Chapter 4 - Entailment

Entailment (Deductive Reasoning)

Refers to the relationship between statements where one statement logically follows from another. In other words, if one statement (or proposition) is true, then it necessarily implies that another statement must also be true.

• It is not possible (given the logical formal relationship) for the premises to be true and the conclusion to be false.

If the conclusion or premises are false, this does not violate the structure of entailment. Meaning that the entailment can be false.

Additional Implicit Premise

A further assumption or unstated idea that is necessary for the argument to hold together, but is not explicitly mentioned. It supports the argument's structure and makes the reasoning complete.

Argument:

1. All dogs bark.

2. Max is a dog.

3. Therefore, Max barks.

Implicit Premise:

• If something is a dog, it will bark (This assumption connects the two premises, but it's not explicitly stated).

Flipping the Argument (Testing for Entailment)

Assuming the conclusion is false, both the premises cannot both be true at the same time. In other words, if the conclusion is false, it should force at least one of the premises to also be false, making it impossible for both premises to be true while the conclusion is false. If so, the premises do not entail the conclusion.

1. Assume the conclusion is false.

2. Check whether the premises can both still be true without the conclusion being true (valid argument).

3. If both premises can be true while the conclusion is false it is invalid.

Implicit Premise

An assumption or statement that is not explicitly stated in an argument but is still necessary for the argument to hold true or make sense.

Argument:

1. All humans need food to survive.

2. John is a human.

3. Therefore, John needs food to survive.

Implicit Premise:

• John is a living being.

Conditional

A conditional statement expresses a relationship between two things, where one thing (the antecedent) depends on another thing (the consequent).

The general structure of a conditional statement is:

• If A, then B.

• A is the antecedent (the condition or premise).

• B is the consequent (the result or conclusion that follows).

• If it rains (A), then the ground will be wet (B).

• A (it rains) is the antecedent (the condition).

• B (the ground will be wet) is the consequent (the result).

Antecedent (Conditional Statement)

In a conditional statement (like "If A, then B"), the antecedent is the part that comes after the word "if". It’s the condition or the thing that is assumed to happen first.

• The "if" part of the conditional statement, what happens or is assumed to happen first. It helps you think about why something might happen or be true.

It is the cause or event that happens first, leading to or influencing the next thing that happens.

Consequent

In a conditional, the clause that expresses what is said to follow if the antecedent is true. The consequent usually comes right after then.

Logical Form

You replace parts of an argument with variables (like A, B, or X) so that you can see the overall pattern. This helps you understand how different arguments might follow the same logic, even if the content is different.

Examples include modus ponens, modus tollens, disjunctive syllogism and hypothetical syllogism

Modus ponens (Affirming the Antecedent)

A valid form of deductive reasoning. Start with a general statement and then use it to make a conclusion.

It follows this structure:

1. If P, then Q (If P happens, then Q will happen).

2. P is true.

3. Therefore, Q is true.

Example:

1. If it rains, the ground will be wet. (If P, then Q)

2. It is raining. (P is true)

3. Therefore, the ground is wet. (Q is true)

This is a valid Modus Ponens argument because, based on the first premise, if it's raining (P), then the ground must be wet (Q).

Used to affirm the truth of the consequent (the result) of a conditional statement by confirming the truth of the antecedent (the condition)

Deductively valid logical form

A way to describe an argument where, if the premises (the statements or facts you start with) are true, the conclusion must also be true. Deductively valid arguments can be untrue.

Here’s an easy example:

• Premise 1: All dogs are animals.

• Premise 2: Fido is a dog.

• Conclusion: Therefore, Fido is an animal.

This argument is deductively valid because if both premises are true, then the conclusion must be true. It’s not possible for the premises to be true and the conclusion to be false in this case.

Hypothetical syllogism (Valid Argument Form)

A logical argument that uses "if... then..." statements. It has two parts and connects them in a way that the conclusion follows logically from the premises.

• If P then Q

• If Q than R

• Therefore, if P then R

Here’s a simple example:

1. If it rains, then the ground will get wet.

2. If the ground gets wet, then people will need umbrellas.

3. Therefore, if it rains, people will need umbrellas.

Negation

Taking a statement and flipping its truth value. If the original statement is true, the negation is false, and if the original statement is false, the negation is true. Flipping the truth of the sentence by adding "not" to it.

In terms of logic:

• If the original statement is true, its negation (with "not") will be false.

• If the original statement is false, its negation will be true.

Disjunctive syllogism (Deductively Valid)

A logical argument that follows a specific pattern that is in valid form. The structure is:

1. Either P or Q is true (This is the disjunction);

2. Not P is true (or not Q);

3. Therefore, Q must be true (or not Q)

Here's an example:

1. Either it is raining or it is snowing. (Disjunction)

2. It is not raining.

3. Therefore, it must be snowing.

In this example, since one of the two possibilities (raining or snowing) must be true, and we know that it is not raining, the conclusion is that it must be snowing. This is a valid disjunctive syllogism.

Counterexample

An example that shows a statement or claim is false. It’s like finding one situation where the rule or idea doesn’t work, proving that the claim isn’t always true.

How it works:

• You have a general statement or claim.

• You find one example where the statement doesn’t hold true.

• That one example is a counterexample, and it shows that the statement isn’t true in all cases.

Example:

Statement: "All birds can fly."

• Counterexample: A penguin is a bird that cannot fly.

Affirming the Consequent

(Invalid Argument)

A logical fallacy (a mistake in reasoning) that happens when someone assumes that just because the second part of an "If... then..." statement is true, the first part must also be true.

If p then q

Q is true

Therefore, P

Example:

1. If it is raining, then the ground will be wet.

2. The ground is wet.

3. Therefore, it is raining.

This reasoning is flawed because the ground could be wet for other reasons, like someone watering the garden, and it doesn't necessarily follow that it is raining.

Denying the antecedent (Invalid Form)

A fallacy (a mistake in reasoning) that happens when someone incorrectly assumes that if the first part of a statement isn't true, then the second part can't be true either.

• If p then q

• Not p

• Therefore, not q

Example:

1. If it rains, the ground will be wet. (Conditional statement)

2. It did not rain. (Denial of the antecedent)

3. Therefore, the ground is not wet. (Conclusion)

In this example, the reasoning is flawed because the fact that it didn’t rain doesn’t guarantee that the ground isn’t wet. There could be other reasons for the ground being wet, such as someone watering the plants, for example.

Modus tollens (Denying the Consequent)

A type of conditional argument (if-then statement) that helps establish the truth of a conclusion based on the negation (denial) of a consequence.

Works by denying the second part (Q) to conclude the first part (P) is also false,

1. If P, then Q.

2. Not Q.

3. Therefore, not P.

If something leads to a certain conclusion, but that conclusion is false, then the initial assumption must also be false.

Used to negate the consequent (the result) of a conditional statement to deny the antecedent (the condition). It allows us to conclude that the initial condition must be false if the consequence is not true.

Deductive Argument

A type of reasoning where you start with general premises (statements or facts) and use them to draw a specific conclusion.

• The key thing with a deductive argument is that if the premises are true, the conclusion must also be true (guaranteed).It is a logical guarantee.

• There's no room for uncertainty everything is guaranteed to work out logically.

• These arguments can be untrue. The only requirement is that the premises guarantee the conclusion.

Disjunction

A disjunction is just a statement that says, "Either this or that is true."The disjunction of two statements is true if at least one of the statements is true.

• If one of the two things is true, then the whole statement is true.

• If both things are false, then the statement is false.

So, a disjunction is like saying, "One of these things has to be true for the statement to be true." If neither is true, then the whole statement is false.

Conjunction

A combination of two statements, where the statement is only true if both A and B are true.

Example:

• Statement 1: "It is raining."

• Statement 2: "I have an umbrella."

When combined using the word "and":

• "It is raining and I have an umbrella."

For this conjunction to be true, both parts of the statement must be true. So, it must actually be raining, and you must actually have an umbrella. If either part of the conjunction is false, then the whole conjunction is false.