Audi, Robert. 1983. “The Applications of Conceptual Analysis.” Metaphilosophy 14: 87-105.
Bealer, George. 1996. “A Priori Knowledge and the Scope of Philosophy.” Philosophical Studies 81: 121-42.
Braddon-Mitchell, David and Robert Nola. 2009. Conceptual Analysis and Philosophical Naturalism. Cambridge, MA: MIT Press.
Copi, Irving, Carl Cohen and Daniel Flage. 2007. Essentials of Logic. 2nd ed. Upper Saddle River, NJ: Pearson Prentice-Hall.
Copi, Irving, Carl Cohen and Kenneth McMahon. 2014. Introduction to Logic. 14th ed. Pearson Education Limited.
Fisher, Alec. 2001. Critical Thinking. Cambridge: Cambridge University Press.
Hempel, Carl. 1952. Fundamentals of Concept Formation in Empirical Science. Chicago: University of Chicago Press.
Jackson, Frank. 1998. From Metaphysics to Ethics: A Defense of Conceptual Analysis. New York: Oxford University Press.
Kahane, Howard and Paul Tidman. 2002. Logic and Philosophy: A Modern Introduction. 9th ed. Wadsworth Cengage Learning.
Lakoff, George. 1987. Women, Fire, and Dangerous Things. Chicago: University of Chicago Press.
Lemmon, Edward John. 1971. Beginning Logic. Chapman and Hall.
MacFarlane, John. 2000. What Does it Mean to Say that Logic is Formal? PhD thesis. University of Pittsburgh.
Manwarring, Max G. and Court Prisk, eds. 1988. El Salvador at War: An Oral History of the Conflict from the 1979 Insurrection to the Present. Washington, D.C.: National Defense University Press.
McGinn, Colin. 2012. Truth by Analysis. Oxford: Oxford University Press.
Moore, G. E. 1968. “A Reply to my Critics, §11: Analysis.” In The Philosophy of G. E. Moore, ed. Paul Schilpp, 660-667. 3rd ed. La Salle, IL: Open Court.
Oliver, Alex. 2010. “The Matter of Form: Logic's Beginnings.” In The Force of Argument: Essays in Honor of Timothy Smiley, eds. Timothy J. Smiley, John Lear, and Alex Oliver, 165-185. New York: Routledge.
Ramsey, William. 1998. “Prototypes and Conceptual Analysis.” In Rethinking Intuition, eds. Michael Depaul and William Ramsey, 161-78. Lanham, MD: Rowman & Littlefield.
Rosch, Eleanor and Carolyn Mervis. 1998. “Family Resemblances: Studies in Internal Structure of Categories.” In Rethinking Intuition, eds. Michael Depaul and William Ramsey, 17-44. Lanham, MD: Rowman & Littlefield.
Shaffer, Michael. Forthcoming. The Experimental Turn and the Methods of Philosophy. Routledge.Shaffer, Michael. 2015. “The Problem of Necessary and Sufficient Conditions and Conceptual Analysis.” Metaphilosophy 46: 555-63.
Smiley, Timothy. 1982. “The Schematic Fallacy.” Proceedings of the Aristotelian Society 83: 1-17.
Smith, Nicholas J. J. 2012. Logic: The Laws of Truth. Princeton: Princeton University Press.
Strawson, Peter Frederick. 1952. An Introduction to Logical Theory. London: Methuen.
Velasco, P. Del Nero. 2010. Educando para a Argumentação. Belo Horizonte, Brazil: Autêntica Editora.
Walton, Douglas. 1995. A Pragmatic Theory of Fallacy. Tuscaloosa: University of Alabama Press.Wittgenstein, Ludwig. 1953. Philosophical Investigations, trans. G. E. M. Anscombe. Oxford: MacMillan.
Wittgenstein, Ludwig. 1958. The Blue and Brown Books. New York: Harper.
Bowell, Tracey and Gary Kemp. 2002. Critical Thinking: A Concise Guide. New York/Oxford: Routledge.
Groarke, Leo. 2017. “Informal Logic.” In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/spr2017/entries/logic-informal/
Priest, Graham. 2017. Logic: A Very Short Introduction. 2nd ed. Oxford: Oxford University Press.
Sinnott-Armstrong, Walton and Robert Fogelin. 2015. Understanding Arguments: An Introduction to Informal Logic. 9th ed. Stamford, CT: Cengage Learning.
Thomson, Anne. 2008. Critical Reasoning: A Practical Introduction. London/New York: Routledge.
Douven, Igor. 2017. “Abduction.” In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/sum2017/entries/abduction/
Russel, Bertrand. 2001. “On Induction.” In The Problems of Philosophy. Oxford University Press.
Vickers, John. 2016. “The Problem of Induction.” In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/spr2016/entries/induction-problem/
Pietroski, Paul. 2016. “Logical Form.” In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/spr2016/entries/logical-form/
Sainsbury, Mark. 2001. Logical Forms: An Introduction to Philosophical Logic. 2nd ed. Oxford: Blackwell.
Walton, Douglas. 1995. A Pragmatic Theory of Fallacy. Tuscaloosa, AL: The University of Alabama Press.
Walton, Douglas. 2008. Informal Logic: A Pragmatic Approach. 2nd ed. Cambridge/New York: Cambridge University Press.
Bealer, George. 1998. “Intuition and the Autonomy of Philosophy.” In Rethinking Intuition, eds. Michael Depaul and William Ramsey, 201-240. Lanham, MD: Rowman & Littlefield.
Brennan, Andrew. 2017. “Necessary and Sufficient Conditions.” In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/sum2017/entries/necessary-sufficient/
Fodor, Jerry. 1998. Concepts. Oxford: Oxford University Press.
]]>Argument = a set of propositions, one of which, the conclusion, is (supposed to be) supported by the others, the premises.
If we’re reasoning by making claims and backing them up with reasons, then the claim that’s being backed up is the conclusion of an argument; the reasons given to support it are the argument’s premises. If we’re reasoning by drawing an inference from a set of statements, then the inference we draw is the conclusion of an argument, and the statements from which it’s drawn are the premises. We include the parenthetical hedge—“supposed to be”—in the definition to make room for bad arguments. A bad argument, very roughly speaking, is one where the premises fail to support the conclusion; a good argument’s premises actually do support the conclusion.First, explicate the following arguments, paraphrasing as necessary and only including tacit premises when explicitly instructed to do so. Next, diagram the arguments.
Quality of Inference | Deductive | Inductive | Abductive |
---|---|---|---|
Bad inference | Invalid | Weak | Weak |
Good inference | Valid | Strong | Strong |
Good inference + true premises | Sound | Cogent | Cogent |
$latex A$ | $latex \neg A$ |
---|---|
T | F |
F | T |
$latex A$ | $latex B$ | $latex A \rightarrow B$ |
---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
$latex A$ | $latex B$ | $latex \neg A$ | $latex \neg B$ | $latex A \rightarrow \neg B$ |
---|---|---|---|---|
T | T | F | F | F |
T | F | F | T | T |
F | T | T | F | T |
F | F | T | T | T |
$latex C$ | $latex D$ | $latex C\rightarrow D$ | $latex \neg C$ | $latex \neg D$ |
---|---|---|---|---|
T | T | T | F | F |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | T | T |
Form can thus be studied independently of subject-matter, and it is mainly in virtue of their form, as it turns out, rather than their subject-matter that arguments are valid or invalid. Hence it is the forms of argument, rather than actual arguments themselves, that logic investigates. (Lemmon 1971, 4)According to this conception of logic, logicians are in a position to evaluate the validity of an argument, even if they do not strictly understand the content of the claims within the argument, nor under what conditions they would be true. Whether or not the claims within arguments are true, therefore, is not a matter for logic. Instead, what logic does is to explore the logical forms of arguments, and thereby establish their (in)validity.
Question for Reflection
$latex A$ | $latex B$ | $latex A \wedge B$ |
---|---|---|
T | T | T |
T | F | F |
F | T | F |
F | F | F |
$latex A$ | $latex B$ | $latex A \vee B$ |
---|---|---|
T | T | T |
T | F | T |
F | T | T |
F | F | F |
A Question for You!
A Question for You!
A Question for You!
A Question for You!
A Question for You!
(D1) $latex \mathrm{S}(p, q) \equiv (p \rightarrow q)$
(D2) $latex \mathrm{N}(q, p) \equiv (p \rightarrow q)$
In effect, D1 and D2 are then intended to be the standard logical interpretations of our ordinary language concepts of necessary and sufficient conditions framed in terms of classical propositional logic.[footnote]See, for example, Copi, Cohen and Flage (2007, 196, 446, 449) and Fisher (2001, 241).[/footnote] They are based on the idea that necessary and sufficient conditions can be exhaustively defined in terms of the conditional understood as material implication and represented by the “→” of classical propositional logic with the following familiar truth conditions:[footnote]The concept of the material conditional introduced here is just a formalization of what we were previously and informally calling “conditionals”.[/footnote]$latex A$ | $latex B$ | $latex A \rightarrow B$ |
---|---|---|
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Affirming the Antecedent (Modus Ponens)
Denying the Consequent (Modus Tollens)
Affirming the Consequent
Denying the Antecedent
(D3) $latex \mathrm{NS}(p, q) \equiv [(p \rightarrow q) \,\& \,(q \rightarrow p)]$
However, since the formula $latex (p \rightarrow q) \,\&\, (q \rightarrow p)$ is equivalent to the formula $latex (p \equiv q)$ in classical propositional logic, sets of such necessary and sufficient conditions can be more compactly defined in terms of logical equivalence as follows:(D4) $latex \mathrm{NS}(p, q) \equiv (p \equiv q)$
This concept is just the idea that the truth values of p and q are always the same, and the notion of logical equivalence has the following truth conditions:$latex A$ | $latex B$ | $latex A \equiv B$ |
---|---|---|
T | T | T |
T | F | F |
F | T | F |
F | F | T |
(1) Conceptual analyses take the form of proposed definitions (i.e. sets of necessary and sufficient conditions) of analysanda.
(2) The adequacy of any analysandum can be tested against concrete and/or imagined cases.
(3) Whether or not a proposed analysandum is adequate with respect to a given case can be determined by the use of a priori intuition, with a priori intuition being a distinct, reliable and fallible non-sensory mental faculty.[footnote]A priori knowledge is knowledge totally independent of any experience.[/footnote]
(4) Intuition allows us to reliably access knowledge about concepts.
(5) The method of reflective equilibrium is the particular method by which intuitions can be used to confirm/disconfirm analysanda.[footnote]Recent defenses of SPM include: Bealer (1996), Jackson (1998), and McGinn (2012). For closely related views, see Braddon-Mitchell and Nola (2009). See Shaffer (forthcoming) for extensive discussion of this view. Reflective equilibrium is the method of bringing intuitively true cases into conformity with a rule or principle.[/footnote]
According to the defenders of SPM, this is essentially the orthodox methodology of analytic philosophy, and it has been assumed to be adequate for the solution of philosophical problems by a significant number of both practicing and prominent philosophers throughout the recent history of philosophy. For example, this is the contention made by Colin McGinn in a recent book. McGinn is not in the least bit tentative in his blanket defense of SPM as the one and only method of philosophy. With this aim in mind, early in his 2012 book he makes the following extended declaration about philosophy:… it is not a species of empirical enquiry, and it is not methodologically comparable to the natural sciences (though it is comparable to the formal sciences). It seeks the discovery of essences. It operates “from the armchair”: that is, by unaided (usually solitary) contemplation. Its only experiments are thought-experiments, and its data are possibilities (or “intuitions” about possibilities). Thus philosophy seeks a priori knowledge of objective being—of non-linguistic and non-conceptual reality. We are investigating being as such, but we do so using only a priori methods. (McGinn 2012, 4)As should be immediately apparent, this is a clear, straightforward, and ringing endorsement of SPM as it has been understood here. To buttress this contention we need only take note of his other claims that “…the proper method for uncovering the essence of things is precisely conceptual analysis,” (McGinn 2012, 4) and that “philosophy, correctly conceived, simply is conceptual analysis” (McGinn 2012, 11). In effect, he believes then that we arrive at such analyses by considering possible cases and asking ourselves whether the concept applies or not in those cases—that is by consulting our “intuitions” about such cases (McGinn 2012, 5). What is also important for the purposes at hand is his acknowledgment that this account of philosophical methodology “was really the standard conception for most of the history of the subject, in one form or another” (McGinn 2012, 7). So, not only does McGinn endorse SPM as the sole methodology of contemporary philosophy, but he also claims that it is the enduring methodology of philosophical inquiry throughout its history.[footnote]See McGinn (2012, 4-11) for a summary of significant historical examples of the use of SPM, including some of those discussed here in more detail.[/footnote] One important clarification regarding McGinn’s version of SPM concerns the nature of the object of analysis (the analysans) and, more importantly, the nature of the analysandum itself as they are typically understood (i.e. as definitions of a particular sort framed as sets of necessary and sufficient conditions). Carl Hempel usefully makes a crucial distinction in this regard, which we can use to illuminate the standard view of such definitions:
The word “definition” has come to be used in several different senses....A real definition is conceived of as a statement of the “essential characteristics” of some entity, as when man is defined as a rational animal or a chair as a separate moveable seat for one person. A nominal definition, on the other hand, is a convention which merely introduces an alternative—and usually abbreviated—notation for a given linguistic expression, in the manner of a stipulation. (Hempel 1952, 2)Moreover, he tells us further that some real definitions are to be understood as meaning analyses, or as analytic definitions, of the term in question. The validation of such claims requires only that we know the meanings of the constituent expressions, and no empirical investigation is necessary to determine the correctness of the analysandum (Hempel 1952, 8). This is, of course, precisely what McGinn has in mind with respect to conceptual analysis. It is, then, worth making the obvious point that conceptual analysis is the operation of analyzing concepts via proposing definitions, but to point that out is not enough to fully grasp the view. It is true that SPM is a method that takes as inputs our concepts, but it involves the clear recognition that the definitions involved are to be understood as meaning analyses rather than as nominal or stipulative (i.e. “dictionary”) definitions. So, for example, the question of whether knowledge is justified true belief is just the question of the analysis of the concept of knowledge in terms of definitions constituted by sets of necessary and sufficient conditions understood as a meaning analysis. Conceptual analysis is then a method of doing something with concepts that we already possess—wherever they have ultimately come from.[footnote]Strictly speaking, conceptual analyses can also involve some degree of alteration in the content of the pre-theoretical concepts, as often happens when such analysis involves making a concept more precise.[/footnote] It is defining a pre-theoretical concept by offering a synonymous expression. It then appears to be the case that the defenders of SPM must believe that concepts have the form of sets of necessary and sufficient conditions, that such analyses are meaning analyses, and that analyses of our pre-analytic concepts are informative. Typical analysanda are thus kinds of decompositions of pre-analytic concepts. They are conceptual truths with the form of analytic definitions. So, for McGinn and other like-minded thinkers, analysanda have a very simple logical form, and we can see this via the example of the analysis of the concept of knowledge. Where Kx is “x is knowledge”, Jx is “x is justified”, Tx is “x is true” and Bx is “x is believed”, the standard analysis of knowledge looks like this:
x is Kx $latex \equiv$ x is Jx & x is Tx & x is Bx
This analysis is supposed to tell us the true nature, or essence, of the concept of knowledge in terms of a finite set of defining essential features, with the logical form of a set of jointly necessary and sufficient conditions. So, providing such an analysis involves decomposing the analysans into a list of features, thus exposing in some important sense the content of the concept.We are unable to clearly circumscribe the concepts we use; not because we don’t know their real definition, but because there is no real “definition” to them. (Wittgenstein 1958, 25)Second, he sought to replace the notion of concepts understood as sets of necessary and sufficient conditions with an alternative theory of concepts. This alternative account of concepts is based on the notion of a “family resemblance relation.” To see the first point more clearly, let us look at Wittgenstein’s favorite example from his Philosophical Investigations. Wittgenstein specifically argued that the concept of a game cannot be correctly analyzed in terms of a set of necessary and sufficient conditions. This is because games do not share some set of defining features in common. Rather, the members of the set of games are only similar to one another in some respects, and it is these relations of similarity that constitute the family of games. As we have seen, SPM assumes the following principle:
(CON) For any concept C, there exists a set of necessary and sufficient conditions that constitutes the content of C.
Wittgenstein’s attack on SPM is mounted via an attack on CON, and this is the fundamental ground of the potential vacuity problem. Essentially, the gist of the problem is that if there are no (or even just very few) concepts that can be correctly regimented as sets of necessary and sufficient conditions, there can be no (or very few) correct conceptual analyses in the sense of SPM. The basis of Wittgenstein’s criticism then can be understood as follows: it is clear from the consideration of examples across the history of philosophy that most or all philosophical attempts to analyze concepts by providing sets of necessary and sufficient conditions have failed. This is because, for any proposed set of necessary or sufficient conditions intended to be the correct analysis of a concept, there are instances of that concept that do not meet the set of proposed defining conditions. Think back to Wittgenstein’s favorite example of the concept of a game. Poker and soccer are both plausibly taken to be games and so we might, for example, posit that something is a game, if and only if, that activity involves a winner and a loser. But, the game patty cake is another plausible case of a game and does not have a winner and a loser. So, this definition of a game in terms of a set of necessary and sufficient conditions fails. Wittgenstein claims that this example generalizes, and the presumptive best explanation for the failed philosophical attempts to articulate the contents of concepts in terms of sets of necessary and sufficient conditions is that the contents of concepts are not captured by sets of necessary and sufficient conditions (i.e. the denial of CON). In other words, Wittgenstein holds that for any (or, at least most) attempt(s) to specify the contents of concepts in terms of necessary and sufficient conditions, we will find counter-examples. As a replacement for CON, Wittgenstein introduces the notion of a family resemblance class. The central idea is that the cases that fall under a concept are related to one another not by a defining set of necessary and sufficient conditions, but rather by complex overlapping similarity conditions that relate groups of members of the total set of cases that fall under the concept. However, no one set of conditions holds for all and only the members that exhibit that concept. Thus, if Wittgenstein is correct, the reason that there are no correct conceptual analyses is due to the fact that concepts cannot be analysed in terms of necessary and sufficient conditions. SPM is, thus, potentially (if not actually) vacuous.Bowell, Tracey and Gary Kemp. 2002. Critical Thinking: A Concise Guide. New York/Oxford: Routledge.
Groarke, Leo. 2017. “Informal Logic.” In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/spr2017/entries/logic-informal/
Priest, Graham. 2017. Logic: A Very Short Introduction. 2nd ed. Oxford: Oxford University Press.
Sinnott-Armstrong, Walton and Robert Fogelin. 2015. Understanding Arguments: An Introduction to Informal Logic. 9th ed. Stamford, CT: Cengage Learning.
Thomson, Anne. 2008. Critical Reasoning: A Practical Introduction. London/New York: Routledge.
Douven, Igor. 2017. “Abduction.” In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/sum2017/entries/abduction/
Russel, Bertrand. 2001. “On Induction.” In The Problems of Philosophy. Oxford University Press.
Vickers, John. 2016. “The Problem of Induction.” In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/spr2016/entries/induction-problem/
Pietroski, Paul. 2016. “Logical Form.” In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/spr2016/entries/logical-form/
Sainsbury, Mark. 2001. Logical Forms: An Introduction to Philosophical Logic. 2nd ed. Oxford: Blackwell.
Walton, Douglas. 1995. A Pragmatic Theory of Fallacy. Tuscaloosa, AL: The University of Alabama Press.
Walton, Douglas. 2008. Informal Logic: A Pragmatic Approach. 2nd ed. Cambridge/New York: Cambridge University Press.
Bealer, George. 1998. “Intuition and the Autonomy of Philosophy.” In Rethinking Intuition, eds. Michael Depaul and William Ramsey, 201-240. Lanham, MD: Rowman & Littlefield.
Brennan, Andrew. 2017. “Necessary and Sufficient Conditions.” In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. https://plato.stanford.edu/archives/sum2017/entries/necessary-sufficient/
Fodor, Jerry. 1998. Concepts. Oxford: Oxford University Press.
]]>$latex A$ | $latex B$ | $latex A \rightarrow B$ | $latex B$ | $latex A$ |
---|---|---|---|---|
T | T | T | T | T |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | F | F |
$latex A$ | $latex B$ | $latex C$ | $latex A \rightarrow B$ | $latex B \rightarrow C$ | $latex A \rightarrow C$ |
---|---|---|---|---|---|
T | T | T | T | T | T |
T | T | F | T | F | F |
T | F | T | F | T | T |
T | F | F | F | T | F |
F | T | T | T | T | T |
F | T | F | T | F | T |
F | F | T | T | T | T |
F | F | F | T | T | T |
Evaluate whether the following arguments are valid or invalid. First, identify their logical form, and then use truth-tables to establish their (in)validity.
1. We now know the situation. The Yankees either have to beat the Red Sox or they won’t make it to the World Series, and they won’t do the former.$latex A$ | $latex B$ | $latex A \vee \neg B$ | $latex \neg A$ | $latex \neg B$ |
---|---|---|---|---|
T | T | T | F | F |
T | F | T | F | T |
F | T | F | T | F |
F | F | T | T | T |
$latex A$ | $latex B$ | $latex A \rightarrow B$ | $latex B$ | $latex A$ |
---|---|---|---|---|
T | T | T | T | T |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | F | F |
$latex A$ | $latex B$ | $latex \neg (A \wedge B)$ | $latex \neg B \vee \neg A$ |
---|---|---|---|
T | T | F | F |
T | F | T | T |
F | T | T | T |
F | F | T | T |
$latex A$ | $latex B$ | $latex C$ | $latex (A \vee B) \rightarrow C$ | $latex A$ | $latex C$ |
---|---|---|---|---|---|
T | T | T | T | T | T |
T | T | F | F | T | F |
T | F | T | T | T | T |
T | F | F | F | T | F |
F | T | T | T | F | T |
F | T | F | F | F | F |
F | F | T | T | F | T |
F | F | F | T | F | F |
Category | Item | Status |
---|---|---|
Organizing Content | Content is organized under headings and subheadings | Yes |
Organizing Content | Headings and subheadings are used sequentially (e.g. Heading 1, Heading 2, etc.) as well as logically (if the title is Heading 1 then there should be no other Heading 1 styles as the title is the uppermost level) | Yes |
Images | Images (including graphs and diagrams) that convey information include Alternative Text (alt-text) descriptions of the image's content or function | Yes |
Images | Graphs, charts, and maps also include contextual or supporting details in the text surrounding the image | Yes |
Images | Images do not rely on colour to convey information | Yes |
Images | Images that are purely decorative contain empty alternative text descriptions. (Descriptive text is unnecessary if the image doesn't convey contextual content information) | N/A |
Tables | Tables include row and column headers | Truth tables have only column headers |
Tables | Tables include a title or caption | Yes |
Tables | Tables do not have merged or split cells | Yes |
Tables | Tables have adequate cell padding | Yes |
Weblinks | The weblink is meaningful in context, and does not use generic text such as "click here" or "read more" | Yes |
Weblinks | Where URLs are spelled out, such as in lists of references or footnotes, ARIA labels are used | Yes |
Weblinks | Weblinks do not open new windows or tabs | Yes |
Embedded Multimedia | A transcript has been made available for a multimedia resource that includes audio narration or instruction | N/A |
Embedded Multimedia | Captions of all speech content and relevant non-speech content are included in the multimedia resource that includes audio synchronized with a video presentation | N/A |
Embedded Multimedia | Audio descriptions of contextual visuals (graphs, charts, etc.) are included in the multimedia resource | N/A |
Formulas | Formulas have been created using LaTeX and MathJax | Yes |
Font Size | Font size is 12 point or higher for body text | Yes |
Font Size | Font size is 9 point for footnotes or endnotes | Yes |
Font Size | Font size can be zoomed to 200% | Yes |
Version | Date | Change | Affected Page(s) |
---|---|---|---|
1.0 | Nov. 18, 2020 | Original | |
1.1 | Nov. 19, 2020 | Added link to another openly licensed logic book to a list of such books in a footnote. | Footnote in the Series Introduction. |
1.2 | August 11, 2021 | Updated list of books in the series to reflect recent additions. | Introduction to the Series |
1.2.1 | August 22, 2021 | Updated list of books in the series to add the Epistemology book. | Introduction to the Series |
$latex A$ | $latex B$ | $latex A \rightarrow B$ | $latex B$ | $latex A$ |
---|---|---|---|---|
T | T | T | T | T |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | F | F |
$latex A$ | $latex B$ | $latex C$ | $latex A \rightarrow B$ | $latex B \rightarrow C$ | $latex A \rightarrow C$ |
---|---|---|---|---|---|
T | T | T | T | T | T |
T | T | F | T | F | F |
T | F | T | F | T | T |
T | F | F | F | T | F |
F | T | T | T | T | T |
F | T | F | T | F | T |
F | F | T | T | T | T |
F | F | F | T | T | T |
Evaluate whether the following arguments are valid or invalid. First, identify their logical form, and then use truth-tables to establish their (in)validity.
1. We now know the situation. The Yankees either have to beat the Red Sox or they won’t make it to the World Series, and they won’t do the former.$latex A$ | $latex B$ | $latex A \vee \neg B$ | $latex \neg A$ | $latex \neg B$ |
---|---|---|---|---|
T | T | T | F | F |
T | F | T | F | T |
F | T | F | T | F |
F | F | T | T | T |
$latex A$ | $latex B$ | $latex A \rightarrow B$ | $latex B$ | $latex A$ |
---|---|---|---|---|
T | T | T | T | T |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | F | F |
$latex A$ | $latex B$ | $latex \neg (A \wedge B)$ | $latex \neg B \vee \neg A$ |
---|---|---|---|
T | T | F | F |
T | F | T | T |
F | T | T | T |
F | F | T | T |
$latex A$ | $latex B$ | $latex C$ | $latex (A \vee B) \rightarrow C$ | $latex A$ | $latex C$ |
---|---|---|---|---|---|
T | T | T | T | T | T |
T | T | F | F | T | F |
T | F | T | T | T | T |
T | F | F | F | T | F |
F | T | T | T | F | T |
F | T | F | F | F | F |
F | F | T | T | F | T |
F | F | F | T | F | F |
— Berta Grimau, Institute of Information Theory and Automation (Czech Academy of Sciences), Prague, Czech Republic
]]>