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Definition
| The ability of people to reason correctly most of the time, and, when they fail to reason correctly, on the ability of others to point out the gaps in their reasoning. |
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| One claim is a logical consequence of another if there is no way the latter could be true without the former also being true. |
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| To prove that one claim is a logical consequence of others. |
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| A sentence formed by a predicate followed by the right number of names. |
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Term
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Definition
Symbols which are used to refer to some fixed individual object.
-They are FOL analogue of names, though in FOL they are not capitalized.
-Each individual constant must name an actually existing object.
-No individual constant can name more than one object.
-An object can have more than one name, or no name at all. |
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Term
| Predicate Symbol (relation symbols) |
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Definition
Symbols used to express some property of objects or some relation between objects.
-Also referred to as relation symbols. |
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Definition
| The "arguments" of the predicate; there are often two or more in FOL and the predicate refers to whatever remains. |
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| A number which tells you how many individual constants the predicate symbol needs in order to form a sentence. |
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Definition
| Predicates taking one argument. |
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| Predicates taking two arguments. |
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| Predicates taking three arguments. |
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| A property for which, given any object, there is a definite fact of the matter whether or not the object has the property. |
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| When the predicate symbol "=" appears in between its two arguments. |
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| When the predicate precedes the arguments. |
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Definition
| Something that is either true or false. |
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| The value of a claim which is either true or false. |
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Definition
| Anything which a claim can be made about. |
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Term
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Definition
Permit the formation of name-like terms from names and other name-like terms in FOL.
-Examples include Max's father "father(max)" or Claire's mother "mother(claire)."
-Unlike predicates, function symbols are always written in lower case.
-Should always refer to existing individuals. |
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Term
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Definition
| Constants which refer to people or individuals. |
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Term
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Definition
| Combine the use of function symbols and terms to create a complex term. |
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Term
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Definition
When atomic sentences are always formed by placing individual constants on either side of one of the two predicates.
-Allows us to make identity claims, of the form a=b
-Membership claims, of the form a(e |
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Term
| Formal System of Deduction |
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Definition
| A system that allows us to show that a sentence of FOL is a consequence of others. |
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Term
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Definition
| Any series of statements in which one, called the conclusion, is meant to be a consequence of the others, called the premises. |
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Definition
An argument is valid if the conclusion must be true in any circumstances in which the premises are true. We say that the conclusion of a logically valid argument is a logical consequence of its premises.
-Guarantees the truth of its conclusion on the assumption that the premises are true. |
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Term
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Definition
| When an argument is valid and the premises are all true. |
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Term
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Definition
Step-by-step demonstration that a conclusion (say S) follows from some premises (say P, Q, R).
-Establishes a series of intermediate conclusions, each of which is an obvious consequence of the original premises and the intermediate conclusions established. |
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Term
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Definition
| A proof which begins by stating the premises or assumptions of the proof, and then explains in a step-by-step fashion how we can get from the assumptions to the desired conclusion. |
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Term
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Definition
Employs a fixed stock of rules and highly stylized method of presentation.
-Displays the logical structure of a proof in a form that can be mechanically checked.
-Allows us to prove things about probability. |
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Term
| Indiscernibility of Identicals (Substitution) |
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Definition
| If b = c, then anything true of b is also true of c, for b is c. |
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Term
| = Elim (Identity Elimination) |
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Definition
Eliminates a use of the identity symbol when we move from the premises of the argument to its conclusion.
-If we have proven a sentence containing n (which we indicate by writing P(n)) and a sentence of the form n = m, then we are justified in asserting any sentence which results from P(n) by replacing some or all of the occurrences of n by m. |
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Term
| = Intro (Reflexivity of Identity / Identity Introduction) |
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Definition
Permits the introduction of an identity statement into proofs.
-Any sentence of the form a = a can be validly inferred from whatever premises are at hand, or from no premises at all.
-Allows you to introduce, for any name (or complex term) n in use in the proof, the assertion n = n.
~Can be done at any step
~Needs no steps to be cited earlier as justification. |
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Term
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Definition
| Permits the conclusion of b = a from a = b. |
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Term
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Definition
| If a = b and b = c are both true, then so is a = c. |
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Term
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Definition
Refer to the same relation but point in opposite directions.
-Larger(b,c) = Smaller(c,b) |
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Term
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Definition
Format developed by Frederic Fitch for presenting proofs.
-The vertical line that runs on the left of the steps draws our attention to the fact that we have a single purported proof consisting of a sequence of several steps.
-The horizontal Fitch bar indicates the division between the claims which are assumed and those that allegedly follow from them. |
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Term
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Definition
| Indicates which rule allows us to make the step, and which earlier steps (if any) the rule is applied to. |
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Term
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Definition
| Allows the repetition of an earlier step if desired. |
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Term
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Definition
| Proof that a particular piece of information must be true if the given information, the premises of the argument, are correct. |
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Term
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Definition
| The purported conclusion is not a consequence of the premises provided in the argument. |
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Term
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Definition
| Circumstances that make the premises true but the conclusion false. |
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Term
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Definition
| Possible situation or circumstance in which the premises are all true while the conclusion is false. |
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Term
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Definition
| The truth value of a complex sentence built up using these connectives depends on nothing more than the truth values of the simpler sentences from which it is built. |
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Term
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Definition
| A table which shows how the truth value of a sentence formed with the connective depends on the truth values of the sentence's immediate parts. |
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Term
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Definition
The negation symbol is used to express negation in FOL, the notion we commonly express in English using terms like not, it is not the case that, non, and un.
-In FOL, this symbol is always applied to the front of the sentence to be negated.
-Can be applied to complex and atomic sentences. |
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Term
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Definition
| An atomic or the negation of an atomic sentence. |
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Term
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Definition
The conjunction symbol is used to express conjunction in FOL, the notion we normally express in English using terms like and, moreover, and but.
-In FOL, this connective is always placed between two sentences, whereas in English we can also conjoin other parts of speech, such as nouns. |
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Term
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Definition
The disjunction symbol is used to express disjunction in FOL, the notion we express in English using or.
-Is always placed between two sentences, whereas in English we can also disjoin nouns, verbs, and other parts of speech. |
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Term
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Definition
| Neither is displayed using the negation sign in front of an atomic sentence which features or, such as: NOT(Home(john) V Home(mary)) |
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Term
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Definition
1. ¬¬P = P
2. ¬(P ^ Q) = ¬P v ¬Q
3. ¬(P v Q) = ¬P ^ ¬Q |
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Term
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Definition
| If two sentences are true in exactly the same circumstances. |
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Term
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Definition
A logically necessary sentence in which it is impossible for the initial premises to be true and the conclusion to be false, because it is impossible for the conclusion to be false.
-Example: a = a |
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Term
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Definition
A sentence is logically possible if it could be (or could have been) true on logical grounds.
-It is logically possible to go faster than the speed of light (done on Star Trek.) |
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Term
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Definition
| When a sentence is true in every logically possible circumstance. |
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Term
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Definition
A sentence which is true in some world that can be built using Tarski's World
-Every TW sentence is logically possible. |
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Term
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Definition
| An instance of the principle P v ¬P, which cannot be false. |
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Term
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Definition
A complex sentence that is true no matter what the truth values of the atomics are |
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Term
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Definition
| If a sentence is true in every world in which it has a truth table. |
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Term
| TT-Necessary (Tautological) |
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Definition
| If a sentence comes out true in every row of its truth table. |
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Term
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Definition
| If a sentence is true in at least one row of its truth table. |
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Term
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Definition
Sentences that have the same truth values in every possible circumstance.
-Have the same truth conditions, since the conditions under which they come out or true or false are identical. |
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Term
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Definition
Two sentences are tautologically equivalent if they can be seen to be equivalent simply in virtue of the meanings of the truth-functional connectives.
-All tautologically equivalent sentences are logically equivalent; however, the reverse does not hold. |
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Term
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Definition
| Reveals if Q is a consequence of P through the use of a joint truth table, in which if every row that is true for P is true for Q, then Q is said to be a tautological consequence of P. |
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Term
| Substitution of Logical Equivalents |
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Definition
If S(P) is an FOL sentence that contains the sentence P as a component part, and S(Q) is the result of substituting Q for P in S(P), then P and Q are logically equivalent as:
P = Q & S(P) = S(Q) |
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Term
| Disjunction Normal Form (DNF) |
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Definition
| Using the first distributive law, a sentence in negation normal form can be translated into a sentence that is a disjunction of conjunctions of literals. |
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Term
| Conjunction Normal Form (CNF) |
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Definition
| Using the distribution of v over ^, we can turn any negation normal form sentence into one that is a conjunction of one or more sentences, each of which is a disjunction of one or more literals. |
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Term
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Definition
| Building blocks; symbols used in language. |
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Term
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Definition
| Ways of putting the blocks together to get good / bad strings |
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Term
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Definition
| Theory of how strings come to be true or false |
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Term
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Definition
| Theory of what makes strings of sentences valid or invalid |
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