LOGIC CHEAT SHEET
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DEFINITION OF ‘VALID ARGUMENT’:
An argument is valid if and only if, if the premises of the argument are all true ,then the truth
of the conclusion is guaranteed
The validity of the argument rules out that the premises of the argument could all be true
but the conclusion false
NB: A valid argument is valid because of its logical form- because of the logical ‘shape’ of the
relationship between the premises and the conclusion
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DEFINITION OF ‘SOUND ARGUMENT’
An argument is sound if and only if the argument is both valid and all its premises are true
PROCESS OF ANALYSIS
1. Identify the conclusion of the argument:
(a) Are there any conclusion-indicating words in the argument?
(b) If there are, mark out the conclusion and write it up in the analysis at the bottom of the
list of propositions
(c) If there aren’t any conclusion-indicating words, can you identify the conclusion easily
anyway? Tey to do so!! Ask yourself, what is the take-home message of the passage that the
author wants to convince you of? That is probably the conclusion.
2. Identify the premises of the argument:
(a) are there any premise-indicating word in the argument?
(b) if there are, mark out the premises they indicate and write them up in the analysis above
the argument stroke above the conclusion
(c) If there aren’t, can you distinguish things said in the argument that are germane to
establishing the conclusion of the argument from things which things said are not germane
to the logic of the argument for the conclusion you have identified? Read the argument very
carefully. Remember, the point of logical analysis of the argument is to spell out as explicitly
as possible what reasons the argument provides for believing the conclusion.
(d) If you can make this distinction among what are said in the argument, as you should be
able to in most cases, write down the premises relevant to establishing the conclusion of the
argument, above the argument stroke of the analysis
(e) If you’ve got some premises written up in the analysis of the argument because of
premise-indicating words, look at the other propositions in the argument and identify any
that are relevant establishing the conclusion
Write these in the analysis
(f) Now that you have the premises written into the analysis and the conclusion; ask yourself
whether there are logical relationships between premises: is there a sub-argument in the
argument- what is ITS conclusion and the premises to establish it?
Rewrite the analysis of the argument so that the order of the premises expressed in the
analysis reflects the logical relationships, sub-arguments etc, in the original argument
(g) number the premises and conclusion in the analysis
Review your analysis and make sure you’re happy that it expresses what you take the
argument to be arguing;
3. Make any necessary adjustments and finish the write up of the analysis
In the following table “P”, “Q”, “R” and “S” are all complete sentences, that is, propositions (and can
be nothing else) (more exactly, they are declarative sentences (not imperatives, questions etc or
partial, incomplete sentences, nouns, adjectives or such sub-sentential expressions)
COMMON VALID LOGICAL FORMS
FORMAL FALLACIES to avoid while applying the valid
forms
Putting the Consequent
Modus Ponens
1. If P, then Q
2. P
3. Therefore, Q
1. If P, then Q
2. Q
3. Therefore, P
Modus Tollens
Denying Antecedent
1. If P, then Q
2. not Q
Therefore, not P
Disjunctive Syllogism
1. either P or Q
2. not Q
3. therefore P
1. If P, then Q
2. Not P
3. Therefore, not Q
Putting the Disjunct
1*. Either Q or P
2*. Not P
3*. Therefore, Q
1. Either P or Q
2. P
3. Therefore, not Q
Conjunctive Syllogism
1. not (both P and Q)
2. P
3. Therefore, not Q
1*. Either P or Q
2*. Q
3*. Therefore not P
Denying the Conjunct
1*. Not(both P and Q)
2*. Q
3*. Therefore, not P
1. Not (both P and Q)
2. not P
3. Therefore, Q
1*. Not (both P and Q)
2*. not Q
3*. Therefore, P
Hypothetical Syllogism
1. If P, then Q
2. If Q, then R
3. Therefore, If P, then R
Constructive Dilemma
1. If P, then Q
2. If R, then S
3. Either P or R
4. Therefore, either Q or S
SOME POINTS ABOUT “If … then ….” and “ …. only if …..” and “ …. If and only if …..” sentences
If P then Q,
Q if P
and
P only if Q
say the same thing: they all say: If P then Q
If P then Q
says the same thing as
If not Q, then not P
P only if Q
says the same thing as
If not Q, then not P
If P then Q
DOES NOT say the same thing as
P if and only if Q
P if and only if Q
says the same thing as
(If P then Q) and (If Q then P)
If P then Q
says the same thing as
Either not P or Q
A COUPLE OF OTHER USEFUL EQUIVALENCES
Not (both P and Q)
says the same thing as
either not P or not Q
Neither P nor Q
says the same thing as
Not (either P or Q)
Neither P nor Q
says the same thing as
both not P and not Q