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Instructions Exercise 1 Write each of the following arguments into standard form

Instructions

Exercise 1
Write each of the following arguments into standard form:

  1. Plato will graduate only if he bribes his instructors. But Plato does not bribe, never has, never will. Therefore, Plato will not graduate.
  2. In the absence of interesting activities in the world life would be boring. But I don’t find life boring—in fact—life is not boring. Therefore, there must interesting activities in the world.
  3. If the washing machine is disconnected, nothing will get washed. But it is connected, I plugged it in myself. So, we will be able to wash.
  4. Your gas bill will be acceptable provided that you do not use more that X many units a month. But you never do use more than X units a month, so your bill will be acceptable.
  5. If Socrates is a dogbomber, then he will have trouble holding down a job. Knowing Socrates, he definitely will have trouble holding down a job, so he must be a dogbomber.


Label the premises (P) and the conclusion (C); you can use the line (“——–“) to separate the conclusion from the premises.
Identify which inference form is used in each of these (affirming the antecedent, affirming the consequent, denying the antecedent, or denying the consequent).
And then, state whether the inference (the argument) is valid or invalid


Additional Information

Conditional Statements

If … the cat is on the mat … then … the dog is in the yard

(P) (Q)

The antecedent the consequent

Different ways of representing the conditional:

  • “P implies Q”
  • “if P is true, then Q is true”
  • “P → Q”
  • “P Ↄ Q”

Note: The statement “if P then Q” is a false statement only
when its antecedent statement P is true but its consequent Q is false; otherwise the statement “If P then Q” is true.
Therefore,

  • When P is false, then “If P then Q” is a true statement
  • When Q is true, then “If P then Q” is also a true statement

Two Valid Inference Forms (and two invalid forms.)

Valid


A. Modus Ponens (affirming the antecedent)

1. P → Q If P is true then Q is true (P implies Q)

2. P P is true (affirms the antecedent of premise 1)

—————————————-

3. Q Q is true

Given that, assuming that, 1 and 2 are true then it is not possible for Q to be false

also Valid

B. Modus Tollens (denying the consequent)

1. P → Q If P is true then Q is true (P implies Q)

2. ~ Q Q is not true (denies the consequent of premise 1)

———————————
3. ~ P P is not true

If P truly implies Q, but Q is false, then P must be false

Note that these two inference rules come right out of the meaning of the statement “If P is true, then Q is true”.

Negating a conditional

If statement S is true, then…
not-S, or ~ S, or ⌐ S are false

If S is a conditional statement, If P then Q, then to negate it it must be negated this way:

not- (If P then Q). The “not” must go on the outside of the parenthesis so that the whole expression is negated.
The statement ” if not-P then Q” with “not” inside says something different.

Note also that asserting ” Not- (if P then Q) ” is logically equivalent to asserting “P and not-Q”,
i.e. asserting that P can be true while Q is not

Invalid – These are Formal Fallacies (they violate logical form)

Denying the antecedent is a fallacy

1. If P is true then Q is true

2. P is false (denies the antecedent of premise 1)

—————
Q is false

This argument form is invalid
Even though P implies Q, so might other things. P being false does nothing to Q’s truth status, one way or the other

also a Formal Fallacy

Affirming the consequent is a fallacy
1. If P implies Q

2. Q is true (affirms the consequent of premise 1)

————-

P is true No, this is an invalid inference

Stylizations of conditional statements: If P then Q
These are different ways of stating “If P then Q”, different ways of translating “If P then Q”.

P is a sufficient condition for Q
Q is a necessary condition for P
Q is so if P is so (‘so’ roughly meaning true)
Q provided that P
P only if Q is so
Only if Q is P so
Given that P is so, then Q
Not-P unless Q
Assuming P, then Q
Note: “if” dictates antecedent. For example, “P if J” means “If J then P”. It has nothing to do with the letters. It is the logic of “if”.
But the word “only” alters things. The statement “if P then Q” means the same thing as “P only if Q”., or “only if Q is the case, is P also the case”. Thus “if” and “only if” are different. “If” indicates a sufficient condition and logically will identify the antecedent. “Only if”, on the other hand, indicates a necessary condition and logically identifies the consequent.

The above are all variations (translations) for saying “If P then Q” or “P implies Q”

Before starting homework Exercise 1` be sure to review the above translation rules (or stylizations) for conditional statements (right above); and also especially review the material starting at the top about conditionals and the inference forms (some are valid and some are not).

For the homework start off with this exercise – call it Exercise 1
Do the exercise offline first so that you have it ready to copy and paste it into the submission device.

These are exercises on argument forms involving conditional propositions.
Re-write each of the following arguments into standard form:

  1. Plato will graduate only if he bribes his instructors. But Plato does not bribe, never has, never will. Therefore, Plato will not graduate.
  2. In the absence of interesting activities in the world life would be boring. But I don’t find life boring—in fact—life is not boring. Therefore, there must interesting activities in the world.
  3. If the washing machine is disconnected, nothing will get washed. But it is connected, I plugged it in myself. So, we will be able to wash.
  4. Your gas bill will be acceptable provided that you do not use more that X many units a month. But you never do use more than X units a month, so your bill will be acceptable.
  5. If Socrates is a dogbomber, then he will have trouble holding down a job. Knowing Socrates, he definitely will have trouble holding down a job, so he must be a dogbomber.


Each argument will have the premises listed first, and then the conclusion at the bottom. Label the premises P for each premise and C for the conclusion. You can use the line (“——–“) to separate the conclusion from the premises. Identify (state) which inference form is used in each of these (is it affirming the antecedent, affirming the consequent, denying the antecedent, or denying the consequent?)
Explain whether the inference (the argument) is valid or invalid. (remember, these are only exercises in deduction—do not get involved with inductive assessments about the truth of the premises.

HOMEWORK ASSIGNMENT


Exercise 1 above), Display each of those five arguments into standard form. Identify which inference form is used in each, and explain whether the inference (the argument) is valid or invalid.

Again, just repeating the exercise specifications:
Re-write each of those five arguments into standard form: that is a list with one sentence per line; the premises listed first, and then the conclusion at the bottom.

Label the premises (P) and the conclusion (C); you can use the line (“——–“) to separate the conclusion from the premises.

Identify (state) which inference form is used in each of these (affirming the antecedent, affirming the consequent, denying the antecedent, or denying the consequent).

And then, state whether the inference (the argument) is valid or invalid. (remember, these are only exercises in deduction—do not get involved with inductive assessments about the truth of the premises.





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