Deductive Reasoning: Reasoning with Syllogisms

Question 1

Compound Propositions

Answer 1

A proposition that combines two or more component propositions by means of a connective

Question 2

Propositional Logic

Answer 2

The branch of deductive reasoning that deals with the logical relationships among propositions

Question 3

Disjunctive Syllogism

Answer 3

Disjunctive propositions are propositions of the for: p or q

• the disjuncts in a disjunctive syllogism are p and q
• the disjunctive proposition asserts neither p nor q, but does express both
• “OR” is interpreted in the inclusive sense so that ‘p or q’ means ‘either p is true or q is true or both are true’

p or q
Not-p
therefore q.

Question 4

Hypothetical Syllogism

Answer 4

Form: If p, then q.

• Have two parts: p, which is the antecedent, and q, which is the consequent.
• A hypothetical proposition ASSERTS neither the antecedent nor consequent, but does EXPRESS both
• Like ‘A’ categorical statements, a hypothetical proposition is not equivalent to its convers, but is equivalent to its contrapositive.

Question 5

Nonstandard hypothetical proposition forms:

Answer 5

p if q = if q, then p
p provided that q = if q, then p
p only if q = if p, then q
p if and only if q = if p, then q, and if q, then p
p unless q = if not-q, then p
p so long as q = if p, then q
Without X, then q = if X does not occur, then q

Question 6

Pure Hypothetical Syllogism

Answer 6

If p, then q
If q, then r
Therefore: If p, then r
Ex:
If Zach is there, then Jack will be there
If Jack is there Sarah will be there
Therefore: If Zach is there then Sarah will be there

Question 7

Modus Ponens (Affirming the Antecedent)

Answer 7

If p, then q
p
therefore: q

ex:
If it is raining, then it is wet
It is raining
Therefore it’s wet

Question 8

Modus Tollens (Denying the Consequent)

Answer 8

If p, then q
Not-q
therefore: Not-p

Ex:
If God exists, then there is no evil
There is evil
Therefore God does not exist

Question 9

Two Forms that look like modus ponens and modus tollens but are invalid

Answer 9

Denying the antecedent:
If p, then q
not-p
not-q
&
Affirming the consequent: 
If p, then q
q
p

Question 10

Strategies for Identifying the Form of a Syllogism

Answer 10

1. If any statement in the argument is compound, then try to cast the argument as either a disjunctive or hypothetical syllogism.
2. Look for repetition. If propositions are repeated, it is likely a disjunctive or hypothetical syllogism. If terms are repeated, it is likely a categorical syllogism.
3. Look for the presence of explicit connectives such as ‘or’ and ‘if’. If such connectives are used, then it is likely a hypothetical or disjunctive syllogism.
4. Look for the presence of explicit quantifiers such as ‘all’ or ‘some’, which are signs of categorical syllogisms.
5. If the argument applies a generalization to a particular case, it is probably a categorical syllogism.

Question 11

Nonstandard quantifiers: Variants on ‘all’

Answer 11

A, every, each, any, whenever

Question 12

Nonstandard quantifiers: Variants on ‘no’

Answer 12

not a, none, nothing, never

Question 13

Nonstandard quantifiers: Variants on ‘some’

Answer 13

there is, a, many, most, a few

Question 14

Method for analyzing extended arguments

Answer 14

1. Identify the conclusion (may be implicit)
2. Number (or letters for implicit premises/conclusions) as many of the premises in the passage as you can
3. Put into standard form as much of the argument as you can
4. If this is not sufficient to reveal the whole structure of the argument, then use what you know about syllogisms to identify the missing premises/ conclusions
5. Evaluate each step for validity in order to determine whether the whole argument is valid: the whole argument is valid only if each part is valid