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Such inferences are neither deductive (which assumes application of a valid inference rule) nor inductive (which assumes a generalization from a pattern of cases). This type of inference is called abductive, or "inference to the best explanation", see abductive reasoning. "Clearly" indicates that the explanation inferred is the best available. According to ...


10

In mathematical logic P → Q must be read : "if P, then Q", as well as : "P only if Q". Thus "Q only if P" is Q → P, which is not equivalent to P → Q. You can see also this post as well as this one.


10

First, your premises are inconsistent: your second premise implies that turtles do not see other turtles, or themselves, yet, according to the first premise, they see everything. So, taking y=x, we can deduce both ∀xSxx and ∀x¬Sxx. After that you can deduce whatever you want directly using the law of explosion, contradiction implies anything. If you fix ...


8

This argument could hardly be rendered into a valid form without all kinds of additional assumptions and clarifications. For example, Assumes we know what God wants and what he/she might or might not do to satisfy those wants. It is difficult enough to speculate about what other people might want or might do about their wants, without trying to speculate ...


8

If the question is raised in an intro to philosophy course (like critical thinking or scientific reasoning), the answer should be that the above inference is an example of inductive logic. There are two kinds of inductive reasoning. One is generalization, as Conifold suggests. But there is also an inductive inference of John Stuart Mill whose purpose is to ...


8

When it comes to justification there is indeed a symmetric problem of deduction. But forming general opinions or laws is not part of deduction, it is abductive (or in older terminology inductive), when it comes to science it is the "hypothetical" part of the hypothetico-deductive method, see Are “if smoke then fire” arguments deductive or inductive? for more ...


7

In formal logic this is called a Disjunctive Syllogism (sometimes called 'the process of elimination' in informal logic) - grandchild of the syllogism a la Aristotle you've got going on with the green man. As logically valid as they come, this one! Formally if one knows: 'X or Y' and 'not X', one may conclude 'Y' It's often difficult to find the ...


6

Deductive arguments aren't non-falsifiable because arguments aren't either true or false. Deductive arguments are either sound, valid but unsound, or invalid. Here's an example: (1) All men are mortal. (2) Socrates is a man. (3) Therefore, Socrates is mortal. It's only the conclusion, or one of the premises that could meaningfully be said to be ...


6

One issue is that different authors use "argument" and "inference" in ways different from each other, and from the colloquial meaning. For example, your source interprets "argument" as just the list of premises and the conclusion, whereas in the colloquial sense it is the sequence of intermediate logically elementary steps that lead from premises to the ...


6

The point is that we should use bayesian reasoning to infer a cause from its consequences. The probability that an hypothesis is true given the evidence is not the same as the probability of the evidence given the hypothesis. For example: the probability that the floor is wet if it has rained is 1 but the probability that it has rained given that the floor ...


6

The following truth table shows that ~(A↔B) → ~(A→B) is not a tautology: If A is False and B is True then the antecedent is True but the consequent is False making the conditional False. Because the truth table does not show a tautology, one should not be able to derive a natural deduction proof of the result. Michael Rieppel. Truth Table Generator. ...


5

This is known as "the process of elimination."


5

An argument is deductively valid if and only if it's impossible for all its premises to be true and its conclusion to be false at the same time. If it's impossible for its premises to be true at the same time, then that is itself sufficient to meet that definition, and make the argument valid, independent of the truth-value of the conclusion. Whether this ...


5

In logic, a tautology (from the Greek word ταυτολογία) is a formula that is true in every possible interpretation. – Wikipedia There is an 'interpretation' possible in which snow does not melt during the day in the Sahara / a human lives without oxygen / photons have no mass. That is because these statements can only be verified with a posteriori ...


5

The argument is not logically valid. You need a sixth step, along the line of A method of knowing God without interpretation exists. As to the point about the circularity; It's incorrect to reduce the argument to 'I believe he exists because he exists' The correct reduction is 'It's impossible for God to exist without faith in his existence also ...


5

The two different symbols on the page you link to are indeed different. The first is the turnstile symbol Ⱶ which may be read as 'proves', while the arrow → is material implication. These are very different. Material implication is a symbol in the object language defined by the truth table that you give, i.e. T/F/T/T. Turnstile is a symbol in the ...


5

See Enthymeme : An enthymeme is a logical fallacy in which a categorical syllogism omits a premise that is necessary for the conclusion to be true or omits the conclusion itself. The missing proposition is considered to be implied. The fallacy is a syllogistic fallacy and a formal fallacy. Formal fallacy because a formal deductive arguments is a ...


4

We're given two premises: A; (A ∧ B) ∨ (C ∧ D). The goal is to assume (1-2) and derive: ¬(C ∧ D) → (A ∧ B). I should start by saying that premise (1) is useless and in what follows we'll simply use premise (2). Here is one strategy you can take. It's pretty system-neutral, so try to implement it in your particular proof system ...


4

Your articulation: "An argument is valid IFF the premises are false or the conclusion is true". misses an important feature in the textbook's definition. Namely, you've lost the must, but the must is crucial. The validity of an argument does not hinge on the truth or falisty of its premises or the truth of its conclusion. Instead, validity looks at the ...


4

One easy, and perfectly general, such premise X is: All A are x From this premise one can deductively infer that all A are x, unsurprisingly. One problem with this maneuver, of course, is that we have no better evidence for the extra premise than we have for the conclusion -- both being the same proposition. Therefore, as it is sometimes said, there is ...


4

We need first-order logic with equality. We have : 1) ∀x (Fx ∨ x=c) 2) ¬Fb ∧ Gb 3) ¬Fa → Ga and we want to derive 3) from 1) and 2). I think that the "trick" is to rewrite 1) as : a) ∀x (¬Fx → x=c) b) ¬Fb --- from 2) by ∧-elim c) ¬Fb → b=c --- from a) by ∀-elim d) b=c --- from b) and c) by →-elim (modus ponens) e) ¬Fa --- assumed f) ¬...


4

1) ¬∀xFx --- premise 2) ¬∃x¬Fx --- assumed [a] 3) ¬Fy --- assumed [b] 4) ∃x¬Fx --- from 3) 5) ⊥ --- contradiction: from 2) and 3) 6) Fy --- from 3) and 5) by Double Negation, discharging [b] 7) ∀xFx --- from 7): no y free in "open" assumptions (i.e. [a]) 8) ⊥ --- contradiction: from 1) and 7) 9) ∃x¬Fx --- from 2) and 8) by Double Negation, ...


4

The quoted passage is part of an exposition of Hume's original argument. One of the previous paragraphs explains what "deductively" meant to Hume: "The deductive system that Hume had at hand was just the weak and complex theory of ideas in force at the time, augmented by syllogistic logic. His ‘demonstrations’ rather than structured deductions are often ...


4

welcome to PSE. The concept of truth is one of the head-spinners of philosophy. Some philosophers believe that truth is a property possessed by all and only true propositions, statements, sentences, claims, &c. Just as a square has the property of being a figure in plane geometry with four equal angles, so a true statement has has the property of being ...


4

It seems you don't understand the terms. The essence of deductive reasoning: If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. https://en.wikipedia.org/wiki/Deductive_reasoning Inductive reasoning, on the other hand: Inductive reasoning is a method of ...


4

...it seems that any inductive reasoning can be done with deductive reasoning by adding in some assumption that a particular pattern continues to hold. You got it exactly right. The assumption is the Uniformity Principle. Some philosophers have accepted the principle as supported by common observation; others have dropped wet blankets all over it. ...


4

You can't derive ~(A→B) from ~(A↔B). Consider: A = I'm in Paris. B = I'm in France. ~(A↔B) is true, because being in Paris is not equivalent to being in France (I could be in France but not in Paris). But ~(A→B) is false, because if I'm in Paris then necessarily I'm in France. So you can't derive ~(A→B) from ~(A↔B).


3

You can find a decent introduction here. Is there any particular reason why you didn't do a quick search in the basic encyclopedias first? Imagine what would happen if everyone behaved that way...


3

Fact 1. ¬(A ∨ B) |= ¬(B ∨ A) Proof. The negated form of the conclusion hints at an obvious way of proceeding: assume (B ∨ A) with the hope of deriving a contradiction. The disjunctive form of this assumption suggests the second step (proof by case analysis): assume B, derive some sentence Γ, then assume A and derive that Γ ...


3

Two usually discussed challenges are by Nozick and Dretske. Nozick's analysis of knowledge includes the following condition for S to know that p: (sensitivity) Were p false, S would not believe p. Truth conditions for subjunctive conditionals are given in the usual way: Sensitivity requires that in the close worlds to the actual world in which not-p ...


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