
  • Simple arguments
  • 100
  • 75
  • 46
  • 58
  • Categorical information about both alternatives
  • 100
  • 88
  • 79
  • 75
  • Categorical information about both additionals
  • 100
  • 67
  • 83
  • 71

Experiment 2 summary

  • Combined additional antecedents didn’t suppress MP or MT inferences, and combined alternative antecedents didn’t suppress DA or AC inferences.
  • Presumably, the conditionals were interpreted correctly, but nonetheless both fallacies and correct inferences were made in the presence of categorical information.

What if…

  • Specific alternatives or additionals are suggested simply by general subject knowledge?

Example (no duration)

  • During the student protest, the policeman said to the student: “If you enter the building then I will arrest you.”
  • (Note: this inference is not strictly valid as it is embedded within a description.)

Example (short duration)

  • During the 15-minute student protest, the policeman said to the student: “If you enter the building then I will arrest you.”
    • The student entered the building, so…
      • Nothing follows – what if the protest was no longer in progress when the student entered the building? [The policeman arrested the student.]

Example (long duration)

  • During the 2-week student protest, the policeman said to the student: “If you enter the building then I will arrest you.”
    • The student did not enter the building, so…
      • Nothing follows – what if other actions caused the student to be arrested?

Suppressing valid inferences

  • Hypothesis 3: when accompanied by assertions of short-duration, the plausibility of additional antecedents will be suggested, and people will systematically reject valid MP and MT inferences; when accompanied by assertions of long-duration, the plausibility of alternative antecedents will be suggested, and people will reject invalid DA and AC inferences.

Experiment 3 setup

  • 24 subjects, 3 groups (no logic tuition)
  • 3 argument types
    • Simple conditional (no duration), conditional + alternatives (long-duration), conditional + additionals (short-duration)
  • 4 inference types
    • MP, MT, DA, AC
  • Is the conclusion true? (Yes/No/Maybe/Can’t Say)

Experiment 3 results

  • Inference Type
  • Argument Type
  • MP
  • MT
  • DA
  • AC
  • Simple arguments
  • 72
  • 66
  • 44
  • 28
  • Alternative arguments (i.e. long-duration)
  • 56
  • 59
  • 19
  • 31
  • Additional arguments (i.e. short-duration)
  • 41
  • 25
  • 25
  • 34

Experiment 3 summary

  • Additional (short-duration) arguments suppressed MP and MT inferences, while alternative (long-duration) antecedents suppressed DA, but not AC, inferences.
  • Perhaps the inconsistency is due to the “considerable amount of contextual information already given in the descriptions.”

Labs vs. conversations

  • In labs, it is common for subjects to assume they are being given all the information they need.
  • In conversations, this is not always the case, so alternative options are often supposed.

What does this tell us?

  • Context can suppress both valid and invalid inferences.
  • For formal theories, if suppression of an invalid inference implies no corresponding rule, then suppression of a valid inference should imply the same. Yet this is not the case.
  • Suppression alone tells us nothing about the existence of the mental rules of inference proposed by formal theorists.

Interpretation in formal theories

  • Let us consider that the problem lies in interpretation.
  • For alternative antecedents:
    • If P then Q (P  Q)
    • If R then Q (R  Q)
    • If P or R then Q (P v R  Q)
  • For additional antecedents:
    • If P then Q (P  Q)
    • If R then Q (R  Q)
    • If P and R then Q (P & R  Q)

Semantics in formal theories

  • P v R  Q
    • Blocks both DA and AC inferences
  • P & R  Q
    • Blocks both MP and MT inferences
  • Semantic content plays a key role in developing an interpretation of these representations.
  • Formal rules need additional information on comprehension, how premises with the same logical form are represented in different ways.

Other explanations

  • People rely on content-specific/domain-dependent rules instead of uninterpreted abstract rules.
    • How do people reason in unfamiliar areas?
  • People use a “general semantic procedure.” (i.e. mental models)

Mental models

  • Construct a model of a state of affairs.
  • Attempt to formulate a conclusion.
  • Search for alternative models that refute the conclusion.
  • (Johnson-Laird, 1983)

Simple example

  • If P then Q:
  • P Q
  • P Q
  • o Q
  • Wherever P exists, Q also exists, so given P, Q follows (MP). MT requires additional information to be added to the model.

Extended example

  • Adding a second conditional (if R then Q) depends on the meaning and general knowledge of how to integrate it in.
  • For additionals:
  • P R Q
  • P R Q
  • o Q
  • An assertion of P will no longer suffice to conclude Q.

Extended example

The point

  • Interpretation (and therefore context) plays a critical role in the interaction among statements of the same logical form.
  • Theories based on mental models seem to better account for this process.

The end

  • Questions?

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