{"id":49,"date":"2016-12-16T18:14:05","date_gmt":"2016-12-16T18:14:05","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/9-if-and-only-if-using-theorems\/"},"modified":"2025-10-13T18:14:53","modified_gmt":"2025-10-13T18:14:53","slug":"9-if-and-only-if-using-theorems","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/9-if-and-only-if-using-theorems\/","title":{"raw":"\u201c\u2026 if and only if \u2026\u201d, Using Theorems","rendered":"\u201c\u2026 if and only if \u2026\u201d, Using Theorems"},"content":{"raw":"

9.1 \u00a0A historical example<\/h2>\r\nThe philosopher David Hume (1711-1776) is remembered for being a brilliant skeptical empiricist. \u00a0A person is a skeptic about a topic if that person both has very strict standards for what constitutes knowledge about that topic and also believes we cannot meet those strict standards.\r\n\r\nEmpiricism is the view that we primarily gain knowledge through experience, particular experiences of our senses. In his book, An Inquiry Concerning Human Understanding<\/em>,<\/span>\u00a0Hume lays out his principles for knowledge, and then advises us to clean up our libraries:\r\n
When we run over libraries, persuaded of these principles, what havoc must we make? If we take in our hand any volume of divinity or school metaphysics, for instance, let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames, for it can contain nothing but sophistry and illusion.[11]<\/a><\/sup><\/blockquote>\r\nHume felt that the only sources of knowledge were logical or mathematical reasoning (which he calls above \u201cabstract reasoning concerning quantity or number\u201d) or sense experience (\u201cexperimental reasoning concerning matter of fact and existence\u201d). \u00a0Hume is led to argue that any claims not based upon one or the other method is worthless.\r\n\r\nWe can reconstruct Hume\u2019s argument in the following way. \u00a0Suppose t<\/span>\u00a0is some topic about which we claim to have knowledge. \u00a0Suppose that we did not get this knowledge from experience or logic. \u00a0Written in English, we can reconstruct his argument in the following way:\r\n

We have knowledge about t<\/span>\u00a0<\/strong><\/em>if and only if our claims about t<\/span>\u00a0<\/strong><\/em>are learned from experimental reasoning or from logic or mathematics.<\/p>\r\n

Our claims about t<\/span>\u00a0<\/strong><\/em>are not learned from experimental reasoning.<\/p>\r\n

Our claims about t<\/span>\u00a0<\/strong><\/em>are not learned from logic or mathematics.<\/p>\r\n

_____<\/p>\r\n

We do not have knowledge about t<\/span><\/strong><\/em>.<\/span><\/p>\r\nWhat does that phrase \u201cif and only if\u201d mean? \u00a0Philosophers think that it, and several synonymous phrases, are used often in reasoning. \u00a0Leaving \u201cif and only\u201d unexplained for now, we can use the following translation key to write up the argument in a mix of our propositional logic and English.\r\n

P<\/span><\/strong>: \u00a0We have knowledge about t<\/span><\/strong><\/em>.<\/span><\/p>\r\n

Q<\/span><\/strong>: \u00a0Our claims about t<\/span>\u00a0are learned from experimental reasoning.<\/p>\r\n

R<\/span><\/strong>: \u00a0Our claims about t<\/span>\u00a0are learned from logic or mathematics.<\/p>\r\nAnd so we have:\r\n

P<\/span>\u00a0<\/strong>if and only if (QvR)<\/span><\/strong><\/p>\r\n

\u00acQ<\/span><\/strong><\/p>\r\n

\u00acR<\/span><\/strong><\/p>\r\n

_____<\/strong><\/p>\r\n

\u00acP<\/span><\/strong><\/p>\r\nOur task is to add to our logical language an equivalent to \u201cif and only if\u201d. \u00a0Then we can evaluate this reformulation of Hume\u2019s argument.\r\n

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Check for Understanding<\/p>\r\n\r\n<\/header>\r\n

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  1. Critical Thinking Task:\u00a0<\/strong>Create a story\/song\/poem you could use to explain the concept of empiricism to a family member.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

    9.2 \u00a0The biconditional<\/h2>\r\n
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    Pre-Reading Questions<\/p>\r\n\r\n<\/header>\r\n

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    1. What do you think the phrase \"if and only if\" means in everyday language? Can you give an example of a statement that uses this phrase?<\/li>\r\n \t
    2. How would you express the meaning of \"P if and only if Q\" by breaking it down into simpler logical statements?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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      Key Terms<\/p>\r\n\r\n<\/header>\r\n

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