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| The most basic sentences in FOL, those formed by a predicate followed by the right number (arity) of names (or complex terms, if the language contains function symbols). |
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| the antecedent of a conditional is its first component clause. In p --> q, p is the antecedent and q is the consequent |
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the arity of a predicate indicates the number of arguments it takes. unary- 1 argument place binary- 2 argument places |
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| The logical connectives conjunction, disjunction. and negation allow us to form complex claims from simpler claims. |
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| the gramatical expressions of FOL... defined inductively |
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two kinds 1. a sequence of statements in which one (the conclusion) is supposed to follow from or be supported by the others (premises) 2. Another use refers to the term(s) taken by a predicate in an atomic wff or the argument places (correspondent to the arity). |
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Valid Validity Valid argument |
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is an argument where the conclusion follows from the premises (regardless of the truth of the premises). If this occurs then the conclusion is a logical consequence of the premises. Logical consequence= it is impossible for all the premises to be true and the conclusion false |
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Sound Soundness Sound Argument |
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An argument is sound only if it is valid (that is its conclusion is a logical consequence of its premises) and all the premises are true ex. if it rains, the ground gets wet it is raining right now therefore, the ground will become wet |
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a sentence that is logically true in virtue of its truth-functional structure. This can be checked using truth tables since S is a tautology iff every row of the truth table for S assigns true to the maine connective. ex. (A v ~A) |
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Two sentences are tautologically equivalent if they are equivalent simply in virtue of the meanings of the truth-functional connectives. This can be checked since Q and S are taut equiv iff every row of their joint truth table assigns the same value to teh main connectives of Q and S |
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| exists between two sentences that are equivalent in virtue of their truth functional connectives, identity, and quantifiers. |
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| Exists when a sentence S follows from some premises simply in virtue of the meaning of the truth-functional connectives. |
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| A sent S is a FO Consequence of some premise if S follows from the premises simply in virtue of the meanings of the truth-functional connectives, identity, and the quantifiers |
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Syntax=grammer Semantics= meaning of a sentence and its truth conditions |
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Conjunction Elimination (^ Elim) |
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Disjunction Introduction ( v Intro) |
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Pi . . . P1v....vPi....vPn
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Disjunction Elimination (v Elim) |
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Definition
P1v...vPn Pn . ----
. .
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P1 .
---- S
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. S (Goal)
. S
-->
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| Does the conclusion have a negation in it? |
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Definition
| if yes--> use proof by contradiction ( negation introduction) |
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Do the premises have a disjunction? |
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| if yes --- use proof by cases |
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Does the conclusion have a conditional in it?
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use a conditional proof. Do this by starting a subproof assuming the antecedent and deriving the consequent. |
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| Do you have an existential quantifier in the premises? |
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| if yes-- start a subproof assume a random object a has whatever characteristic is attached to the existiential quantifier and then derive some quality Q. Then you can just conclude Q from the existential elim rule. |
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| Do you have a Universal Quantifier as the conclusion? |
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if yes-- use the universal intro rule assume random object a in a subproof conclude something about a and then you can conclude that all objects posess that something. |
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| the procedure in which you replace all constituent parts with letters to determine their truth function |
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| a quantified sentence of FOL is said to be a tautology iff its truth-functional form is a tautology |
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| Every tautology is a logical truth, but among quantified sentences there are many logical truths that are not tautologies. |
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