Flashcards Logic and Critical Thinking: Arguments, Truth Tables, and Inference Rules

Deductive reasoning

Reasoning where the conclusion must be true if the premises are true. Example: All dogs are animals. Max is a dog. Therefore, Max is an animal.

Inductive reasoning

Reasoning where premises make the conclusion likely, but do not guarantee it. Example: The sun has risen every morning, so it will probably rise tomorrow.

Argument

A set of statements where premises are used to support a conclusion. Example: All humans are mortal. Socrates is human. Therefore, Socrates is mortal.

Premise

A statement that provides evidence or support for a conclusion. Example: All birds have feathers.

Conclusion

The statement an argument is trying to prove. Example: Therefore, a penguin has feathers.

Validity

An argument is valid when it is impossible for all its premises to be true while its conclusion is false. Example: All cats are mammals. Luna is a cat. Therefore, Luna is a mammal.

Invalid argument

An argument is invalid when its premises could all be true while its conclusion is false. Example: All dogs are animals. Max is an animal. Therefore, Max is a dog.

Truth of premises

Whether the statements in an argument actually match reality. Example: "All fish can fly" is false because fish cannot fly.

Validity vs. truth

Validity is about whether the conclusion logically follows from the premises, not whether the premises are actually true. Example: All fish are birds. All birds can fly. Therefore, all fish can fly. The argument is valid even though its premises are false.

Soundness

An argument is sound when it is both valid and has all true premises. Example: All humans are mortal. Socrates is human. Therefore, Socrates is mortal.

Unsound argument

An argument is unsound when it is invalid or has at least one false premise. Example: All dogs can fly. Max is a dog. Therefore, Max can fly.

Counterexample

A possible situation where all the premises are true but the conclusion is false. One counterexample proves an argument is invalid. Example: All dogs are animals. Max is an animal. But Max could be a cat.

Logical possibility

Something that could be true without creating a logical contradiction. Example: It is logically possible for a penguin to be blue, even if most penguins are not blue.

Logical impossibility

Something that cannot be true because it creates a contradiction. Example: A square cannot have exactly three sides.

Consistency

Statements are consistent when they could all be true at the same time. Example: "Max is a dog" and "Max is an animal" are consistent.

Inconsistency

Statements are inconsistent when they cannot all be true at the same time. Example: "Max is alive" and "Max is not alive" are inconsistent.

Monotonicity

Deductive reasoning is monotonic because adding more premises to a valid argument cannot make it invalid. Example: All dogs are animals. Max is a dog. Therefore, Max is an animal. Adding "Max is brown" does not change the validity.

Why deductive reasoning is monotonic

Once a conclusion logically follows from the original premises, adding another premise cannot take away that logical connection. Example: If all dogs are animals and Max is a dog, Max must still be an animal even if we add that Max is brown.

Nonmonotonic reasoning

Reasoning where new information can change or weaken an earlier conclusion. Example: You think a bird can fly, but then learn it is a penguin.

Truth-functional statement

A statement whose truth depends entirely on the truth values of its component statements and the logical operators connecting them. Example: "It is raining AND it is cold" is true only if both parts are true.

Simple statement

A statement that does not contain another statement connected by a logical operator. Example: "It is raining."

Compound statement

A statement made by combining two or more statements with logical operators. Example: "It is raining AND it is cold."

Logical operator

A word or symbol that connects or changes statements and determines the truth of a compound statement. Examples: NOT, AND, OR, IF...THEN.

Negation (NOT)

Reverses the truth value of a statement. If the original is true, its negation is false, and vice versa. Example: If P means "It is raining," NOT P means "It is not raining."

Conjunction (AND)

A compound statement that is true only when both statements are true. Example: "I am studying AND I am listening to music" is true only if both are happening.

Disjunction (OR)

A compound statement that is true when at least one of the statements is true. Example: "I will have coffee OR tea" is true if I have coffee, tea, or both.

Inclusive OR

An OR statement that allows either option or both options to be true. Example: "You can take math or science" allows you to take both.

Exclusive OR

An OR statement that allows exactly one option to be true, but not both. Example: "You can choose either coffee or tea, but not both."

Conditional (IF...THEN)

A statement saying that if one statement is true, another statement must also be true. Example: If it rains, then the ground gets wet.

Antecedent

The first part of a conditional statement, the part after IF. Example: In "If it rains, then the ground gets wet," the antecedent is "it rains."

Consequent

The second part of a conditional statement, the part after THEN. Example: In "If it rains, then the ground gets wet," the consequent is "the ground gets wet."

Material implication

A conditional statement that is false only when the antecedent is true and the consequent is false. Example: "If I study, then I pass" is false only if I study but do not pass.

Necessary condition

A condition that must be present for something else to happen. Example: Having oxygen is necessary for a typical human fire to burn.

Sufficient condition

A condition that guarantees something else will happen. Example: Being a square is sufficient for being a rectangle.

Necessary vs. sufficient condition

A necessary condition must be present, while a sufficient condition guarantees the result. Example: Oxygen is necessary for fire, while being a square is sufficient for being a rectangle.

Biconditional (IF AND ONLY IF)

A statement that is true when both statements have the same truth value. It means each statement is both necessary and sufficient for the other. Example: A shape is a square if and only if it has four equal sides and four right angles.

Logical equivalence

Two statements are logically equivalent when they always have the same truth value. Example: "It is not true that it is raining" is equivalent to "It is not raining."

Truth table

A chart that shows the truth value of a compound statement for every possible combination of truth values of its component statements. Example: A truth table for P AND Q shows the result when both are true, one is true, or both are false.

Tautology

A statement that is true under every possible combination of truth values. Example: "P OR NOT P" is always true.

Contradiction

A statement that is false under every possible combination of truth values. Example: "P AND NOT P" is always false.

Contingent statement

A statement that is true in some situations and false in others. Example: "It is raining" could be true or false depending on the weather.

Modus Ponens (Affirming the Antecedent)

A valid argument form: If P then Q. P. Therefore, Q. Example: If I study, I pass. I study. Therefore, I pass.

Modus Tollens (Denying the Consequent)

A valid argument form: If P then Q. Not Q. Therefore, not P. Example: If it rains, the ground gets wet. The ground is not wet. Therefore, it did not rain.

Affirming the Consequent

An invalid argument form: If P then Q. Q. Therefore, P. Example: If it rains, the ground gets wet. The ground is wet. Therefore, it rained. The ground could be wet for another reason.

Denying the Antecedent

An invalid argument form: If P then Q. Not P. Therefore, not Q. Example: If I study, I pass. I did not study. Therefore, I did not pass. I could still pass without studying.

Hypothetical Syllogism

A valid argument form: If P then Q. If Q then R. Therefore, if P then R. Example: If I study, I pass. If I pass, I graduate. Therefore, if I study, I graduate.

Disjunctive Syllogism

A valid argument form: P OR Q. Not P. Therefore, Q. Example: I will have coffee or tea. I will not have coffee. Therefore, I will have tea.

Constructive Dilemma

A valid argument form where two conditional statements and a disjunction support a conclusion containing both possible outcomes. Example: If I study, I pass. If I exercise, I get stronger. I will study or exercise. Therefore, I will pass or get stronger.

Destructive Dilemma

A valid argument form where two conditional statements and the denial of both possible outcomes support the denial of the starting possibilities. Example: If I study, I pass. If I exercise, I get stronger. I do not pass or I do not get stronger. Therefore, I did not study or I did not exercise.

Universal statement

A statement about every member of a group. Example: All dogs are mammals.

Existential statement

A statement claiming that at least one member of a group has a certain characteristic. Example: Some dogs are brown.

Universal generalization

Using observations about a sample to make a claim about an entire population. Example: Surveying students from different years and concluding that most BU students prefer online textbooks.

Inductive generalization

Using a sample to make a broader conclusion about a population. Example: Asking 500 randomly selected voters who they support and estimating the preferences of all voters.

Population

The entire group that a researcher wants to draw a conclusion about. Example: All BU undergraduate students.

Sample

A smaller group selected from a population to study. Example: 500 BU students surveyed about campus dining.

Representative sample

A sample that accurately reflects the important characteristics of the population. Example: A survey of BU students that includes students from different years, majors, and backgrounds.

Biased sample

A sample that does not accurately represent the population because certain groups are overrepresented or underrepresented. Example: Surveying only Questrom students and using the results to describe all BU students.

Random sample

A sample in which members of the population have an equal chance of being selected. Example: Randomly selecting 100 names from a list of all BU students.

Sample size

The number of individuals or observations included in a sample. Example: A survey of 1,000 people has a sample size of 1,000.

Larger sample size

Generally makes an inductive conclusion more reliable because it includes more observations, assuming the sample is properly selected. Example: Surveying 1,000 randomly selected students is generally more reliable than surveying 10.

Sample quality

The degree to which a sample represents the population. A large sample can still be misleading if it is biased. Example: Surveying 10,000 people from one neighborhood does not necessarily represent an entire country.

Statistical syllogism

Using information about a group to make a probability-based conclusion about an individual member of that group. Example: Most BU students drink coffee. Jenna is a BU student. Therefore, Jenna probably drinks coffee.

Inductive strength

The degree to which the premises make the conclusion likely. Example: A survey of 5,000 randomly selected voters provides stronger support for an election prediction than a survey of 5 voters.

Strong inductive argument

An argument where the premises make the conclusion highly likely, even though they do not guarantee it. Example: 95% of students in a representative sample passed the exam. Jenna is in that group. She probably passed.

Weak inductive argument

An argument where the premises provide little support for the conclusion. Example: Two students in one class like economics, so all college students probably like economics.

Cogency

A strong inductive argument with true premises. Example: A well-designed national survey finds that 70% of people support a policy, and the sample was actually representative.

Uncogent argument

An inductive argument that is weak or has at least one false premise. Example: A survey of five people is used to claim that 90% of the country supports a policy.

Induction by confirmation

Supporting a hypothesis by finding evidence that is consistent with it. Example: You think a plant needs sunlight, and you observe that plants receiving sunlight grow well.

Induction by disconfirmation

Testing a hypothesis by looking for evidence that could show it is false. Example: You think all swans are white, so you search for a swan that is not white.

Confirmation bias

The tendency to look for or favor evidence that supports what you already believe while ignoring conflicting evidence. Example: You believe a stock will rise and only read positive articles about it.

Disconfirming evidence

Evidence that goes against a hypothesis or belief. Example: Finding a black swan disconfirms the claim that all swans are white.

Falsifying evidence

Evidence that demonstrates a hypothesis is false. Example: One non-white swan falsifies the claim that all swans are white.

Analogy

A comparison between two things based on similarities. Example: Comparing the human brain to a computer because both process information.

Analogical reasoning

Reasoning that uses similarities between two things to draw a conclusion about another shared feature. Example: Two medications have similar ingredients and effects, so one may have a similar side effect to the other.

Analogical argument strength

An analogical argument is stronger when the similarities are relevant to the conclusion and there are fewer important differences. Example: Two cars with the same engine and design are more likely to have similar fuel efficiency than two cars that only share the same color.

Relevant similarity

A similarity that actually supports the conclusion being made. Example: Two medications having the same active ingredient is relevant when predicting similar effects.

Irrelevant similarity

A similarity that does not provide meaningful support for the conclusion. Example: Two people both liking pizza does not mean they will have the same medical history.

Disanalogy

A relevant difference between two things being compared that weakens an analogical argument. Example: Two medications may be similar, but one has a different active ingredient that changes its effects.

Hume's Problem of Induction

The problem that past experiences cannot logically guarantee that the future will resemble the past. Example: The sun has risen every day, but that does not logically prove it must rise tomorrow.

Uniformity of Nature

The assumption that the future will generally resemble the past and that nature will behave consistently. Example: We expect water to freeze at the same temperature tomorrow because it has done so in the past under the same conditions.

Problem of induction

Inductive reasoning assumes patterns will continue, but that assumption cannot be proven through deduction without circular reasoning. Example: Saying the sun will rise tomorrow because it has always risen assumes the future will resemble the past.

Circular reasoning

An argument that uses its conclusion as part of the support for that same conclusion. Example: The future will resemble the past because the future has always resembled the past.

Causation

A relationship where one event or factor produces or contributes to another event. Example: Smoking increases the risk of lung cancer.

Correlation

A relationship where two variables change together, but this does not necessarily mean one causes the other. Example: Ice cream sales and drowning incidents both increase in summer.

Correlation does not imply causation

Just because two things happen together does not prove one causes the other. Example: Ice cream sales and drowning both rise in summer, but warmer weather may explain both.

Third variable problem

A situation where a separate factor influences two variables and makes them appear causally connected. Example: Hot weather increases both ice cream sales and swimming, which can increase drowning incidents.

Reverse causation

The possibility that the supposed effect actually causes the supposed cause. Example: Researchers find that unhappy people sleep poorly, but poor sleep could also contribute to unhappiness.

Causal hypothesis

A proposed explanation that one factor causes another. Example: Getting more sleep improves exam performance.

Causal inference

Using evidence to determine whether one factor actually causes another. Example: Comparing students who receive different amounts of sleep while controlling for other factors.

Controlled experiment

An experiment where researchers manipulate one variable while keeping other relevant conditions as similar as possible. Example: Giving one group a new study method and another group the usual method, then comparing exam results.

Independent variable

The factor that the researcher changes or manipulates in an experiment. Example: The amount of fertilizer given to a plant.

Dependent variable

The outcome that the researcher measures to see whether the independent variable had an effect. Example: The plant's height after receiving fertilizer.

Control group

The group that does not receive the treatment being tested and provides a comparison. Example: Plants that receive no fertilizer.

Experimental group

The group that receives the treatment being tested. Example: Plants that receive fertilizer.

Random assignment

Placing participants into experimental groups by chance to reduce differences between groups. Example: Randomly assigning students to either a new teaching method or the regular teaching method.

Confounding variable

A factor other than the independent variable that could affect the dependent variable and distort the results. Example: One plant group receives more sunlight as well as more fertilizer.

Placebo

A treatment with no active ingredient or intended effect, used to help control for expectations. Example: Giving a control group a sugar pill instead of the real medication.

Placebo effect

When a person experiences a change because they believe they are receiving a treatment, even if the treatment itself has no active effect. Example: Someone feels better after taking a sugar pill because they believe it is medicine.

Blind experiment

An experiment where participants do not know which treatment they are receiving, reducing the influence of expectations. Example: Patients do not know whether they received the real medication or a placebo.

Double-blind experiment

An experiment where neither the participants nor the researchers interacting with them know who received the treatment. Example: Neither patients nor the doctors evaluating them know which patients received the real drug.