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A proposition is a statement, a sentence which has truth value. A single proposition can be expressed by many different sentences.
God loves the world.
The world is loved by God.
Dues mundum amat. |
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| Branch of formal, deductive logic in which the basic unit of thought is the proposition. |
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| Truth-Functional Propositions |
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| When a proposition's truth value depends upon the truth values of its component parts. |
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| A proposition with only one component part. |
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| A proposition with more than one component part or is modified in some other way. |
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| Words which combine or modify simple propositions to make compound propositions. |
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| Uppercase letter that represents a single, given proposition. |
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| Lowercase letter that represents any proposition. |
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| Logical operator that denies or contradicts a proposition. |
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| A listing of the possible truth values for a set of one or more propositions. |
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| Displays the truth values produced by a logical operator modifying a minimum number of variables. |
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| Logical operator that joins two propositions and is true if and only if both the propositions and true. |
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| Logical operator that joins two propositions and is true if and only if one or both of the propositions is true. |
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Conditional
(also called Hypothetical or material implication) |
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Definition
| Logical operator that asserts one component (the antecedent) implies the other (the consequent). It is false if and only if the antecedent is true and the consequent is false. |
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| Proposition following the "if" of a conditional proposition. |
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| Proposition following the "then" of a conditional |
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| If p then q is equivalent to If not q then not p |
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Definition
Logical operator representing if and only if propositions. It is true when both component propositions have the same truth value, and is false when their truth values differ.
Example: Skyscrapers are buildings if and only if it is false that skyscrapers are not buildings. |
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| 3 Tools used in Propositional Logic |
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Definition
1. Truth Tables
2. Formal Proofs
3. Truth Trees |
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| Two propositions are logically equivalent if and only if they have identical truth values |
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| a proposition that is always true due to its logical structure (every row of its truth table is true) |
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| a proposition that is always false due to its logical structure (every row of its truth table is false) |
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| in a valid argument, if the premises are true, then the conclusion MUST be true |
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| a set of propositions which CAN all be true at the same time |
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| a valid argument which presents a choice between 2 conditionals |
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Definition
Extended (double) Modus Ponens
(if p then q)*(if r then s)
P v r
therefore q v s |
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Definition
extended (double) Modus Tollens
(if p then q) * (if r then s)
not q v not s
therefore not p v not r |
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| 3 Main oppositions to escape the "Horns of a Dilemma" |
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Definition
1. Go between the horns
2. Grasp it by the horns
3. Rebut the horns |
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Definition
Provide a 3rd alternative
(Deny the disjunctive premise) |
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Definition
| Reject at least one of the two conditionals in the conjunctive premise. |
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| Construct a counter-dilemma using the same or similar components as the original dilemma - another way of looking at the facts in order to arrive at a different conclusion. |
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Definition
quod erat deomonstrandum
"what was to be demonstrated"
used to show that a proof is completed |
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| valid argument forms which can be used to justify steps in a proof |
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| Hypothetical Syllogism - HS |
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Definition
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| Disjunctive Syllogism - DS |
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Definition
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| Constructive Dilemma - CD |
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Definition
(p > q) * (r > s)
p v r
% q v s |
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Formal Proof Method
for validity |
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Definition
1. Write premises in symbols and number them
2. deduce intermediate conclusions using the rules of inference, justifying each step by writing the steps used as premises and the abbreviation for the rule
3. continue until the desired conclusion is reached |
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3 Hints for recognizing
Rules of Inference |
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Definition
1. a very complicated compound proposition can represent one variable
2. variables in the rules can represent propositions which are similar to those represented by other variables
3. variables from the rules change value from one step to another within a proof |
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